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If the parent-ground-space dimension is eventually constant, then for every \(N{\gt}2\) it equals the number of positive loops in the sector graph. In particular, this conclusion holds under the BNT and all-chain nearest-neighbor parent-ground-space hypotheses of Theorem 12.1.68.
Let \(H_n,X_n\) act on finite-dimensional complex Hilbert spaces which may depend on \(n\). Suppose that, for all sufficiently large \(n\), \(H_n\geq 0\), \(\| \psi _n\| =1\), \(\ker H_n=\mathbb {C}\psi _n\), and \(H_n\) has norm gap at least the same constant \(\gamma \geq 0\) on \((\ker H_n)^\perp \). If \(\| X_n\psi _n\| ^2\to a\), \(\langle \psi _n,X_n\psi _n\rangle \to 0\), and \(\langle \psi _n,X_n^*[H_n,X_n]\psi _n\rangle \to e\), then \(\gamma a\leq e\). Finite-volume uniqueness is a hypothesis of this corollary.
Suppose that \(A\) is equipped with Appendix B structural data \((X,\Lambda ,U)\): the matrix \(X\) is invertible, the diagonal entries of \(\Lambda \) are strictly positive, \(A^i=X\Lambda U^iX^{-1}\), and \(U\) satisfies the pair-index isometry equation below. Let \(\mathcal H_a\) and \(\mathcal H_b\) have bases indexed by \(\alpha ,\beta \in \{ 0,\ldots ,D-1\} \). The coefficients \(U^i_{\alpha ,\beta }\) define maps
and
For every \(N\geq 0\), the coefficients of \(U\) define
with
The conjugate coefficient map is
This is the tensor power occurring in the basic-vector expression [ 13 , Equation (3.17) ] .
The \(AX\) coefficient-space representative is, before the \(AX\)-lift, the canonical two-site parent interaction of the core tensor, \(\widehat Q_{AX}=q_2(A_{\mathrm{core}})\). The source projector \(Q_{AX}\) of Definition D.2 acts on \(\mathcal H_A\otimes \mathcal H_X\). The displayed operator is the two-site coefficient representative that will be placed on the \(AX\) face; it is not by itself the construction of that source projector. The \(XB\) coefficient-space representative is the same operator, \(\widehat Q_{XB}=\widehat Q_{AX}\), later placed on the \(XB\) face; the source projector \(Q_{XB}\) acts on \(\mathcal H_X\otimes \mathcal H_B\).
Put \(|\varphi \rangle =\sum _b\lambda _b|b,b\rangle \). For outer-boundary coefficients \(v_{a,c}\), insertion of this vector on the middle virtual bond is the map \(I_\varphi \) defined by
The two-site physical map is \(U^{\otimes 2}I_\varphi \). A boundary matrix \(Y\) determines outer-boundary coefficients by \((v_Y)_{a,c}=\lambda _aY_{c,a}\). Opening the outer virtual bond in the two-site contraction of [ 13 , Equations (3.17)–(3.18) ] gives these coefficients.
Write \(s=\langle \varphi ,\varphi \rangle =\sum _b\lambda _b^2\). The orthogonal projector onto \(\mathbb {C}\varphi \) has pair-index coefficients
On three virtual site pairs, let \(P_{01}\) place this projector on the bond joining the right index of site zero to the left index of site one, and let \(P_{12}\) place it on the next bond.
Let \(a\to a\) be a positive loop and choose a nonzero vector \(\varphi _a\in \operatorname{ran}P_{a,a}\). In sector coordinates define the matrix product tensor \((C_a^{(r,l)})_{x,y}=\delta _{x,l}\varphi _a(r,y)\) and set it to zero outside sector \(a\). The physical loop tensor is the image of \(C_a\) under the one-site unitary of the spatial decomposition. Its associated cyclic product pairs the right factor at site \(n\) with the left factor at site \(n+1\). The corresponding \(N\)-site product state is denoted by \(\Psi _a^{(N)}\in (\mathbb {C}^d)^{\otimes N}\).
- MPSTensor.BeigiSectorGraphData.loopBondVector
- MPSTensor.BeigiSectorGraphData.loopBondVector_mem
- MPSTensor.BeigiSectorGraphData.loopBondVector_ne_zero
- MPSTensor.BeigiSectorGraphData.loopCoordinateTensor
- MPSTensor.BeigiSectorGraphData.loopTensor
- MPSTensor.BeigiSectorGraphData.loopCyclicProduct
- MPSTensor.BeigiSectorGraphData.loopProductState
- MPSTensor.BeigiSectorGraphData.loopProductStateES
Let \(\Phi _a=(\varphi _a(r,l))_{r,l}\) and write \(r_a=\operatorname{rank}\Phi _a\). Choose a rank factorization
The tensor on the minimal Schmidt support is
again extended by zero outside sector \(a\). Its physical tensor is its image under the same one-site unitary.
- MPSTensor.BeigiSectorGraphData.loopSchmidtRank
- MPSTensor.BeigiSectorGraphData.LoopSchmidtFactorization
- MPSTensor.BeigiSectorGraphData.loopSchmidtFactorization
- MPSTensor.BeigiSectorGraphData.minimalLoopCoordinateTensor
- MPSTensor.BeigiSectorGraphData.minimalLoopTensor
- MPSTensor.BeigiSectorGraphData.minimalLoopCoordinateTensor_sector_apply
- MPSTensor.BeigiSectorGraphData.minimalLoopCoordinateTensor_sector_ne
Let \(\nu _a=\lVert \varphi _a\rVert _2{\gt}0\), where the norm is the Hilbert-space norm of the bond vector, or equivalently the Frobenius norm of \(\Phi _a\). In either coordinate system, the normalized representative is \(\widehat C_a=\nu _a^{-1}\widetilde C_a\). The raw coordinate and physical tensors are retained separately.
Suppose that a one-site unitary identifies the physical Hilbert space with a finite direct sum
and that in these coordinates the complementary two-site interaction is \(\bigoplus _{a,b}\mathbb {1}_{a,l}\otimes P_{a,b}\otimes \mathbb {1}_{b,r}\), where each \(P_{a,b}\) is an orthogonal projector. The directed edge \(a\to b\) is present precisely when \(\operatorname{ran}P_{a,b}\) is nonzero. An ordered cycle of length \(N\) is a sequence \((a_0,\ldots ,a_{N-1})\) for which every cyclically adjacent edge \(a_n\to a_{n+1}\) is present. Cyclic rotations are not identified. This is the graph following equation (2) of [ 5 , Section III ] .
Let \(A^j\) be a finite family of tensors with fixed physical dimension and virtual dimensions \(D_j\). On the diagonal virtual boundary space define
Equip the domain with the direct sum of the Hilbert–Schmidt inner products and the codomain with the usual configuration inner product. Its range is \(\sum _jG_L(A^j)\).
For a range-\(R\) matrix \(h\), let \(H_h(n)\) be its finite open interaction sum. Group \(L\) sites into one site and let \(\mathcal R_{K,L}\) decode both indices of a matrix on \(K\) grouped sites. Define
Decoding \(h_B\) recovers \(H_h(KL)\). The definition allows \(d,L,K\geq 0\). This is a regrouping construction for [ 225 , Section 3, equations (3.12)–(3.14) ] .
Let \(A\) have physical dimension \(d\) and bond dimension \(D\), let \(B\) have blocked physical dimension \(d^L\) and bond dimension \(E\), and let \(V\colon \mathbb {C}^E\to \mathbb {C}^D\). For \(N,r\geq 0\), define a linear boundary map on \(M_{D\times E}(\mathbb {C})\) by
where \(u\) is the \(N\)-letter blocked prefix and \(\tau \) is the \(r\)-letter original tail. Its physical realization lies on \(NL+r\) original sites. The Euclidean realization uses Hilbert–Schmidt virtual coordinates, with the column index preceding the row index. All dimensions and lengths may be zero. These are the residual word contractions under the compression in [ 191 , Lemma blocking-arbitrary ] and the local supports of [ 225 , Section 3, lines 1504–1538 ] .
Given a tensor \(A\) and a blocking length \(L\), the \(L\)-blocked tensor is the tensor with physical index set \(\{ 0,\ldots ,d{-}1\} ^L\) (hence physical dimension \(d^L\)) and the same bond dimension \(D\), whose matrices are indexed by words \((i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\):
Blocking coarse-grains \(L\) neighbouring sites into one tensor: the inner virtual bonds are contracted, the \(L\) physical legs merge into a single composite index, and the two outer bonds remain as the bond of \(A^{[L]}\).
![\begin{tenkz}[cols=4, boundary=open, physical=up]
\tn[skin=box, ports={90:physical:$i_1$}]{A} & \tn[skin=box]{A} &
\tn[skin=dots]{} & \tn[skin=box, ports={90:physical:$i_L$}]{A}
\tnmark[form=enclosure, label pos=w]{(1,1) .. (1,4)}{$A^{[L]}$}
\end{tenkz}](tenkz_svg/tenkz-ba04f50f368d5028.svg)
For a fixed blocked length \(m\), the fixed-length form of the block-injective canonical-form condition is
Equivalently,
This is the fixed-length form of the block-injective canonical-form condition of [ 17 , lines 2113–2117 ] .
A basis of normal tensors (BNT) for a total tensor \(A_{\mathrm{tot}}\) consists of \(g\) block tensors \((A_1, \ldots , A_g)\) with bond dimensions \((D_1, \ldots , D_g)\) such that each \(A_j\) is normal (Definition 2.5.5); at each system size \(N\geq 1\) the MPV of \(A_{\mathrm{tot}}\) lies in the span of the block MPVs \(\{ |V^{(N)}(A_j)\rangle \} _{j=1}^g\); and for sufficiently large \(N\) those block MPVs are linearly independent [ 17 , Definition IV.2 ] . In the canonical-form characterization of [ 13 , Section 2.3 ] , a BNT is a minimal family of representatives for \(\widetilde A_k^i=e^{i\phi _k}X_kA_j^iX_k^{-1}\) among the normal blocks.
Let \(A\) be an MPS tensor, let \(\Omega _K\) be the set of length-\(K\) words in the physical alphabet, and let \(\mathcal B_K=\bigoplus _{u\in \Omega _K}M_{D}(\mathbb {C})\) with its Hilbert–Schmidt direct-sum norm. Define
by applying the one-site left virtual map independently to each prefix fiber. Define
by applying the length-\(K\) tail virtual map independently to each final-site fiber. These definitions make sense without any nonemptiness assumption on the virtual dimension or on either configuration set.
When \(D\geq 1\) and \(\rho \in M_{D}(\mathbb {C})\) is positive definite, set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and define the centered virtual residual by
Assume \(D\geq 1\) and let \(\rho \in M_{D}(\mathbb {C})\) be positive definite. Set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and \(S_\infty =\mathcal K_\infty ^{-1}\). Suppose that \(\Gamma _{l+1}^{\mathrm{ES}}\), \(\Gamma _{K+l}^{\mathrm{ES}}\), and \(\Gamma _{K+l+1}^{\mathrm{ES}}\) are injective, and write
The three correction sandwiches in Theorem 12.3.5.25 are
Let \(h_R\) be the configuration-basis matrix of the orthogonal projection onto \(G_R(A)^\perp \). This matrix is positive semidefinite and represents the same projection on the Euclidean realization of the physical configuration space. It is the canonical choice in [ 225 , Lemma existenceinteraction ] .
For \(N{\gt}0\) and \(L\le N\), define the periodic parent-Hamiltonian ground space
Here the condition \(\psi |_{[i,i+L-1]}\in G_L(A)\) means that, for every choice of physical indices outside the window, the restriction of \(\psi \) to that window lies in \(G_L(A)\). The one-index notation \(G_L(A)\) denotes the local ground space \(\mathcal G_L\) of [ 17 , Section IV.C ] . The two-index notation \(\mathcal G_{N,L}(A)\) used here denotes the periodic intersection of those local constraints over all translated length-\(L\) windows on the \(N\)-site ring. When \(N=0\) or \(L{\gt}N\), set \(\mathcal G_{N,L}(A):=\top \) by convention.
For an MPS tensor \(A\), a projection \(P\), and \(L\geq 0\), let \(G_A^P(L)\) be the set of open-boundary vectors \(\Gamma _L^A(Y)\) with \(PYP=Y\). For \(P=I\) it is \(G_A(L)\).
Let \(Y\in M_{D\times E}(\mathbb {C})\), let \(V\colon \mathbb {C}^E\to \mathbb {C}^D\), and let \(A\) be an original tensor. Define \(C_rY=(V^\dagger A^\tau Y)_\tau \), where \(\tau \) runs over the original words of length \(r\). For \(\operatorname{tr}\rho \ne 0\), set \(K_\rho Z=Z\rho /\operatorname{tr}\rho \) on the square prefix boundary space. The residual limiting metric is \(K_{\infty ,r}=C_r^\dagger (\bigoplus _\tau K_\rho )C_r\). If \(Q_r=\sum _\tau A^{\tau \dagger }VV^\dagger A^\tau \), then
Here \(K_N^B=(\Gamma _N^B)^\dagger \Gamma _N^B\). The rectangular coordinates place the column index before the row index. All lengths may be zero. No letter intertwining or normalization is assumed.
- MPSTensor.correlatedTailMapES
- MPSTensor.boundaryFamilyEquiv_correlatedTailMapES_apply
- MPSTensor.correlatedTailMapES_adjoint_apply
- MPSTensor.inner_correlatedTailMapES
- MPSTensor.residualBoundaryGramLimit
- MPSTensor.residualBoundaryGramLimit_frobeniusEquivEuclidean_apply
- MPSTensor.blockedResidualBoundaryMapES_adjoint_comp_self
Let \(W\leq N\) and fix a site \(i\). Splitting the cyclically ordered sites into the first \(W\) sites and their complement gives
If \((\sigma ,\tau )\) is an active configuration and a complementary configuration, then their inverse image has value \(\sigma _r\) at the site \(i+r\) for \(0\leq r{\lt}W\), and value \(\tau _r\) at the site \(i+W+r\) for \(0\leq r{\lt}N-W\). The isometry and its inverse act by composition with this configuration bijection.
- MPSTensor.cyclicActiveBlockConfigEquiv
- MPSTensor.cyclicActiveBlockConfigLinearIsometryEquiv
- MPSTensor.cyclicActiveBlockConfigEquiv_symm_apply_window
- MPSTensor.cyclicActiveBlockConfigLinearIsometryEquiv_apply_apply
- MPSTensor.cyclicActiveBlockConfigLinearIsometryEquiv_symm_apply_apply
- MPSTensor.cyclicActiveBlockConfigEquiv_symm_apply_spectator
For matrices \(P_u\) indexed by \(\mathbb Z/m\mathbb Z\), define
If \(P_uA_i=A_iP_{u+1}\), both \(A_iX\) and \(XA_i\) belong to \(V_n\) whenever \(X\in V_{n+1}\). When the \(P_u\) are Hermitian, \(X\in V_n\) implies that \(X^*\) has the word degree \(n\), namely \(P_uX^*=X^*P_{u+n}\). These are the cyclic coordinates of [ 191 , Lemma 5 ] .
For an \(N\)-site periodic chain with \(N{\gt}0\), index the range-\(R\) local interactions by \(\mathbb Z/N\mathbb Z\). If \(s\in \mathbb Z/N\mathbb Z\) and \(i\in \{ 0,\ldots ,N-1\} \) is its standard representative, write
If \(L\leq N\) and \(i\in \{ 0,\ldots ,N{-}1\} \), identify the sites of the periodic chain with the disjoint union of the cyclic window \(i,i+1,\ldots ,i+L-1\) and its ordered complement \(i+L,i+L+1,\ldots ,i+N-1\), with all indices taken modulo \(N\).
For a family \((P_u)_{u\in \mathbb Z/m\mathbb Z}\), set
If \(P_uA_i=A_iP_{u+1}\), every word of length \(n\) belongs to \(W_n\). The family need not consist of projections for this conclusion. These degrees describe the cyclic sectors of [ 191 , Lemma 5 ] .
For \(\Lambda \in M_{d}(\mathbb {C})\) and an operator \(h\) on \(\ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\), the deformed interaction is
This is the operator written in [ 17 , Section IV.C ] as \(h'=(A^{-1}\otimes A^{-1})^\dagger h(A^{-1}\otimes A^{-1})\) for the two-site interaction of an isometric tensor network, and as \(h'=({\Lambda ^{-1}}^\dagger )^{\otimes k}h(\Lambda ^{-1})^{\otimes k}\) for a general physical perturbation.
Let \(S,I,J\) be finite sets and let the scalar field be \(\mathbb {R}\) or \(\mathbb {C}\). For a vector \(x\in \ell ^2(S\times I)\), its \(s\)th fiber is \(x_s=(x_{s,i})_{i\in I}\). For a linear operator \(G:\ell ^2(I)\to \ell ^2(J)\), define \(G^{\oplus S}\) by \((G^{\oplus S}x)_s=Gx_s\). The fiber projection itself is defined without a finiteness assumption.
Given a word \(w = (i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\), the word evaluation is the matrix product
The empty word evaluates to the identity, \(A^\varnothing = \mathbb {1}_D\). In tensor-network notation,
![\begin{tenkz}[cols=4, boundary=open, physical=up]
\tn[label pos=135, ports={90:physical:$i_1$}]{A} &
\tn[label pos=135]{A} & \tn[skin=dots]{} &
\tn[label pos=135, ports={90:physical:$i_L$}]{A}
\tnmark[form=bracket]{(1,1) .. (1,4)}{$L\text{ copies}$}
\end{tenkz}](tenkz_svg/tenkz-a034a758a71b7980.svg)
in which each black node denotes the same local tensor \(A\), the virtual legs remain open, and the physical legs are labelled by the word \((i_1,\ldots ,i_L)\).
For an arbitrary range-\(R\) matrix \(h\), let \(H_N(h)=\sum _{a=0}^{N-R}\iota _a(h)\), where \(\iota _a\) places \(h\) on the sites \(a,\ldots ,a+R-1\) of an \(N\)-site chain, with the sum empty when \(N{\lt}R\). For \(R\geq 1\) and \(N\geq R\), its Euclidean matrix action is the open interaction Hamiltonian. If \(h\) is the complement projection of \(G_R(A)\), it is the canonical open parent Hamiltonian. This is the nonwrapping interaction sum in [ 225 , equation (3.12) ] .
Fix an \(N\)-site Hilbert space and let \(h_i\) denote the nonwrapping interaction of length \(L\) starting at \(i\). For every \(m\geq 0\), set
All operators act on the same \(N\)-site space. The local interval ground projection in the proof of Theorem 2.1(i) is
Let \(R,k\geq 1\), put \(t=R-1\) and \(w=2t+k\), and let \(X\) be supported on \(k\) sites. On \(w\) sites set \(B=I_t\otimes X\otimes I_t\) and
Neither positivity nor self-adjointness of \(h\) is needed for this definition. It is the fixed local expression used in the limiting argument of [ 225 , Section 6 ] .
Let \(\rho \in M_{D}(\mathbb {C})\) satisfy \(\operatorname{tr}(\rho )\neq 0\). The normalized rank-one map associated with \(\rho \) is
If \(\operatorname{tr}(\rho )=1\), this becomes \(E_\infty ^\rho (B)=\operatorname{tr}(\rho B)I_D\), which is the limiting map in [ 194 , Equation (5.1) ] .
Suppose that \(\rho \) is positive definite and \(\operatorname{tr}(\rho )=1\). For \(n\in \mathbb {N}\), define
This is the trace-one transfer-remainder quantity formed from the factor and weighted norm in [ 194 , Equations (5.9)–(5.10) ] .
Let \(A\) be an MPS tensor, with \(A^\mu =v(\mu )^\dagger \) in the notation of [ 194 , Equations (5.2)–(5.3) ] . Its FNW transfer map is
Let \(\rho \in M_{D}(\mathbb {C})\) be positive definite and let \(\operatorname{tr}(\rho )\neq 0\). Equip \(M_{D}(\mathbb {C})\) with \(\langle X,Y\rangle _\rho =\operatorname{tr}(\rho X^\dagger Y)\). For a linear map \(F\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), write
The weighted remainder spectral radius is
where the remainder is viewed as a bounded operator on this weighted Hilbert space.
Let \(U\) and \(V\) be subspaces of a finite-dimensional complex Hilbert space. Their Friedrichs overlap is at most \(\eta \) when
for every \(x\in U\cap (U\cap V)^\perp \) and \(y\in V\cap (U\cap V)^\perp \).
Two tensors \(A\) and \(B\) of the same bond dimension \(D\) are gauge equivalent if there exists an invertible matrix \(X \in \mathrm{GL}_D(\mathbb {C})\) such that, for every \(i \in \{ 0,\ldots ,d{-}1\} \),
In tensor-network notation, (8) is represented by
![\begin{tenkz}[boundary=open]
\tn[skin=ring]{X} & \tn[ports={90:physical:$i$}]{A} &
\tn[skin=ring]{X^{-1}}
\end{tenkz}](tenkz_svg/tenkz-813ccc23975a5e90.svg)
where the black node denotes the tensor \(A\), the upper leg is the physical index \(i\), and the two red side nodes denote the gauge matrices acting on the virtual legs.
Two tensors \(A\) and \(B\) are gauge-phase equivalent if there exist \(X \in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \in \mathbb {C}\setminus \{ 0\} \) such that, for every physical index \(i\),
For a complex-linear map \(\mathcal L\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), let \(R(\mathcal L)\) be the matrix with entries
The operator \(\mathscr R(\mathcal L)\) on \(\ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\) is represented by the matrix \(R(\mathcal L)\) in the standard orthonormal basis. Thus the row–column pair \(((b,a),(e,c))\) is sent explicitly to the Choi row–column pair \(((c,e),(a,b))\).
Let \(\iota _L\colon (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\simeq \ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\) be the canonical computational-basis identification. The local ground space under this \(\ell ^2\) identification is \(G_L^{\mathrm{ES}}(A):=\iota _L(G_L(A)) \subseteq \ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\).
The map \(\Gamma _L\) in (1) is a \(\mathbb {C}\)-linear map from \(M_{D}(\mathbb {C})\) to \((\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\). In tensor-network notation,
![\begin{tenkz}[rows={wire}, cols=5, boundary=periodic]
\tn[skin=ring]{X} &
\tn[label pos=135, ports={90:physical:$\sigma_1$}]{A} &
\tn[label pos=135, ports={90:physical}]{A} & \tn[skin=dots]{} &
\tn[label pos=135, ports={90:physical:$\sigma_L$}]{A}
\tnmark[form=bracket]{(1,2) .. (1,5)}{$L$}
\end{tenkz}](tenkz_svg/tenkz-850da2de312b2b1e.svg)
The chain denotes the word tensor \(A^\sigma \) with open physical indices \(\sigma _1,\ldots ,\sigma _L\), and the red capsule denotes the boundary matrix \(X\) inserted on the closing virtual bond before taking the trace. The brace records the chain length \(L\).
Let \(U\colon M_{D}(\mathbb {C})\to \ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\) be Frobenius vectorization in column–row coordinates. The Hilbert-space boundary map is
Thus \(\Gamma _L^{\mathrm{ES}}\) has a Euclidean coordinate-space domain, whereas the original map \(\Gamma _L\) has matrix domain \(M_{D}(\mathbb {C})\).
Fix \(p\geq 1\), \(M\geq 2\), and an open chain of length \(N=pM\). For \(0\leq n\leq M\), let \(G_n\) be the ground projection of the range-\(2p\) parent Hamiltonian on \([0,pn)\), extended by the identity on the remaining sites. For \(0\leq n{\lt}M\), let \(Q_n\) be the ground projection of the same interaction restricted to \([\max \{ 0,p(n-1)\} ,p(n+1))\). Thus \(Q_0=I\), while for \(n\geq 1\) the interval Hamiltonian is a single parent interaction.
The structure used below records only the idempotent-product consequence of the parent commuting Hamiltonian condition of [ 13 , Appendix D.2 ] . It consists of commuting idempotent endomorphisms \(P_{AX}\) and \(P_{XB}\) such that, for the idempotent \(P_K\) associated to \(K\), \(P_{AX}\circ P_{XB}=P_K\). In the source, this is the equation obtained from local projectors \(Q_{AX}\) and \(Q_{XB}\) satisfying \([Q_{AX},Q_{XB}]=0\) and \(K_{AXB}=(K_{AX}\otimes H_B)\cap (H_A\otimes K_{XB})\).
An MPS tensor \(A\) has an invariant projection if there exists an orthogonal projection \(P\) with \(P\neq 0\), \(P\neq \mathbb {1}\), and \((\mathbb {1}-P)A^iP=0\) for all \(i\). A tensor is irreducible if it has no nontrivial invariant projection (Definition 7.1.1.1).
For a fixed chain length \(N\), this condition is the conjunction of the nearest-neighbor commutation equation, the zero-energy equation for the MPS vector, and the ground-space spanning equation:
The source statement in [ 13 , Theorem 3.10(iii) ] quantifies this condition over all \(N{\gt}2\).
This is the source quantification in [ 13 , Theorem 3.10(iii) ] . For every \(N{\gt}2\), the fixed-chain condition of Definition 12.1.16 holds:
This condition records the ground-space spanning equation, but does not prove it for a concrete canonical-form tensor.
The parent Hamiltonian with block length \(L\) on \(N\) sites has commuting local terms if \(h_i h_j=h_j h_i\) for all \(i,j\in \{ 0,\ldots ,N{-}1\} \). This records the translated commutation equation \([\tau _j(P_L),P_L]=0\) from [ 13 , Definition 3.9 ] ; the same source definition also requires the ground space to be spanned, for \(N{\gt}L\), by the periodic MPS vectors of the BNT components.
Given an endomorphism \(P_K\) and sets of observables \(\mathcal{O}_A\), \(\mathcal{O}_B\), regions \(A\) and \(B\) are decorrelated w.r.t. \(K\) when \(P_K\circ O_A\circ (\mathbb {1}-P_K)\circ O_B\circ P_K=0\) for all \(O_A\in \mathcal{O}_A\) and \(O_B\in \mathcal{O}_B\).
The nearest-neighbor commuting condition is the length-\(2\) specialization of the preceding commutation equation. For every \(i,j\in \{ 0,\ldots ,N{-}1\} \), one has \(h_i h_j=h_j h_i\). The source definition [ 13 , Definition 3.9 ] also requires the corresponding parent Hamiltonian to have \(L=2\) and to have the prescribed ground-space spanning property for \(N{\gt}2\).
This condition records the commutation equation and the zero-energy equation for the nearest-neighbor local terms:
This is the zero-energy part of the condition appearing in [ 13 , Theorem 3.10(iii) ] : the MPS vector has zero energy for the translated two-site local terms, and those terms commute. The missing source assertion is that the whole parent-Hamiltonian ground space is spanned by the corresponding periodic MPS vectors from the BNT components, as recorded in Definition 12.1.15.
A family \((U_j)_j\) satisfies the joint isometry condition if
Diagrammatically, contracting \(U_j\) with the conjugate of \(U_{j'}\) over the physical index identifies the virtual legs pairwise:
![\begin{tenkz}[rows={ket,bra}, cols=1, west=open, east=open]
\tn[skin=box, label pos=n]{U_j} \\
\tn[skin=box, label pos=s, conjugate]{\overline{U_{j'}}}
\end{tenkz}](tenkz_svg/tenkz-af772fecc88fb745.svg)
\( = \delta _{j,j'}\, \) ![\begin{tenkz}[rows={wire,wire}, cols=1, west=open, east=cup,
bonds=none]
\tn[skin=none]{} \\ \tn[skin=none]{}
\end{tenkz}](tenkz_svg/tenkz-0b7c3272ff1f9862.svg)
\( \otimes \) ![\begin{tenkz}[rows={wire,wire}, cols=1, west=cup, east=open,
bonds=none]
\tn[skin=none]{} \\ \tn[skin=none]{}
\end{tenkz}](tenkz_svg/tenkz-94361ab6b0bcff3a.svg)
Each block satisfies unit pair-index orthonormality (\(j=j'\)), and the cross-block sums between distinct blocks vanish (\(j\ne j'\)), as in [ 13 , Theorem 3.11 ] .
A tensor \(A\) is \(L\)-block injective if
An injective tensor is \(1\)-block injective.
For \(K,L\in \mathbb {N}\), the left boundary map is the \(\mathbb {C}\)-linear map
defined for every family \(Z\colon (\{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} )\to M_{D}(\mathbb {C})\) by
where \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) is the prefix and \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \) is the suffix. When \(Z_\tau =A^\tau X\) the output equals the standard \(\Gamma _{K+L}(X)\).
For \(a\in \mathbb Z\) and \(N\geq 0\), put \(I(a,N)=\{ a,a+1,\ldots ,a+N-1\} \). The site numbering \(i\mapsto a+i\) identifies configurations on \(I(a,N)\) with length-\(N\) configurations. Reindexing both matrix coordinates gives a star-algebra equivalence \(C_{a,N}:\mathcal A_{I(a,N)}\to M_{d^N}(\mathbb C)\). Explicitly, \(C_{a,N}(X)_{\sigma \tau }=X_{\widetilde\sigma \widetilde\tau }\), where \(\widetilde\sigma (a+i)=\sigma _i\). This includes \(N=0\) and the one-dimensional algebra on the empty interval.
For each site index \(i\in \{ 0,\ldots ,N{-}1\} \), the translated local term is determined by
For tensors \(A\) and \(B\) with a common physical alphabet and possibly different bond dimensions, define the insertion of a \(k\)-site matrix \(X\) by
Here \(Y\) is a rectangular matrix from the virtual space of \(A\) to that of \(B\). For \(A=B\), this is the ordinary observable insertion.
For MPS tensors \(A\) of bond dimension \(D_1\) and \(B\) of bond dimension \(D_2\), possibly with \(D_1\neq D_2\), the rectangular mixed transfer operator is the map \(F_{AB}^{\mathrm{rect}}: M_{D_1\times D_2}(\mathbb {C})\to M_{D_1\times D_2}(\mathbb {C})\) defined by
Suppose \(\sum _i A_i^\dagger A_i=I\), \(\rho \geq 0\), \(E_A(\rho )=\rho \), and \(\operatorname {tr}\rho \neq 0\). The functionals \(\eta _{A,\Lambda }\) form a compatible family of normalized positive contractive local states. In particular, \(\eta _{A,I(a,N)}(X)=\eta _A(C_{a,N}X)\) for every \(N\geq 0\). This is the compatible local interpretation of [ 225 , Section 3, equations (3.1)–(3.2b) ] .
For a finite region \(\Lambda \subset \mathbb Z\), include an observable \(X\in \mathcal A_\Lambda \) into a consecutive interval \(I(a,N)\) containing \(\Lambda \) and put
To define the functional before proving independence, one may use \(I(-r,2r+1)\) with \(r=\max _{x\in \Lambda }|x|\) and \(r=0\) when \(\Lambda =\varnothing \). This gives a complex-linear functional without a normalization assumption on \(A\) or \(\rho \).
For an MPS tensor \(A\), let \(p_A(N)\) be the matrix of the orthogonal projection onto \(G_A(N)\) in numbered coordinates on \(N\) sites. It is a self-adjoint idempotent. Its inclusion \(j_{a,N}(p_A(N))\) into the quasi-local algebra on the interval beginning at \(a\) is also a self-adjoint idempotent, including when \(N=0\).
The matrix product vector (MPV) of a tensor \(A\) at system size \(N\) is the vector
Equivalently, for a configuration \(\sigma = (i_1, \ldots , i_N) \in \{ 0,\ldots ,d{-}1\} ^N\), we write
The coefficient function \(\sigma \mapsto V^{(N)}(A)_\sigma \) gives the components of the vector in (2); the displayed ket is the corresponding vector in \((\mathbb {C}^d)^{\otimes N}\). For a general word \(w\), we also write \(c_w(A) := \operatorname{tr}(A^w)\). The MPV family generated by \(A\) is the collection \(\mathcal{V}(A) = \bigl\{ |V^{(N)}(A)\rangle \bigr\} _{N \ge 1}\). The coefficient \(V^{(N)}(A)_\sigma \) is the periodic contraction
![\begin{tenkz}[cols=4, boundary=periodic, physical=up]
\tn[label pos=135, ports={90:physical:$i_1$}]{A} &
\tn[label pos=135]{A} & \tn[skin=dots]{} &
\tn[label pos=135, ports={90:physical:$i_N$}]{A}
\tnmark[form=bracket]{(1,1) .. (1,4)}{$N\text{ copies}$}
\end{tenkz}](tenkz_svg/tenkz-b2f26c946b5d0c00.svg)
of \(N\) copies of the local tensor \(A\), with the outer virtual legs closed by the trace.
Let \(\mathcal H\) be a complex Hilbert space. A nested ground-projection family is a sequence \((G_n)_{n\geq 0}\) of orthogonal projections such that
Its \(n\)th martingale difference is
A tensor \(A\) of physical dimension \(d\) and bond dimension \(D\) is a normal tensor if, after the spectral-radius normalization of [ 13 ] , (i) \(A\) admits no nontrivial invariant orthogonal projection and (ii) the associated CPM \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) has spectral radius \(r(\mathcal{E}_A)=1\) and peripheral spectrum \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).
Let \(g\mapsto U_g\) be a linear representation of a monoid on the physical space. A tensor \(A\) is on-site symmetric up to a phase if for every \(g\) and every chain length \(N\) there is a scalar \(\lambda _{g,N}\) with \(U_g^{\otimes N}|\psi _N\rangle =\lambda _{g,N}|\psi _N\rangle \), where \(|\psi _N\rangle \) is the matrix product vector of \(A\) of length \(N\). The scalars may depend on both \(g\) and \(N\). Exact on-site symmetry is the case \(\lambda _{g,N}=1\). This is the symmetry of the state in [ 236 , lines 655–657, 2590–2593 ] .
For every \(L\in \mathbb {N}\), let \(\mathcal U_L(A)\) be the subspace of the \((L+1)\)-site Hilbert space whose corresponding functions satisfy the left ground condition: every fixed-final-site restriction belongs to \(G_L(A)\). For every \(K,L\in \mathbb {N}\), let \(\mathcal V_{K,L}(A)\) be the subspace of the \((K+L)\)-site Hilbert space whose corresponding functions satisfy the tail ground condition: every fixed-prefix restriction belongs to \(G_L(A)\). These definitions include \(L=0\). Let \(P^{\mathrm{left}}_L(A)\) and \(P^{\mathrm{tail}}_{K,L}(A)\) be the orthogonal projectors onto \(\mathcal U_L(A)\) and \(\mathcal V_{K,L}(A)\), respectively.
Assume \(D\geq 1\), \(l{\gt}0\), and that \(A\) is \(l\)-block-injective. Fix \(K{\gt}0\), set \(n=K+l\) and \(n_l=l+1\), and work in the Hilbert space on \(\Lambda _{n+1}\). The physical open-chain martingale difference is
The first projector imposes the ground condition on the left \(n\) sites while acting on all \(n+1\) sites. Block injectivity identifies these MPS spaces with the corresponding open-parent Hamiltonian kernels. The condition \(K{\gt}0\) excludes the undersized volume, where an interaction of length \(l+1\) has no nonwrapping window and its physical ground projector is the identity.
Let \(R,P{\gt}0\). On the Hilbert space of an \(N\)-site chain, index the nonwrapping canonical interactions by \(s\in \mathbb {Z}/P\mathbb {Z}\). Writing \(\bar s\) for the representative in \(\{ 0,\ldots ,P-1\} \), define
The period indexes the interactions; the physical chain still has \(N\) sites.
For \(L,N\in \mathbb {N}\), let
The compatible length-\(N\) open-chain parent Hamiltonian of interaction length \(L\) is
Thus no summand crosses the two endpoints of the interval. For \(L=l+1\), the starting sites are \(0,\ldots ,N-l-1\), in agreement with [ 225 , Equation (3.12) ] .
For \(L,l,n,N\in \mathbb {N}\), the Hamiltonian on the last \(l\) sites of the prefix \([0,n)\), regarded as an operator on the \(N\)-site chain, is
Its local windows lie in the nonwrapping interval \([n-l,n)\).
On a three-site space \(\mathcal{H}_A\otimes \mathcal{H}_X\otimes \mathcal{H}_B\), let \(Q_{AX}\) and \(Q_{XB}\) be two-site projectors. Their local actions on the three-site space are the operators
The algebraic part of the overlapping nearest-neighbor commutation condition is
after these two local actions are viewed on the common three-site space. This is the local-support part of the parent commuting Hamiltonian condition in [ 13 , Definition D.2 ] . The cyclic windows beginning at \(i\) and \(i+1\) realize the two supports \(AX\) and \(XB\), and hence overlap at the middle site.
The source local condition also fixes a three-site subspace \(K_{AXB}\) and imposes the kernel-intersection equation \(K_{AXB}=\ker ((Q_{AX})_{AX})\cap \ker ((Q_{XB})_{XB})\). Forgetting this intersection equation gives the algebraic overlapping two-site commutation condition displayed above.
The two lift maps also distribute over complementary projectors:
Consequently, if the projectors \(Q_{AX}\) and \(Q_{XB}\) commute on the three-site space, then their complementary projectors \(P_{AX}=1-Q_{AX}\) and \(P_{XB}=1-Q_{XB}\) commute after the same lifts. This is the elementary algebraic passage used when one moves between the \(Q\)-projectors of [ 13 , Appendix D.2 ] and the complementary ground-space projectors \(P\). Since the original two projectors are idempotent, the complementary projectors are idempotent as well; hence the pair \(P_{AX},P_{XB}\) satisfies the same idempotent-and-commutation condition on the three-site space.
- MPSTensor.axPairCfg
- MPSTensor.xbPairCfg
- MPSTensor.replaceAXCfg
- MPSTensor.replaceXBCfg
- MPSTensor.leftPairLift
- MPSTensor.rightPairLift
- MPSTensor.leftPairLift_sub
- MPSTensor.rightPairLift_sub
- MPSTensor.leftPairLift_one_sub
- MPSTensor.rightPairLift_one_sub
- MPSTensor.HasOverlappingTwoSiteCommutation
- MPSTensor.HasAppendixD2ParentCommutingHamiltonian
- MPSTensor.HasOverlappingTwoSiteCommutation.left_complement_idempotent
- MPSTensor.HasOverlappingTwoSiteCommutation.right_complement_idempotent
- MPSTensor.HasOverlappingTwoSiteCommutation.commute_complement_lifts
- MPSTensor.HasOverlappingTwoSiteCommutation.complement
- MPSTensor.adjacent_twoSite_cyclicWindowsOverlap
The transfer map is primitive in the sense of [ 233 ] if there is a positive integer \(q\) such that
for every nonzero \(\varphi \in \mathbb {C}^D\). The same \(q\) works for all nonzero vectors.
Let \(B\) be a canonical-form tensor with BNT components \(A_1,\ldots ,A_g\). The source parent-Hamiltonian spanning condition is that, for every \(N{\gt}L\), \(\ker H_L^{(N)}(B)=\operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \). This is the ground-space clause in [ 13 , Definition 3.9 ] . It is separate from the commutation equation \([\tau _j(P_L),P_L]=0\) and from the zero-energy equation for a single periodic MPS vector.
Let \(h\) act on the physical space of \(R\) consecutive sites. For \(R\leq N\), let \(U_i\) be the unitary which places the cyclic interval of length \(R\) starting at \(i\) first and the remaining sites second. Define
When \(R{\gt}N\), set each translated interaction, and hence their sum, to zero. This agrees with the convention for the canonical parent Hamiltonian.
For an observable \(O\) on a block of \(L\) physical spins, define the linear map on the virtual matrix space by
This transfer map also depends linearly on the observable \(O\). Diagrammatically, \(O\) is inserted between the two layers of the doubled chain on its \(L\) sites:
![\begin{tenkz}[rows={ket,op,bra}, cols=3, bonds=none]
\tn[name=k1, label pos=n,
ports={180:virtual, 0:virtual, 270:physical}]{A} &
\tn[name=kd, skin=dots, ports={180:virtual, 0:virtual}]{} &
\tn[name=k3, label pos=n,
ports={180:virtual, 0:virtual, 270:physical}]{A} \\
\tn[name=O, skin=pill, wide=3,
ports={90@1:physical, 90@3:physical,
270@1:physical, 270@3:physical}]{O} \\
\tn[name=b1, label pos=s, conjugate,
ports={180:virtual, 0:virtual, 90:physical}]{\overline{A}} &
\tn[name=bd, skin=dots, ports={180:virtual, 0:virtual}]{} &
\tn[name=b3, label pos=s, conjugate,
ports={180:virtual, 0:virtual, 90:physical}]{\overline{A}}
\tnwire{k1.0}{kd.180}
\tnwire{kd.0}{k3.180}
\tnwire{b1.0}{bd.180}
\tnwire{bd.0}{b3.180}
\tnwire{k1.270}{O.90@1}
\tnwire{k3.270}{O.90@3}
\tnwire{O.270@1}{b1.90}
\tnwire{O.270@3}{b3.90}
\tnwire{k1.180}{open w}
\tnwire{b1.180}{open w}
\tnwire{k3.0}{open e}
\tnwire{b3.0}{open e}
\end{tenkz}](tenkz_svg/tenkz-b03436980ea2acd9.svg)
This is the observable transfer map of [ 13 , lines 490–496 ] .
Let \(\varphi \) be a continuous complex-linear functional on the quasi-local algebra that is nonnegative on every element \(Y^*Y\). Its restriction to the numbered matrix algebra on an interval is the positive functional \(f_{a,N}(X)=\varphi (j_{a,N}(X))\).
For \(K,L\in \mathbb {N}\) and a fixed suffix configuration \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \), the prefix restriction is the \(\mathbb {C}\)-linear map \(P_\tau \colon \mathbb {C}^{\{ 0,\ldots ,d{-}1\} ^{K+L}}\to \mathbb {C}^{\{ 0,\ldots ,d{-}1\} ^K}\) defined by \((P_\tau \psi )(\sigma )=\psi (\sigma ,\tau )\) for \(\sigma \colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \). This is the left-sided analogue of the suffix restriction of Definition 11.5.5.5.
For a tensor \(A\), a matrix \(\rho \) with nonzero trace, and an observable \(X\) on \(k\) sites, write \(E_X(Y)=\sum _{\sigma ,\tau }X_{\sigma \tau }A^\tau Y(A^\sigma )^*\) and \(\eta _A(X)=\operatorname {tr}(E_X(\rho ))/\operatorname {tr}\rho \). This is the local expectation appearing in [ 225 , Section 6 ] .
Assume \(D{\gt}0\). A tensor \(A\) together with a matrix \(\rho \in M_{D}(\mathbb {C})\) is primitive if
where \(P\) is the fixed-point projection associated to \(\rho \). When the choice of \(\rho \) is irrelevant, we simply say that \(A\) is a primitive MPS tensor. This condition combines a complementary spectral gap with a nonzero This condition combines a complementary spectral gap with a nonzero positive semidefinite fixed point. When \(\rho {\gt}0\), this implies strong irreducibility and uniform vector spreading (Definition 6.1.1.3) by Theorem 6.1.1.15.
Let \(d,L{\gt}0\) and let \(\mathcal B_L:\mathcal A_{d^L}\to \mathcal A_d\) be the site-blocking isomorphism. A continuous functional \(\omega \) on the blocked chain defines the original-lattice functional \(T_L\omega =\omega \circ \mathcal B_L^{-1}\). This is the state transport associated with the regrouping of sites in [ 225 , lines 825–836 ] .
A compatible family of finite-region states consists of complex-linear functionals \(f_\Lambda \colon \mathcal A_\Lambda \to \mathbb {C}\), indexed by finite regions \(\Lambda \Subset \mathbb {Z}\), such that, for every inclusion \(\Lambda \subseteq \Gamma \) and every \(X\in \mathcal A_\Lambda \),
Relabelling configurations defines a star-algebra equivalence
This is the finite-region translation associated with the lattice translation \(i\mapsto i+a\).
If \(\Lambda \subseteq \Gamma \) are finite regions, define
Here product configurations are identified with configurations on \(\Gamma \). This is a unital star-algebra homomorphism.
For \(d{\gt}0\) and every \(a\in \mathbb Z\), algebraic-local translation extends to a star-algebra automorphism
of the quasi-local algebra. On the dense algebraic-local subalgebra and on finite-region observables it satisfies
Let \(d{\gt}0\), and let \(\mathcal A_d\) denote the quasi-local observable algebra. Its state space consists of all continuous complex-linear functionals \(\omega \) such that \(\| \omega \| =1\), \(\omega (I)=1\), and \(\omega (X^*X)\ge 0\) for every \(X\in \mathcal A_d\). A state is pure if it is an extreme point of this set, viewed as a real convex set. Translation invariance is not required of these functionals. These are the state and purity conventions in [ 225 , Section 1 and Section 3 ] .
For a finite region \(\Lambda \subset \mathbb Z\) and \(a\in \mathbb Z\), define
Thus
If \(\Lambda \subseteq \Gamma \), then \(\Lambda +a\subseteq \Gamma +a\).
Let \(d{\gt}0\) and let \(h\) be an observable on \(R\) consecutive sites. Define \(\mathcal F_h\) to be the set of all states \(\phi \) of the quasi-local algebra such that \(\phi (j_{a,R}(h))=0\) for every \(a\in \mathbb {Z}\). Translation invariance of \(\phi \) is not required. For a positive parent interaction, this is its zero-energy state set, as in [ 225 , Section 1 and Theorem 1.1 ] .
Decode a blocked configuration \(v\) of length \(M\) into its \(ML\) original sites and append an original tail \(\tau \) of length \(r\). This gives a bijection \(e_{M,r}\) from the pairs \((v,\tau )\) to the original configurations of length \(ML+r\), including zero lengths.
For a rectangular map \(V:\mathbb C^E\to \mathbb C^D\), let \(J_V(Y)=YV^*\) on the Hilbert space of rectangular matrices, with the Hilbert–Schmidt norm. Suppose that \(V^*(A^{[L]})^i=B^iV^*\) for every blocked letter \(i\). Then the residual boundary map satisfies \(\mathcal F^{N,r}_{A,B,V}=\Gamma ^A_{NL+r}J_V\) for every \(N,r\geq 0\). In particular, the boundary trace uses \(YV^*\) in the original site order.
Decode the blocked prefix \(u\) of length \(K\) and blocked middle \(v\) of length \(M\) into configurations of lengths \(KL\) and \(ML\). Concatenation with the original tail \(\tau \) gives a configuration on \(KL+ML+r\) sites. This bijection is the full residual-window configuration bijection followed by the identification \((K+M)L=KL+ML\), and it sends \((u,v,\tau )\) to the concatenation of the three configurations in that order. All zero lengths are included. These are the physical covering coordinates of [ 225 , Section 6 ] .
With \(G_{N,r}=\Gamma _{N,r}^\dagger \Gamma _{N,r}\), define
Whenever \(Q_r\) is the exact tail Gram matrix, \(F_{N,r}-R_\rho =G_{N,r}-K_{\infty ,r}\). The inverse equations for \(H_{N,r}\) hold under the hypotheses of the following theorem.
Let \(\mathcal V=\bigsqcup _j\{ j\} \times [D_j]\times [D_j]\) be the disjoint union of the within-sector virtual pairs, and let \(V\colon \mathcal V\to \mathbb {C}^d\) be the common physical map whose matrix elements are \(V_{i,(j,\alpha ,\beta )}=(U_j^i)_{\alpha ,\beta }\). For each sector insert the vector \(|\varphi _j\rangle =\sum _b\lambda _{j,b}|b,b\rangle \) on the middle virtual bond, apply \(V\) on both sites, and sum the resulting physical vectors. This defines the equal-sector two-site embedding
Unequal-sector virtual bonds do not occur. The corresponding physical support operator \(P_{2,\mathrm{res}}\) is the orthogonal projector onto \(G_2(\bigoplus _j B_j)\).
On \(M_{D\times E}(\mathbb {C})\) with its Hilbert–Schmidt inner product, define \(R_\rho Y=Y\rho /\operatorname{tr}\rho \) and \(\mathcal QY=QY\). If \(E{\gt}0\) and \(\rho {\gt}0\), then \(R_\rho \) is self-adjoint and invertible, with \(R_\rho ^{-1}Y=(\operatorname{tr}\rho )Y\rho ^{-1}\). If \(Q\) is an orthogonal projection, then so is \(\mathcal Q\). Right multiplication commutes with every row operator \(\mathcal Q\).
- MPSTensor.rectangularRightGramMetric
- MPSTensor.rectangularRightGramMetric_apply
- MPSTensor.rectangularRightGramMetric_isUnit
- MPSTensor.rectangularRightGramMetric_inverse_apply
- MPSTensor.rectangularRightGramMetric_commute_rowProjection
- MPSTensor.rectangularRightGramMetric_isSelfAdjoint
- MPSTensor.rectangularRowProjection
- MPSTensor.rectangularRowProjection_apply
- MPSTensor.rectangularRowProjection_isStarProjection
Let \(A\) have physical dimension \(d\) and bond dimension \(D\), let \(B\) have physical dimension \(d^L\) and bond dimension \(E\), and let \(V\in \mathbb {C}^{D\times E}\). For nonnegative integers \(K,M,r\), use physical configurations \((u,v,\tau )\) of respective lengths \(K,M,r\), with the first two in the blocked alphabet. Write \(\mathbf x=(X_u)_u\) for a family in \(\mathbb {C}^{D\times E}\) and \(\mathbf y=(Y_\tau )_\tau \) for a family in \(M_{E}(\mathbb {C})\). Define
The family coordinates are ordered as \((u,\text{column},\text{row})\). Decoding the blocked letters and concatenating \(u,v,\tau \) gives a bijection with the original configurations of length \((K+M)L+r\), including zero lengths. These are the residual coordinates for the overlap argument in [ 225 , Lemma 6.1(ii) ] .
For lengths \(K,M,r\geq 0\), the left-window coordinate bijection sends a pair \((\tau ,\sigma )\) of an original tail \(\tau \) of length \(r\) and a blocked configuration \(\sigma =u\mathbin {\frown }v\) of length \(K+M\) to \((u,(v,\tau ))\).
For the right, left, and full window maps \(T,L_1,C\) and the two virtual factors \(J_1,C_r\), set
where \(H_{K+M,r}\) is the supported full-boundary Gram inverse.
The two maps to the common family of square virtual boundaries have fibers \(\mathcal A\mathbf x=(V^*A^\tau X_u)_{u,\tau }\) and \(\mathcal B\mathbf y=(Y_\tau B^u)_{u,\tau }\). They are finite direct sums followed by isometric coordinate rearrangements.
Let \(P\) be a sector decomposition. In addition to spanning and eventual linear independence, the canonical form specifies irreducible left-canonical representatives, self-overlap normalization, pairwise gauge-phase separation, and weight normalization. Explicitly, each basis block has positive bond dimension, is irreducible, and is left-canonical; the normalized self-overlap of every basis block converges to \(1\); the basis MPV family is eventually linearly independent; distinct same-dimension basis blocks are not gauge-phase equivalent after identifying equal bond dimensions; and the [ 13 , Section 2.3 ] canonical-form normalization holds:
The MPV expansion takes the two-layer form
matching [ 13 , Section 2.3 ] and [ 17 , Definition IV.2 ] . The canonical-form normalization accommodates equal-modulus and repeated-copy sectors, such as \(C \oplus (-C)\) and \(C \oplus (1/2) C\).
A sector-weight tuple over \(g\) basis blocks consists of multiplicities \(r_j {\gt} 0\), nonzero weights \(\mu _{j,q}\) for \(q \in \{ 1,\ldots ,r_j\} \), and the sector coefficients
A sector decomposition consists of the basis blocks \((A_j)_{j=1}^g\) together with these multiplicities and weights. If \(A_j\) has bond dimension \(D_j\), then the flattened copy family has total bond dimension
- MPSTensor.SectorWeightData
- MPSTensor.SectorDecomposition
- MPSTensor.SectorDecomposition.toTensor
- MPSTensor.SectorDecomposition.totalDim
- MPSTensor.SectorDecomposition.flatDim_flatIndexEquiv
- MPSTensor.SectorDecomposition.flatWeight_flatIndexEquiv
- MPSTensor.SectorDecomposition.flatBasis_flatIndexEquiv_heq
For a finite list of one-site observables \((X_1,\ldots ,X_n)\), define \(O_{\mathrm{rev}}=X_n\otimes \cdots \otimes X_1\). The empty product is the identity on the zero-site space. Equivalently, \(O_{\mathrm{rev}}(X_1,X_2,\ldots ,X_n) =O_{\mathrm{rev}}(X_2,\ldots ,X_n)\otimes X_1\). This records the physical order required by trace duality of the source-written Heisenberg composition in [ 225 , equations (3.5)–(3.6) ] .
Transport the tail and left boundary maps through the canonical Euclidean identifications on their virtual matrix families and physical coefficient spaces. Denote the resulting continuous maps by \(\Gamma ^{\mathrm{tail}}_{K,L}\) and \(\Gamma ^{\mathrm{left}}_{K,L}\).
Let \(V:\mathbb C^D\to \mathbb C^d\otimes \mathbb C^D\) be a linear map, with the physical coordinate written first. Define its physical matrix slices by
This is the coordinate convention for the supplied representation in [ 225 , equation (3.6) ] .
For \(K,L\in \mathbb {N}\), the tail boundary map is the \(\mathbb {C}\)-linear map
defined for every family \(Y\colon (\{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} )\to M_{D}(\mathbb {C})\) by
where \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) is the prefix and \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \) is the suffix. When \(Y_u=X A^u\) the output equals the standard \(\Gamma _{K+L}(X)\).
For every type \(\alpha \), evaluation at the three coordinates gives the canonical equivalence
Its forward map is
Writing \(\widehat M_R\) for the lift of an operator \(M_R\) supported on \(R\in \{ AX,XB,A,B\} \), the four lifts to \(H_A\otimes (H_X\otimes H_B)\) are
Identities (21) and (22) use the canonical identification \((H_A\otimes H_X)\otimes H_B\cong H_A\otimes (H_X\otimes H_B)\).
For a Hermitian operator \(P\), let \(P_{AX}\) and \(P_{XB}\) be the support projectors of \(\operatorname{tr}_B P\) and \(\operatorname{tr}_A P\), respectively, as in [ 13 , Appendix D.2, lines 2225–2229 ] . Write \(\widehat P_{AX}\) and \(\widehat P_{XB}\) for their lifts to the tripartite space.
The two reduced operators are \(\rho _{AX}=\operatorname{tr}_B P\) and \(\rho _{XB}=\operatorname{tr}_A P\).
There are orthogonal projectors \(P_{AX}\) and \(P_{XB}\) on the two local factors whose lifts commute and whose ranges satisfy the source ground-space intersection condition \(\operatorname{ran}P=\operatorname{ran}\widehat P_{AX}\cap \operatorname{ran}\widehat P_{XB}\). Thus \(Q_{AX}=\mathbb {1}-P_{AX}\) and \(Q_{XB}=\mathbb {1}-P_{XB}\) are exactly the commuting parent terms of [ 13 , Definition D.2, lines 2205–2218 ] .
Given a group representation \(U : G \to \mathrm{GL}_{d}(\mathbb {C})\) on the physical index, the \(g\)-twisted tensor is defined, for each \(i \in \{ 0,\ldots ,d{-}1\} \), by
Let \(t=(t_{ij})\) and \(z=(z_{ij})\) be real matrices indexed by a finite set of \(k\) vertices. Set \(A^{ij}=t_{ij}E_{ij}\). For endpoints \(a,f\), define the four-site vector
Write \(\mathbf1\) for the mark with all entries one and, for a real function \(u\) on the vertices, \(u\otimes \mathbf1\) and \(\mathbf1\otimes u\) for the marks \(z_{be}=u_b\) and \(z_{be}=u_e\). For \(X\in M_{k}(\mathbb {C})\), the four-site boundary vector of \(A\) has the coordinates
The map
is a complex linear isometric isomorphism. Its inverse is
Thus the weighted matrix space has complex Hilbert dimension \(D^2\). This Euclidean realization is not asserted as a result of Fannes, Nachtergaele, and Werner.
Assume \(\operatorname {tr}(\rho )\neq 0\) and let \(\mathcal K_\infty =\mathscr R(P_\rho )\). Define
Let \(P_i\) be a finite family of symmetric projections and let \(H=\sum _iP_i\). Suppose the anticommutator forms obey
with row sums \(\sum _{j\ne i}c_{ij}\le 1\), column sums \(\sum _{i\ne j}c_{ij}\le 1\), and \(\gamma \le 1\). Then \(H^2\ge \gamma H\) as a quadratic form.
Let \(Q_{AX}\) and \(Q_{XB}\) be the two-site coefficient-space transports of source projectors satisfying the Definition D.2 local condition on \(A X B\). If these transported projectors agree with the Appendix B coefficient representatives,
then \(\widehat Q_{AX}\) and \(\widehat Q_{XB}\) satisfy the algebraic overlapping two-site condition of Definition 12.1.2. Consequently, \(h_0^{(3)}(A,2)h_1^{(3)}(A,2) = h_1^{(3)}(A,2)h_0^{(3)}(A,2)\).
If the Appendix B \(AX\) and \(XB\) coefficient representatives satisfy the lifted Definition D.2 commutator
then the corresponding first two translated length-two local terms on the three-site window commute: \(h_0^{(3)}(A,2)h_1^{(3)}(A,2) = h_1^{(3)}(A,2)h_0^{(3)}(A,2)\). The same conclusion holds if one supplies a Definition D.2-shaped local condition whose two operators are these coefficient-space representatives, rather than the source projectors themselves.
Suppose the lifted \(AX\) and \(XB\) representatives commute:
Then \(\widehat Q_{AX}\) and \(\widehat Q_{XB}\) satisfy the algebraic overlapping two-site condition of Definition 12.1.2.
If the Appendix B \(AX\) and \(XB\) coefficient representatives satisfy the overlapping two-site commutation datum
then the corresponding first two translated length-two local terms on the three-site window commute: \(h_0^{(3)}(A,2)h_1^{(3)}(A,2) = h_1^{(3)}(A,2)h_0^{(3)}(A,2)\).
On the three-site \(A,X,B\) window, the two adjacent length-two local terms are the two local lifts
Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), where \(\mu _j\ne 0\). Suppose that, for every \(N{\gt}L\), \(G_L^{(N)}(B) =\bigvee _{j=0}^{r-1}\operatorname{span}\{ V^{(N)}(A_j)\} \). Then the source parent-Hamiltonian spanning equation holds: \(\ker H_N(B,L) =\operatorname{span}\{ V^{(N)}(A_j):j=0,\ldots ,r-1\} \). It is enough to prove the two equations
Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\). Suppose \(N{\gt}0\), \(L\le N\), and the periodic chain constraints for \(B\) split into the block constraints:
If every block chain space is contained in its periodic MPS line,
then the zero-energy space of the parent Hamiltonian is contained in the BNT vector span:
Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), with every \(\mu _j\ne 0\). Suppose the physical alphabet and all block dimensions are nonempty. Assume that the block family is irreducible, left-canonical, has normalized self-overlap, and has no two blocks gauge-phase equivalent. Suppose also that each \(A_j\) is \(L_0\)-block-injective, that \(L_0{\gt}0\), and that \(\sum _a A^j_a(A^j_a)^\dagger =I \qquad (0\le j{\lt}r)\). Let \(N{\gt}0\), \(L{\gt}0\), \(L\le N\), \(L+L_0\le N\), and \((L_0+1) +(r-1)((L_0+1)+((L_0+1)+(L_0+1)))+1 \le L\). Assume that, for every boundary-crossing interval beginning at \(i\), the simultaneous products \(w\longmapsto (A^1_w,\ldots ,A^r_w), \qquad |w|=N-i\), span the product algebra \(\prod _jM_{D_j}(\mathbb {C})\). If every block chain space is contained in its periodic MPS line,
then
Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), with every \(\mu _j\ne 0\). Suppose the physical alphabet and all block dimensions are nonempty. Assume that the block family is irreducible, left-canonical, has normalized self-overlap, and has no two blocks gauge-phase equivalent. Suppose also that each \(A_j\) is \(L_0\)-block-injective, that \(L_0{\gt}0\), and that \(\sum _a A^j_a(A^j_a)^\dagger =I \qquad (0\le j{\lt}r)\). Let \(N{\gt}0\), \(L{\gt}0\), \(L\le N\), \(L+L_0\le N\), and \((L_0+1) +(r-1)((L_0+1)+((L_0+1)+(L_0+1)))+1 \le L\). Assume the opened-boundary \(C^j,D^j\) comparison: whenever \(\psi \in \mathcal G_{N,L}(B)\) and \(\psi =\Gamma _N^B(\bigoplus _j X_j)\), there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every \(\rho \), and every \(\beta \), \(A^j_\beta C^j_{i,\rho } =((\mu _j^N X_j)A^j_\beta )A^j_\rho \). If every block chain space is contained in its periodic MPS line,
then
Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), with every \(\mu _j\ne 0\). Then, for every \(N{\gt}L\), the source parent-Hamiltonian spanning condition \(\ker H_L^{(N)}(B) =\operatorname{span}\{ V^{(N)}(A_j):j=0,\ldots ,r-1\} \) is equivalent to the reverse containment, for the same \(N\):
In the nearest-neighbor case \(L=2\), the all-chain source condition of Definition 12.1.17 is equivalent to the conjunction
Let \(\phi \) be a linear functional on a complex algebra \(\mathcal A\) with \(\phi (1)\neq 0\) such that \(\phi (MN)=0\) for all \(N\) forces \(M=0\), and let \((x_i)_i\) and \((y_i)_i\) be finite families spanning \(\mathcal A\). If \(\phi (y_w)=c_n\phi (x_w)\) for every word \(w\) of length \(n\in \{ 1,2,3\} \), where \(x_w\) and \(y_w\) denote the ordered products along \(w\), then \(\phi (y_w)=c_1^{|w|}\phi (x_w)\) for every word \(w\).
Consequently, let \(A^1,\dots ,A^r\) be blocks of positive bond dimensions whose one-site tuples span \(\bigoplus _kM_{D_k}(\mathbb {C})\), let \(g\mapsto U_g\) be a unitary representation of a monoid, and suppose that for every \(g\) and \(N\) there is a scalar \(\lambda _{g,N}\) with \(U_g^{\otimes N}|\psi _N\rangle =\lambda _{g,N}|\psi _N\rangle \) for the matrix product vectors \(|\psi _N\rangle \) of \(\bigoplus _kA^k\). Then for each \(g\) there is \(c\) with \(|c|=1\) such that \(\sum _k\operatorname{tr}B^k_w=c^{|w|}\sum _k\operatorname{tr}A^k_w\) for every word \(w\), where \(B^k_i=\sum _l(U_g)_{il}A^k_l\). Exact symmetry is the case \(\lambda _{g,N}=1\). A symmetry of the state holds only up to such a phase [ 236 , lines 1313–1314, 2590–2593 ] .
Let \((A^k)_k\) and \((B^k)_k\) be two families of blocks of the same bond dimensions \(D_k\), each of whose one-site tuples \((A^k_i)_k\) and \((B^k_i)_k\) span \(\bigoplus _kM_{D_k}(\mathbb {C})\). Suppose that \(\sum _k\operatorname{tr}A^k_w=\sum _k\operatorname{tr}B^k_w\) for every word \(w\), where \(A^k_w\) denotes the ordered product along \(w\). Then there is an algebra automorphism \(T\) of \(\bigoplus _kM_{D_k}(\mathbb {C})\) with \(T\bigl((A^k_w)_k\bigr)=(B^k_w)_k\) for every word \(w\).
Let \(A\) be an MPS tensor with nonzero physical dimension, let \(p\) be any blocking length, and set \(B=\operatorname {block}_p(A)\). Then
If \(B\) is injective, then the bound on either side
gives the adjacent three-site anticommutator inequality
The same quadratic-form inequality holds for every forward or backward adjacent pair of cyclic range-two terms on every chain of length at least three, including pairs crossing the periodic cut.
With the hypotheses and notation of Lemma 13.3.2.10, one has
This records the ground-space equality conclusion of the conditional crossing-span version above. It is the corresponding part of [ 228 , Theorem 12 ] , but only under the visible tail-word span hypothesis and in the longer stated range. The independence of the \(N\)-site spaces is a separate conclusion of Lemma 13.3.2.10. This intermediate statement remains conditional on the short tail-word spans. The source-range conclusion without them is supplied by the global change-of-cut theorem Theorem 13.3.1.41 under the unrestricted source normalization, with positive definite dual fixed points satisfying
The doubly normalized theorem with \(\Lambda _j=\mathbb {1}\) remains available as a specialization of that result.
With the hypotheses and notation of Lemma 13.3.2.8, one has
This records only the ground-space equality conclusion obtained after the \(C^j,D^j\) boundary-condition comparison input in [ 228 , Theorem 12, proof lines 1446–1456 ] ; the local normalized \(E^j\) calculation is Lemma 13.2.1.9. The independence of the \(N\)-site spaces is a separate conclusion of Lemma 13.3.2.8. The statement is still in the length-\(L_0\) injectivity range displayed in that lemma, not the shorter range in [ 228 , Theorem 12 ] .
Let \(B\) be a canonical-form tensor with BNT components \(A_1,\ldots ,A_g\). If every component vector has zero energy for the parent Hamiltonian, \(H_L^{(N)}(B)V^{(N)}(A_j)=0\) for \(1\le j\le g\), then
It is enough to assume the local frustration-free equations \(h_iV^{(N)}(A_j)=0\) for every \(i\) and \(j\). For \(N{\gt}0\) and \(L\le N\), in the block-diagonal case \(B=\bigoplus _j\mu _jA_j\), with every \(\mu _j\ne 0\), the same inclusion follows because each local space \(G_L(A_j)\) is contained in \(G_L(B)\). This establishes the forward inclusion \(\operatorname{span}\{ V^{(N)}(A_j)\} _j \subseteq \ker H_L^{(N)}(B)\) of [ 13 , Definition 3.9 ] .
If two tensors have the same \(R\)-site boundary space, their canonical interaction matrices agree. For \(R{\gt}0\), their canonical open Hamiltonians consequently agree at every volume. Their bond dimensions may differ.
Let \(L\le N\). Suppose that \(h_i h_j=h_j h_i\) whenever the length-\(L\) cyclic supports of the two local terms meet. Then the parent Hamiltonian has commuting local terms: \(h_i h_j=h_j h_i\) for all \(i,j\). In the case \(L=2\), this gives the nearest-neighbor commuting condition.
Let \(S\) be a compact subset of a topological space and \(P(\delta ,N,x)\) a property of a real \(\delta \), a natural number \(N\) and a point \(x\) that is preserved by decreasing \(\delta \) and by increasing \(N\). If every \(x\in S\) has \(\delta {\gt}0\) and \(N\) such that \(P(\delta ,N,y)\) holds for all \(y\) near \(x\), then one \(\delta {\gt}0\) and one \(N\) satisfy \(P(\delta ,N,x)\) for all \(x\in S\).
If \(L\le N\) and the cyclic supports \(S_i\) and \(S_j\) do not meet, then the Hilbert-space local terms have non-negative ordered cross term: \(0\le \operatorname{Re}\langle h_{i,\mathrm{ES}}v,h_{j,\mathrm{ES}}v\rangle \).
If \(A\) and \(B\) are length-\(L\) block-injective, the map \(Z\mapsto (t\mapsto \operatorname{tr}(A_tZ))\) has image dimension \(D_A^2\), and similarly for \(B\). Therefore an inclusion \(\mathcal G_L^A\subseteq \mathcal G_L^B\), together with \(D_B\le D_A\), forces \(D_A=D_B\) and \(\mathcal G_L^A=\mathcal G_L^B\). In particular, the three-block trace relation from Lemma 13.2.4 gives this equality whenever the ordering \(D_B\le D_A\) is available. If the inclusion comes from the strictly larger block, \(D_B{\lt}D_A\), then the same dimension step already contradicts the existence of the three-block trace relation. More generally, let \(H\) be a source hypothesis which excludes the simultaneous conclusion
Then the three-block intersection vanishes:
With injectivity at the three-block length, the strict-size branch gives
One such source hypothesis is non-proportionality of the periodic MPS vectors at a sufficiently long chain length: \(\mathbb {C}\, V^{(N)}(A)\ne \mathbb {C}\, V^{(N)}(B)\). Indeed,
by equality of the periodic-boundary constraint spaces, followed by injective uniqueness. For same-dimensional normalized BNT representatives that are not gauge-phase equivalent, the non-equivalence condition supplies this non-proportionality at arbitrarily large chain lengths; the direct-sum injectivity hypotheses remain explicit. Combining the equal-dimension and unequal-dimension branches over all ordered block pairs gives one common positive length \(S_0\) with
For an ordered pair of blocks, the trace-separation conclusion has the form
Equivalently, the homogeneous pair span is full:
Concatenating words propagates this equality to positive multiples of \(S\), and hence to a finite period window for every ordered pair. Combining that window with the conditional period-window homogenization theorem gives a single length \(S_0\) such that
The same conclusion applies to the corresponding normal canonical-form hypotheses with a basis of normal tensors once one-site injectivity is supplied explicitly, since those hypotheses carry the irreducibility and normalization assumptions used in the direct-sum comparison.
- MPSTensor.leftTraceWordMap_injective_of_isNBlkInjective
- MPSTensor.leftTraceWordMap_range_finrank_eq_of_isNBlkInjective
- MPSTensor.leftTraceWordMap_range_eq_of_range_le_of_isNBlkInjective_of_dim_ge
- MPSTensor.leftTraceWordMap_range_eq_of_three_block_trace_relation_left_of_dim_ge
- MPSTensor.not_three_block_trace_relation_left_of_isNBlkInjective_of_dim_gt
- MPSTensor.groundSpace_finrank_eq_of_isNBlkInjective
- MPSTensor.bondDim_eq_of_groundSpace_eq_of_isNBlkInjective
- MPSTensor.bondDim_eq_and_groundSpace_eq_of_groundSpace_le_of_isNBlkInjective_of_dim_ge
- MPSTensor.bondDim_eq_and_groundSpace_eq_of_three_block_trace_relation_left_of_dim_ge
- MPSTensor.groundSpace_inf_eq_bot_of_not_bondDim_eq_and_groundSpace_eq_of_dim_ge
- MPSTensor.pairTraceSeparatingAt_threeBlock_of_isNBlkInjective_of_dim_gt
- MPSTensor.pairTraceSeparatingAt_threeBlock_of_isNBlkInjective_of_dim_ne
- MPSTensor.mpvSubmodule_ne_of_not_exists_mpv_eq_smul
- MPSTensor.not_exists_mpv_eq_smul_of_not_exists_forall_mpv_eq_mul
- MPSTensor.not_bondDim_eq_and_groundSpace_eq_of_mpvSubmodule_ne
- MPSTensor.groundSpace_inf_eq_bot_of_exists_not_forall_mpv_eq_mul_of_dim_ge
- MPSTensor.pairTraceSeparatingAt_of_groundSpace_inf_eq_bot_of_injective_groundSpaceMap
- MPSTensor.pairTraceSeparatingAt_of_groundSpace_inf_eq_bot_of_isNBlkInjective
- MPSTensor.pairTraceSeparatingAt_threeBlock_of_exists_not_forall_mpv_eq_mul_of_dim_ge
- MPSTensor.groundSpace_inf_eq_bot_of_blocksNotGaugePhaseEquiv_same_dim_of_dim_ge
- MPSTensor.pairTraceSeparatingAt_threeBlock_of_blocksNotGaugePhaseEquiv_same_dim_of_dim_ge
- MPSTensor.forall_pairTraceSeparatingAt_threeBlock_of_blocksNotGaugePhaseEquiv
- MPSTensor.exists_forall_pairTraceSeparatingAt_threeBlock_of_blocksNotGaugePhaseEquiv
The Hilbert-space local projector \(P_L^{\mathrm{ES}}(A)\) is positive. Consequently, every summand \(R_{i,\tau }^{\dagger }P_L^{\mathrm{ES}}(A)R_{i,\tau }\) is positive, and so is the finite sum over \(\tau \).
If \(w = (J_1, \ldots , J_m)\) is a word in the blocked alphabet, where each \(J_k\) is an \(L\)-tuple in \(\{ 0, \ldots , d{-}1\} ^L\), then concatenating the block words gives a word \(\widetilde{w}\) of length \(mL\) in the original alphabet and \((A^{[L]})^w = A^{\widetilde{w}}\).
Let \(P_i\) be a finite family of symmetric projections and let \(\gamma \le 1\). Suppose an interaction predicate \(i\sim j\) has at most \(m\) off-diagonal neighbours in each row and in each column, where \(m{\gt}0\). If noninteracting pairs have non-negative anticommutator form and interacting pairs satisfy
then \(H=\sum _iP_i\) satisfies \(H^2\ge \gamma H\) as a quadratic form.
Let \(P_i\) be a finite family of symmetric projections and let \(\gamma \le 1\). Suppose an interaction predicate has at most \(m{\gt}0\) off-diagonal neighbours in each row, noninteracting ordered cross terms are non-negative, and interacting pairs satisfy the norm-compression estimate
Then \(H=\sum _iP_i\) satisfies \(H^2\ge \gamma H\) as a quadratic form. The same conclusion holds from a separate compression coefficient \(\eta \) whenever \(\eta \le (1-\gamma )/m\). In particular, if \(0\le \eta \) and \(\eta m{\lt}1\), then \(H^2\ge (1-\eta m)H\) and \(1-\eta m{\gt}0\).
Let \(P_i\) be a finite family of symmetric projections and let \(\gamma \le 1\). Suppose an interaction predicate \(i\sim j\) has at most \(m\) off-diagonal neighbours in each row for some positive integer \(m\). If noninteracting ordered cross terms are non-negative and, on each interacting pair,
then \(H=\sum _iP_i\) satisfies \(H^2\ge \gamma H\) as a quadratic form.
The prefix and interval Hamiltonians are positive and symmetric:
If \(\psi \in G_{K+L}(A)\), then every fixed-prefix suffix restriction of \(\psi \) lies in \(G_L(A)\). For a vector in \(G_{L+1}(A)\), fixing either endpoint gives the expected left or right multiplication of its boundary matrix.
Set \(E_n=G_n-G_{n+1}\). For \(0\leq m,n{\lt}M\) with \(m{\lt}n-1\) or \(m{\gt}n\), one has
The initial differences satisfy \(E_0=0\) and \(Q_1E_1=0\).
For operators \(X,X'\) on \((\mathbb {C}^d)^{\otimes L_1}\) and \(Y,Y'\) on \((\mathbb {C}^d)^{\otimes L_2}\), with \(X\otimes Y\) acting on \(L_1+L_2\) consecutive sites,
For an operator \(O\) on \((\mathbb {C}^d)^{\otimes N}\) and the unnormalized vector \(|\phi _N(A)\rangle \) of (1), \(\langle \phi _N(A)|O|\phi _N(A)\rangle =\operatorname {tr}\mathcal{E}_O\), the trace of a linear map on \(M_{D}(\mathbb {C})\).
- MPSTensor.physicalObservableTransfer_apply
- MPSTensor.appendObservable
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- MPSTensor.appendObservable_append
- MPSTensor.appendObservable_mul
- MPSTensor.appendObservable_one_one
- MPSTensor.physicalObservableTransfer_appendObservable
- MPSTensor.physicalObservableTransfer_one
- MPSTensor.physicalObservableTransfer_conjTranspose
- MPSTensor.inner_mpvState_toEuclideanLin
Let \(A\) and \(B\) be left-canonical irreducible tensors of common positive bond dimension satisfying \(B^i=\zeta XA^iX^{-1}\) for an invertible \(X\) and a nonzero scalar \(\zeta \). There is a unitary matrix \(U\), with the same scalar \(\zeta \), such that \(|\zeta |=1\) and \(B^i=\zeta UA^iU^\dagger \) for every \(i\).
The Hilbert-space parent Hamiltonian is the sum of the Hilbert-space local terms, \(H_{N,\mathrm{ES}}=\sum _i h_{i,\mathrm{ES}}\), and each local term admits the exact cyclic-averaging formula
For \(L\le N\), a Hilbert-space local term \(h_{i,\mathrm{ES}}\) annihilates a vector \(v\) if and only if every restriction of \(v\) to the cyclic length-\(L\) window starting at \(i\), with the outside configuration fixed, lies in \(G_L^{\mathrm{ES}}(A)\).
Let \(A=(A_i)_i\) and \(B=(B_i)_i\) be two tensors, possibly with different bond dimensions, and let \(u\) be unitary on the physical index. If
then
Let \(U,A\) and \(V,B\) be pairs of tensors with respective common bond dimensions. Suppose that \(\zeta _A,\zeta _B\ne 0\) and that \(X_A,X_B\) are invertible matrices such that, for every physical index \(i\),
Then
Identifying equal bond dimensions on either tensor leg preserves this vanishing condition in both directions.
For \(M\in \mathbb {C}^{m\times d}\) and a length-\(N\) physical configuration \(s=(s_0,\ldots ,s_{N-1})\in \{ 0,\ldots ,m-1\} ^N\), physical rotation satisfies
Let \(P\) be an orthogonal projection on a complex inner-product space, and let \(Q\) be a bounded linear operator from that space to another. If
then, for every \(v\),
This includes \(\epsilon =0\).
Let \(x\) and \(y\) belong to a complex inner-product space. For every \(c{\gt}0\),
Let \(N\ge 2\). Suppose that, for every \(i\in \mathbb {Z}/N\mathbb {Z}\), the translated length-two local term commutes with its cyclic right neighbor:
Then all translated length-two local terms commute.
Let \(H\) be a positive operator on a finite-dimensional complex Hilbert space and let \(\gamma \geq 0\). If
then every vector \(v\) satisfies
Let \(S_m\subseteq (\mathbb {C}^d)^{\otimes m}\) be subspaces, and let \(0{\lt}L\le N\). Suppose that every non-wrapping length-\(L\) interval of \(\psi \), for every fixed choice of the complementary sites, lies in \(S_L\). Suppose also that, for every \(m\ge L\), \((\mathbb {C}^d\otimes S_m)\cap (S_m\otimes \mathbb {C}^d) = S_{m+1}\). Then \(\psi \in S_N\).
Suppose \(W\leq N\), fix a cyclic start \(s\), and let \(q{\lt}W\) satisfy \(q+R\leq W\). Replacing the range-\(R\) subwindow beginning at \(s+q\) changes only the active configuration. Restriction to this subwindow agrees with restriction at \(q\) inside the active block. When \(R{\gt}0\), any local operator, extended by the identity, satisfies
Its action is obtained by applying \(h\) to the restricted window with outside coordinates fixed. If \(0{\lt}R\leq W\leq n\leq N\), translation \(q\mapsto n-W+q\) also identifies the nonwrapping starts on \(W\) sites with the range-\(R\) starts contained in \([n-W,n)\).
Suppose \(0{\lt}R\leq W\leq n\leq N\). The suffix Hamiltonian is unitarily equivalent to \(H_W^{\mathrm{open}}(h)\) acting independently on the complementary configurations. If \(\kappa p_W(A)\leq H_W^{\mathrm{open}}(h)\), then the same inequality holds after embedding both sides on the suffix \([n-W,n)\). Here \(p_W(A)\) is the canonical complementary ground-space projection; no sign condition on \(\kappa \) or positivity condition on \(h\) is needed for this transport of order.
Open interaction sums preserve order and positivity, commute with scalar multiplication, and vanish for the zero interaction. When \(R{\gt}0\), the sum of the canonical parent projection is the canonical open parent Hamiltonian.
Let \(0{\lt}R\leq W\leq N\). If \(\kappa \, h_W\leq H^{\mathrm{open}}_{W,R}\) for the range-\(W\) parent interaction \(h_W\) and some \(\kappa {\gt}0\), then \(\kappa H^{\mathrm{open}}_{N,W}\leq (W-R+1)H^{\mathrm{open}}_{N,R}\). For a tensor for which both open kernels equal \(G_N^{\mathrm{ES}}\), a gap \(\gamma \lVert v\rVert \leq \lVert H^{\mathrm{open}}_{N,W}v\rVert \) on \((G_N^{\mathrm{ES}})^\perp \) gives \(\frac{\kappa \gamma }{W-R+1}\lVert v\rVert \leq \lVert H^{\mathrm{open}}_{N,R}v\rVert \) on the same complement. The assertion may be used at one volume or uniformly over every \(N\geq W\). For a one-site injective tensor with \(R=2\) and \(W\geq 2\), the kernel hypotheses hold automatically.
Let \(P_i\) be a finite family of symmetric projections and let \(H=\sum _iP_i\). If the ordered off-diagonal forms obey
with row sums \(\sum _{j\ne i}c_{ij}\le 1\) and \(\gamma \le 1\), then \(H^2\ge \gamma H\) as a quadratic form.
Let \(A\) be an MPS tensor, and let \(R,W\) be interaction and interval lengths. Write \(K_{R,W}=H^{\mathrm{open}}_{W,R}(A)\) for the open parent Hamiltonian of interaction length \(R\) on \(W\) sites, and suppose that its kernel is \(\mathcal G_W(A)\). Writing \(h_W\) for the orthogonal projection onto \(\mathcal G_W(A)^\perp \), there are positive constants \(\kappa ,C\) such that
Assume uniformly for every \(N\ge 2L\) that the Hilbert-space local terms satisfy the anticommutator martingale estimate
for \(i\ne j\), with row and column coefficient sums bounded by one. If \(L{\gt}1\), then \(1/(4L){\gt}0\) and, for every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\frac{1}{4L}\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).
Fix \(L{\gt}1\) and let \(i\sim _L j\) mean that the cyclic supports of the length-\(L\) windows at \(i\) and \(j\) meet. Suppose uniformly for every \(N\ge 2L\) that overlapping off-diagonal terms satisfy
Then the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).
Fix \(L{\gt}1\) and let \(i\sim _L j\) mean that the cyclic supports of the length-\(L\) windows at \(i\) and \(j\) meet. Suppose uniformly for every \(N\ge 2L\) that overlapping off-diagonal terms satisfy
Then \(1/(4L){\gt}0\), and the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).
Let \(H_{N,L}^{\mathrm{ES}}(A)\) be the Hilbert-space representative of the parent Hamiltonian. If a constant \(\gamma {\gt}0\) satisfies the uniform quadratic-form estimate
for every \(N\ge 2L\) and every vector \(v\), then \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \) for every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \). In particular, with \(L{\gt}1\) and \(\gamma =1/(4L)\), the explicit gap-bound conclusion follows once this same quadratic-form estimate (37) is supplied.
Fix \(L{\gt}1\) and set \(m=2(L-1)\). Suppose uniformly for every \(N\ge 2L\) that the overlap relation on local windows has at most \(m\) off-diagonal neighbours in each row and in each column. If non-overlapping local terms have non-negative anticommutator form, and overlapping local terms satisfy
then the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).
Fix \(L{\gt}1\) and set \(m=2(L-1)\). Suppose uniformly for every \(N\ge 2L\) that the hypotheses of the fixed-chain finite-overlap martingale reduction hold with \(\gamma =1/(4L)\) and this value of \(m\). Then \(1/(4L){\gt}0\), and the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).
For a fixed chain length \(N\), let \(m\) be a positive integer and let \(\gamma \le 1\). Suppose an overlap predicate on local windows has at most \(m\) overlapping off-diagonal windows in each row. If non-overlapping local terms have non-negative ordered cross terms, and overlapping local terms satisfy the one-sided ordered cross-term estimate
and the Hilbert-space local terms are symmetric projections, then the Hilbert-space Hamiltonian satisfies \(H_{N,L}^{\mathrm{ES}}{}^2\ge \gamma H_{N,L}^{\mathrm{ES}}\) as a quadratic form.
Assume uniformly for every \(N\ge 2L\) that the Hilbert-space local terms satisfy the ordered row-sum estimate (45) with \(\gamma =1/(4L)\). If \(L{\gt}1\), then \(1/(4L){\gt}0\) and, for every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\frac{1}{4L}\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).
For a fixed chain length \(N\), suppose the Hilbert-space local terms \(h_{i,\mathrm{ES}}\) satisfy the ordered row-sum estimate
with \(\sum _{j\ne i}c_{ij}\le 1\) and \(\gamma \le 1\). Then the Hilbert-space parent Hamiltonian satisfies
for every vector \(v\).
Let \(B\) and \(C\) have the same physical alphabet and satisfy \(G_L(B)=G_L(C)\). Their periodic chain spaces agree for every \(N\):
Their parent-Hamiltonian kernels agree for every \(N{\gt}L\):
Consequently, any fixed BNT-vector family satisfying the parent-Hamiltonian spanning equation for \(C\) also satisfies it for \(B\).
A nonzero irreducible tensor \(A\), rescaled by a nonzero \(\zeta \) and gauged, becomes a periodic tensor \(B\) of some period \(m\), with \(|\phi _B\rangle =\zeta ^N|\phi _A\rangle \) on \(N\) sites. A tensor is periodic with period one exactly when it is irreducible, left canonical and its transfer channel is primitive.
If \(0\leq P\leq Q\), then \(\ker Q\subseteq \ker P\). Consequently, if \(P,Q\) are positive operators and \(a,b,c{\gt}0\) satisfy
their kernels agree. In finite dimension, a norm gap \(\gamma {\gt}0\) for \(P\) on its kernel complement implies a norm gap \(a\gamma /b\) for \(Q\) on the same subspace.
Let \(H\) be a positive operator on a finite-dimensional complex Hilbert space. If \(P\) is the orthogonal projection onto \((\ker H)^\perp \), then \(aP\leq H\leq bP\) for some \(a,b{\gt}0\). More precisely, a norm gap \(\gamma \geq 0\) on \((\ker H)^\perp \) implies \(\gamma P\leq H\).
Let \(A\) be a tensor and let \(R{\gt}0\). Suppose the canonical open Hamiltonian has kernel \(\mathcal G_A(N)\) for every \(N\ge R\). Then the range-\(R\) canonical orthogonal-complement projection is a positive interaction \(h\), and the open interaction generated by \(h\) has those same exact kernels. The interaction is literally the configuration-basis matrix of that canonical projection.
Let \(P\) and \(Q\) be finite-dimensional complex endomorphisms, and let \(\gamma \geq 0\). Suppose that \(P\) is positive and
for every \(v\in (\ker P)^\perp \). Then
on \((\ker P)^\perp \). If also \(P\leq Q\) and \(\ker P=\ker Q\), then
for every \(v\in (\ker Q)^\perp \).
Let \(R\leq W\). For any positive operators \(T_i\), indexed by the nonwrapping range-\(R\) starts in an \(N\)-site chain,
In particular, for a positive interaction \(h\), \(\sum _{n=W}^{N}H_{[n-W,n)}(h) \leq (W-R+1)H_N^{\mathrm{open}}(h)\). This is the counting argument of condition C1 in [ 225 ] , applied to arbitrary positive terms.
Suppose \(\operatorname{span}\{ (A^1_w,\ldots ,A^r_w):|w|=n\} = \prod _jM_{D_j}(\mathbb {C})\). Then the sum of the local spaces \(G_n(A^j)\) is direct: if
then \(\phi _j=0\) for every \(j\). In particular, for the normalized BNT hypotheses above, this directness holds for every \(n\ge (L_0+1)+(r-1)3(L_0+1)\).
Let \(d,L{\gt}0\). If \(\overline{\mathcal B}_L\) denotes the quasi-local blocking equivalence, then for every \(a\in \mathbb Z\),
This is the completion of the translation rule induced by the site grouping in [ 16 , lines 2308 and 2313–2320 ] .
For every finite region \(\Lambda \), the map \(j_\Lambda \) is an injective isometry. If \(\Lambda \subseteq \Gamma \), then
Let \(H\) be a positive semidefinite self-adjoint operator on a finite-dimensional complex Hilbert space, and suppose that, for all \(v\), \(\gamma \langle v,Hv\rangle \le \langle Hv,Hv\rangle \), i.e., \(H^2\ge \gamma H\) as a quadratic form, for some \(\gamma {\gt}0\). Then, for every \(v\) in the orthogonal complement of \(\ker H\), \(\gamma \| v\| \le \| Hv\| \). In particular, every positive eigenvalue of \(H\) is at least \(\gamma \). The positive-semidefiniteness hypothesis is essential: without it, \(H=-\mathbb {1}\) with \(\gamma =2\) satisfies the quadratic-form inequality vacuously but violates the norm bound.
A virtual similarity preserves the physical blocking relation and hence transfer idempotence. It also preserves every local MPS space, so the canonical parent interaction and all its translated local terms are unchanged.
Let \(T\) and \(U\) be operators on a finite-dimensional active Hilbert space, and let \(S\) be a finite set of right-spectator configurations. The operator \(T^{\oplus S}\), defined by applying \(T\) independently at every \(s\in S\), satisfies
and
with equality in (80) whenever \(S\) is nonempty.
- ContinuousLinearMap.rightFiber
- ContinuousLinearMap.rightFiberwiseMap
- ContinuousLinearMap.rightFiberwiseMap_apply_apply
- ContinuousLinearMap.rightFiber_rightFiberwiseMap
- ContinuousLinearMap.rightFiberwiseMap_zero
- ContinuousLinearMap.singleRightFiber
- ContinuousLinearMap.norm_singleRightFiber
- ContinuousLinearMap.rightFiberwiseMap_add
- ContinuousLinearMap.rightFiberwiseMap_sub
- ContinuousLinearMap.rightFiberwiseMap_sum
- ContinuousLinearMap.rightFiberwiseMap_comp
- ContinuousLinearMap.norm_conj_linearIsometryEquiv
- ContinuousLinearMap.norm_rightFiberwiseMap_le
- ContinuousLinearMap.norm_rightFiberwiseMap
For \(n,r\in \mathbb {N}\), restriction to the first \(n\) sites and the final \(r\) sites gives a canonical equivalence
and the corresponding Euclidean linear isometry. A vector belongs to \(\ker (T^{\oplus S})\) exactly when each spectator fiber belongs to \(\ker T\). If \(P_{\ker T}\) is the orthogonal projection onto \(\ker T\), then
- MPSTensor.rightSpectatorConfigEquiv
- MPSTensor.rightSpectatorConfigLinearIsometryEquiv
- ContinuousLinearMap.mem_ker_rightFiberwiseMap_iff
- ContinuousLinearMap.isSymmetric_rightFiberwiseMap
- ContinuousLinearMap.isSymmetricProjection_rightFiberwiseMap
- ContinuousLinearMap.ker_starProjection_rightFiberwiseMap
- MPSTensor.ker_starProjection_conj_linearIsometryEquiv
Let \(G\) act on a finite active configuration space. If \(G^2\geq \gamma G\) as a quadratic form, the same inequality holds when \(G\) acts independently over any finite right spectator coordinate. In particular, positivity and a norm gap \(\gamma \) for \(G\) imply this fiberwise quadratic-form inequality.
The direct sum of the three bond-one qubit tensors \((1,0)\), \((0,1)\) and \((1/\sqrt2,1/\sqrt2)\), with weights one, has a basis of three normalized, normal, pairwise inequivalent blocks. Its simultaneous injectivity length is exactly two. Its two-site local ground space is \(\operatorname {Sym}^2(\mathbb C^2)\), and its canonical parent interaction is the singlet projection \(h_2=(I-F)/2\), where \(F\) exchanges the two physical sites.
- MPSTensor.shortRangeHeisenbergBlock
- MPSTensor.shortRangeHeisenbergTensor
- MPSTensor.wordTupleSpanTop_shortRangeHeisenbergBlock_two
- MPSTensor.not_wordTupleSpanTop_shortRangeHeisenbergBlock_one
- MPSTensor.shortRangeHeisenbergBlock_normalization
- MPSTensor.shortRangeHeisenbergBlock_isNormalTensor
- MPSTensor.shortRangeHeisenbergBlock_isNormal
- MPSTensor.shortRangeHeisenbergBlock_blocksNotGaugePhaseEquiv
- MPSTensor.isCPSVBasisOfNormalTensors_shortRangeHeisenbergTensor
- MPSTensor.mem_groundSpaceES_shortRangeHeisenbergTensor_two_iff
- MPSTensor.groundSpaceES_shortRangeHeisenbergTensor_two_starProjection
- MPSTensor.parentInteractionES_shortRangeHeisenbergTensor_two_apply
The Hilbert-space local projector \(P_L^{\mathrm{ES}}(A)\) is a symmetric projection, and every Hilbert-space local term \(h_{i,\mathrm{ES}}\) on the \(N\)-site chain is a symmetric projection.
Let \(L\le N\) and let \(h_{i,\mathrm{ES}},h_{j,\mathrm{ES}}\) be Hilbert-space local terms whose cyclic \(L\)-windows are site-disjoint: no site of the periodic chain has cyclic offset \({\lt}L\) from both \(i\) and \(j\). Then the terms commute pointwise: for every vector \(v\), \(h_{i,\mathrm{ES}}(h_{j,\mathrm{ES}}v) =h_{j,\mathrm{ES}}(h_{i,\mathrm{ES}}v)\). The same commutation identity holds for the corresponding local terms before passing to the Hilbert-space model. Their ordered cross term is non-negative: \(0\le \operatorname{Re}\langle h_{i,\mathrm{ES}}v,h_{j,\mathrm{ES}}v\rangle \).
Each Hilbert-space local term \(h_{i,\mathrm{ES}}\) is positive, hence the full Hilbert-space parent Hamiltonian \(H_{N,\mathrm{ES}}\) is positive.
The lifts from \(AX\) and \(XB\) preserve multiplication,
and each of the four lifts from \(AX\), \(XB\), \(A\), and \(B\) preserves adjoints:
Let \(P_U\) be the orthogonal projection onto \(U\). If the Friedrichs overlap of \(U\) and \(V\) is at most \(\eta \), where \(\eta \ge 0\), then
for every \(v\in V\cap (U\cap V)^\perp \).
Let \(P_U\) and \(P_V\) be the orthogonal projections onto \(U\) and \(V\). If the Friedrichs overlap of \(U\) and \(V\) is at most \(\eta \), where \(\eta \ge 0\), then every \(v\in (U\cap V)^\perp \) satisfies
Let \(P\) and \(Q\) be the orthogonal projections onto \(K^\perp \) and \(L^\perp \). If the Friedrichs overlap of these two ranges is at most \(\eta \ge 0\), then every \(v\in K^\perp \cap (K^\perp \cap L^\perp )^\perp \) satisfies
For \(i,j\in \mathbb {Z}/N\mathbb {Z}\), if the two length-two cyclic supports starting at \(i\) and \(j\) overlap, then
For a non-wrapping interval \([s,s+M-1] \subseteq \{ 0,\ldots ,N-1\} \) and a full configuration \(\rho \) supplying the complementary values, the restriction map is
These identities include the boundary cases \(s=0\) and \(M=N\), the deletion of the first or last site, and the composition of fixing \(K\) prefix sites followed by restricting to \([s+K,s+K+L-1]\).
On the periodic chain, for a full configuration \(\rho \) supplying the complementary values, put
The restriction to the cyclic interval beginning at \(i\) is
with all site labels read modulo \(N\). If \(L\le N\), this is the same configuration as \(\rho ^{[i,i+L-1]\leftarrow \omega }\). If two boundary conditions agree whenever \(\delta _i(k)\ge L\), their restrictions are equal. In particular, filling the interval inside the outside configuration leaves the restriction unchanged:
For every \(r{\lt}L\), the inserted word is recovered as \(\rho ^{\mathrm{cyc}}_{i,L,\omega }(i+r)=\omega (r)\) with site labels taken modulo \(N\). Reconstructing a configuration from its local restriction returns the original configuration: taking \(\rho =\sigma \) and \(\omega (r)=\sigma (i+r)\) for \(r{\lt}L\),
Modifying a single inserted site commutes with the cyclic extension: for \(r{\lt}L\) and a letter \(x\), writing \(\omega [r\mapsto x]\) for the word \(\omega \) with its value at \(r\) replaced by \(x\),
Deleting the final site of an \((L+1)\)-interval gives the \(L\)-interval beginning at \(i\); for \(L+1\le N\), deleting the first site gives the \(L\)-interval beginning at \(i+1\); and the cyclic formula agrees with the contiguous restriction when the interval does not wrap around the ring.
- MPSTensor.cyclicCfg
- MPSTensor.cyclicRestrictₗ
- MPSTensor.cyclicRestrictₗ_apply
- MPSTensor.cyclicRestrictₗ_congr_outside
- MPSTensor.cyclicRestrictₗ_cyclicCfg_outside
- MPSTensor.cyclicCfg_cyclicForwardSite
- MPSTensor.eq_cyclic_site_of_offset_eq
- MPSTensor.cyclicCfg_cyclicForwardSite_apply
- MPSTensor.cyclicCfg_extractWindow
- MPSTensor.update_cyclicCfg
- MPSTensor.cyclicForwardSite_one_offset
- MPSTensor.cyclicRestrictₗ_restrictLast
- MPSTensor.cyclicRestrictₗ_restrictFirst
- MPSTensor.cyclicCfg_eq_contiguousCfg
- MPSTensor.cyclicRestrictₗ_eq_contiguousRestrictₗ
Let \(P_L^{\mathrm{ES}}(A)\) denote the orthogonal projector onto \((G_L^{\mathrm{ES}}(A))^\perp \) on the canonical \(\ell ^2\) \(L\)-site Hilbert space. For a cyclic window starting at \(i\) and an outside configuration \(\tau \), the map \(R_{i,\tau }\) restricts an \(N\)-site vector to that length-\(L\) cyclic window with the outside configuration fixed by \(\tau \). The local term \(h_{i,\mathrm{ES}}\) is the corresponding \(N\)-site local projector, and \(R_{i,\tau }^{\dagger }P_L^{\mathrm{ES}}(A)R_{i,\tau }\) is the basic positive summand in its cyclic-averaging formula.
The nearest-neighbor condition is the length-\(2\) specialization of the general commuting condition. The ground-vector condition combines this commuting equation with the frustration-free equation for \(V^{(N)}(A)\). For \(N\ge 2\), Lemma 11.4.47 with \(L=2\) gives \(h_iV^{(N)}(A)=0\) for every site \(i\). The fixed-chain ground-space condition also contains \(\ker H_2^{(N)}(B) = \operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \), and quantifying this condition over all \(N{\gt}2\) gives the nearest-neighbor case of the spanning condition above. Conversely, the all-chain condition is exactly the conjunction of the all-chain ground-vector equations and the nearest-neighbor instance of the spanning condition above. This is precisely the condition named in Definition 12.1.17. Since the displayed zero-energy equation follows from the parent-Hamiltonian construction for \(N{\gt}2\), independently of the commutation and spanning equations, this is equivalently
- MPSTensor.IsNNCPH.isCommutingParentHam
- MPSTensor.IsNNCPHGroundState.isNNCPH
- MPSTensor.IsNNCPHGroundState.isFrustrationFree
- MPSTensor.IsNNCPH.isNNCPHGroundState
- MPSTensor.HasNNCPHGroundSpace.isNNCPHGroundState
- MPSTensor.HasNNCPHGroundSpace.isNNCPH
- MPSTensor.HasNNCPHGroundSpace.isFrustrationFree
- MPSTensor.HasNNCPHGroundSpace.groundSpaceSpanning
- MPSTensor.HasNNCPHGroundSpace.hasParentHamiltonianGroundSpaceSpanning
- MPSTensor.HasNNCPHGroundSpaces.hasNNCPHGroundSpace
- MPSTensor.HasNNCPHGroundSpaces.hasParentHamiltonianGroundSpaceSpanning
- MPSTensor.hasNNCPHGroundSpaces_iff_forall_isNNCPHGroundState_and_groundSpaceSpanning
- MPSTensor.hasNNCPHGroundSpaces_iff_forall_isNNCPH_and_groundSpaceSpanning
- MPSTensor.IsCommutingParentHam.symm
- MPSTensor.IsCommutingParentHam.ham_comm_localTerm
The projection-geometry reduction uses the following elementary Hilbert-space identities: the diagonal identity \(\operatorname{Re}\langle Pv,Pv\rangle =\operatorname{Re}\langle Pv,v\rangle \) for symmetric projections; nonnegativity of this quadratic form; nonnegativity of \(\operatorname{Re}\langle Pv,Qv\rangle \) for commuting symmetric projections \(P,Q\); the Cauchy–Schwarz conversion from \(\| PQv\| \le c\| Pv\| \) to \(\operatorname{Re}\langle Pv,Qv\rangle \ge -c\operatorname{Re}\langle Pv,v\rangle \); the symmetric variant which, for \(c\ge 0\), converts the operator-product bound \(\| PQv\| \le c\| Qv\| \) to the anticommutator estimate \(\operatorname{Re}\langle Pv,Qv\rangle +\operatorname{Re}\langle Qv,Pv\rangle \ge -c(\operatorname{Re}\langle Pv,v\rangle +\operatorname{Re}\langle Qv,v\rangle )\); finite-sum expansions of \(\operatorname{Re}\langle (\sum _iP_i)v,v\rangle \) and \(\operatorname{Re}\langle (\sum _iP_i)v,(\sum _iP_i)v\rangle \); the conclusion that a vector annihilated by a finite sum of symmetric projections is annihilated by each projection; the diagonal/off-diagonal split of the ordered double sum; the aggregate off-diagonal estimate obtained from row sums \(\sum _{j\ne i}c_{ij}\le 1\); and the corresponding aggregate estimate for the anticommutator martingale condition.
- LinearMap.IsSymmetricProjection.re_inner_apply_apply_self
- LinearMap.IsSymmetricProjection.re_inner_apply_apply_nonneg_of_commute
- LinearMap.IsSymmetricProjection.re_inner_apply_apply_ge_neg_of_norm_apply_le
- LinearMap.IsSymmetricProjection.re_inner_anticommutator_ge_neg_of_norm_apply_le
- ProjectionGeometry.re_inner_sum_apply_left
- ProjectionGeometry.apply_eq_zero_of_sum_apply_eq_zero
- ProjectionGeometry.re_inner_sum_apply_apply
- ProjectionGeometry.sum_sum_eq_diag_add_offdiag
- ProjectionGeometry.crossTerm_sum_bound_of_ordered_rowSum
- ProjectionGeometry.crossTerm_sum_bound_of_anticommutator_coeffSum
- ProjectionGeometry.crossTerm_sum_bound_of_anticommutator_rowCol
If \(A\) is injective and \(L\ge 2\), then on an \((L+1)\)-site chain the kernels of the two adjacent Hilbert-space \(L\)-site local terms have common kernel equal to the Hilbert-space \((L+1)\)-site MPS ground space:
Let \(\iota _L\) denote the canonical identification of the \(L\)-site state space with its Hilbert-space realization, and let \(R^{\mathrm{cyc}}_{i,\tau }\) denote cyclic restriction to the length-\(L\) window beginning at \(i\), with background configuration \(\tau \). For every \(L\in \mathbb {N}\), every \((L+1)\)-site Hilbert-space vector \(v\), and every background configuration \(\tau \),
On three sites, for every one-site prefix \(u\) and every state \(\psi \),
Finally, if \(L{\gt}0\), then for every \((L+1)\)-site Hilbert-space vector \(v\) and every background configuration \(\tau \),
No hypothesis on the physical or virtual dimension is required.
For a tensor \(B\), let \(h_0\) and \(h_1\) be the two translated length-two parent interactions on the three-site Hilbert space. Then
Consequently, the orthogonal projectors onto the complementary excitation subspaces satisfy
These identities require neither injectivity nor a nonzero bond dimension.
Let \(d\geq 0\), \(K,L{\gt}0\), and let \(Y\geq 0\) be an operator on \(KL\) original sites. For every \(N\geq 0\),
The coefficient is one. This is a derived positive-term counting consequence of [ 225 , equation (3.12) ] .
Let \(A\) be any tensor and \(R,L{\gt}0\). Assume that its canonical open Hamiltonian has kernel \(\mathcal G_A(N)\) for every \(N\geq R\) and one positive norm gap at every sufficiently large length divisible by \(L\). Assume also that, for every choice of exterior lengths \(K(M),r(M)\), the projection defect of the cover with left length \((K(M)+M)L\), right length \(ML+r(M)\), and overlap length \(ML\) tends to zero as \(M\to \infty \). Then there is \(\delta {\gt}0\) such that
All three estimates are supplied hypotheses of this criterion. This is the covering argument of [ 225 , Section 6 ] .
Assume the preceding hypotheses and \(R,L{\gt}0\). Suppose that the canonical range-\(R\) open Hamiltonian has kernel \(\mathcal G_A(N)\) for every \(N\ge R\), and has a common positive norm gap at all sufficiently large lengths divisible by \(L\). Then there is \(\delta {\gt}0\) such that, for every \(N\) and every \(v\perp \ker H_{R,N}^{\mathrm{open}}(A)\),
Exact kernels and the aligned gap remain explicit assumptions of this criterion. The conclusion uses the actual kernel also at volumes shorter than the interaction range.
Let \(h\) be a matrix on an \(R\)-site interval. Its matrix sum over all nonwrapping windows and the corresponding linear operator both vanish when the chain length satisfies \(N{\lt}R\). If \(R{\gt}0\), the matrix and linear operator representations of the open interaction sum agree for every \(N\geq 0\). No positivity assumption on \(h\) is needed. This is the empty-sum convention in [ 225 ] , equation (3.12).
Let \(L{\gt}1\). Suppose that for every \(N\ge 2L\) and every overlapping off-diagonal pair of cyclic length-\(L\) windows,
in quadratic form. Then the parent Hamiltonian has the explicit gap constant \(\gamma _L:=1/(4L){\gt}0\). Concretely, if \(N\ge 2L\) and \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), then \(\gamma _L\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).
Let \(A\) be injective and let an Appendix B structural datum be fixed. Put
with the two copies of \(q_2(A)\) lifted to the indicated faces of the three-site coefficient space. Then
These are the coefficient-space analogs of the idempotence, commutation, and kernel-intersection equations in [ 13 , Definition D.2 ] . The hatted operators are not identified here with the source projectors on \(\mathcal H_A\otimes \mathcal H_X\) and \(\mathcal H_X\otimes \mathcal H_B\). For a normal left-canonical RFP tensor, injectivity follows from the RFP structural theorem and is not an additional hypothesis.
The two Appendix B coefficient representatives satisfy
and their adjacent actions on the common three-site coefficient space commute:
Thus they satisfy the algebraic overlapping two-site condition of Definition 12.1.2, providing the coefficient-space analog of the commutation condition in [ 13 , Definition D.2 ] .
Let \(A\) be equipped with Appendix B structural data \((X,\Lambda ,U)\) as in Definition 12.1.34. Its pair-index isometry equation
implies \(U^*U=1\). In particular, \(U\) is injective.
For every \(N\geq 0\), one has \(U^{*\otimes N}U^{\otimes N}=1\). Hence \(U^{\otimes N}\) is injective.
On the homogeneous three-single-site coefficient space \((\mathbb {C}^d)^{\otimes 3}\), the hatted Appendix B (AX) representative acts as \([\widehat Q_{AX}]_{AX} =[\widehat Q_{AX}]\otimes I_d\). This equation gives the coefficient-factor specialization of the left local action in [ 13 , Definition D.2 ] . It does not construct a projector for an arbitrary factorization \(\mathcal H_A\otimes \mathcal H_X\otimes \mathcal H_B\).
On the homogeneous three-single-site coefficient space \((\mathbb {C}^d)^{\otimes 3}\), the hatted Appendix B (XB) representative acts as \([\widehat Q_{XB}]_{XB} =I_d\otimes [\widehat Q_{XB}]\). This equation gives the coefficient-factor specialization of the right local action in [ 13 , Definition D.2 ] . It does not construct a projector for an arbitrary factorization \(\mathcal H_A\otimes \mathcal H_X\otimes \mathcal H_B\).
Opening the outer virtual bond in the two-site contraction of [ 13 , Equations (3.17)–(3.18) ] gives the following local-support family. For every boundary matrix \(Y\), \(U^{\otimes 2}I_\varphi (v_Y) = \Gamma _2(\Lambda U)(Y)\). Consequently, \(\operatorname{ran}(U^{\otimes 2}I_\varphi )=G_2(\Lambda U)\).
The projector \(P_2\) satisfies
In particular, \(q_2(\Lambda U)\) annihilates the image of \(U^{\otimes 2}I_\varphi \), while \(P_2\) fixes that image pointwise.
Let \((P_2)_{01}\) and \((P_2)_{12}\) be the two adjacent placements of the two-site basic support projector on three physical sites. Then \((P_2)_{01}(P_2)_{12} =(P_2)_{12}(P_2)_{01}\). This is the local three-site consequence of the parent-space construction and the basic-vector form in [ 13 , Definition 3.9, Equations (3.16)–(3.18), and Appendix B ] .
The two adjacent placements commute: \(P_{01}P_{12}=P_{12}P_{01}\). The following support-location schematic marks the two bonds on which the projectors act; it does not depict the doubled operator network for the equality above.
![\begin{tenkz}[rows={wire}, cols=5, west=open, east=open, physical=up]
\tn[skin=box]{U_0} &
\tn[physical=none, label pos=s]{\varphi_{01}} &
\tn[skin=box]{U_1} &
\tn[physical=none, label pos=s]{\varphi_{12}} &
\tn[skin=box]{U_2}
\tndeclare{species}{support}{hue=source:red}
\tnmark[form=enclosure, species=support, tint, label pos=s]
{(1,2)}{$\supp(P_{01})$}
\tnmark[form=enclosure, species=support, tint, label pos=s]
{(1,4)}{$\supp(P_{12})$}
\end{tenkz}](tenkz_svg/tenkz-783c6927ca8e52d6.svg)
Let \(V_i(n)\) be a finite family of subspaces of Hilbert spaces that may depend on \(n\). Suppose that for every \(\varepsilon {\gt}0\), eventually
Then the family \((V_i(n))_i\) is eventually independent.
Suppose that \(A\) is equipped with Beigi sector data, with one-site spatial unitary \(U\). For \(N\geq 2\), let \(Q_N\) be the product of the translated complementary parent interactions and let \(\widetilde Q_N\) be the corresponding product in sector coordinates. Then \((U^*)^{\otimes N}Q_NU^{\otimes N} = \widetilde Q_N\).
Suppose that \(A\) is equipped with Beigi sector data. For \(N\geq 2\), let \(h_0,\ldots ,h_{N-1}\) be its translated length-two parent interactions and set \(Q_N=\prod _{n=0}^{N-1}(\mathbb {1}-h_n)\). Every vector fixed by \(Q_N\) belongs to the kernel of the periodic parent Hamiltonian \(H_N=\sum _n h_n\).
Let \(G\) be a finite directed graph whose edge weights are non-negative integers, and let
be the weighted sum over ordered cyclic vertex sequences. Suppose that \(s_N\) is constant for all sufficiently large \(N\). Then every edge that belongs to a directed cycle has weight one, every directed cycle is a repetition of a single loop, and, for every \(N{\gt}0\), \(s_N=\# \{ a:w(a,a){\gt}0\} \). This is the comparison in [ 5 , Section IV ] .
Let \(a\to a\) be a positive loop. For every \(N\geq 2\), its physical cyclic product-of-pairs state belongs to the kernel of the length-two periodic parent Hamiltonian. Equivalently, its Euclidean representative belongs to the parent ground space.
Suppose that the parent-ground-space dimension is eventually constant. For every \(N{\gt}2\), the positive-loop product states form a nonzero pairwise orthogonal family and
In particular, the same conclusion holds under the BNT and all-chain nearest-neighbor parent-ground-space hypotheses.
- MPSTensor.BeigiSectorGraphData.loopCyclicProduct_ne_zero
- MPSTensor.BeigiSectorGraphData.loopCyclicProduct_inner_eq_zero_of_ne
- MPSTensor.BeigiSectorGraphData.loopProductState_ne_zero
- MPSTensor.BeigiSectorGraphData.loopProductStateES_ne_zero
- MPSTensor.BeigiSectorGraphData.loopProductStateES_inner_eq_zero_of_ne
- MPSTensor.BeigiSectorGraphData.loopProductStateES_linearIndependent
- MPSTensor.BeigiSectorGraphData.span_loopProductStateES_eq_parentHamiltonianGroundSpaceES
- MPSTensor.BeigiSectorGraphData.span_loopProductStateES_eq_parentHamiltonianGroundSpaceES_of_hasBNTSectorData
For every \(N{\gt}0\), the periodic matrix product vector of \(C_a\) has coefficient \(\prod _{n=0}^{N-1}\varphi _a(r_n,l_{n+1})\) on a cyclic configuration in sector \(a\), and coefficient zero on every other sector configuration. Consequently, the periodic vector of the physical loop tensor is exactly the sitewise-unitary image of this cyclic product-of-pairs state. The same identities hold for the tensor on the minimal Schmidt support. For the normalized minimal tensor it is \(\nu _a^{-N}\Psi _a^{(N)}\).
The tensor \(\widetilde C_a\) on the Schmidt support is injective, as is its image under the one-site unitary. In particular, the physical tensor is normal.
The transfer map of the unnormalized coordinate tensor is
Hence the transfer map of \(\widehat C_a\) is idempotent. This remains true after the physical unitary rotation. The normalized physical tensor is injective and therefore normal.
- MPSTensor.BeigiSectorGraphData.loopBondNorm_pos
- MPSTensor.BeigiSectorGraphData.loopBondNorm_ne_zero
- MPSTensor.BeigiSectorGraphData.loopBondNorm_eq_frobeniusNorm
- MPSTensor.BeigiSectorGraphData.transferMap_minimalLoopCoordinateTensor
- MPSTensor.BeigiSectorGraphData.transferMap_minimalLoopCoordinateTensor_comp_self
- MPSTensor.BeigiSectorGraphData.transferMap_normalizedMinimalLoopCoordinateTensor
- MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopCoordinateTensor_isTransferIdempotent
- MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopTensor_isTransferIdempotent
- MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopTensor_isInjective
- MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopTensor_isNormal
For every \(N\geq 2\), the periodic ground-space dimension is
where the sum is over ordered cycles and the indices are read modulo \(N\). This is equations (3)–(4) of [ 5 , Section III ] for chains of length at least two. The one-site periodic interaction requires a separate convention.
At period two, the ground-space dimension is
The sum is over ordered pairs: if \(a\ne b\), then \((a,b)\) and \((b,a)\) are distinct summands. This is the length-two convention stated after equation (4) of [ 5 , Section III ] .
Let \(A^0,\ldots ,A^{r-1}\) have positive physical and bond dimensions. Suppose their word tuples of a supplied positive length \(S\) span the full product matrix algebra. Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA^j\), where every \(\mu _j\) is nonzero. For each interaction range \(R\geq S+1\), there is a constant \(\gamma _R{\gt}0\) such that
for every chain length \(N\in \mathbb N\) and every \(v\perp \ker H_{R,N}\), where \(H_{R,N}\) is the canonical periodic parent Hamiltonian of \(B\). This is the block-injective gap assertion of [ 17 , Section IV.C ] at the simultaneous injectivity range.
Let \(A^0,\ldots ,A^{r-1}\) be tensors with positive bond dimensions. Suppose that their simultaneous word evaluations at length \(S\) span the product of their matrix algebras. Then no two distinct blocks are gauge-phase equivalent. This is a consequence of the simultaneous block-injectivity condition in [ 17 , Section IV.C ] .
Let \(x\mapsto (A_x^j)_j\) be a continuous family of blocks. At every length \(L\) at which the length-\(L\) word tuples span the product of the block matrix algebras for every parameter, the projection onto the sum of the local block spaces on \(L\) sites, and the projection onto the local space of the weighted block sum, are continuous in the parameter. Suppose moreover that the physical dimension and all bond dimensions are positive, and that the word tuples span at a length \(S{\gt}0\) for every parameter. For the weighted block sum, the periodic kernel projection is continuous whenever \(S+1\leq R\leq N\). Nonzero block weights may vary arbitrarily. Thus each such fixed volume has a positive gap uniform on any compact parameter set. The overlaps of two local block spaces are also continuous at any common injectivity length.
- MPSTensor.blockGroundSpaceMapES
- MPSTensor.blockGroundSpaceMapES_apply
- MPSTensor.continuous_blockGroundSpaceMapES_family
- MPSTensor.blockGroundSpaceMapES_injective_of_wordTupleSpanTop
- MPSTensor.range_blockGroundSpaceMapES
- MPSTensor.continuous_iSup_groundSpaceES_starProjection_family
- MPSTensor.continuous_groundSpaceES_toTensorFromBlocks_starProjection_family
- MPSTensor.continuous_norm_groundSpaceES_overlap_family
- MPSTensor.blockPeriodicMpvMapES
- MPSTensor.blockPeriodicMpvMapES_apply
- MPSTensor.blockPeriodicMpvMapES_eq_blockGroundSpaceMapES
- MPSTensor.continuous_blockPeriodicMpvMapES_family
- MPSTensor.ker_parentHamiltonianES_toTensorFromBlocks_eq_range_blockPeriodicMpvMapES
- MPSTensor.blockPeriodicMpvMapES_injective_of_wordTupleSpanTop
- MPSTensor.range_blockPeriodicMpvMapES
- MPSTensor.continuous_parentHamiltonianES_toTensorFromBlocks_kernelProjection_family
- MPSTensor.exists_uniform_parentHamiltonianES_toTensorFromBlocks_gap_fixed_volume_of_wordTupleSpanTop
The tuples of length-\(L\) word products span \(\prod _j\operatorname {Mat}_{D_j}(\mathbb C)\) if and only if \(\Gamma _L^{\oplus }\) is injective.
For a continuous family of fixed-dimensional blocks, the parameters where \(\Gamma _L^{\oplus }\) is injective form an open set, as do the parameters where the length-\(L\) word tuples span the full product matrix algebra. If \(L{\gt}0\), either condition implies the corresponding condition at every length \(n\geq L\). No normalization is required.
Let \(A\) and \(B\) be trace-preserving tensors with positive-definite fixed matrices \(\rho \) and \(\sigma \). Suppose their complementary transfer maps have spectral radius less than one, and that \(\rho _{\operatorname{spec}}(F_{BA}^{\mathrm{rect}}){\lt}1\). For every \(\varepsilon {\gt}0\), there is \(n_0\) such that for every \(n\geq n_0\), \(x\in G_n(A)\), and \(y\in G_n(B)\),
This is the bound on inner products of ground-space vectors obtained from the spectral-radius estimate in the proof of the lemma disjoint in [ 225 ] .
Let \(x\mapsto A_x^j\) be continuous, with fixed block dimensions, and suppose that their length-\(L\) word tuples span the product matrix algebra for every \(x\). Then the projection onto \(\sum _jG_L(A_x^j)\) is continuous. The same holds for the ground space of any weighted block sum with nowhere-zero weights.
Let \(J_j\) be the coordinate inclusion of the \(j\)th summand of a finite direct sum, and let \(T_i=\bigoplus _j\mu _j A_j^i\). Then
The statements permit summands of dimension zero and an empty family. They are the coordinate identities underlying the canonical direct-sum decomposition in [ 13 ] , Section II.
Let \(A^0,\ldots ,A^{r-1}\) be tensors with bond dimensions \(D_0,\ldots ,D_{r-1}\), and let \(L\in \mathbb N\). Suppose that their simultaneous word evaluations at length \(L\) span \(\prod _{k=0}^{r-1}M_{D_k}(\mathbb {C})\). Then every \(A^j\) is \(L\)-block injective.
In the setting of Theorem 12.10.35, assume \(\sum _iA^{k\dagger }_iA^k_i=I\) for every block. For the joint physical map \(P_{i,(k,a,b)}=A^k_i(a,b)\) there is a unitary \(X_g\) on \(\bigoplus _k\mathbb {C}^{D_k}\otimes \mathbb {C}^{D_k}\) with \(U_gP=PX_g\). Consequently \(U_g\) commutes with the identity-extended positive polar factor \(Q\), and the same \(X_g\) satisfies \(U_gP_\gamma =P_\gamma X_g\) along the constructed deformation \(P_\gamma =Q_\gamma W\). The block-diagonal normalization is exactly \(\operatorname {tr}_{\mathrm{left}}(P^\dagger P)=I\) of [ 236 , line 653 ] , and the conclusion is the covariance \(U_gP=PX_g\) of [ 236 , lines 652–660 ] for several blocks, under symmetry of the matrix product vectors up to a phase; see [ 160 ] .
In the setting of Theorem 12.10.35, \(U_g^{\otimes L}G_L(\bigoplus _kA^k)=G_L(\bigoplus _kA^k)\) for every \(L\). If every block satisfies \(\sum _iA^{k\dagger }_iA^k_i=I\), then along the constructed deformation \(P_\gamma =Q_\gamma W\) both the canonical two-site parent projection and the positive interaction
commute with \(U_g^{\otimes 2}\), and their open and periodic sums commute with \(U_g^{\otimes N}\) for every \(N\geq 2\). This is the parent-symmetry conclusion of [ 236 , lines 672–676 ] for several blocks in the printed standard form, for symmetry of the matrix product vectors up to a phase.
- MPSTensor.groundSpaceES_toTensorFromBlocks_invariant_of_isOnSiteSymmetricUpToPhase
- MPSTensor.parentInteractionES_isometricDeformationBlocks_commute_of_isOnSiteSymmetricUpToPhase
- MPSTensor.isometricDeformationInteractionES_commute_of_isOnSiteSymmetricUpToPhase
- MPSTensor.isometricDeformation_parentHamiltonianES_commute_of_isOnSiteSymmetricUpToPhase
- MPSTensor.isometricDeformation_openParentHamiltonianES_commute_of_isOnSiteSymmetricUpToPhase
- MPSTensor.isometricDeformation_periodicInteractionHamiltonianES_commute_of_isOnSiteSymmetricUpToPhase
- MPSTensor.isometricDeformation_openInteractionHamiltonianES_commute_of_isOnSiteSymmetricUpToPhase
Let \(A^1,\dots ,A^r\) be blocks of positive bond dimensions \(D_k\) whose one-site tuples \((A^k_i)_k\) span \(\bigoplus _kM_{D_k}(\mathbb {C})\). Let \(g\mapsto U_g\) be a unitary representation of a monoid on the physical space, and suppose that \(U_g^{\otimes N}\) maps every matrix product vector \(|\psi _N\rangle \) of the block-diagonal tensor \(\bigoplus _kA^k\) to a scalar multiple \(\lambda _{g,N}|\psi _N\rangle \). Then for each \(g\) there are a scalar \(c\) with \(|c|=1\), a permutation \(\sigma \) of the blocks with \(D_{\sigma (k)}=D_k\) and invertible matrices \(X_k\) such that
for every block \(k\) and letter \(i\). If in addition every block satisfies \(\sum _iA^{k\dagger }_iA^k_i=I\), the matrices \(X_k\) may be taken unitary. The joint one-site span is the standard form of [ 236 , lines 325–352 ] . This is the several-block form of the virtual action in [ 236 , lines 652–660 ] .
Let \(A^0,\ldots ,A^{r-1}\) have positive bond dimensions and common physical dimension \(d\geq 1\). Suppose that their simultaneous length-\(S\) word tuples span the product matrix algebra, where \(S{\gt}0\). Let \(B=\bigoplus _j\mu _jA^j\), with every \(\mu _j\ne 0\). For every interaction range \(R\geq S+1\), there exists \(\delta {\gt}0\) such that
for every \(N\geq R\) and every \(v\perp G_N(B)\). No normalization or inequivalence assumption on the original blocks is required. This is the finite-interval conclusion of [ 225 , Theorem 1.2 and Section 6 ] at the sufficient interaction ranges supplied by block injectivity.
Let \(A^0,\ldots ,A^{r-1}\) have positive physical and bond dimensions. Suppose that their simultaneous length-\(S\) word tuples span the product matrix algebra, where \(S{\gt}0\). Set \(B=\bigoplus _j\mu _jA^j\), with every \(\mu _j\neq 0\), and fix \(R\geq S+1\). There are \(p{\gt}0\), with \(R\leq 2p\), and \(\delta {\gt}0\) such that
for every \(M\geq 2\) and every \(v\perp G_{pM}^{\mathrm{ES}}(B)\). This combines the block-injective local spaces of [ 17 , Section IV.C ] with the martingale estimate of [ 225 , Theorem 2.1(ii) and Section 6 ] .
Let \(A^0,\ldots ,A^{r-1}\) have positive physical and bond dimensions. Suppose that their simultaneous length-\(S\) word tuples span the product matrix algebra, where \(S{\gt}0\). Set \(B=\bigoplus _j\mu _jA^j\), with every \(\mu _j\neq 0\). For every \(S+1\leq R\leq N\), \(\ker H^{\mathrm{open}}_{N,R}(B)=G_N^{\mathrm{ES}}(B)\). This sufficient range follows from the inversion and regrowth argument in [ 17 , Section IV.C, lines 2114–2129 ] .
Let \(A^0,\ldots ,A^{r-1}\) have positive physical and bond dimensions. Suppose that their simultaneous length-\(S\) word tuples span the product matrix algebra, where \(S{\gt}0\). Set \(B=\bigoplus _j\mu _jA^j\), with every \(\mu _j\neq 0\), and fix \(R\geq S+1\). There are \(m\geq R\) and \(\gamma {\gt}0\) such that \((R-1)^2{\lt}m\gamma \) and
for every \(v\perp G_{m+R-1}^{\mathrm{ES}}(B)\). The strict inequality is the finite-range threshold of Theorem 12.3.7.81; the coefficient is discussed in docs/paper-gaps/knabe88_finite_range_coefficient.tex.
Let \(B=\bigoplus _{j=1}^g\mu _jA_j\), where \(\mu _j\ne 0\) for \(1\le j\le g\). Then, for every length \(L\),
The second identity is the image of the first under the canonical identification \(\iota _L\), using \(G_L^{\mathrm{ES}}(A)=\iota _L(G_L(A))\).
Suppose the length-\(S\) word tuples of a finite family \(A^0,\ldots ,A^{r-1}\) span the full product matrix algebra. If every \(\zeta _j\) is nonzero, the word tuples of \(\zeta _0A^0,\ldots ,\zeta _{r-1}A^{r-1}\) also span that algebra at the same length \(S\).
Let \(S{\gt}0\), and suppose the length-\(S\) word tuples of a finite family \(A^0,\ldots ,A^{r-1}\) with positive bond dimensions span the full product matrix algebra. There are normalized primitive representatives \(B^0,\ldots ,B^{r-1}\), with positive definite fixed points, whose word tuples still span the full product algebra at length \(S\). For every \(0\leq j{\lt}r\), every length \(L\), and every chain length \(N\),
For any nonzero weights \(\mu _j\), the periodic parent Hamiltonians of \(\bigoplus _j\mu _j A^j\) and \(\bigoplus _j\mu _j B^j\) are equal at every interaction range and chain length.
Let \(A^0,\ldots ,A^{r-1}\) have positive bond dimensions and a common physical dimension. Suppose that their simultaneous length-\(S\) word tuples span the product matrix algebra, where \(S{\gt}0\). Then their word tuples span the same algebra at every length \(n\geq S\). This is the propagation of block injectivity used in [ 17 , Section IV.C ] .
Let \(A\) be any MPS tensor, let \(L\geq 1\), and let \(\varphi \) be a positive normalized quasi-local state. Then \(\varphi (P_A([a,a+N)))=1\) for every \(a\in \mathbb Z\) and \(N\geq 1\) if and only if \(\varphi (P_A([aL,(a+N)L)))=1\) for every \(a\in \mathbb Z\) and \(N\geq 1\). No primitivity, translation invariance, or spectral gap is assumed. This gives the support-space consequence of finite blocking in [ 225 , Section 3 ] .
Let \(d,L,R,K{\gt}0\), let \(h\geq 0\) have range \(R\), and assume \(R+L\leq KL+1\). On the grouped chain, let \(h_B\) be the original open interaction sum on \(KL\) sites expressed in grouped coordinates, and put \(M=KL+1-R\). Let \(\varphi \) be positive on squares and vanish on every original translate of \(h\), and let \(\psi =\varphi \circ \mathcal B_L\) be its functional on the grouped quasi-local algebra. For every grouped interval observable \(Y\), with original image \(X=\mathcal B_L(Y)\),
Both commutators denote their stabilized finite-range representatives.
Let \(d,L,R,K{\gt}0\), \(N\geq K\), \(R+L\leq KL+1\), and \(h\geq 0\). Let \(\widehat h\) be the interaction obtained by regrouping \(H_h(KL)\). If \(H_{\widehat h}(N)\) has norm gap \(\gamma {\gt}0\) on its kernel complement, then \(H_h(NL)\) has norm gap \(\gamma /(KL+1-R)\) on its entire kernel complement. This is a finite-volume consequence of [ 225 , Section 3, equations (3.12)–(3.14) ] .
Let \(d,L{\gt}0\) and let \(T_L\omega =\omega \circ \overline{\mathcal B}_L^{-1}\) be functional transport from the blocked chain to the original chain. For every continuous complex-linear functional \(\omega \) on the blocked algebra,
Equivalently, for every continuous complex-linear functional \(\varphi \) on the original algebra,
Both identities include \(N=0\) and arbitrary integer \(a\).
Let \(A\) be any MPS tensor, and let \(P_A(N)\) denote the matrix of the orthogonal projection onto \(G_N(A)\). For \(L,N\geq 0\),
If \(d{\gt}0\) and \(L{\gt}0\), then for every \(a\in \mathbb Z\) and \(N\geq 0\),
No normality or translation invariance is assumed. These are coordinate consequences of the site regrouping in [ 225 , lines 825–836 ] and the blocked local spaces of [ 17 , lines 1815–1820 and 1985–1992 ] .
For \(d,L{\gt}0\), exact support of \(T_L^{-1}\varphi \) in every interval space of \(A^{[L]}\) is equivalent to exact support of \(\varphi \) in every aligned interval space of \(A\):
The same equivalence holds when both quantifiers over lengths are restricted to \(N{\gt}0\). These identities hold for every continuous complex-linear functional; positivity and normalization are not required.
Let \(d,L,R,K{\gt}0\), \(h\geq 0\), \(N\geq K\), and \(R+L\leq KL+1\). Then
The decoded operators have equal kernels. Equivalently, the configuration isometry maps \(\ker H_{h_B}(N)\) onto \(\ker H_h(NL)\). These are derived positive counting consequences of [ 225 , Section 3, equations (3.12)–(3.14) ] ; no local MPS-kernel condition is required.
Let \(d,L,R,K{\gt}0\), \(R+L\leq KL+1\), \(h\geq 0\), and let \(A\) be any MPS tensor. Suppose that \(\ker H_h(n)=\mathcal G_n(A)\) for all sufficiently large \(n\). Then, for all sufficiently large \(N\),
After decoding, this is the equality with the isometric image of \(\mathcal G_N(A^{[L]})\). No normality or primitivity hypothesis is needed. This transports the eventual kernel hypothesis of [ 225 , Theorem 1.2 ] through its regrouping convention.
Let \(d,L,R,K{\gt}0\), \(R\leq KL\), and \(N\geq K\). If \(h_B\) is the grouped interaction, then
where \(W_{n,b}\) places an operator at the nonwrapping start \(b\). This is the aligned interaction sum of [ 225 , equation (3.12) ] with each grouped interaction expanded into its original terms.
Let \(d,L,K{\gt}0\), let \(b+K\leq N\), and let \(h\) be any operator on \(K\) blocked sites. Write \(W_{N,b}(h)\) for its nonwrapping placement starting at \(b\), and \(\mathcal R_{N,L}\) for decoding both matrix indices. Then
This is an operator identity under the site regrouping in [ 225 , lines 825–836 and Section 3 ] and [ 16 , Appendix, lines 2308 and 2313–2320 ] .
The Euclidean local space of the blocked tensor is carried exactly to the original Euclidean local space:
Hence their orthogonal projectors satisfy
These statements include \(N=0\) or \(p=0\); in those cases the same explicit configuration isometry and boundary-map calculation apply. In particular, the projector identity holds for three blocked sites and \(3p\) original sites. These identities concern the full local ground spaces and their orthogonal projectors under blocking.
For \(d,L,K{\gt}0\) and every \(N\geq 0\), write \(H_h(N)\) for the open sum of the range-\(K\) interaction \(h\) on the blocked chain. Then
The sum is zero when \(N{\lt}K\). This is the open interaction convention of [ 225 , equation (3.12) ] transported under blocking.
Let \(d,L{\gt}0\). Suppose \(A^{[L]}\) has an exact joint-space presentation at every length by inequivalent primitive tensors of positive bond dimensions, with faithful invariant matrices and nonzero weights. Let \(h\geq 0\) have positive range \(R\) and eventual original kernel \(G_A(n)\). If \(K{\gt}0\) and \(R+L\leq KL+1\), there is \(\delta {\gt}0\) such that
Only block-divisible original volumes are asserted. This is the corresponding restriction of [ 225 , Theorem 1.2 ] .
Let \(d,L{\gt}0\) and let \(A\) be an MPS tensor. Suppose that a finite, possibly empty, family of pairwise gauge-phase inequivalent normalized primitive tensors \(B_j\) of positive bond dimensions has faithful invariant matrices, nonzero weights \(\mu _j\), and the exact identities
Let \(h\geq 0\) have range \(R{\gt}0\), and suppose \(\ker H_h(n)=\mathcal G_n(A)\) for all sufficiently large \(n\). There is one \(\gamma {\gt}0\) with the following property. For every pure zero-energy state \(\varphi \), every \(a\in \mathbb Z\), and every matrix \(X\) on \(k{\gt}0\) consecutive original sites, put \(X_a=j_{a,k}(X)\) and assume \(\varphi (X_a)=0\). Along every filter for which the two margins \(\ell ,r\) tend to infinity, there is \(e\in \mathbb C\) such that
The energy and variance are real and nonnegative. The same \(\gamma \) works for all states, observables, filters, and diverging margins. No translation invariance of \(\varphi \) or short-range injectivity bound is assumed. This is the exact-presentation version of the blocking reduction in [ 225 , Theorem 1.2 and Section 6 ] ; identification with a general GVBS presentation remains separate, as recorded in [ 61 ] .
Let \(d,L{\gt}0\). Suppose \(B_1,\ldots ,B_b\) are pairwise inequivalent primitive tensors of positive bond dimensions on the blocked alphabet, with faithful invariant matrices, and \(\mu _j\ne 0\). Assume the exact identities
Let \(\omega _j\) denote the normalized sector states on the blocked chain. For any normalized positive state \(\varphi \) on the original chain, support on all nonempty \(A\)-intervals is equivalent to
The pure supported states are exactly the states \(T_L\omega _j\). Purity and the support condition are taken among all states. This is the conclusion of [ 225 , Theorem 1.1, lines 854–926 ] for the specified exact representation; its existence and identification with a general GVBS construction are separate assertions.
Under the preceding hypotheses, suppose that a real constant \(\gamma \) satisfies \(\gamma \psi (Y^*Y)\leq \psi \bigl(Y^*[H_B,Y]\bigr)\) for every positive-length grouped interval observable \(Y\) with \(\psi (Y)=0\). Then every original local observable \(X\) with \(\varphi (X)=0\) satisfies
The original commutator energy is also the limit of the literal finite-volume commutator energy as both boundary distances tend to infinity, along any filter. In particular, a positive grouped gap yields a positive original gap. This is a conditional transport statement; it does not assert the existence of a gap for an arbitrary interaction. The argument is the site-grouping reduction used in [ 225 , Section 6 ] .
If \(V^\dagger (A^{[L]})^i=B^iV^\dagger \) for every blocked letter \(i\), then every \(N,r\geq 0\) and \(Y\in M_{D\times E}(\mathbb {C})\) satisfy
This is the literal boundary-vector identity from compressed words, including empty prefixes and tails.
Let \(B_a,V_a\) be a finite family satisfying \(\sum _aV_aV_a^\dagger =I\) and \(V_a^\dagger (A^{[L]})^i=B_a^iV_a^\dagger \) for every \(a,i\). Then, for every \(N,r\geq 0\),
The same equality holds in the physical Euclidean space. No orthogonality or injectivity of the residual ranges is assumed.
Let \(L{\gt}0\), and let \(B\) be a tensor on the blocked physical alphabet of \(L\) sites with \(\sum _i(B^i)^*B^i=I\), and let \(\rho \geq 0\) be a fixed point of its transfer map with \(\operatorname {tr}\rho \neq 0\). Transport the quasi-local state of \(B\) and \(\rho \) to the original chain through the blocking isomorphism. The transported state is invariant under translations by integer multiples of \(L\). On an aligned interval of \(NL\) sites it is the insertion expectation of \(B\) after reindexing the configurations into blocks, and an observable on \(k\geq 1\) sites placed inside such an interval has the insertion expectation of its nonwrapping placement [ 225 , lines 825–836 and Section 3 ] .
Suppose \(G_{A^{[L]}}(N)=G_C(N)\) for every \(N\geq 0\). Then a state \(\varphi \) is supported on all nonempty original \(A\)-intervals if and only if \(T_L^{-1}\varphi \) is supported on all nonempty \(C\)-intervals. No primitivity assumption is made.
Let \(A_j\) be a finite family of tensors and let \(\mu _j\neq 0\). For all integers \(L,N\geq 0\), the open-boundary space of the blocked weighted sum satisfies
No periodicity, positivity, or nonempty-family assumption is required. These are the boundary spaces of [ 225 , Section 3 ] under the blocking coordinates of [ 191 , Lemma 6 ] .
Let \(P\) be an auxiliary BNT sector decomposition with distinct basis tensors \((A_j)_j\), and let \(A=\bigoplus _j A_j\) be their direct sum with one unit-weight copy of each sector. Then \((A_j)_j\) is a basis of normal tensors for \(A\): every \(A_j\) is a normal tensor, at every positive system length \(N\)
and the states \(\{ |V^{(N)}(A_j)\rangle \} _j\) are linearly independent for all sufficiently large \(N\).
Let \(S\) be a finite spectator index set, let \(V\) be the Euclidean virtual matrix space, and let \(G:V\to V\) be linear. If \(G^{\oplus S}\) acts on a boundary family \(x=(x_s)_{s\in S}\) by \((G^{\oplus S}x)_s=Gx_s\), then
This estimate also holds when \(S\) is empty.
Let \(A,B\) be trace-preserving primitive MPS tensors with positive definite invariant matrices and positive bond dimensions. Suppose that \(A\) and \(B\) are inequivalent under a bond similarity and a phase, including any identification of equal bond dimensions. Then
For any fixed observable \(X\) on \(k\geq 1\) sites, any filter along which \(\ell \longrightarrow \infty \), and any choice of \(r\geq 0\), one also has
No growth condition is required of the right interval. This is a locality consequence of the overlap estimate in [ 225 , proof of the lemma on distinct local support spaces ] .
Let \(A,B\) be arbitrary MPS tensors with the same physical dimension. Let \(X\) act on \(k\geq 1\) sites, and let \(\ell \geq 1\) and \(r\geq 0\). Write \(N=\ell +k+r\) and let \(X_{\ell ,r}\) be the placement of \(X\) after the first \(\ell \) sites. Then
Place a \(k\)-site observable \(X\) between free intervals of lengths \(\ell ,r\), obtaining \(X^{(\ell ,r)}=I_\ell \otimes X\otimes I_r\). Placement preserves products and adjoints. When \(k\geq 1\), it is the usual nonwrapping window inclusion and \(\| X^{(\ell ,r)}\| \leq \| X\| \).
For every \(K\in \mathbb {N}\), the common overlap factors have fiber formulas
and norm bounds
The formulas and bounds remain valid for empty direct sums and incur no factor depending on the number of configurations.
Assume \(D\geq 1\) and that \(\rho \in M_{D}(\mathbb {C})\) is positive definite. The centered residual of Definition 12.3.7.47 factors exactly at the generally nonidentity limiting metric as
and hence
If, in addition, \(A\) is primitive with fixed-point witness \(\rho \), then there are constants \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that, for every \(K\in \mathbb {N}\) and every \(l\geq 1\),
All formulas in this theorem are reconstructed algebra. They are not sourced formulas and assert neither a full projector-defect estimate nor the FNW–Nachtergaele rational coefficient.
- MPSTensor.boundaryFamilyEquiv_c3LeftOverlapFactorES_apply
- MPSTensor.boundaryFamilyEquiv_c3RightOverlapFactorES_apply
- MPSTensor.c3LeftOverlapFactorES_norm_le
- MPSTensor.c3RightOverlapFactorES_norm_le
- MPSTensor.c3_centeredVirtualResidualES_eq_overlapFactors
- MPSTensor.c3_centeredVirtualResidualES_norm_le
- MPSTensor.IsPrimitiveMPS.c3_centeredVirtualResidualES_norm_sq_le_geometric
Under the hypotheses of Definition 12.3.7.55,
Under the hypotheses of Definition 12.3.7.55, the three corrections have the exact factorizations
Let \(\rho \in M_{D}(\mathbb {C})\) be positive definite and set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and \(S_\infty =\mathcal K_\infty ^{-1}\). Write \(J_{\rm t}\) and \(J_{\rm l}\) for the tail and left virtual maps, and write \(S_n=\mathcal K_n^{-1}\) whenever the length-\(n\) boundary map is injective. The virtual residual in the exact tail-after-left C3 projector factorization has the telescoping decomposition
where \(M_l=(\Gamma ^{\rm tail}_{K,l+1})^\dagger \Gamma ^{\rm left}_{K+l,1}\) and the superscript \(\oplus \) denotes the corresponding fiberwise spectator operator. Substituting this equality into the exact inverse-Gram factorization gives the same decomposition of the physical projector residual, sandwiched between the tail boundary map, the two local fiberwise inverse Grams, and the adjoint left boundary map.
The limiting product in the first parenthesis has, without an additional hypothesis, the explicit limiting mixed pairing
Consequently, pairing the first parenthesis against boundary families is exactly the same finite sum with \(\mathcal K_l-\mathcal K_\infty \) in place of \(\mathcal K_l\).
These identities are reconstructed operator algebra. They assert neither the numerical FNW/Nachtergaele contraction nor any identification of a physical ground projector with \(P_\rho \).
Fix \(K,L_0\in \mathbb N\). Suppose that the tail boundary map for \((K,L_0+1)\), the left boundary map for \((K+L_0,1)\), and \(\Gamma _{K+L_0+1}\) are injective. The difference between the product of the first two inverse-Gram range projectors and the third inverse-Gram range projector is exactly the common-factorization residual obtained from the tail and left virtual maps. Their ranges are, respectively, \(\mathcal V_{K,L_0+1}(A)\), \(\mathcal U_{K+L_0}(A)\), and \(\mathcal G_{K+L_0+1}(A)\). This identity asserts no numerical contraction estimate.
Let \(A\) be a tensor in canonical form with positive physical dimension. Fix any canonical decomposition of \(A\), and let \(B_0,\ldots ,B_{g-1}\) be its distinct normal representatives. Let \(S\) be a positive integer such that the simultaneous length-\(S\) word tuples of \(B_0,\ldots ,B_{g-1}\) span the product of the block matrix algebras. Then, for every \(R\geq S+1\), there exists \(\gamma {\gt}0\) such that, for every chain length \(N\) and every vector \(v\perp \ker H_{R,N}(A)\),
Let \(A\) be a tensor in canonical form with positive physical dimension. Fix any canonical decomposition of \(A\), and let \(B_0,\ldots ,B_{g-1}\) be its distinct normal representatives. Suppose that \(S{\gt}0\) and the tuples of length-\(S\) words span the product of the block matrix algebras. For every \(R\geq S+1\) and every fixed positive operator \(h\) on \(R\) sites with kernel \(G_R(A)\), there is \(\gamma {\gt}0\) such that, for every \(N\) and every \(v\perp \ker H_N(h)\),
Let \(A\) be a tensor in canonical form with positive physical dimension. One may choose a basis of normal tensors \(B_0,\ldots ,B_{g-1}\) such that the following holds for every positive integer \(S\): if the simultaneous length-\(S\) word tuples span the product of the block matrix algebras, then, for every \(R\geq S+1\), there exists \(\gamma {\gt}0\) such that, for every chain length \(N\) and every vector \(v\perp \ker H_{R,N}(A)\),
The representatives are chosen once, before \(S\) and \(R\). This includes canonical forms with repeated blocks, as in [ 17 , Section IV.C ] .
Let \(A\) be a tensor in canonical form, with distinct normal representatives \(B_0,\ldots ,B_{g-1}\), and set \(B=\bigoplus _j B_j\). For every positive interaction range \(R\) and every chain length \(N\), the canonical periodic parent Hamiltonians satisfy
This equality includes the multiplicity copies and the ambient reconstruction of the original tensor.
Let \(A\) be a tensor in canonical form with positive physical dimension. One may choose a basis of normal tensors \(B_0,\ldots ,B_{g-1}\) such that the following holds. Suppose that \(S{\gt}0\) and the tuples of length-\(S\) words span the product of the block matrix algebras. For every \(R\geq S+1\) and every fixed positive operator \(h\) on \(R\) sites with kernel \(G_R(A)\), there is \(\gamma {\gt}0\) such that, for every \(N\) and every \(v\perp \ker H_N(h)\),
The representatives are chosen once, independently of \(S\), \(R\), and \(h\).
Let \(A\) be a canonical tensor with positive physical dimension and positive bond dimension \(D\). There are distinct normal representatives \((A_j)_j\) and a positive integer \(R\leq 3D^5\) such that, for every \(R\leq L\leq N\), every collection of positive operators \(H_i\) with the same kernels as the canonical range-\(L\) parent terms satisfies
The equality includes \(N=L=R\), and in particular holds at \(N=L=3D^5\) for the canonical parent Hamiltonian. Repeated copies of the normal representatives are allowed in \(A\).
Let \(A\) be a tensor. Suppose that, for every sufficiently large range \(W\), its canonical open Hamiltonians have kernels \(\mathcal G_A(N)\) for \(N\ge W\) and a positive uniform norm gap above their actual kernels at all volumes. Let \(h\ge 0\) have positive range \(R\), and suppose \(\ker H_h(N)=\mathcal G_A(N)\) for all sufficiently large \(N\). Then there exists \(\gamma {\gt}0\) such that, for every \(N\ge 0\) and \(v\perp \ker H_h(N)\),
No local kernel condition or tensor injectivity is imposed.
Let \(A\) be a canonical tensor with physical dimension \(d\geq 1\), and choose its distinct normal representatives \(B^j\). At every positive interaction range, its open parent Hamiltonian agrees with that of \(\bigoplus _j B^j\), including all multiplicities and the ambient reconstruction of \(A\). If the representatives are simultaneously injective at \(S{\gt}0\) and \(R\geq S+1\), then \(\ker H^{\mathrm{open}}_{N,R}(A)=G_N(A)\) for every \(N\geq R\), and there is \(\delta {\gt}0\) such that
for every \(N\geq R\) and every \(v\perp G_N(A)\). If the bond dimension \(D\) is positive, the conclusion holds at every \(R\geq 3D^5\) without a supplied injectivity length.
- MPSTensor.CPSVCanonicalFormData.openParentHamiltonianES_eq_representatives
- MPSTensor.CPSVCanonicalFormData.exists_openParentHamiltonianES_gap_of_wordTupleSpanTop
- MPSTensor.IsCPSVCanonicalForm.exists_bnt_open_gap_of_wordTupleSpanTop
- MPSTensor.IsCPSVCanonicalForm.exists_openParentHamiltonianES_gap_of_three_bondDim_pow_five_le
- MPSTensor.CPSVCanonicalFormData.ker_openParentHamiltonianES_eq_groundSpaceES_of_wordTupleSpanTop
Let \(d\ge 1\) and let \(A\) be any tensor. Suppose that, for every \(q\ge R\),
If \(R{\gt}0\), the canonical open Hamiltonian of range \(R\) has kernel \(\mathcal G_A(N)\) for every \(N\ge R\). Consequently, if the one-step identity holds at every sufficiently large length, there is \(R_0{\gt}0\) such that the same kernel identity holds for every \(N\ge R\ge R_0\). The intersection identity is an explicit hypothesis of this criterion.
Let \(A\) be a tensor in canonical form with positive physical dimension and positive bond dimension \(D\). For every \(R\geq 3D^5\), there is \(\gamma {\gt}0\) such that
for every \(N\in \mathbb N\) and every \(v\perp \ker H_{R,N}(A)\). The constant may depend on \(A\) and \(R\), but not on \(N\). This sufficient range combines the blocking estimate of [ 13 , lines 317–345 ] with the gap assertion of [ 17 , Section IV.C ] . It need not be optimal; the sharper sufficient condition \(R\geq S+1\) is recorded in [ 131 ] . The unrestricted assertion fails at some shorter ranges; see [ 27 ] .
Let \(d\geq 1\), let \(\sum _iA_i^*A_i=I\), and let \(\rho \geq 0\) have nonzero trace and satisfy \(E_A(\rho )=\rho \). For every range \(R\geq 0\), the quasi-local state defined by \(A,\rho \) belongs to the zero-energy face of the canonical interaction determined by \(G_A(R)\). No faithfulness or purity is assumed.
Let \(A\) be a tensor in canonical form with positive physical dimension and positive bond dimension \(D\). Let \(R\geq 3D^5\) and let \(h\) be any fixed positive operator on \(R\) sites with \(\ker h=G_R(A)\). Then there is \(\gamma {\gt}0\) such that
for every \(N\in \mathbb N\) and every \(v\perp \ker H_N(h)\). The constant may depend on \(A\), \(R\), and \(h\), but not on \(N\). This is the positive-interaction comparison of [ 17 , Section IV.C ] at the sufficient dimension bound of [ 13 , lines 317–345 ] .
If every eigenvalue \(z\) of a Hermitian matrix \(M\) has \(\operatorname {Re}z=0\) or \(\operatorname {Re}z\geq \delta \), then \(\delta \lVert x\rVert ^2\leq \operatorname {Re}\langle x,Mx\rangle \) for all \(x\perp \ker M\). Conversely, if \(M\geq 0\) satisfies this quadratic-form bound, then every eigenvalue \(z\) of \(M\) has \(\operatorname {Re}z\geq 0\), and \(\operatorname {Re}z=0\) or \(\operatorname {Re}z\geq \delta \). Let \(0\leq A\leq B\) be matrices with \(\ker A\subseteq \ker B\) and such a gap \(\delta \) for \(A\), and let \(t\geq 0\). Then \(C_t=A+t(B-A)\) is positive semidefinite, \(\ker C_t=\ker A\), and every eigenvalue \(z\) of \(C_t\) satisfies \(\operatorname {Re}z=0\) or \(\operatorname {Re}z\geq \delta \). If \(\ker A\neq 0\), then \(0\) is an eigenvalue of every \(C_t\).
Let \(H_n,X_n\) act on finite-dimensional complex Hilbert spaces which may depend on \(n\). Suppose that, for all sufficiently large \(n\), \(H_n\geq 0\), \(\psi _n\in \ker H_n\), and \(H_n\) has norm gap at least the same constant \(\gamma \geq 0\) on \((\ker H_n)^\perp \). Let \(a,e\in \mathbb {R}\). If \(\| X_n\psi _n\| ^2\to a\), \(\| P_{\ker H_n}X_n\psi _n\| ^2\to 0\), and \(\langle \psi _n,X_n^*[H_n,X_n]\psi _n\rangle \to e\), then \(\gamma a\leq e\).
If \(P_{AX}\) and \(P_{XB}\) are idempotent endomorphisms satisfying \(P_{AX}\circ P_{XB}=P_K\) and \(P_{AX}\circ P_{XB}=P_{XB}\circ P_{AX}\), and \(A\)-observables commute with \(P_{XB}\) while \(B\)-observables commute with \(P_{AX}\), then regions \(A\) and \(B\) are decorrelated w.r.t. \(K\). This is the idempotent-product form of the backward direction of [ 13 , Appendix D, Section D.2, Proposition D.3 ] .
Let \(B\geq 0\) be a two-site operator whose overlapping placements commute: \((B\otimes \mathbb {1})(\mathbb {1}\otimes B) = (\mathbb {1}\otimes B)(B\otimes \mathbb {1})\). There are positive integers \(d_{q,l},d_{q,r}\), a one-site unitary \(U\), and positive semidefinite operators \(\eta _{q,h}\) on \(\mathcal H_{q,r}\otimes \mathcal H_{h,l}\) such that
This is the symmetric two-sided form obtained from [ 5 , Lemma 2.1 and Section III, equation (2) ] .
Let \(S\) be compact and let \(x\mapsto A_x^j\) be continuous, with fixed block dimensions. Suppose that every block is trace preserving on \(S\) and that, for each \(x\in S\), the simultaneous word products span the full product matrix algebra at some length. Then there is one \(L\) such that \(\Gamma _n^{\oplus }(A_x)\) is injective for all \(x\in S\) and all \(n\geq L\). In particular, the spanning hypothesis holds for normalized primitive blocks with positive virtual dimensions and positive definite fixed points that are pairwise inequivalent under similarity and phase. No continuity of the fixed-point matrices is required.
Let \(K\) be compact, and let \(A_x^j\) be continuous block tensors with positive physical and bond dimensions. Suppose their length-\(S\) word tuples span the product matrix algebra for every \(x\), with \(S{\gt}0\). Put \(B_x=\bigoplus _j\mu _j(x)A_x^j\), with every weight nonzero. For each fixed \(R\geq S+1\), there is \(\delta {\gt}0\) such that \(\delta \| v\| \leq \| H^{\mathrm{open}}_{N,R}(B_x)v\| \) for every \(x\in K\), every \(N\geq R\), and every \(v\perp \ker H^{\mathrm{open}}_{N,R}(B_x)\). No continuity of the weights is required. The compactness argument follows [ 236 , Appendix A ] at the stated sufficient range.
Under the hypotheses of Theorem 12.8.1.35, let \(h_x\) be a continuous family of positive range-\(R\) interactions with \(\ker h_x=G_R(B_x)\). Then there is \(\delta {\gt}0\) such that \(\delta \| v\| \leq \| H^{\mathrm{open}}_N(h_x)v\| \) for every \(x\in K\), every \(N\geq R\), and every \(v\perp \ker H^{\mathrm{open}}_N(h_x)\). The same conclusion holds under the normalized normal-block hypotheses of Theorem 12.8.1.36 at \(R\geq 3D_*^5\). Nonzero weights need not be continuous.
Under the preceding simultaneous spanning hypothesis, fix \(R\geq S+1\). On a compact parameter set there is \(\delta {\gt}0\) such that the canonical periodic parent Hamiltonian of the weighted block sum satisfies
In particular, for pairwise inequivalent normalized normal blocks of dimensions \(D_j{\gt}0\), one may choose any \(R\geq 3\max (\sum _jD_j,1)^5\). The simultaneous spanning length is then derived from the dimensions. This includes the two-site interaction of the initially blocked path in [ 236 , Appendix A ] ; see [ 176 ] .
- MPSTensor.eventually_parentHamiltonianES_toTensorFromBlocks_gap_of_strict_openGap
- MPSTensor.exists_uniform_parentHamiltonianES_toTensorFromBlocks_gap_of_compact
- MPSTensor.exists_uniform_parentHamiltonianES_toTensorFromBlocks_gap_of_compact_all_lengths
- MPSTensor.exists_uniform_block_parentHamiltonianES_gap_of_compact_isNormalTensor
Let \(S\) be compact and let \(x\mapsto A_x^j\) be continuous, with fixed physical and block dimensions. Suppose that for every \(x\in S\) there is a positive length \(L_x\) at which the simultaneous word products span the full product matrix algebra. Then there is one \(L{\gt}0\) such that simultaneous spanning holds for every \(x\in S\) and every length \(n\geq L\). No normalization or positivity of the block dimensions is required.
Let \(K\) be a compact parameter set, and let \(A_x^j\) be continuous block tensors with positive physical and bond dimensions. Suppose that their length-\(S\) word tuples span the product matrix algebra for every parameter, where \(S{\gt}0\). Set \(B_x=\bigoplus _j\mu _j(x)A_x^j\), with all weights nonzero; the weights need not be continuous. Fix \(R\geq S+1\) and a finite upper length \(M\). There is \(\delta {\gt}0\) such that
for every \(x\in K\), every \(R\leq N\leq M\), and every \(v\perp \ker H^{\mathrm{open}}_{N,R}(B_x)\). The constant may depend on \(M\).
Let \(E\) be a finite-dimensional complex Hilbert space, let \(x\mapsto H_x\colon E\to E\) be continuous, and suppose the orthogonal projections onto \(\ker H_x\) are continuous in \(x\). For every compact parameter set \(S\), there is \(\delta {\gt}0\) such that \(\delta \lVert v\rVert \leq \lVert H_xv\rVert \) for every \(x\in S\) and \(v\perp \ker H_x\). No positivity or self-adjointness is required.
Let \(S\) be a compact subset of a topological parameter space. For each \(N\in \mathbb N\), let \(E_N,F_N\) be seminormed additive groups, and let \(H_N(x):E_N\to F_N\) and \(V_N(x)\subseteq E_N\) be families of maps and admissible vectors. Suppose that every \(x\in S\) has a neighborhood \(U_x\), a constant \(\delta _x{\gt}0\), and a threshold \(N_x\) such that \(\delta _x\| v\| \leq \| H_N(y)v\| \) whenever \(y\in U_x\), \(N\geq N_x\), and \(v\in V_N(y)\). Then there are \(\delta {\gt}0\) and \(N_0\) such that \(\delta \| v\| \leq \| H_N(x)v\| \) for every \(x\in S\), \(N\geq N_0\), and \(v\in V_N(x)\). Linearity of the maps is unnecessary. This is the norm-bound form of the compactness argument in [ 236 , Appendix A ] .
Let \(A_x^j\) be continuous normalized normal blocks, pairwise inequivalent under virtual similarity and phase at every parameter, with physical dimension \(d\geq 1\) and fixed bond dimensions \(D_j\). Put \(D_* =\max \{ \sum _jD_j,1\} \) and \(B_x=\bigoplus _j\mu _j(x)A_x^j\), with every weight nonzero. For every compact \(K\) and every fixed \(R\geq 3D_*^5\), one positive open norm gap holds uniformly for \(x\in K\) and all \(N\geq R\).
Let \(x\mapsto A_x\) be continuous with nonzero fixed bond dimension, and let \(S\) be compact. Suppose every \(A_x\) is injective after the same positive length \(p\). For every fixed interaction range \(R{\gt}p\), there is \(\delta {\gt}0\) such that
If each \(A_x\) is merely normal, the same assertion holds at the ranges \(R\geq D^4+1\). No gap estimate or normalized gauge is supplied as a hypothesis.
- MPSTensor.exists_openParentHamiltonianES_uniform_gap_of_isNBlkInjective
- MPSTensor.exists_strict_openParentHamiltonianES_window_of_isNBlkInjective
- MPSTensor.eventually_parentHamiltonianES_gap_of_strict_openGap_of_isNBlkInjective
- MPSTensor.exists_uniform_parentHamiltonianES_gap_of_compact_isNBlkInjective
- MPSTensor.exists_uniform_parentHamiltonianES_gap_of_compact_isNBlkInjective_all_lengths
- MPSTensor.exists_uniform_parentHamiltonianES_gap_fixed_volume_of_isNBlkInjective
- MPSTensor.isNBlkInjective_pow_four_of_isNormal
- MPSTensor.periodicMpvLineMap_injective_of_isNBlkInjective
- MPSTensor.ker_parentHamiltonianES_eq_range_periodicMpvLineMap_of_isNBlkInjective
- MPSTensor.continuous_parentHamiltonianES_kernelProjection_family_of_isNBlkInjective
- MPSTensor.exists_uniform_parentHamiltonianES_gap_of_compact_isNormal_all_lengths
- MPSTensor.exists_uniform_parentHamiltonianES_gap_of_compact_isNBlkInjective_at_range
- MPSTensor.exists_uniform_parentHamiltonianES_gap_of_compact_isNormal_at_range
Let \(K\) be a compact parameter set. Suppose the local ground projections of \(A_x\) are continuous, and let \(h_x\) be a continuous family of positive interactions with kernel \(G_R(A_x)\), where \(R{\gt}0\). If the canonical open parent Hamiltonians have one positive gap for every \(x\in K\) and every \(N\geq N_0\), then the interaction sums \(H^{\mathrm{open}}_N(h_x)\) also have one positive gap for those parameters and lengths.
For a fixed canonical tensor \(A\), this conclusion requires no supplied canonical gap: simultaneous injectivity at \(S{\gt}0\) and \(R\geq S+1\) give one gap uniform over \(x\in K\) and every \(N\geq R\). When its bond dimension \(D\) is positive, \(R\geq 3D^5\) suffices without a supplied injectivity length.
Let \(x\mapsto A_x\) be a continuous family of one-site injective tensors of nonzero bond dimension, and let \(S\) be a compact parameter set. Then there is \(\delta {\gt}0\) such that, for every \(x\in S\), every \(N\geq 2\) and every \(v\perp \ker H_{N,2}(A_x)\),
This is the uniform-gap conclusion of [ 236 , Appendix A, lines 2575–2580 ] for one-site injective tensors and the two-site parent interaction; the source also treats non-injective normal forms [ 176 ] .
Suppose the local ground-space projection and a positive parent interaction \(h_x\) depend continuously on \(x\), with the same local kernel. On a compact parameter set there is \(\kappa {\gt}0\) such that \(h_x\geq \kappa P_{G_R(x)^\perp }\). Consequently every canonical periodic gap uniform in the parameter and chain length transfers to the chosen interactions. This applies, in particular, to compact normal families at every fixed range \(R\geq D^4+1\).
- MPSTensor.exists_uniform_pos_parentInteractionES_le_of_compact
- MPSTensor.exists_uniform_periodicInteractionHamiltonianES_gap_of_compact_canonical_gap
- MPSTensor.exists_uniform_parentInteraction_gap_of_compact_isNormal
- MPSTensor.exists_uniform_block_parentInteraction_gap_of_compact
- MPSTensor.exists_uniform_parentInteraction_gap_of_compact_isNormal_at_range
Let \(e:I\to J\) be a bijection of finite configuration sets, and let \(H\) be any square complex matrix on \(I\). The associated Euclidean isometry \(U_e\) satisfies
Every estimate \(\gamma \| v\| \leq \| Hv\| \) on \((\ker H)^\perp \) holds with the same constant for \(H^e\) on \((\ker H^e)^\perp \). Positivity is unnecessary.
Let \((A^j)_j\) be simultaneously one-site injective blocks of positive bond dimensions. Define the joint physical map by \(P_{i,(j,a,b)}=A^j_i(a,b)\) and write \(P=QW\) as in Theorem 12.10.20. Reading the columns of \(W\) as block matrices \(W^j_i\), set \(A^j_i(\gamma )=\sum _l(Q_\gamma )_{il}W^j_l\), where \(Q_\gamma =\gamma Q+(1-\gamma )I\). This continuous path has endpoints \(W^j\) and \(A^j\), preserves simultaneous spanning at every length, and has coisometric initial physical map on \(\operatorname {ran}P\).
Both its canonical two-site parent Hamiltonians and the positive interactions \(h_\gamma =(Q_\gamma ^{-1})^{\otimes 2} h_0(Q_\gamma ^{-1})^{\otimes 2}\) have a positive periodic gap uniform in \(\gamma \in [0,1]\) and all \(N\geq 2\). Here \(h_0\) is the canonical two-site parent projection of the unweighted block sum of \((W^j)_j\). The interactions are continuous and have exactly the deformed local ground spaces as their kernels.
For pairwise inequivalent normal blocks, there is a positive initial blocking length \(L\) satisfying \(L+1\leq 3\max (\sum _jD_j,1)^5\), after which the constructed path has the canonical uniform gap just stated. The path construction and gap assertion are those of [ 236 , lines 575–625, Appendix A ] .
- MPSTensor.blockPhysicalMatrix
- MPSTensor.leftPolarBlocks
- MPSTensor.leftPolarPhysicalFactor
- MPSTensor.leftPolarPhysicalFactor_posDef
- MPSTensor.rotatePhysical_leftPolarPhysicalFactor_leftPolarBlocks
- MPSTensor.wordTupleSpanTop_leftPolarBlocks_iff
- MPSTensor.blockPhysicalMatrix_leftPolarBlocks
- MPSTensor.blockPhysicalMatrix_leftPolarBlocks_mul_conjTranspose
- MPSTensor.isometricDeformationBlocks
- MPSTensor.isometricDeformationBlocks_zero
- MPSTensor.isometricDeformationBlocks_one
- MPSTensor.continuous_isometricDeformationBlocks
- MPSTensor.wordTupleSpanTop_isometricDeformationBlocks_iff
- MPSTensor.isometricDeformationInteractionES
- MPSTensor.continuous_isometricDeformationInteractionES
- MPSTensor.isParentInteraction_isometricDeformationInteractionES
- MPSTensor.exists_uniform_isometricDeformation_parent_gap
- MPSTensor.exists_uniform_isometricDeformation_interaction_gap
- MPSTensor.exists_blocked_isometricDeformation_parent_gap_of_isNormalTensor
Let \((A^j)_j\) be simultaneously one-site injective blocks of positive bond dimensions, and form their joint polar decomposition \(P=QW\). Along the constructed path \(A^j(\gamma )=Q_\gamma W^j\), both the canonical two-site open parents and the open Hamiltonians with interaction \(h_\gamma =(Q_\gamma ^{-1})^{\otimes 2} h_0(Q_\gamma ^{-1})^{\otimes 2}\) have a positive gap uniform in \(\gamma \in [0,1]\) and every \(N\geq 2\); here \(h_0\) is the canonical two-site parent projection of the unweighted sum of the polar blocks.
For pairwise inequivalent normalized normal blocks, each gap assertion applies after finite initial blocking at a positive length \(L\) with \(L+1\leq 3\max (\sum _jD_j,1)^5\). The chain length then counts blocked sites. The path is that of [ 236 , lines 575–625, Appendix A ] .
Let \(x\mapsto A_x^j\) be continuous for every block \(j\), with fixed physical and virtual dimensions, and suppose that the joint boundary map at length \(L\) is injective for every parameter \(x\). For arbitrary nowhere-zero weights \(\mu _j(x)\), set \(B_x=\bigoplus _j\mu _j(x)A_x^j\). Then the projector onto \(G_L(B_x)\) and the canonical interaction \(h_L(B_x)\) depend continuously on \(x\). At every fixed volume \(N\geq L\), the corresponding open and periodic parent Hamiltonians are continuous as well. Continuity of the weights is unnecessary. These are the finite-window continuity inputs in [ 236 , Appendix A, lines 2575–2578 ] .
- MPSTensor.continuous_iSup_groundSpaceES_starProjection_of_injective_family
- MPSTensor.continuous_groundSpaceES_toTensorFromBlocks_starProjection_of_injective_family
- MPSTensor.continuous_parentInteractionES_toTensorFromBlocks_family
- MPSTensor.continuous_openParentHamiltonianES_toTensorFromBlocks_family
- MPSTensor.continuous_parentHamiltonianES_toTensorFromBlocks_family
Let \(x\mapsto A_x\) be a continuous family of tensors. For fixed \(L\), the boundary map \(X\mapsto \sum _\sigma \operatorname {tr}(A_{x,\sigma }X)\lvert \sigma \rangle \) on \(L\) sites is continuous in \(x\). If every \(A_x\) is \(L\)-block injective, the orthogonal projection onto the local ground space on \(L\) sites is continuous in \(x\). Hence, for \(L\leq N\), the open and periodic parent Hamiltonians with interaction range \(L\) on \(N\) sites are continuous in \(x\).
Let \(x\mapsto A_x^j\) be continuous and injective after blocking \(L\) sites for every \(x\) and \(j\). For each pair of blocks \(i,j\), the norm \(\| P_{G_L(A_x^i)}P_{G_L(A_x^j)}\| \) is continuous in \(x\).
Let \(E\) be a finite-dimensional Hilbert space, \(F\) a Hilbert space, and \(T_x\colon E\to F\) a continuous family of injective bounded maps. Then the orthogonal projection \(T_x(T_x^\dagger T_x)^{-1}T_x^\dagger \) onto the range of \(T_x\) depends continuously on \(x\).
Suppose that \(\sum _iA_i^\dagger A_i=I\), that the \(P_a\) are orthogonal projections, and that \(P_aA_i=A_iP_{a+1}\) on a positive cyclic index. If \(VV^\dagger =P_a\) and \(\operatorname {tr}\rho \neq 0\), then every \(r\geq 0\) satisfies
More generally, \(C_r\) is contractive whenever its exact Gram matrix \(Q_r\) is an orthogonal projection. Blocking by the period itself leaves each cyclic projection as its own step-orbit projection. These are the cyclic word consequences of [ 191 , Lemma blocking-arbitrary ] .
Whenever \(\| C_r\| \leq 1\),
Under the cyclic hypotheses of Theorem 12.9.194, if the positive-dimensional prefix tensor \(B\) is primitive with invariant matrix \(\rho \), there are \(C{\gt}0\) and \(0{\lt}q{\lt}1\) such that every \(N\geq 1\) and every \(r\geq 0\) satisfy
No independent tail condition is assumed. This is an intermediate Gram estimate for the projection comparison in [ 225 , Lemma commutation(i), lines 2442–2531 ] .
Let \(B\) and \(C\) be blocked-prefix tensors, and form their residual boundary ranges using arbitrary original tails and rectangular support maps. If \(\lvert \langle u,v\rangle \rvert \leq \varepsilon \lVert u\rVert \lVert v\rVert \) for \(u\in \mathcal G_N(B)\) and \(v\in \mathcal G_N(C)\), with \(\varepsilon \geq 0\), the same bound holds between their residual ranges at length \(NL+r\), for every \(r\geq 0\).
Let \((A_j)_{j=1}^g\) be a basis of normal tensors, with bond dimensions \(D_j\). There is a positive integer \(L\) such that the one-letter evaluations of the blocked tensors span the full product algebra:
Moreover, there is \(L_0\) such that, for every \(L\geq L_0\), the unblocked word tuples have the simultaneous span
This establishes simultaneous block injectivity following [ 13 , lines 317–345 ] .
Let \((A_j)_{j=1}^{g}\) be a basis of normal tensors with bond dimensions \(D_j\), or more generally pairwise inequivalent normalized normal tensors. If \(K{\gt}0\) and \(\sum _jD_j\leq K\), then there is a positive integer \(L\) with \(L+1\leq 3K^5\), in particular \(L\leq 3K^5\), such that
Let \(A\) and \(B\) be normal tensors of positive, possibly different, bond dimensions. Suppose that, for some nonzero \(\zeta \in \mathbb {C}\), \(|V^{(N)}(B)\rangle =\zeta ^N|V^{(N)}(A)\rangle \) for \(N\geq 0\). Then the bond dimensions are equal and there is an invertible matrix \(X\) and a scalar \(\eta \) of unit modulus such that \(B^i=\eta X A^iX^{-1}\) for every physical letter \(i\). This is the gauge recovery used in [ 13 , Appendix A ] .
Let \(A\) be a normal tensor in the sense of [ 13 , Section 2.2 ] . Its bond dimension is necessarily positive, and there is a positive definite matrix \(\sigma \) such that the gauge obtained from \(\sigma \) is trace-preserving, primitive, and irreducible. No scalar rescaling is needed.
Suppose that \(W\leq N\) and \(q+R\leq W\). Under the isometry \(U_i\) in (303), the range-\(R\) interaction beginning at the cyclic site \(i+q\) becomes the corresponding nonwrapping interaction on the active \(W\)-site interval, acting independently on the complementary configuration:
Let \(m{\gt}0\), let \(P_uA_i=A_iP_{u+1}\), and suppose that the tensor \(A^{[m]}\) has an isometric sector resolution \(B_j,V_j\), with \(j\in \{ 0,\ldots ,m-1\} \), satisfying
Suppose each support \(V_jV_j^*\) is one of the matrices \(P_u\); their cyclic labels need not be prescribed. If the length-\(q\) simultaneous word span of \(B_j\) is full, then
This supplies the initial graded spanning step for the intersection construction in [ 225 , equations (3.10)–(3.11) and Section 6 ] .
Let \(m{\gt}0\), let \(P_uP_v=\delta _{uv}P_u\), and assume \(\sum _uP_u=I\). If every length-\(n\) word satisfies \(P_uA^\sigma =A^\sigma P_{u+n}\), then
In particular every vector in \(G_n(A)\) has a representative in \(V_n\). This is the cyclic-boundary preparation for the intersection property in [ 225 , equations (3.10)–(3.11) ] .
Suppose \(P_uA_i=A_iP_{u+1}\) and \(\sum _iA_i^*A_i=I\). If every vector of \(G_{n+1}(A)\) has a boundary representative in \(V_{n+1}\), and \(\Gamma _n\) is injective on \(V_n\), then
In particular these hypotheses follow when \(m{\gt}0\), the \(P_u\) are Hermitian orthogonal projections summing to \(I\), and the length-\(n\) word span equals \(\{ Z:\ P_uZ=ZP_{u+n}\text{ for every }u\} \). This is an algebraic criterion for [ 225 , equations (3.10)–(3.11) and Lemma existenceinteraction ] ; eventual spanning for a periodic tensor is a separate step.
Let \(1\leq R\leq m\) and \(2m\leq N\), and put
For every \(s\in \mathbb Z/N\mathbb Z\), let \(A_s=\sum _{q=0}^{m-1}h_{s+q}^{(N,R)}\). The \(m\) interaction windows occupy exactly the \(W\) cyclic sites beginning at \(s\). Under the active block isometry \(U_s\),
For \(Q_q\) as in Theorem 12.3.7.71,
Suppose that \(h_i\) and \(h_{i+d}\) commute whenever \(R\leq d\leq N-R\). Then the cross term is nonnegative for these offsets. If \(1\leq m\), \(R\leq m\), and \(2m\leq N\), it follows that
Let \(h_i\), with \(i\in \mathbb Z/N\mathbb Z\), be orthogonal projections. Define
Then, for every vector \(v\),
- ProjectionGeometry.cyclicWindowSum
- ProjectionGeometry.cyclicCrossTerms
- ProjectionGeometry.reInnerCyclicCrossTerms
- ProjectionGeometry.reInnerCyclicCrossTerms_eq_re_inner_cyclicCrossTerms
- ZModCyclicSums.sum_comp_addRight
- ZModCyclicSums.sum_range_Ico_triangular
- ZModCyclicSums.sum_range_Ico_triangular_lower
- ZModCyclicSums.sum_cyclic_window_eq
- ProjectionGeometry.cyclicWindowSum_sq_sum_eq
- ProjectionGeometry.re_inner_cyclicWindowSum_sum_eq
- ProjectionGeometry.re_inner_cyclicWindowSum_sum_apply
Let \(R\leq e\) and \(e+R\leq N\). Addition by \(e\) in \(\mathbb Z/N\mathbb Z\) moves the standard representative forward by \(e\) sites. The two length-\(R\) cyclic windows beginning at \(s\) and \(s+e\) are disjoint. Consequently,
Let \(A\) be a normalized irreducible periodic tensor of period \(m\). There are positive sector dimensions, primitive normalized tensors \(B_j\) on blocks of \(m\) sites, faithful invariant matrices \(\rho _j\), and isometries \(V_j\) whose mutually orthogonal support projections sum to the identity. Distinct \(B_j\) are inequivalent up to similarity and phase. In the inverse cyclic labelling, the projections satisfy \(P_{j+1}A_i=A_iP_j\). Both rectangular intertwiners hold: \(A^{[m]}_iV_j=V_jB_{j,i}\) and \(V_j^*A^{[m]}_i=B_{j,i}V_j^*\). This retains the cyclic coordinates of [ 191 , Lemma 5 ] used in the grouping argument of [ 225 , Section 6 ] .
Let \(M\) be a commutative monoid. If \(L\leq N\), \(i\in \{ 0,\ldots ,N{-}1\} \), and \(f\colon \{ 0,\ldots ,N{-}1\} \to M\), then the product of \(f\) over the chain equals the product over the cyclic window beginning at \(i\) times the product over its ordered complement.
Suppose that every \(P_u\) is Hermitian, \(P_uA_i=A_iP_{u+1}\), and \(\sum _iA_i^*A_i=I\). If the words of length \(q\) span \(W_q\), then the words of length \(N\) span \(W_N\) for every \(N\ge q\). This algebraic implication supplies the later lengths in the intersection argument of [ 225 , equations (3.10)–(3.11) ] ; the initial spanning equality is assumed here.
Suppose every bond dimension is positive. The transfer map of \(\bigoplus _k B_k\) is idempotent if and only if, for every pair \(j,j'\), the mixed transfer operator is idempotent:
Let \(A,A'\) be gauge and phase inequivalent normalized irreducible periodic tensors, with possibly different periods and bond dimensions. Fix \(L{\gt}0\) and tensors \(B,C\) on \(L\)-site blocks, together with isometries \(V,W\) satisfying \(V^*(A^{[L]})_i=B_iV^*\) and \(W^*(A'^{[L]})_i=C_iW^*\) for every blocked letter. Suppose that \(B\) is a primitive normalized tensor of positive bond dimension with a faithful invariant matrix. Then \(B,C\) are inequivalent up to similarity and phase whenever their bond dimensions agree. The conclusion follows from original-chain separation; separation of compressed corners is not assumed.
Let \(C,B_j\) be tensors on the same physical alphabet and let \(V_j\) be rectangular matrices satisfying \(\sum _jV_jV_j^*=I\) and \(C_iV_j=V_jB_{j,i}\). Let \(\mathcal E\) be the sector embedding of Definition 12.9.188. Then every simultaneous word tuple maps to the corresponding word of \(C\). If the length-\(q\) simultaneous word span is the full product matrix algebra, \(\operatorname {ran}\mathcal E\) is contained in the length-\(q\) word span of \(C\). If also \(V_j^*V_j=I\), every matrix \(X\) commuting with all \(V_jV_j^*\) satisfies
and hence belongs to that word span. No positivity of the sector dimensions or tensor normalization is assumed. These are the embedded cyclic-sector identities of [ 191 , Lemma 5 ] .
Let \(H\) be a linear operator on a complex Hilbert space. If
for every \(v\in (\ker H)^\perp \), then \(\kappa \lVert v\rVert \leq \lVert Hv\rVert \) on the same subspace.
Finite direct sums preserve adjoints, composition, identity operators, and operator differences. They satisfy \(\lVert G^{\oplus S}\rVert \leq \lVert G\rVert \) and \(\langle y,G^{\oplus S}x\rangle =\sum _s\langle y_s,Gx_s\rangle \). These statements include empty coordinate sets.
Let \(S,I,J\) be finite coordinate sets and let \(T\colon \mathbb {C}^I\to \mathbb {C}^J\). Write \(\mathcal F_S(T)\) for the operator applying \(T\) to every fiber of \(\mathbb {C}^{S\times I}\). Then
For any sector family \(T_j\colon \mathbb {C}^{I_j}\to \mathbb {C}^J\) with varying finite domains and common finite target, the joint-range projection satisfies
The sector index in this identity may be infinite, with each sum interpreted as the linear span. For a finite sector family, \(\mathcal F_S\) preserves operator sums and
Empty spectator sets and empty sector families are included. These finite-dimensional identities supply the spectator projection transport used in [ 225 , Lemma commutation(i) ] .
- EuclideanSpace.mem_range_fiberwiseMap_iff
- EuclideanSpace.mem_orthogonal_range_fiberwiseMap_iff
- EuclideanSpace.fiberwiseMap_range_starProjection
- EuclideanSpace.iSup_range_fiberwiseMap_eq
- EuclideanSpace.fiberwiseMap_iSup_range_starProjection
- EuclideanSpace.fiberwiseMap_sum
- EuclideanSpace.norm_sum_fiberwiseMap_range_starProjection_sub_iSup_le
Let \(A\) be any MPS tensor of positive physical dimension, and let \(h\geq 0\) be a local interaction of range \(R{\gt}0\). Suppose that \(\ker H_N(h)=G_N(A)\) for every sufficiently large \(N\). For every continuous complex-linear functional \(\varphi \),
These are the eventual finite-volume kernel and local support conditions of [ 225 , Theorems 1.1–1.2 and Section 3 ] . No translation invariance of \(\varphi \) or injectivity bound on \(R\) is assumed.
Let \(A_j\) be a finite family of normalized primitive MPS tensors with faithful fixed density matrices, pairwise inequivalent under gauge and phase, and let \(B=\bigoplus _j\mu _jA_j\), where every \(\mu _j\ne 0\). Let \(h\geq 0\) be an interaction of range \(R\geq 1\). Suppose that \(\ker H^{\mathrm{open}}_{h,N}=\mathcal G_N(B)\) for all sufficiently large \(N\). Then there exists \(\gamma {\gt}0\) such that
for every \(N\in \mathbb N\) and every \(v\perp \ker H^{\mathrm{open}}_{h,N}\). No condition on the local kernel of \(h\), or lower bound on \(R\) beyond \(R\geq 1\), is required. This is the finite-chain conclusion of [ 225 , Theorem 1.2 ] , under its eventual finite-interval kernel hypothesis, for the primitive-sector representation; the passage from more general periodic presentations is separate [ 61 ] .
Let \(T=\bigoplus _j\mu _jA_j\) be a finite weighted family of inequivalent trace-preserving primitive tensors with faithful invariant density matrices and nonzero coefficients. Let \(h\geq 0\) have range \(R\geq 1\), and suppose \(\ker H_N^{\mathrm{open}}(h)=G_N(T)\) for every sufficiently large \(N\). Let both free margins tend to infinity. Then there is one \(\gamma {\gt}0\) such that
and, for every pure \(\eta \in \mathcal F_h\) and every local observable \(X\) with \(\eta (X)=0\), the literal limit exists and satisfies
The margins may diverge along an arbitrary filter. This is the pair of conclusions in [ 225 , Theorem 1.2 ] for the stated primitive family; the original range has no injectivity lower bound. General periodic presentations remain separate [ 61 ] .
Let \(h\geq 0\) have range \(R{\gt}0\), and suppose that the kernels of its sufficiently long open-chain Hamiltonians equal the joint boundary spaces of a weighted tensor family with nonzero weights. Let \(\omega _\alpha \) be a constructed sector state, with a trace-preserving tensor and a positive invariant matrix of nonzero trace. For \(X\) on \(k{\gt}0\) sites,
Neither primitivity nor faithfulness of the invariant matrix is needed for this assertion. The local commutator patch has real expectation.
Under the primitive-family hypotheses of Theorem 12.9.110, choose the same \(\gamma {\gt}0\) for the finite-chain bound. Let \(\ell ,r\) tend to infinity along any filter. For every constructed sector state and centered local \(X\) on \(k{\gt}0\) sites, the literal energies
converge to a real number \(e\) satisfying \(\gamma \omega _\alpha (X^*X)\leq e\). Thus one constant gives both the all-volume finite-chain norm gap and the infinite-volume sector bound. This states the primitive-sector case explicitly; classification of all pure zero-energy states is a separate assertion.
Let \(A_1,\ldots ,A_b\) be pairwise gauge-phase inequivalent normalized primitive tensors with faithful invariant matrices, and let all weights be nonzero. Let \(h\geq 0\) have arbitrary range \(R{\gt}0\). Suppose that \(\ker H_N^{\mathrm{open}}(h)\) equals the joint boundary-condition space for all sufficiently large \(N\). Then there is \(\gamma {\gt}0\) such that, for every \(N\) and every \(v\perp \ker H_N^{\mathrm{open}}(h)\),
and, for every sector, position, and centered observable \(X\) on \(k{\gt}0\) sites,
The same inequality holds for the finite-interval insertion expectations. No local parent-kernel equality or lower bound on \(R\) in terms of an injectivity length is assumed. This is the primitive-sector specialization of the argument in [ 225 , Theorem 1.2 and Section 6 ] ; the more general GVBS presentations are recorded in [ 61 ] .
Let \(A\) be any MPS tensor of positive physical dimension, and let \(\varphi \) be a normalized positive norm-one quasi-local state. If \(\varphi (j_{-n,2n+1}(p_A(2n+1)))=1\) for every sufficiently large \(n\), then \(\varphi (j_{a,N}(p_A(N)))=1\) for every \(a\in \mathbb Z\) and every \(N{\gt}0\). This is the finite-support restriction used in [ 225 , Theorem 1.1 and Section 3 ] .
Assume \(D\geq 1\), \(l{\gt}0\), and that \(A\) is \(l\)-block-injective. Fix \(K{\gt}0\). With \(n=K+l\) and \(n_l=l+1\), the MPS parent chain satisfies Nachtergaele’s identity following condition C3:
Let \(A\) have positive bond dimension, satisfy \(\sum _iA_i^*A_i=I\), and have a faithful stationary matrix \(\rho {\gt}0\). There exists \(R_0{\gt}0\) such that its canonical open Hamiltonians satisfy
In particular, the canonical local projection of range \(R_0\) defines a positive parent interaction with these exact kernels. The tensor need not be irreducible or periodic, and no interaction or kernel identity is supplied. This concerns the normalized generating data in [ 225 , Section 3 ] ; their construction from an arbitrary GVBS boundary limit remains separate.
Under the preceding hypotheses, there are \(R{\gt}0\), a positive interaction \(h\) of range \(R\), and constants \(\delta ,\gamma {\gt}0\) such that
The generated stationary state belongs to the zero-energy face of \(h\). Every pure state \(\varphi \) in this face and every observable \(X\) on an interval of positive length with \(\varphi (X)=0\) satisfy
The limit holds along every interval exhaustion for which both margins around the support of \(X\) diverge. The choices of \(R,h,\delta ,\gamma \) precede the state, observable, and exhaustion. No purity of the generated stationary state is assumed. This is the supplied generating-data case of [ 225 ] , Theorems 1.1–1.2 and Section 6.
Under the preceding hypotheses, there exist \(R{\gt}0\), a positive interaction \(h\), and \(\gamma {\gt}0\) with exact open kernels \(G_A(N)\) for all \(N\geq R\). The state defined by \(A,\rho \) has zero energy for \(h\). Every pure normalized zero-energy state \(\varphi \) and centered local observable \(X\) on a positive-length interval satisfy
The literal limit holds along every interval exhaustion with both margins diverging. The interaction and constant precede the state, observable, filter, and margins. No translation invariance is imposed on the competing pure states.
Under the preceding hypotheses, the same given interaction \(h\) has a uniform finite-volume gap at every original chain length and one common literal infinite-volume commutator gap in every pure zero-energy state. The infinite-volume limit holds along every interval exhaustion with both margins around the local observable diverging. Both positive constants precede the state, observable, and exhaustion. This is the given-interaction conclusion of [ 225 ] , Theorem 1.2, in the supplied faithful generating-data setting.
Let \(A\) be normalized with positive bond dimension and a faithful stationary matrix, as in Theorem 12.9.315. Let \(h\geq 0\) be any given interaction of positive range \(R\) such that \(\ker H_h(N)=G_A(N)\) for all sufficiently large \(N\). Then there is \(\delta {\gt}0\) such that
The local kernel \(\ker h\) need not equal \(G_A(R)\). This is the finite-volume conclusion of [ 225 ] , Theorem 1.2, in the supplied faithful generating-data setting.
Let \(A\) have positive bond dimension and satisfy \(\sum _i(A^i)^*A^i=I\), with a positive definite fixed point \(\rho \) of its transfer map, and let \(h\geq 0\) be an interaction of positive range whose open-chain kernels eventually equal \(G_N(A)\). The state generated by \(A\) and \(\rho \) has zero local energy, and there are a blocking length \(L{\gt}0\) and a nonempty finite family of pairwise inequivalent normalized primitive tensors \(B_j\) with positive definite fixed points such that the blocked tensor of \(A\) and the unit-weight direct sum of the \(B_j\) have the same boundary spaces at every length. The zero-energy states of \(h\) are exactly the convex combinations of the states generated by the \(B_j\), transported back through the blocking, and the pure zero-energy states are exactly these transported states [ 225 , Theorems 1.1–1.2 ] .
Let \(A\) have positive bond dimension, assume \(\sum _iA_i^*A_i=I\), and let \(\rho {\gt}0\) satisfy \(E_A(\rho )=\rho \). Let \(h\geq 0\) have range \(R{\gt}0\), and suppose that \(\ker H_h(N)=G_A(N)\) for every sufficiently large \(N\). There is one \(\gamma {\gt}0\) with the following property. For every pure zero-energy state \(\varphi \), every observable \(X\) supported on a positive-length interval \([a,a+k)\) with \(\varphi (X)=0\), and every filter along which \(\ell _n,r_n\to \infty \), put \(\Lambda _n=[a-\ell _n,a+k+r_n)\). Then the literal finite-volume limit exists and satisfies
Neither irreducibility nor periodicity of \(A\), nor translation invariance of \(\varphi \), is assumed. This is a consequence for supplied faithful normalized generating data in the setting of [ 225 , Theorem 1.2 and Sections 3, 6 ] . Constructing such data from an arbitrary GVBS boundary limit remains separate; see [ 61 ] .
Let \(A\) have positive bond dimension and satisfy \(\sum _i A_i^*A_i=I\). Suppose that \(\rho {\gt}0\) and \(\sum _i A_i\rho A_i^*=\rho \). There is \(R_0{\gt}0\) such that, for each \(R\geq R_0\), the canonical open Hamiltonians satisfy
and there is \(\delta _R{\gt}0\) such that
The constant is independent of the original chain length, including its residue modulo every blocking length. This is the supplied generating-data case of [ 225 ] , Theorems 1.1–1.2.
Let \(A\) be an MPS tensor of positive bond dimension, let \(\rho {\gt}0\), and let \(X\) act on \(k\geq 0\) sites. Define \(\eta _A\) by the trace insertion formula, without assuming transfer invariance of \(\rho \). Then
Tensor normalization, injectivity, and primitivity are unnecessary. This is the finite support characterization used for the faithful generators in [ 225 , Section 3 and the proof of the overlap lemma in Section 4 ] ; the construction of minimal faithful generators from a general GVBS presentation remains separate [ 61 ] .
Let \(A\) have positive bond dimension, assume \(\sum _iA_i^*A_i=I\), and let \(\rho {\gt}0\) satisfy \(E_A(\rho )=\rho \). There exist a nonempty finite family of positive-dimensional periodic tensors \(B_j\) with periods \(m_j\geq 1\) and support isometries \(V_j\) such that
For every \(N\geq 0\),
This is a consequence for supplied normalized faithful generating data in [ 225 , Section 3 ] . It does not construct such data from a general GVBS boundary limit or impose minimality; that passage remains separate [ 61 ] .
Let \(A\) have positive bond dimension, assume \(\sum _iA_i^*A_i=I\), and let \(\rho {\gt}0\) satisfy \(E_A(\rho )=\rho \). Write \(\omega _{A,\rho }\) for the resulting quasi-local state. For every \(k\geq 0\), there exists a density matrix \(\sigma _k\geq 0\), of trace one, such that
Thus the same density describes every translated interval of length \(k\), and its support is exactly the tensor boundary space. The generating data are supplied, as in [ 225 , Section 3 and the overlap proof in Section 4 ] ; their construction from a general GVBS presentation remains separate [ 61 ] .
Let \(A\) be left-canonical, and suppose its transfer map has a positive-definite fixed point \(\rho \). Every orthogonal projection \(P\) satisfying \((\mathbb {1}-P)A^iP=0\) for all \(i\) commutes with every \(A^i\). Consequently \(A\) has invariant-projector closure. This is a sufficient stationary-weight criterion, obtained from [ 244 , Proposition 6.11 and Theorem 6.13 ] .
Let \(J\) be finite, and let \(w_j(n)\) be real weights that eventually lie in \([0,1]\), with \(\sum _jw_j(n)\to 1\). There are a strictly increasing subsequence \(n_s\) and weights \(p_j\geq 0\), \(\sum _jp_j=1\), such that \(w_j(n_s)\to p_j\) for every \(j\). For any family of tests \(x\), if \(f(x)-\sum _jw_j(n)g_j(x)\longrightarrow 0\) for every test \(x\), then one choice of normalized weights satisfies \(f(x)=\sum _jp_jg_j(x)\) for all tests. The test family need not be countable.
Let \(R,k\geq 1\) and \(X_N=I_\ell \otimes X\otimes I_r\), where \(\ell ,r\geq t=R-1\) and \(N=\ell +k+r\). Then
Equivalently, both identities hold for an arbitrary placement of the enlarged interval inside a longer chain. These are locality identities; no spectral-gap assumption is made.
For fixed \(R,W\) and \(m{\gt}0\), the finitely many canonical open Hamiltonians at lengths \(W+r\), \(0\le r{\lt}m\), have a common positive gap on their actual kernel complements. A gap valid at all lengths \(Km+Mm+r\) with \(K\ge K_0\) and \(r{\lt}m\) is valid at every \(N\ge (K_0+M)m\).
Let \(\varphi \) be pure among all states of the quasi-local algebra. Suppose that \(\omega _i\) are states indexed by a finite set, that \(w_i\geq 0\) and \(\sum _iw_i=1\), and that \(\varphi =\sum _iw_i\omega _i\). Then \(\varphi =\omega _i\) for some \(i\). No translation-invariance condition is imposed on these states.
Let \(1\leq R\leq m\), \(2m\leq N\), and let \(h_i\) be cyclically indexed orthogonal projections such that \(h_i\) commutes with \(h_{i+d}\) whenever \(R\leq d\leq N-R\). Suppose that, for every \(s\) and \(v\),
If \((R-1)^2{\lt}m\gamma \), then, with
one has
The coefficient for general \(R\) is derived here and is not attributed to the cited papers.
Let \(U,V,W\) and \(S_i,T_i,C_i\) be subspaces of a finite-dimensional real or complex Hilbert space, with \(i\) ranging over a finite set \(I\). Write \(P_Z\) for the orthogonal projection onto \(Z\), and set \(e_U=\| P_U-\sum _iP_{S_i}\| \), \(e_V=\| P_V-\sum _iP_{T_i}\| \), and \(e_W=\| \sum _iP_{C_i}-P_W\| \). Then
No orthogonality of the sector subspaces is required. This finite-sum estimate assembles the projection comparisons in [ 225 , Lemmas commutation(ii) and disjoint ] .
Suppose that the orthogonal projection onto \(G_L(A_x)\) is continuous in \(x\). Then its complementary canonical interaction is continuous. For each fixed \(N\geq L\), every translated local term and the open and periodic parent Hamiltonians are continuous as well.
Suppose that \(0{\lt}l\) and \(l+1\leq N\). Take interaction length \(L=l+1\) and define \(E_n=G_n^{(N)}-G_{n+1}^{(N)}\). Then the two boundary ranges in the fixed \(N\)-site space are explicit:
At the remaining boundary index,
At \(n=l\), both \(Q_l\) and \(G_{l+1}^{(N)}\) are the kernel projection of the unique interaction on \([0,l+1)\). Thus
Hence
- MPSTensor.openPrefixParentHamiltonianES_eq_zero_of_lt
- MPSTensor.openPrefixGroundProjectionES_eq_one_of_lt
- MPSTensor.fixedAmbient_martingaleDifference_eq_zero_of_lt
- MPSTensor.openPrefixParentHamiltonianES_eq_openSuffixParentHamiltonianES_at_endpoint
- MPSTensor.openIntervalGroundProjectionES_at_endpoint
- MPSTensor.openIntervalGroundProjectionES_comp_martingaleDifference_at_endpoint
Let \(K,l,n,r\in \mathbb {N}\). Suppose that \(A\) has nonzero bond dimension, is block injective at length \(L_0{\gt}0\), and satisfies \(L_0+1\leq n\). The kernel of the range-\((L_0+1)\) prefix Hamiltonian on the first \(n\) sites of the \((n+1)\)-site volume is the left open-chain ground space. When that prefix instead fills its own \(n\)-site volume, its kernel is the whole \(n\)-site ground space. Independently, the kernel of the full final length-\((l+1)\) suffix window beginning at \(K\) is the tail open-chain ground space. Let \(U_{\mathrm{pre}}\) and \(U_{\mathrm{tail}}\) be the canonical right-spectator isometries for the active-volume decompositions \((n+1+r)=(n+1)+r\) and \((K+l+1+r)=(K+l+1)+r\), respectively. Their orthogonal projections satisfy
- MPSTensor.openPrefixParentHamiltonianES_self_eq_openParentHamiltonianES
- MPSTensor.ker_openPrefixParentHamiltonianES_eq_openChainLeftGroundSpaceES
- MPSTensor.openSuffixParentHamiltonianES_full_window_eq_localTermES
- MPSTensor.ker_openSuffixParentHamiltonianES_eq_openChainTailGroundSpaceES
- MPSTensor.ker_openPrefixParentHamiltonianES_self_eq_groundSpaceES
- MPSTensor.openPrefixGroundProjectionES_conj_rightSpectatorConfigLinearIsometryEquiv
- MPSTensor.openPrefixWholeGroundProjectionES_conj_rightSpectatorConfigLinearIsometryEquiv
- MPSTensor.openIntervalGroundProjectionES_conj_rightSpectatorConfigLinearIsometryEquiv
If \(m\leq n\), then
Hence \((G_m^{(N)})_{m\geq 0}\) is a nested ground-projection family.
Let \(\rho \in M_{D}(\mathbb {C})\) have nonzero trace and define \(P_\rho (X):=(\operatorname{tr}X/\operatorname{tr}\rho )\rho \). Then
If \(D{\gt}0\) and \(\rho \in M_{D}(\mathbb {C})\) is positive definite, then \(R(P_\rho )\) is positive definite and \(\mathscr R(P_\rho )\) is invertible, hence injective.
For the transition matrix \(P\) and weighted matrix-unit tensor \(A^{ij}=\sqrt{P_{ij}}E_{ij}\) defined above, let \(G_I\) denote the orthogonal projection onto the ground space on the interval \(I\), extended by the identity on the remaining sites. Then
For the weighted matrix-unit tensor of the Markov example, the matrix \(\rho =I/4\) is positive definite, has trace one, and is stationary. The tensor has a complementary transfer spectral gap at \(\rho \). Every nonunit eigenvalue of its FNW observable transfer map is zero.
- MPSTensor.FNWDimensionConstant.markovDensity
- MPSTensor.FNWDimensionConstant.markovDensity_posDef
- MPSTensor.FNWDimensionConstant.markovDensity_trace
- MPSTensor.FNWDimensionConstant.markovDensity_fixed
- MPSTensor.FNWDimensionConstant.markovTensor_isPrimitiveMPS
- MPSTensor.FNWDimensionConstant.markovTensor_fnwTransfer_eigenvalue_eq_zero
There exists an injective matrix-product tensor of physical dimension \(16\) and auxiliary dimension \(4\) with a faithful primitive stationary density matrix of trace one such that every eigenvalue other than \(1\) of its observable transfer map has modulus smaller than \(\lambda =1/1000\). Nevertheless, at overlap length \(m=2\),
Thus the dimension-only coefficient and the prescribed-rate assertion following Nachtergaele’s bound cannot hold simultaneously at this fixed overlap length; see [ 135 ] .
Let \(A\) be primitive with nonzero bond dimension and a positive-definite trace-one fixed point \(\rho \). Suppose \(\lambda {\gt}0\) exceeds the modulus of every nonunit transfer eigenvalue. For every \(c{\gt}0\), there is an integer \(M\) such that, for every \(n\geq M\),
The onset \(M\) may depend on the state, rate, and prescribed prefactor.
Under the preceding hypotheses, suppose additionally that \(\lambda {\lt}1\). For every \(c{\gt}0\), there exists an onset \(M\) such that, for every overlap \(m\geq M\), every prefix length \(r\geq 0\), and every suffix length \(\ell \geq 1\), we have \(c\lambda ^m{\lt}1\) and
The onset depends on the state, rate, and prefactor; it is not asserted to be the injectivity threshold. This is the version of [ 225 , Section 6, equation (6.1) ] with the additional onset stated explicitly [ 135 ] .
For every real function \(u\) on the vertices and all endpoints \(a,f\),
For a real weighted matrix-unit tensor, fix external vertices \(a,f\) and write \(p_{bce}=t_{ab}t_{bc}t_{ce}t_{ef}\). The path vector with mark \(z_{be}\) has norm one if \(\sum _{b,c,e}z_{be}^2p_{bce}^2=1\). It is orthogonal to the full four-site ground space if \(\sum _{b,c,e}z_{be}p_{bce}^2=0\).
Let \(U,V,W\) be subspaces of a complex Hilbert space with orthogonal projections \(P_U,P_V,P_W\). If \(x\in V\cap W^\perp \) and \(y\in U\) have norm one, then
Let \(A\) be primitive with nonzero bond dimension and fixed point \(\rho \). Write \(E=E_{\mathrm{FNW},A}\) for its observable transfer map and \(P=E_\infty ^\rho \) for its stationary projection. Every nonzero eigenvalue of \(E-P\) is an eigenvalue of \(E\).
Let \(A\) be primitive with nonzero bond dimension and positive-definite fixed point \(\rho \). Suppose \(\lambda {\gt}0\) and every eigenvalue \(\nu \neq 1\) of its observable transfer map satisfies \(|\nu |{\lt}\lambda \). Then the spectral radius of the transfer remainder in the \(\rho \)-weighted norm is less than \(\lambda \). If \(\operatorname{tr}\rho =1\), there is a constant \(C_\lambda {\gt}0\) such that, for every integer \(n\geq 1\),
This establishes the eigenvalue prescription following [ 225 , Section 6, equation (6.1) ] .
For the four-dimensional matrix \(P\) above, let \(A_{ij}=\sqrt{P_{ij}}E_{ij}\). Then \(A\) is injective at length one, its transfer map \(\mathcal E_A\) is unital and trace preserving, and every matrix \(X\) satisfies
These are the channel properties of the explicit construction used to examine [ 225 , Section 6, equation (6.1) ] .
Let \(e=(1,1,1,1)^{\mathsf T}\) and let \(u=(1,1,-1,-1)^{\mathsf T}\), \(v=(1,-1,1,-1)^{\mathsf T}\), \(w=(1,-1,-1,1)^{\mathsf T}\). Put \(Q=ee^{\mathsf T}/4\) and \(P=Q+(uv^{\mathsf T}+vw^{\mathsf T})/16\). Then \(P\) is strictly positive and doubly stochastic, and
This is an explicit construction for examining the prescribed-rate coefficient in [ 225 , Section 6, equation (6.1) ] .
- MPSTensor.FNWDimensionConstant.hadamardU
- MPSTensor.FNWDimensionConstant.hadamardV
- MPSTensor.FNWDimensionConstant.hadamardW
- MPSTensor.FNWDimensionConstant.uniformMatrix
- MPSTensor.FNWDimensionConstant.markovMatrix
- MPSTensor.FNWDimensionConstant.squareMatrix
- MPSTensor.FNWDimensionConstant.remainderMatrix
- MPSTensor.FNWDimensionConstant.markovMatrix_pos
- MPSTensor.FNWDimensionConstant.markovMatrix_sum_identities
- MPSTensor.FNWDimensionConstant.markovMatrix_pow_two
- MPSTensor.FNWDimensionConstant.markovMatrix_pow_three
- MPSTensor.FNWDimensionConstant.remainderMatrix_nilpotent
Regard \(P\), \(u\) and \(w\) above as real. Then \(P_{ij}{\gt}0\), \(\sum _jP_{ij}=\sum _jP_{ji}=1\) and \(u_i^2=w_i^2=1\) for every \(i\), and the identities \(Pu=0\), \(w^{\mathsf T}P=0\), \(P^2=Q+uw^{\mathsf T}/64\) and \(P^3=Q\) of Theorem 12.8.1.9 hold over \(\mathbb {R}\). For every pair of endpoints \(a,f\), set \(p_{abcef}=P_{ab}P_{bc}P_{ce}P_{ef}\). Then
The amplitudes \(t_{ij}=\sqrt{P_{ij}}\) are strictly positive and satisfy \(t_{ij}^2=P_{ij}\).
- MPSTensor.FNWDimensionConstant.realMarkovMatrix
- MPSTensor.FNWDimensionConstant.realHadamardU
- MPSTensor.FNWDimensionConstant.realHadamardW
- MPSTensor.FNWDimensionConstant.markovAmplitude
- MPSTensor.FNWDimensionConstant.realMarkovMatrix_pos
- MPSTensor.FNWDimensionConstant.realMarkovMatrix_sum_row
- MPSTensor.FNWDimensionConstant.realMarkovMatrix_sum_col
- MPSTensor.FNWDimensionConstant.realHadamardU_sq
- MPSTensor.FNWDimensionConstant.realHadamardW_sq
- MPSTensor.FNWDimensionConstant.realMarkovMatrix_mul_hadamardU
- MPSTensor.FNWDimensionConstant.realHadamardW_mul_markovMatrix
- MPSTensor.FNWDimensionConstant.realMarkovMatrix_pow_two
- MPSTensor.FNWDimensionConstant.realMarkovMatrix_pow_three
- MPSTensor.FNWDimensionConstant.realMarkovMatrix_path_sum
- MPSTensor.FNWDimensionConstant.realMarkovMatrix_path_hadamardU_sum
- MPSTensor.FNWDimensionConstant.realMarkovMatrix_path_hadamardUW_sum
- MPSTensor.FNWDimensionConstant.markovAmplitude_sq
- MPSTensor.FNWDimensionConstant.markovAmplitude_pos
Let \(t=(t_{ij})\) be a real matrix and define physical letters \(A_{ij}=t_{ij}E_{ij}\). The associated transfer map satisfies
Let \(A\) be primitive with nonzero bond dimension and positive-definite trace-one fixed point \(\rho \). Let \(\lambda \geq 0\) satisfy \(r_\rho (A){\lt}\lambda \). There is a constant \(C_\lambda {\gt}0\) such that, for every integer \(n\geq 1\),
This proves the prescribed-rate transfer-remainder estimate in [ 194 , Lemma 5.2 ] . The condition \(\lambda {\lt}1\) is needed only when one subsequently deduces decay to zero.
Under the hypotheses of Theorem 11.3.4.6, suppose in addition that \(a(m)\leq c\lambda ^m{\lt}1\). Then
Let \(A\) be a primitive MPS tensor of nonzero bond dimension with fixed-point witness \(\rho \), and set \(R_A^\rho =E_{\mathrm{FNW},A}-E_\infty ^\rho \). If \(\nu \) is an eigenvalue of \(R_A^\rho \), then
This is the norm-independent spectral step in [ 194 , Lemma 5.2 ] .
Let \(U\) and \(V\) be ground-space ranges in a finite-dimensional complex Hilbert space, let \(W=U\cap V\), and let \(q_U\) and \(q_V\) be the orthogonal projectors onto \(U\) and \(V\). Suppose that \(\eta \ge 0\) and
whenever \(x\in U\cap W^\perp \) and \(y\in V\cap W^\perp \). Define the complementary excitation projections by \(h_U=\mathbb {1}-q_U\) and \(h_V=\mathbb {1}-q_V\). Then, for every vector \(v\),
Let \(V\) be a complex Banach space and let \(T\colon V\to V\) be a bounded linear operator. If \(\rho _{\mathrm{spec}}(T){\lt}1\), then there are real constants \(C,r\) with \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that
for every \(n\in \mathbb {N}\).
Let \(A\) be an MPS tensor, let \(1\leq L\leq N\), and denote its open ground projection on \(n\) sites by \(P_A(n)\). If \(Q_A(L,N)\) is \(P_A(L)\) placed on the first \(L\) sites and extended by the identity, then
No injectivity or normalization is assumed.
Let \(H\geq 0\) act on a finite-dimensional complex Hilbert space, let \(K=\ker H\), and suppose \(\gamma \geq 0\) and \(\gamma \| v\| \leq \| Hv\| \) for every \(v\perp K\). Then \(\langle Hv,v\rangle \geq \gamma (\| v\| ^2-\| P_Kv\| ^2)\) for every \(v\). For every \(\psi \in K\) and linear operator \(X\),
If \(K=\mathbb {C}\psi \) and \(\| \psi \| =1\), the subtracted term is \(|\langle \psi ,X\psi \rangle |^2\). This is the finite-volume estimate for the limiting argument in [ 225 , Section 6 ] ; the full ground-space projection is retained as described in [ 61 ] .
Let \(\psi \) be a vector on \(K+1\) sites and let \(Y_i,Z_j\in M_{D}(\mathbb {C})\) satisfy \(\psi (i,\tau )=\Gamma _{K}(Y_i)(\tau )\) for all letters \(i\) and words \(\tau \) of length \(K\), and \(A^jY_i=Z_jA^i\) for all letters \(i,j\). If matrices \(R_j\in M_{D}(\mathbb {C})\) satisfy \(\sum _jR_jA^j=\mathbb {1}\), then \(\psi \in G_{K+1}(A)\).
Let \(U\colon M_{D}(\mathbb {C})\to \ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\) be Frobenius vectorization, with the target indexed in column–row order, and let \(\iota _L\) be the computational-basis identification on the physical coefficient space. Then, for every \(X\in M_{D}(\mathbb {C})\), \(\Gamma _L^{\mathrm{ES}}(U(X))=\iota _L(\Gamma _L(X))\).
If \(A\) is \(L\)-block injective, then the injective range projector of \(\Gamma _L^{\mathrm{ES}}\) is the physical orthogonal projector onto the local ground space:
In particular, this is the physical range projector on the local Hilbert space, not the virtual transfer fixed-point projector.
Suppose that \(\Gamma _n^{\mathrm{ES}}(A)\) and \(\Gamma _n^{\mathrm{ES}}(B)\) are injective, and put \(\mathcal K_n(A)=\Gamma _n^{\mathrm{ES}}(A)^*\Gamma _n^{\mathrm{ES}}(A)\) and \(\mathcal K_n(B)=\Gamma _n^{\mathrm{ES}}(B)^*\Gamma _n^{\mathrm{ES}}(B)\). For vectors \(x\in \operatorname {ran}\Gamma _n^{\mathrm{ES}}(A)\) and \(y\in \operatorname {ran}\Gamma _n^{\mathrm{ES}}(B)\),
For every \(\Lambda \in M_{d}(\mathbb {C})\) and \(L\ge 0\),
and the same identity holds for the Hilbert-space realizations of the two local ground spaces.
Let \(\psi \) be a vector on \(K+2\) sites and let \(Y_i,Z_j\in M_{D}(\mathbb {C})\) satisfy \(\psi (i,\tau )=\Gamma _{K+1}(Y_i)(\tau )\) and \(\psi (\tau ,j)=\Gamma _{K+1}(Z_j)(\tau )\) for all letters \(i,j\) and all words \(\tau \) of length \(K+1\). Then
for all letters \(i,j\).
For every tensor \(A\) and every \(L\), the local ground space \(G_{L}(A)\) is spanned by the open-boundary states
Here \((l|\) and \(|r)\) are the boundary vectors on both sides of [ 17 , line 1174 ] , written with the bilinear pairing.
Let \(p\geq 1\). Suppose that the range-\(2p\) open parent Hamiltonian of a tensor has its boundary-condition space as kernel at every length at least \(2p\). For every \(K\geq 0\), let \(P_K\), \(T_K\), and \(F_K\) be the boundary-space projections on the first \(K+p\) sites, the last \(2p\) sites, and all \(K+2p\) sites, respectively, with free sites elsewhere. Suppose that \(\epsilon \geq 0\) and that, for every \(K\geq 0\),
Then on every chain of length \(pM\) with \(M\geq 2\), the grouped martingale projections satisfy
for every \(0\leq n{\lt}M\).
Let \(p\geq 1\), let \(N\geq 2p\), and let \(M\geq 0\). Define \(G_n\), \(Q_n\), and \(E_n=G_n-G_{n+1}\) on the \(N\)-site chain by the formulas of Definition 12.8.19. Let \(\epsilon \geq 0\) and suppose that \(\| Q_nE_n\| \leq \epsilon \) for \(2\leq n{\lt}M\). Then \(\| Q_nE_n\| \leq \epsilon \) for \(0\leq n{\lt}M\).
Let \(p\geq 1\), \(M\geq 0\), and \(N=pM\). For every vector \(v\) on the \(N\)-site chain,
Let \(p\geq 1\), \(M\geq 2\), and \(N=pM\). Consider the range-\(2p\) open parent Hamiltonian \(H_N\) and the grouped projections above. Let \(0\leq \epsilon {\lt}1/\sqrt2\) and suppose that \(\| Q_nE_n\| \leq \epsilon \) for \(0\leq n{\lt}M\). Then, for every \(v\perp \ker H_N\),
Let \(p\geq 1\), \(0\leq n{\lt}M\), and \(N=pM\). Write \(Q_n\) for the orthogonal projection onto the kernel of \(H^{\mathrm{suffix}}_{N,2p;2p,p(n+1)}(A)\). Then
for every vector \(v\) on the \(N\)-site chain.
Let the interaction range satisfy \(R{\gt}0\), let \(p\leq n\), and add \(r\geq 0\) sites to the right of an active interval of length \(n\). In each ambient volume \(s\in \{ n,n+r\} \), write \(P^{(s)}_a\) for the kernel projection of the prefix Hamiltonian ending at \(a\), and write \(Q^{(s)}\) for the kernel projection of the suffix Hamiltonian of length \(l\) ending at \(n\). Then
If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then
where \(Q_{AX}=\mathbb {1}-P_{AX}\) and \(Q_{XB}=\mathbb {1}-P_{XB}\) are the complementary projectors. This is the algebraic identity obtained from the complementary projectors in [ 13 , Appendix D, Section D.2 ] ; it is not the full source parent commuting Hamiltonian condition.
Let \((W_i)_{i\in I}\) be an independent family of subspaces, with \(I\) finite, and let \(U_i,V_i\subseteq W_i\). Then
More generally, suppose a family of linear restrictions \(f_t:E\to F\) determines every vector of \(E\). For any independent family \((G_i)\) in \(F\), the subspaces \(\bigcap _t f_t^{-1}(G_i)\) are independent in \(E\). In particular, adjoining the first and last physical indices preserves independence of middle support spaces. Independence is also unchanged by passing from the coefficient functions to their Euclidean realization.
Let \(A\) and \(B\) be trace-preserving tensors with positive-definite fixed matrices and complementary transfer spectral radii less than one. Suppose that, if their bond dimensions agree, they are not gauge-phase equivalent. Then, for every \(\varepsilon {\gt}0\) and all sufficiently large \(n\), every \(x\in G_n(A)\) and \(y\in G_n(B)\) satisfy
This is the normalized tensor form of equation limitepsilonm in [ 225 ] .
Let \(T:E\to F\) be an injective bounded linear map between real or complex Hilbert spaces, with \(E\) finite-dimensional. Write \(K=(T^*T)^{-1}\) and let \(P\) be the orthogonal projection onto \(\operatorname {ran}T\). For every bounded operator \(X\) on \(F\) and every scalar \(c\),
For a family \(T_i,X_i,c_i\) indexed by an arbitrary filter, suppose \(T_i\) is eventually injective and its inverse Gram operators are eventually bounded by one constant. Then \(\| T_i^*X_iT_i-c_iT_i^*T_i\| \to 0\) implies \(\| P_iX_iP_i-c_iP_i\| \to 0\). The Hilbert spaces may vary with \(i\). For any finite-dimensional ground subspace \(G\) and unit vector \(v\in G\), with orthogonal projection \(P_G\), one also has
- ContinuousLinearMap.norm_starProjection_comp_le_inverseGram
- ContinuousLinearMap.norm_starProjection_comp_sub_smul_le_inverseGram
- ContinuousLinearMap.tendsto_starProjection_comp_sub_smul_zero_of_inverseGram_bound
- ContinuousLinearMap.norm_starProjection_apply_sub_smul_le_compression
- ContinuousLinearMap.norm_inner_sub_scalar_le_compression
Let \(A\) be a one-site injective tensor of positive bond dimension \(D\), in the unital canonical orientation \(\mathcal{E}_A(I)=I\). Let \(g\mapsto U_g\) be a unitary representation of a monoid \(G\), and suppose that the matrix product vectors of \(A\) are unchanged by \(U_g^{\otimes N}\) for every \(g\) and every chain length \(N\). For each \(g\), there is a virtual unitary \(V_g\) such that \(\sum _l(U_g)_{il}A_l=V_gA_iV_g^\dagger \) for every \(i\).
For the single-block physical map \(P_{i,(a,b)}=A_i(a,b)\), the induced unitary \(X_g=V_g^T\otimes V_g^\dagger \) satisfies \(U_gP=PX_g\). Consequently, \(U_g\) commutes with the extended positive polar factor \(Q=(PP^\dagger )^{1/2}+I-E\), and the same \(X_g\) satisfies \(U_gP_\gamma =P_\gamma X_g\) at every \(\gamma \in [0,1]\). No faithful adjoint fixed point or virtual covariance is supplied as an additional hypothesis.
With the displayed coordinates for \(P\), unital normalization means \(\sum _iA_iA_i^\dagger =I\). The condition \(\operatorname {tr}_{\mathrm{left}}(P^\dagger P)=I\) printed in [ 236 , line 653 ] instead means \(\sum _iA_i^\dagger A_i=I\); these are the two dual canonical orientations. This result treats the already-blocked, single-block unital orientation of the argument in [ 236 , lines 645–676 ] , for exact invariance of the matrix product vectors. Several blocks are treated in Theorem 12.10.36.
Let \(A\) be a one-site injective tensor of positive bond dimension, in the unital canonical orientation \(\mathcal{E}_A(I)=I\). Let \(g\mapsto U_g\) be a unitary representation of a monoid, and suppose that \(U_g^{\otimes N}\) fixes the matrix product vector of \(A\) for every \(g\) and \(N\). Then \(U_g^{\otimes L}G_L(A)=G_L(A)\) for every \(L\). Along the constructed single-block polar deformation \(P_\gamma =Q_\gamma W\), both the canonical two-site parent projection and the positive interaction
commute with \(U_g^{\otimes 2}\). Their open and periodic sums commute with \(U_g^{\otimes N}\) for every \(N\geq 2\). The initial local-space invariance follows already from injectivity and exact state symmetry, without the normalization assumption. This proves the parent-symmetry conclusion of [ 236 , lines 645–676 ] for the stated single-block unital case, dual to the printed partial-trace normalization as explained in Theorem 12.10.28. Several blocks are treated in Theorem 12.10.37.
- MPSTensor.groundSpaceES_singletonBlocks_invariant_of_isOnSiteSymmetric
- MPSTensor.parentInteractionES_isometricDeformationBlocks_commute_of_isOnSiteSymmetric
- MPSTensor.isometricDeformationInteractionES_commute_of_isOnSiteSymmetric
- MPSTensor.isometricDeformation_parentHamiltonianES_commute_of_isOnSiteSymmetric
- MPSTensor.isometricDeformation_openParentHamiltonianES_commute_of_isOnSiteSymmetric
- MPSTensor.isometricDeformation_periodicInteractionHamiltonianES_commute_of_isOnSiteSymmetric
- MPSTensor.isometricDeformation_openInteractionHamiltonianES_commute_of_isOnSiteSymmetric
Let \(A\) be a one-site injective tensor of positive bond dimension, with \(\sum _i A_i^\dagger A_i=I\). Let \(g\mapsto U_g\) be a unitary representation of a monoid, and suppose that \(U_g^{\otimes N}\) fixes the matrix product vector of \(A\) for every \(g\) and \(N\). For its constructed single-block polar deformation \(P_\gamma =Q_\gamma W\), both the canonical two-site parent projection and the positive interaction
commute with \(U_g^{\otimes 2}\). Their open and periodic sums commute with \(U_g^{\otimes N}\) for every \(N\geq 2\). In the coordinates \(P_{i,(a,b)}=A_i(a,b)\), the normalization is precisely \(\operatorname {tr}_{\mathrm{left}}(P^\dagger P)=I\). This is the already-blocked single-block parent-symmetry conclusion of [ 236 , lines 645–676 ] in the printed canonical orientation. Several blocks are treated in Theorem 12.10.37.
- MPSTensor.parentInteractionES_isometricDeformationBlocks_commute_of_isOnSiteSymmetric_of_isTP
- MPSTensor.isometricDeformationInteractionES_commute_of_isOnSiteSymmetric_of_isTP
- MPSTensor.isometricDeformation_parentHamiltonianES_commute_of_isOnSiteSymmetric_of_isTP
- MPSTensor.isometricDeformation_openParentHamiltonianES_commute_of_isOnSiteSymmetric_of_isTP
- MPSTensor.isometricDeformation_periodicInteractionHamiltonianES_commute_of_isOnSiteSymmetric_of_isTP
- MPSTensor.isometricDeformation_openInteractionHamiltonianES_commute_of_isOnSiteSymmetric_of_isTP
Let \(A\) be one-site injective, of positive bond dimension, with \(\sum _iA_i^\dagger A_i=I\). Let \(g\mapsto U_g\) be a unitary representation of a monoid, and suppose that \(U_g^{\otimes N}\) fixes the matrix product vector of \(A\) for every \(g\) and \(N\). For each \(g\) there is a virtual unitary \(V_g\) such that \(\sum _l(U_g)_{il}A_l=V_gA_iV_g^\dagger \) for every \(i\). For \(P_{i,(a,b)}=A_i(a,b)\), the induced unitary \(X_g=V_g^T\otimes V_g^\dagger \) satisfies \(U_gP=PX_g\). Hence \(U_g\) commutes with the identity-extended positive polar factor \(Q\), and the same \(X_g\) intertwines \(U_g\) with every \(P_\gamma =Q_\gamma W\) in the constructed deformation.
The normalization here is exactly \(\operatorname {tr}_{\mathrm{left}}(P^\dagger P)=I\) in [ 236 , line 653 ] . These conclusions cover the already-blocked single injective block with exact vector symmetry. Symmetry up to a phase, for one or several blocks, is treated in Theorem 12.10.36; see [ 160 ] .
- MPSTensor.exists_unitary_virtual_of_isOnSiteSymmetric_of_isTP
- MPSTensor.exists_blockPhysicalMatrix_unitary_covariance_of_isOnSiteSymmetric_of_isTP
- MPSTensor.commute_leftPolarPhysicalFactor_of_isOnSiteSymmetric_of_isTP
- MPSTensor.exists_isometricDeformationBlocks_unitary_covariance_of_isOnSiteSymmetric_of_isTP
Let \(A\) be a one-site injective tensor of nonzero bond dimension. Then the open-chain Hamiltonian \(H^{\mathrm{open}}_{N,2}(A)\) built from the canonical two-site parent interaction has a positive gap above \(G_N^{\mathrm{ES}}(A)\), uniform in all sufficiently large \(N\). In particular, there are \(m\geq 2\) and \(\gamma \) with \(m\gamma {\gt}1\) such that \(\gamma \lVert v\rVert \leq \lVert H^{\mathrm{open}}_{m+1,2}(A)v\rVert \) for all \(v\perp G_{m+1}^{\mathrm{ES}}(A)\).
Suppose \(\varphi \) is a normalized positive norm-one quasi-local state. For an interval projection \(p=p^*=p^2\), its expectation \(\varphi (j_{a,N}(p))\) is real and belongs to \([0,1]\). Every interval observable satisfies \(|\varphi (j_{a,N}(X))|\leq \| X\| \). If \(\varphi (j_{a,N}(p))=1\), then
No translation invariance is assumed.
Let \(\mathbb K\) be a real or complex scalar field, let \(E\) be a finite-dimensional complete \(\mathbb K\)-Hilbert space, and let \(F\) be a complete \(\mathbb K\)-Hilbert space. If \(T\colon E\to F\) is an injective continuous linear map, then the inverse Gram operator is a two-sided inverse of the Gram endomorphism:
Consequently, \(S_T\) is both the canonical inverse and the ring inverse of \(T^\dagger T\). The two pseudoinverse factors satisfy
- ContinuousLinearMap.inverseGram_comp_adjoint_comp_self
- ContinuousLinearMap.adjoint_comp_self_comp_inverseGram
- ContinuousLinearMap.inverseGram_eq_inverse
- ContinuousLinearMap.inverseGram_eq_ringInverse
- ContinuousLinearMap.norm_T_comp_inverseGram_eq_sqrt
- ContinuousLinearMap.norm_inverseGram_comp_adjoint_eq_sqrt
Assume \(D\geq 1\) and let \(\rho \in M_{D}(\mathbb {C})\) be positive definite. Set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and \(S_\infty =\mathcal K_\infty ^{-1}\). If \(\Gamma _n^{\mathrm{ES}}\) is injective and \(S_n=\mathcal K_n^{-1}\), then
Let \(V^*V=I\) and \(A_iV=VB_i\) for every \(i\). Then for every \(L\geq 0\),
Let \(A=(A^j)_j\) be a finite block family, assembled with unit coefficients, and let \(P=QW\) be its constructed joint left polar decomposition. Suppose that \(U\) is unitary, \(UQ=QU\), and \(U^{\otimes 2}G_2(A)=G_2(A)\). For every \(\gamma \in [0,1]\), put \(Q_\gamma =\gamma Q+(1-\gamma )I\) and \(P_\gamma =Q_\gamma W\). Then \(G_2(W)\) and \(G_2(P_\gamma )\) are invariant under \(U^{\otimes 2}\). Both the canonical parent projection of \(P_\gamma \) and the positive interaction
commute with \(U^{\otimes 2}\). Their open and periodic Hamiltonians commute with \(U^{\otimes N}\) for every \(N\geq 2\). More generally, supplied invariance of \(G_L(A)\) implies invariance of \(G_L(W)\) and \(G_L(P_\gamma )\) for any fixed \(L\). This is the conditional parent-symmetry implication of [ 236 , lines 672–676 ] . The original local-space invariance and commutation of \(Q\) are explicit hypotheses here. Theorem 12.10.37 derives both from symmetry up to a phase of jointly injective blocks in the source’s standard form.
- MPSTensor.groundSpaceES_invariant_leftPolarBlocks
- MPSTensor.groundSpaceES_invariant_isometricDeformationBlocks
- MPSTensor.parentInteractionES_isometricDeformationBlocks_commute
- MPSTensor.isometricDeformationInteractionES_commute
- MPSTensor.isometricDeformation_parentHamiltonianES_commute
- MPSTensor.isometricDeformation_openParentHamiltonianES_commute
- MPSTensor.isometricDeformation_periodicInteractionHamiltonianES_commute
- MPSTensor.isometricDeformation_openInteractionHamiltonianES_commute
If \(U\) commutes with a physical matrix \(Q\), then it commutes with \(Q_\gamma =\gamma Q+(1-\gamma )I\) for every \(\gamma \in [0,1]\). In the setting of Theorem 12.10.23, the constructed physical map \(P_\gamma =Q_\gamma W\) therefore obeys \(UP_\gamma =P_\gamma X\) throughout the path. This is the covariance implication of [ 236 , lines 672–676 ] ; it does not assert the existence of a virtual unitary from an arbitrary state symmetry.
Let \(C\) be an MPS tensor of bond dimension \(D\), and let \((B_j)_{j\in J}\) be a finite family of tensors with the same physical dimension and bond dimensions \(D_j\). Suppose that \(V_j:\mathbb {C}^{D_j}\to \mathbb {C}^D\) satisfy \(V_j^\dagger V_j=\mathbb {1}_{D_j}\), \(\sum _jV_jV_j^\dagger =\mathbb {1}_D\), and \(C^iV_j=V_jB_j^i\). Then \(G_N(C)=\bigvee _jG_N(B_j)\) for every \(N\ge 0\), both in computational coordinates and in the physical Hilbert space.
Let \((A_i)_{i\in I}\) be a finite family of tensors, possibly empty. Suppose that their ground spaces \(\mathcal G_i(n)\) are independent and that each tensor satisfies
Then the sums \(\mathcal G(q)=\sum _i\mathcal G_i(q)\) satisfy
The individual intersection identities and middle independence remain explicit hypotheses. This criterion does not by itself assert the unrestricted intersection theorem for periodic tensor families.
Let \(G_i\subseteq (\mathbb C^d)^{\otimes n}\) and \(H_i\subseteq (\mathbb C^d)^{\otimes (n+1)}\) be finite families. Assume that the \(G_i\) are independent and that every restriction of a vector in \(H_i\), obtained by fixing either the first or last physical index, lies in \(G_i\). Write \(L_i=H_i\otimes \mathbb C^d\) and \(R_i=\mathbb C^d\otimes H_i\). Then
This is the geometric step of Nachtergaele, arXiv:cond-mat/9410110, the proposition and equivalence lemma of Section 4; no tensor representation is assumed in this statement.
Let \(H,H_0,P,P_0\) be bounded operators on a complex normed space, and suppose \(\delta \lVert v-P_0v\rVert \leq \lVert H_0v\rVert \) for all \(v\), with \(\delta \geq 0\), \(\lVert H-H_0\rVert \leq \varepsilon \) and \(\lVert P-P_0\rVert \leq r\). Then \((\delta (1-r)-\varepsilon )\lVert v\rVert \leq \lVert Hv\rVert \) for every \(v\) with \(Pv=0\). Consequently, if \(x\mapsto H_x\) and the orthogonal projection onto a subspace \(K_x\) are continuous at \(x_0\), \(K_{x_0}\subseteq \ker H_{x_0}\), and \(\delta \lVert v\rVert \leq \lVert H_{x_0}v\rVert \) on \(K_{x_0}^\perp \) with \(\delta {\gt}0\), then every \(\delta '{\lt}\delta \) satisfies \(\delta '\lVert v\rVert \leq \lVert H_xv\rVert \) on \(K_x^\perp \) for all \(x\) near \(x_0\).
Let \(x\mapsto A_x\) be a continuous family of one-site injective tensors of nonzero bond dimension, let \(m\geq 2\), and suppose \(\gamma \lVert v\rVert \leq \lVert H^{\mathrm{open}}_{m+1,2}(A_{x_0})v\rVert \) for \(v\perp G_{m+1}^{\mathrm{ES}}(A_{x_0})\) with \(m\gamma {\gt}1\). Then there is \(\delta {\gt}0\) such that, for all \(x\) near \(x_0\) and all \(N\geq 2m\), the periodic parent Hamiltonian \(H_{N,2}(A_x)\) satisfies \(\delta \lVert v\rVert \leq \lVert H_{N,2}(A_x)v\rVert \) on the orthogonal complement of its kernel.
Let \(M=WQ\) with \(Q\ge 0\), and suppose \(W^\dagger W=E\) for the orthogonal projector \(E\) onto the range of \(Q\). Then \(Q=(M^\dagger M)^{1/2}\), \(E\) is the support projector of \(M\), and \(W\) is the partial isometry of the polar decomposition of \(M\). In particular, every positive semidefinite \(Q\) with \(Q^2=M^\dagger M\) is the positive part of \(M\), and every orthogonal projector with the range of that positive part is the support projector.
Every complex matrix satisfies \(M=VP\) and \(V^\dagger V=\Pi \).
For every prefix \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) and suffix \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \),
The range of the left boundary map is exactly the left ground space:
At \(L=1\), transport of the left boundary map range by the inverse canonical \(\ell ^2\) isomorphism identifies it with the open-chain left ground submodule:
Concretely, a vector \(v\) in the \((K+1)\)-site Euclidean space belongs to \(\mathcal U_K(A)\) exactly when the corresponding function satisfies the left ground condition (every fixed-final-site restriction lies in \(G_K(A)\); Definition 11.5.1.4).
A normal left-canonical renormalization fixed-point tensor of nonzero virtual dimension has pairwise commuting length-two parent interactions on every periodic chain of length \(N{\gt}2\), and its periodic matrix product vector is annihilated by every such interaction. The removal of the left-canonical hypothesis and the ground-space spanning equation in [ 13 , Definition 3.9 ] remain separate.
Suppose that \(\Gamma _K^{\mathrm{ES}}\) is injective and write \(S_K=\mathcal K_K^{-1}\). Then
For arbitrary virtual dimension, the adjoints of the tail and left virtual maps are
If, in addition, \(D{\gt}0\) and \(\rho \in M_{D}(\mathbb {C})\) is positive definite, set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and \(S_\infty =\mathcal K_\infty ^{-1}\). Then
Let \(A\) be a primitive MPS tensor with nonzero bond dimension and a positive-definite fixed point. There are \(l{\gt}1\) and \(\epsilon _l\geq 0\), with \(A\) \(l\)-block injective, such that for every final volume \(N\) and every \(n{\lt}N\),
This is the same overlap length and the same coefficient as in Theorem 12.3.7.62. More generally, for any tensor of positive bond dimension that is \(l\)-block injective with \(l{\gt}1\), a uniform physical open-chain bound \(\epsilon \ge 0\) gives the same bound in every final volume, without requiring \(\epsilon {\lt}1/\sqrt{l+1}\).
- MPSTensor.fixedAmbient_martingaleDifference_conj_rightSpectatorConfigLinearIsometryEquiv
- MPSTensor.openIntervalGroundProjectionES_comp_martingaleDifference_conj_rightSpectator
- MPSTensor.norm_openIntervalGroundProjectionES_comp_martingaleDifference_le_openChain
- MPSTensor.fixedAmbient_martingaleDifference_norm_le_of_openChain
- MPSTensor.IsPrimitiveMPS.exists_fixedAmbient_martingaleDifference_norm_lt_c3_threshold
Let \(A\) be a primitive MPS tensor with nonzero bond dimension and a positive-definite fixed point. There are \(l{\gt}1\) and \(\epsilon _l\geq 0\), with \(A\) \(l\)-block injective, such that for every \(K{\gt}0\), after setting \(n=K+l\) and \(n_l=l+1\),
The coefficient satisfies
The length \(l\) and coefficient \(\epsilon _l\) are exactly those chosen by Theorem 12.3.7.61.
Let \(i+L\leq n\). Under the canonical identification
the nonwrapping length-\(L\) interaction beginning at \(i\) is
Let \(h\) have positive range \(R\), and let \(X,Y\) be observables on positive-length intervals with the same image in the quasi-local algebra. The images of their stabilized local commutator energies \(X^*[H,X]\) and \(Y^*[H,Y]\) are equal.
Let \(A\) be a tensor, let \(\rho \geq 0\) be a virtual matrix, and let \(H\geq 0\) act on \(N\) physical sites with \(G_N(A)\subseteq \ker H\). For every \(N\)-site matrix \(B\),
In particular, the commutator expectation is real. Neither primitivity nor a spectral gap is needed.
Let \(A\) and \(B\) be gauge equivalent. For every \(L\) one has \(G_L(A)=G_L(B)\).
With \(I,J\) as above, \(C_{a-b,b+N+c}(\iota _{I,J}(X))=I_b\otimes C_{a,N}(X)\otimes I_c\). Thus the integer-interval inclusion agrees with the interior placement used in the local GVBS expectations of [ 225 , Section 3, equations (3.1)–(3.2b) ] .
Let \(b,c\geq 0\), \(I=I(a,N)\) and \(J=I(a-b,b+N+c)\). Then \(I\subseteq J\). For configurations \(\sigma ,\tau \) on \(b+N+c\) consecutive sites,
where \(M=\{ b,b+1,\ldots ,b+N-1\} \) and \(\sigma |_M\) denotes the configuration \(i\mapsto \sigma _{b+i}\) on \(N\) sites. This is the identity-tensor inclusion of [ 16 , Appendix ] expressed in increasing integer-site coordinates.
For \(a,b\in \mathbb Z\) and \(N\geq 0\), translation by \(b\) sends \(I(a,N)\) to \(I(a+b,N)\). If \(T_b\) denotes the induced local star-algebra equivalence, then \(C_{a+b,N}(T_bX)=C_{a,N}(X)\). The equality includes the empty interval.
Let \(A\) have positive bond dimension and satisfy \(\sum _i(A^i)^*A^i=I\), and let \(\rho {\gt}0\) be a virtual matrix. For every \(k\geq 0\), there is a density matrix \(\sigma _k\) such that
The trace representation holds for every observable \(X\) on \(k\) sites. No transfer invariance of \(\rho \), injectivity, or primitivity is assumed. These finite densities identify the supports for supplied faithful generating data as in [ 225 , Section 3 and the overlap proof in Section 4 ] . The construction of faithful minimal data from a general GVBS presentation remains separate [ 61 ] .
Put \(q(A)=1-P_{G_2(A)}\) for the canonical two-site parent interaction. On the three-site space \(\mathcal{H}_A\otimes \mathcal{H}_X\otimes \mathcal{H}_B\), with sites \(0,1,2\) labelled \(A,X,B\), the two adjacent translated length-two parent interactions are the two local actions
This identifies the two translated parent interactions with the local actions in [ 13 , Definition D.2 ] .
Suppose the canonical two-site parent interaction \(q(A)=1-P_{G_2(A)}\) satisfies the overlapping \(AX/XB\) commutation condition of Definition 12.1.2. Then, on the three-site chain, \(h^{(3)}_0(A,2)h^{(3)}_1(A,2) = h^{(3)}_1(A,2)h^{(3)}_0(A,2)\). This only identifies the first two translated local terms with the lifted \(AX\) and \(XB\) actions; the construction of the [ 13 , Appendix D.2 ] projectors associated with the [ 13 , Appendix B ] basic-vector form is a separate step.
Suppose the Appendix D.2 projectors \(Q_{AX}\) and \(Q_{XB}\) satisfy the local parent-commuting condition on \(A X B\), and suppose that the canonical two-site parent interaction is their complement on both faces: \(q(A)=1-Q_{AX}=1-Q_{XB}\). Then the first two translated length-two local terms on the three-site chain commute, \(h^{(3)}_0(A,2)h^{(3)}_1(A,2) = h^{(3)}_1(A,2)h^{(3)}_0(A,2)\).
Suppose the projectors \(Q_{AX}\) and \(Q_{XB}\) satisfy the [ 13 , Definition D.2 ] local parent-commuting condition on \(A X B\), and suppose that the canonical two-site parent interaction is identified, as a two-site operator, with the two source projectors: \(q(A)=Q_{AX}\) and \(q(A)=Q_{XB}\). After lifting these two equalities to the \(AX\) and \(XB\) faces, the first two translated length-two local terms on the three-site chain commute. Thus, \(h^{(3)}_0(A,2)h^{(3)}_1(A,2) = h^{(3)}_1(A,2)h^{(3)}_0(A,2)\).
Let \(Q_{AX}\) and \(Q_{XB}\) be two-site projectors whose lifted \(AX\) and \(XB\) actions commute. Suppose that \(q(A)=Q_{AX}\) and \(q(A)=Q_{XB}\). Then the first two translated length-two local terms on the three-site chain commute.
Suppose \(G_0=\mathbf1\). For every \(v\perp \operatorname {ran}(G_N)\),
This is the identity resolutionnormpsi in the proof of [ 225 , Theorem 2.1(i) ] .
For every \(v\in \mathcal H\) and finite \(S\subset \mathbb {N}\),
In particular, for \(S=\{ 0,\ldots ,N-1\} \),
Let \(P_i\) be a finite family of symmetric projections on a finite-dimensional complex Hilbert space, let \(H=\sum _iP_i\), and let \(0{\lt}\gamma \le 1\). Suppose that a real coefficient matrix \((c_{ij})\) has row and column sums at most one away from the diagonal, and that, for \(i\ne j\) and every \(v\),
Then
for every \(v\). Consequently, for every \(v\in (\ker H)^\perp \),
Let \(U:E\to F\) be a surjective complex linear isometry of inner product spaces, and let \(P:E\to E\) and \(Q:F\to F\) be linear maps satisfying \(Q=UPU^{-1}\). Then \(UPx=QUx\), \(\ker Q=U(\ker P)\) and \((\ker Q)^\perp =U((\ker P)^\perp )\). For every real \(\delta \), the estimate \(\delta \| x\| \leq \| Px\| \) on \((\ker P)^\perp \) is equivalent to \(\delta \| y\| \leq \| Qy\| \) on \((\ker Q)^\perp \). Finite dimensionality and positivity are not required.
Let \(A\) be a primitive MPS tensor with nonzero physical and bond dimensions, and suppose that its fixed point is positive definite. There is a blocking length \(p{\gt}0\) such that \(A\) is \(p\)-block injective and, with \(B=\operatorname {block}_p(A)\), for every \(N\geq 4\) and every \(v\in (\ker H_{N,2}(B))^\perp \),
Every complex linear functional on \(\prod _j\operatorname {Mat}_{D_j}(\mathbb C)\) has the form \(M\mapsto \sum _j\operatorname{tr}(\Delta _jM_j)\) for some family \(\Delta \). Consequently, a linear subspace \(W\) is the whole product algebra if the only family satisfying \(\sum _j\operatorname{tr}(\Delta _jM_j)=0\) for every \(M\in W\) is the zero family.
If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then
This is the algebraic consequence corresponding to equation (D.2) from [ 13 ] : \(K_{AXB}=(K_{AX}\otimes H_B)\cap (H_A\otimes K_{XB})\). The source equation itself also carries the tensor-product local-subspace hypotheses.
Under the canonical \(\ell ^2\) identification, a Hilbert-space vector belongs to \(\mathcal U_L(A)\) exactly when the corresponding function satisfies the left ground condition: every fixed-final-site restriction belongs to \(G_L(A)\).
Under the canonical \(\ell ^2\) identification, a Hilbert-space vector belongs to \(\mathcal V_{K,L}(A)\) exactly when the corresponding function satisfies the tail ground condition: every fixed-prefix restriction belongs to \(G_L(A)\).
If \((F_{BA}^{\mathrm{rect}})^n(X)\to 0\) for every boundary matrix \(X\), then \(\lVert \Gamma _n^{\mathrm{ES}}(A)^*\Gamma _n^{\mathrm{ES}}(B)\rVert \to 0\). In particular this holds when the spectral radius of \(F_{BA}^{\mathrm{rect}}\) is less than one.
Let \(A\) and \(B\) have bond dimensions \(D_A\) and \(D_B\), respectively, and let \(\Gamma _n^{\mathrm{ES}}(A)\) and \(\Gamma _n^{\mathrm{ES}}(B)\) be their Hilbert-space boundary maps, writing \(E_{ab}\) also for the coordinate vector of a matrix unit. Let \(F_{BA}^{\mathrm{rect}}\) be the rectangular mixed transfer operator on \(M_{D_B\times D_A}(\mathbb {C})\). For matrix units \(E_{ab}\) and \(E_{ce}\),
Thus \(\Gamma _n^{\mathrm{ES}}(A)^*\Gamma _n^{\mathrm{ES}}(B)\) is the rectangular reshuffling of \((F_{BA}^{\mathrm{rect}})^n\). This is the boundary-coordinate form of equations ip and P12 in the proof of the lemma disjoint in [ 225 ] .
Under the preceding strict mixed spectral-radius hypothesis, let \(N=\ell +k+r\) and \(X_{\ell ,r}=1_\ell \otimes X\otimes 1_r\). As \(\ell ,r\to \infty \) along any common filter,
No injectivity of the boundary maps is required. This is a distinct-sector estimate, rather than an assertion of the infinite-volume spectral gap.
The coefficient of \(\Gamma _{A,k}^*X\Gamma _{B,k}\) between boundary units \((b,a)\) and \((e,c)\) is \(E_{B,A;X}(|c\rangle \langle a|)_{eb}\). Moreover,
and
- MPSTensor.inner_single_mixed_observable_groundSpaceMapES_single
- MPSTensor.adjoint_groundSpaceMapES_observable_groundSpaceMapES_eq_mixedTransfer
- MPSTensor.mixedPhysicalObservableTransfer_one
- MPSTensor.mixedPhysicalObservableTransfer_appendObservable
- MPSTensor.mixedPhysicalObservableTransfer_bulkObservable
- MPSTensor.adjoint_groundSpaceMapES_bulkObservable_groundSpaceMapES_eq_mixedTransfer
Suppose the rectangular mixed transfer map \(E_{B,A}\) has spectral radius less than one. Then \(E_{B,A}^{\, n}\to 0\) in operator norm. For a fixed local matrix \(X\), and along any filter with \(\ell \to \infty \) and \(r\to \infty \),
in operator norm.
Let \((a_m)_{m\geq 0}\) be a nonnegative real sequence. For all \(l,N\in \mathbb {N}\),
The same bound holds after omitting the terms with \(n{\lt}l\). More generally, for a threshold \(n_0\in \mathbb {N}\) only the coefficients indexed by \(m\geq \max \{ 0,n_0-l\} \) occur, so
Suppose \(\sum _i A_i^\dagger A_i=I\) and \(E_A(\rho )=\rho \). Then the preceding definition is independent of the containing interval. For \(\Lambda \subseteq \Gamma \) and \(X\in \mathcal A_\Lambda \), \(\eta _{A,\Gamma }(\iota _{\Lambda ,\Gamma }X)=\eta _{A,\Lambda }(X)\). Neither positivity nor nonzero trace of \(\rho \) is required for these algebraic compatibility identities.
Suppose \(\sum _i A_i^\dagger A_i=I\) and \(E_A(\rho )=\rho \). For every finite region \(\Lambda \subset \mathbb Z\), displacement \(b\in \mathbb Z\), and observable \(X\in \mathcal A_\Lambda \), \(\eta _{A,\Lambda +b}(T_bX)=\eta _{A,\Lambda }(X)\). Positivity and nonzero trace of \(\rho \) are unnecessary for this identity.
Under the hypotheses of the preceding definition, the restriction of \(\omega _A\) to \(\mathcal A_\Lambda \) is \(\eta _{A,\Lambda }\). Thus \(\omega _A(X)=\eta _A(C_{a,N}X)\) for an observable on \(I(a,N)\), including \(N=0\). The extension satisfies \(\| \omega _A\| =1\), \(\omega _A(I)=1\), and \(\omega _A(X^*X)\geq 0\) for every quasi-local observable \(X\). No primitivity or purity is asserted.
Suppose \(d{\gt}0\), \(\sum _i A_i^\dagger A_i=I\), \(\rho \geq 0\), \(E_A(\rho )=\rho \), and \(\operatorname {tr}\rho \neq 0\). For every \(a\in \mathbb Z\), \(k\geq 0\), and \(X\in M_{d^k}(\mathbb C)\), the completed state satisfies \(\omega _A(j_{a,k}(X))=\eta _A(X)\).
Let \(A\) be trace preserving and let \(\rho \ge 0\) be invariant with nonzero trace. If \(G_R(A)\subseteq \ker h\), then the constructed state \(\omega _A\) belongs to \(\mathcal F_h\). Consequently, if \(B=\bigoplus _j\mu _jA_j\), with all \(\mu _j\ne 0\), and \(h\) is a positive parent interaction for \(B\), every sector for which such trace-preserving and invariant-matrix data are given determines a state in \(\mathcal F_h\). Neither primitivity nor faithfulness of the invariant matrix is required.
Suppose \(d{\gt}0\), \(\sum _i A_i^\dagger A_i=I\), \(\rho \geq 0\), \(E_A(\rho )=\rho \), and \(\operatorname {tr}\rho \neq 0\). For every displacement \(b\in \mathbb Z\) and quasi-local observable \(X\), the completed state satisfies \(\omega _A(T_bX)=\omega _A(X)\). This proves translation invariance of the local GVBS state in [ 225 , Section 3, equations (3.1)–(3.2b) ] ; no purity conclusion is asserted.
Let \(A\) be an MPS tensor and let \(X\) act on \(k\) sites. If \(G_k(A)\subseteq \ker X\), then \(\eta _A(X)=0\) for every virtual matrix \(\rho \), without normalization or positivity assumptions. If \(d{\gt}0\), \(\sum _i A_i^\dagger A_i=I\), \(\rho \geq 0\), \(E_A(\rho )=\rho \), and \(\operatorname {tr}\rho \neq 0\), this gives \(\omega _A(j_{a,k}(X))=0\) for every \(a\in \mathbb Z\). In particular, the constructed state has expectation zero for every translate of a positive local parent interaction. This is the local support property in [ 225 , Theorem 1.1 and Section 3 ] .
Let \(d,L{\gt}0\), let \(A\) be a tensor, and let \(\varphi \) be a normalized positive state on the original chain. Write \(T_L\) for transport of states from the blocked chain. Then
This statement requires neither primitivity nor translation invariance. It makes the regrouping in [ 225 , lines 825–836 and Section 3 ] explicit at the level of supported states.
Let \(A\) be any MPS tensor. Let \(P_A(I)\) be the projection onto its open-boundary MPS space on a finite nonempty interval \(I\). If \(I\subseteq J\), the projection \(P_A(I)\), extended by the identity outside \(I\), fixes \(P_A(J)\). Consequently, every positive normalized quasi-local state \(\varphi \) with \(\varphi (P_A(J))=1\) satisfies \(\varphi (P_A(I))=1\). Neither injectivity nor translation invariance is required. This is the restriction direction of [ 225 , Section 3, equation (3.12) ] .
Let \(d{\gt}0\) and let \(A\) be trace preserving. Let \(\rho \ge 0\) be an invariant virtual matrix of nonzero trace. For every \(a\in \mathbb {Z}\) and \(N\ge 0\), the constructed state satisfies \(\omega _A(j_{a,N}(P_{G_N(A)}))=1\). Primitivity and faithfulness of \(\rho \) are not required.
Let \((A_j)_{j\in J}\) be a finite family of trace-preserving primitive tensors of positive bond dimensions, with faithful invariant matrices \(\rho _j\). Suppose distinct tensors are inequivalent up to virtual gauge and phase. Fix a sector \(\alpha \), an observable \(X\) on \(k{\gt}0\) sites, and margins \(\ell _n,r_n\to \infty \) along any filter. Put \(N_n=\ell _n+k+r_n\), let \(P_j(n)=P_{G_{N_n}(A_j)}\), let \(P(n)\) project onto \(\sum _jG_{N_n}(A_j)\), and put \(X_n=I_{\ell _n}\otimes X\otimes I_{r_n}\). Then \(\| P(n)X_nP_\alpha (n)-\eta _{A_\alpha }(X)P_\alpha (n)\| \to 0\).
Let \(A_j\) be finitely many normalized primitive tensors with positive bond dimensions \(D_j\), positive-definite invariant matrices \(\rho _j\), and pairwise gauge-and-phase inequivalence whenever their bond dimensions agree. Fix a sector \(\alpha \) and an observable \(X\) on \(k{\gt}0\) sites with \(\eta _{A_\alpha }(X)=0\). Along an arbitrary filter let \(\ell _n,r_n\to \infty \) and \(N_n=\ell _n+k+r_n\). If \(P_n\) projects onto \(\sum _jG_{N_n}(A_j)\) and \(P_{\alpha ,n}\) onto \(G_{N_n}(A_\alpha )\), then
This supplies the full finite-ground-space projection term in [ 225 , Section 6 ] . No finite-chain uniqueness or purity of a completed state is inferred.
Under the preceding tensor hypotheses, suppose that \(H_n\) are eventually positive operators on chains of length \(N_n=\ell _n+k+r_n\), with \(\ell _n,r_n\longrightarrow \infty \), and eventually
Let \(\psi _n\in G_{A_\alpha }(N_n)\) be eventually unit vectors, and let \(B_n=X_{\ell _n,r_n}\) with \(\eta _\alpha (X)=0\). If
then
Under the same finite-family tensor hypotheses, let \(h\) be a positive interaction on \(R\geq 1\) sites. Put \(t=R-1\), \(N_n=(n+t+k)+(t+n)\), and let \(H_n\) be its open interaction sum. Suppose that these operators eventually have kernel \(G(N_n)\) and norm gap at least \(\gamma \geq 0\). For eventually unit \(\psi _n\in G_{A_\alpha }(N_n)\) and every \(k\geq 1\) observable \(X\) with \(\eta _\alpha (X)=0\),
Let \(A_j\) be trace-preserving primitive tensors of positive bond dimensions, with positive definite invariant matrices \(\rho _j\), and let \(T=\bigoplus _j\mu _jA_j\), where every \(\mu _j\neq 0\). Assume that the simultaneous word span is the full product of the bond matrix algebras at a length \(S\geq 1\). Let \(R\geq S+1\), and let \(h\) be a positive interaction on \(R\) sites with \(\ker h=G_T(R)\). There exists \(\gamma {\gt}0\) such that, for every sector \(\alpha \) and every observable \(X\) on any \(k\geq 1\) sites with \(\eta _\alpha (X)=0\),
In particular this holds for the canonical parent projection \(h=I-P_T(R)\). These are finite-interval sector expectation statements; they do not assert the construction of a GNS Hamiltonian or purity of its state. The range is sufficient and need not be optimal. [ 225 , Theorem 1.2 and Section 6 ] , [ 61 ] .
Let \(A=\bigoplus _{\alpha =0}^{b-1}\mu _\alpha A_\alpha \), with nonzero coefficients, trace-preserving primitive blocks, and faithful invariant density matrices. Suppose that the blocks have simultaneous injectivity length \(S{\gt}0\), and that \(h\geq 0\) is a parent interaction of range \(R\geq S+1\). Then there is \(\gamma {\gt}0\) such that, for every constructed sector state \(\omega _\alpha \) and every local observable \(X\) satisfying \(\omega _\alpha (X)=0\),
Here \(X\) is supported on \([a,a+k)\), with \(k{\gt}0\), and the limit may be taken along any filter for which both margins tend to infinity. The limit is real and nonnegative. This proves the displayed infinite-volume estimate of [ 225 , Theorem 1.2 ] for these constructed sectors at the stated sufficient range; identification of the entire source ground-state face is a separate result.
Let \(d{\gt}0\), and let \(A_j\) be finitely many normalized primitive tensors with positive bond dimensions and positive-definite invariant matrices \(\rho _j\). Suppose that the simultaneous word span is full at a length \(S{\gt}0\). Let \(T=\bigoplus _j\mu _jA_j\) with \(\mu _j\neq 0\), and let \(h\geq 0\) have kernel \(G_R(T)\) at range \(R\geq S+1\). There is a constant \(\gamma {\gt}0\) such that, for every sector \(\alpha \), \(a\in \mathbb Z\), \(k\geq 1\), and \(X\in M_{d^k}(\mathbb C)\) with \(\omega _{A_\alpha }(j_{a,k}(X))=0\),
where \(t=R-1\), \(w=2t+k\), \(B=I_t\otimes X\otimes I_t\), and \(Y=B^*[H_w(h),B]\). The same constant applies to all sectors, positions, and centered observables. The states are the completed expectations of the given primitive blocks; identification of the full pure GVBS ground-state face remains a separate assertion.
Let \((A_j,\rho _j)_{j\in J}\) be a finite family of pairwise inequivalent trace-preserving primitive tensors with positive bond dimensions and positive-definite invariant matrices. Write \(G(N)=\sum _jG_{A_j}(N)\) and let \(P(N)\) be its orthogonal projection. Fix a sector \(\alpha \), an observable \(X\) on \(k\geq 1\) sites with \(\eta _\alpha (X)=0\), and free lengths \(\ell ,r\longrightarrow \infty \). For any varying boundary vectors \(\psi \in G_{A_\alpha }(\ell +k+r)\) with eventually \(\| \psi \| \leq 1\),
The assertion holds along an arbitrary filter.
Let \((A_j)_{j=0}^{b-1}\) be finitely many trace-preserving primitive tensors of positive physical and bond dimensions, with positive definite invariant matrices \(\rho _j\). Suppose distinct sectors are inequivalent up to virtual gauge and phase. Let \(T\) be their weighted block sum with nonzero coefficients, and let \(\omega _j\) be the constructed quasi-local state of the \(j\)th sector. Suppose \(\varphi \) is a normalized positive norm-one continuous complex-linear state on the quasi-local algebra and that, for all sufficiently large \(n\),
Then there are real weights \(w_j\geq 0\), \(\sum _jw_j=1\), such that
The competing state need not be translation invariant. This is the support-to-convex-hull implication in the argument for [ 225 , Theorem 1.1 ] .
Assume the hypotheses of Theorem 12.3.6.4, except that \(l\leq n_0\) is assumed and (130) is replaced by condition C1 on the range \(l\leq n{\lt}M\) of its own window sum at the two volumes at which the source asserts it: the full inequality \(0\leq \sum _{n=l}^{N-1}\operatorname{Re}\langle H_nx,x\rangle \leq d_{l+1}\operatorname{Re}\langle Hx,x\rangle \) at \(M=N\), and its non-negativity clause \(0\leq \sum _{n=l}^{n_0-1}\operatorname{Re}\langle H_nx,x\rangle \) at \(M=n_0\). Assume in addition that \(E_nv=0\) for every \(n{\lt}n_0\). Then
This is the conclusion of Theorem 2.1(i) of [ 225 ] with each of C1, C2, and C3 imposed on the range at which the source imposes it. The added vanishing hypothesis is the minimal repair of that theorem: the two below-threshold quantities in (133) enter (135) with non-negative weights, so both have to vanish, and the vanishing of the omitted martingale mass forces the vanishing of the window correction. Without the added hypothesis the statement is false, by Theorem 12.3.6.7. The source defect and the repair are recorded in [ 134 ] .
Let \((G_n)_{n\geq 0}\) be nested orthogonal projections on a finite-dimensional complex Hilbert space \(\mathcal H\), put \(E_n=G_n-G_{n+1}\), and assume \(G_0=\mathbf1\). Fix \(l,N\in \mathbb {N}\), and let \(Q_n\) be orthogonal projections for \(0\leq n{\lt}N\). Suppose \(E_mQ_n=Q_nE_m\) whenever \(m{\lt}n-l\) or \(n{\lt}m\). Let \(H_n\) be the local Hamiltonians and \(H\) the full Hamiltonian. Assume the quadratic-form version of C1 for every \(x\in \mathcal H\),
and the quadratic-form version of C2 for every \(x\in \mathcal H\) and \(0\leq n{\lt}N\),
and the literal condition C3,
If \(\gamma _{l+1}{\gt}0\), \(d_{l+1}{\gt}0\), and \(v\perp \operatorname {ran}(G_N)\), then
This full-range statement is stronger than the unrestricted source theorem in general. It assumes C1, C2, and C3 for every \(0\leq n{\lt}N\), whereas the source imposes lower thresholds on these conditions. Those thresholds cannot simply be dropped: without separate control of the initial volumes the source hypotheses imply neither the assumptions above nor the conclusion, by Theorem 12.3.6.7. The coefficient is exactly the one in Theorem 2.1(i) of [ 225 ] : \(\gamma _{l+1}/d_{l+1}\) times \((1-\epsilon _l\sqrt{l+1})^2\). This is Theorem 12.3.6.4 at threshold \(n_0=0\), where nothing lies below the threshold.
In the setting of Theorem 12.3.6.8, suppose that \(H\) is positive and \(\operatorname {ran}(G_N)=\ker H\). Then every \(v\in (\ker H)^\perp \) satisfies
Let \((G_n)_{n\geq 0}\) be nested orthogonal projections on a finite-dimensional complex Hilbert space \(\mathcal H\), put \(E_n=G_n-G_{n+1}\), and assume \(G_0=\mathbf1\). Fix \(l,N,n_0\in \mathbb {N}\) with \(n_0\leq N\), and let \(Q_n\) be orthogonal projections for \(n_0\leq n{\lt}N\). Suppose \(E_mQ_n=Q_nE_m\) whenever \(m{\lt}n-l\) or \(n{\lt}m\). Let \(H_n\) be the local Hamiltonians and \(H\) the full Hamiltonian. Assume the quadratic-form version of C1 on the same range, for every \(x\in \mathcal H\),
the quadratic-form version of C2 for every \(x\in \mathcal H\) and \(n_0\leq n{\lt}N\),
and the literal condition C3 for \(n_0\leq n{\lt}N\),
If \(\gamma _{l+1}{\gt}0\), \(d_{l+1}{\gt}0\), and \(v\perp \operatorname {ran}(G_N)\), then
Conditions C2 and C3 have the source forms, with their onset represented by \(n_0\). The displayed C1 inequality follows from source C1 when \(l\leq n_0\) and the omitted local terms are nonnegative. Thus a source-derived application chooses \(n_0\) above both \(l\) and the C2–C3 onset. If \(n_0{\lt}l\), the displayed C1 inequality is instead an independent stronger assumption. The coefficient of the martingale mass above the threshold is exactly the one in Theorem 2.1(i) of [ 225 ] . The correction is an explicit upper bound for the contribution of the at most \(l\) martingale differences immediately below the threshold; it uses multiplicity \(l+1\) uniformly. At \(n_0=0\) the correction is an empty sum.
Let \(A\) be a primitive MPS tensor with nonzero bond dimension and a positive-definite fixed point. There are an overlap length \(l{\gt}1\), measured in sites of \(A\), and a number \(\epsilon _l\geq 0\) such that \(A\) is \(l\)-block injective, the projector defect is bounded uniformly in the left length \(K\) by \(\epsilon _l\), and
Nachtergaele’s projector-defect identity following condition C3 [ 225 , Eq. (2.4) ] , together with the two-projection estimate, gives the corresponding anticommutator bound for every \(K\). If \(A\) is already a blocked representative, one site of \(A\) is one blocked-chain filtration step.
Assume \(D\ge 1\) and let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). Set \(n=K+L_0\) and \(l=L_0\), and write
Thus \(S\) and \(T\) are the left and tail ground spaces in the Hilbert space on \(\Lambda _{n+1}\). Let \(q_S=P_{S^\perp }\) and \(q_T=P_{T^\perp }\) be the complementary excitation projections. If
then every vector \(v\) in the Hilbert space on \(\Lambda _{n+1}\) satisfies
In the notation of Theorem 12.3.6.6: for every \(k\) there are, on a complex Hilbert space of dimension \(k+2\), nested orthogonal projections \((G_n)_{n\geq 0}\) with \(G_0=\mathbf1\), \(G_{N+1}=0\), and \(G_n\neq 0\) for every \(n\leq N\), nonzero orthogonal projections \(Q_n\), local operators \(H_n\), volume Hamiltonians \(H_{\Lambda _M}\), and a vector \(v\perp \operatorname {ran}(G_N)\) such that, with \(l=0\), \(n_0=1\), \(N=k+1\), \(\gamma _1=d_1=1\), and \(\epsilon _0=0\), the required commutations hold, condition C1 holds with the single constant \(d_1\) at every volume \(\Lambda _M\) on the range \(l\leq n{\lt}M\) carried by its own window sum, the gap inequality of condition C2 and condition C3 hold on the range \(n_0\leq n{\lt}N\) from which they are imposed, every ground space is the kernel of its own Hamiltonian, that is \(\ker H_{\Lambda _M}=\operatorname {ran}(G_M)\) for \(M\leq N\) and \(\ker H_n=\operatorname {ran}(Q_n)\) for every window inside \(\Lambda _N\), that is for \(0\leq n{\lt}N\), where the \(Q_n\) are also nonzero orthogonal projections commuting with the martingale differences outside their own windows, and
Hence condition C1 at every volume on the range of its own window sum together with conditions C2 and C3 on \(n_0\leq n{\lt}N\), the ranges at which [ 225 ] imposes them, do not imply the conclusion of its Theorem 2.1(i), even when every ground space is the kernel of its Hamiltonian and is nontrivial, as condition C2 also requires, and even under the conventions the source attaches to the extreme volumes.
Under the hypotheses of Theorem 12.3.7.73 with \(R=2\), if \(1{\lt}m\gamma \), then
Equivalently, the open-window threshold is \(\gamma {\gt}1/m\). This is Knabe’s nearest-neighbor threshold [ 211 ] , quoted in [ 228 , lines 1483–1489 ] and [ 17 , lines 2194–2197 ] .
Suppose that the canonical length-two parent terms of \(A\) commute on a three-site chain. Then the complementary two-site interaction admits a finite sector graph as in Definition 12.1.72.
- MPSTensor.IsNNCPH.nonempty_beigiSectorGraphData
- MPSTensor.IsNNCPH.beigiSectorGraphData
- MPSTensor.twoSiteParentInteractionMatrix_isStarProjection
- MPSTensor.twoSiteParentGroundProjectorMatrix_isStarProjection
- MPSTensor.twoSiteParentGroundProjectorMatrix_posSemidef
- MPSTensor.IsNNCPH.twoSiteParentGroundProjectorMatrix_overlappingLifts_commute
Let \(P\) be a sector decomposition with representatives \(A_1,\ldots ,A_g\) and total tensor \(B\). Suppose that the vectors \(\{ |V^{(N)}(A_j)\rangle \} _{j=1}^g\) are linearly independent for all sufficiently large \(N\), and that \(B\) satisfies the all-chain nearest-neighbor commuting parent-Hamiltonian ground-space condition with these representatives. Then there is \(N_0\) such that, for every \(N{\gt}N_0\), \(\dim _{\mathbb {C}}\ker H_2^{(N)}(B) = g\). Thus the ground-state degeneracy is eventually constant, as required in [ 5 , Section IV ] .
Let \(B\) be normal and in left-canonical form. Suppose that the family \(A_j\) is a BNT for \(B\), and that the nearest-neighbor parent Hamiltonian of \(B\) has the all-chain ground-space property of Definition 12.1.17: for every \(N{\gt}2\), the length-two translated parent interactions commute, \(h_i(B,2)h_j(B,2)=h_j(B,2)h_i(B,2)\), the vector \(V^{(N)}(B)\) has zero energy, and \(\ker H_2^{(N)}(B) = \operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \). Then \(B\) is a renormalization fixed point.
Scope restriction (NNCPH representative): These two reverse-implication results cover only normal left-canonical and multiplicity-one unit-weight representatives, whereas [ 13 , Theorem 3.10 ] is stated for general canonical-form tensors with repeated copies and raw weights; documented in [ 167 ] .
Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and set
Suppose that, for every \(N{\gt}2\), the translated length-two parent interactions of \(B\) commute, every \(V^{(N)}(A_j)\) has zero energy, and
Then \(B\) is a renormalization fixed point. This establishes the reverse implication of [ 13 , Theorem 3.10 ] for multiplicity-one unit-weight representatives.
The conclusion of Theorem 12.9.6 holds for every normal tensor of positive bond dimension, without a normalization hypothesis. Likewise, the conclusion of Theorem 12.9.7 holds for normal blocks with nonzero weights and positive physical and bond dimensions, provided no two blocks are related by similarity and multiplication by any nonzero complex scalar.
Let \((A^j)_j\) be pairwise gauge-phase inequivalent normalized normal blocks, with bond dimensions \(D_j\). There is a positive integer \(L\) satisfying \(L+1\leq 3\max (\sum _jD_j,1)^5\) such that blocking \(L\) physical sites gives a simultaneous one-site span of the direct sum of the block matrix algebras. This is the bounded form of the initial blocking in [ 236 , lines 575–597, Appendix A ] , using the bound in [ 13 , lines 317–345 ] .
Let \(A^j\) be normal tensors with common physical dimension \(d\geq 1\), pairwise inequivalent under virtual similarity and phase. Set \(D_* = \max \{ \sum _j D_j,1\} \) and \(B=\bigoplus _j\mu _jA^j\), with every \(\mu _j\ne 0\). For every \(R\geq 3D_*^5\), there exists \(\delta {\gt}0\) such that \(\delta \| v\| \leq \| H^{\mathrm{open}}_{N,R}(B)v\| \) for every \(N\geq R\) and every \(v\perp G_N(B)\).
Let \(A_0,\ldots ,A_{r-1}\) be normal tensors with common physical dimension \(d\geq 1\) and positive bond dimensions. Suppose no two distinct blocks are related by similarity and multiplication by a nonzero complex scalar, and let \(B=\bigoplus _j\mu _jA_j\), with \(\mu _j\ne 0\) for every \(j\). For every \(0{\lt}\eta {\lt}1\), there is \(p_0\) such that, for every \(p\geq p_0\) and every \(N\geq 2p\),
Thus the canonical open-chain gap approaches one as the even interaction range grows, uniformly over all lengths at least that range. This is a quantitative consequence of the projector decay and martingale estimates of [ 225 , Theorem 2.1(ii) and Section 6 ] .
Let \((A^i)_{i\in I}\) be normal tensors with positive bond dimensions. Suppose that no two distinct blocks are related by a virtual gauge and a nonzero complex scalar. Then there are tensors \((B^i)_{i\in I}\) of the same respective bond dimensions and positive definite matrices \(\rho _i\) such that each \(B^i\) is primitive and trace preserving, \(\rho _i\) is its fixed point, and \(G_L(A^i)=G_L(B^i)\) for every \(i\) and every length \(L\). The tensors \(B^i\) remain pairwise inequivalent under a virtual gauge and a nonzero scalar.
Let \(A\) be a normal tensor. Then \(A\) is primitive in the sense of Definition 6.1.1.3, and \(A\) has no nontrivial invariant projection, so \(A\) is irreducible in the sense of Definition 7.1.1.1. No normalization is imposed on \(A\).
Let \(D{\gt}0\) and let \(A\) be a normal tensor. There exist \(l{\gt}1\) and \(\gamma {\gt}0\) such that \(A\) is \(l\)-block injective and, for every \(N\) and every \(v\in (\ker H_{N,l+1}^{\mathrm{ES}}(A))^\perp \),
This proves the normal-tensor case of the uniform gap assertion in [ 17 , lines 2183–2187 ] , at a suitable interaction range.
Let \(A\) be \(L_0\)-block-injective with \(D \ge 1\) and \(L_0 {\gt} 0\). If an \(N\)-site state satisfies the ground-space constraint on every non-wrapping contiguous interval of length \(L_0+1\), for every fixed choice of the complementary sites, with \(N \ge L_0+1\), then it lies in \(G_N(A)\).
Let \(A\) be injective at a length \(p{\gt}0\), and let \(R{\gt}p\). There is \(\delta {\gt}0\) such that
In particular, for a normal tensor of bond dimension \(D{\gt}0\), every fixed \(R\geq D^4+1\) has this property. No normalization is required.
- MPSTensor.exists_openParentHamiltonianES_uniform_gap_of_isNBlkInjective_all_lengths
- MPSTensor.exists_openParentHamiltonianES_uniform_gap_of_isNBlkInjective_of_lt
- MPSTensor.exists_openParentHamiltonianES_uniform_gap_of_isNormal_of_le
- MPSTensor.exists_openParentHamiltonianES_uniform_gap_of_isNormal_all_lengths
Let \(D{\gt}0\) and let \(A\) be a normal tensor. There exist \(l{\gt}1\), \(\epsilon _l\geq 0\), an integer \(m\geq l+1\), and \(\delta {\gt}0\) such that \(A\) is \(l\)-block injective, \(\epsilon _l{\lt}1/\sqrt{l+1}\),
and, for every \(N\geq 2m\) and every \(v\in (\ker H_{N,l+1}^{\mathrm{ES}}(A))^\perp \),
No normalization is imposed on \(A\). The uniform gap of matrix product state parent Hamiltonians is asserted at
[
17
, lines 2183–2187
]
; the arbitrary finite-range coefficient (359) is the derivation recorded in docs/paper-gaps/knabe88_finite_range_coefficient.tex.
Let \(\zeta \neq 0\) and let \(A\) be a tensor. For every \(N\) the length-\(N\) word spans of \(A\) and of \(\zeta A\) agree, so \(\zeta A\) is \(N\)-block injective exactly when \(A\) is, and \(\zeta A\) is normal exactly when \(A\) is.
Let \(d{\gt}0\), and let \(A_j\) have positive bond dimensions and full simultaneous word span at \(S{\gt}0\). Let \(T=\bigoplus _j\mu _jA_j\) with \(\mu _j\neq 0\), and let \(h\geq 0\) have kernel \(G_R(T)\) at \(R\geq S+1\). There exist normalized primitive representatives \(B_j\) with positive-definite invariant matrices \(\rho _j\) such that \(G_N(A_j)=G_N(B_j)\) for every \(j,N\). Their constructed quasi-local states satisfy the preceding common commutator inequality for the original interaction \(h\), with one positive constant for all sectors, interval positions, and centered local observables. No normalized representatives or invariant matrices are supplied.
Let \(D{\gt}0\) and suppose \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). If \(A\) has eventually full Kraus rank, or equivalently is normal, then \(\mathcal{E}_A\) is strongly irreducible.
Let \(D{\gt}0\) and let \(A\) be a normal tensor. There exist a scalar \(\zeta \neq 0\), a tensor \(B\), and a matrix \(\rho {\gt}0\) such that \(\zeta A\) and \(B\) are gauge equivalent, the pair \((B,\rho )\) is primitive, and
for every chain length \(N\) and every configuration \(\sigma \). The local ground spaces, the parent interactions, and the periodic chain ground spaces of \(A\) and \(B\) also coincide:
for all \(L\) and \(N\), where \(h_L\) is the canonical parent interaction and \(\mathcal G_{N,L}\) the periodic chain ground space.
This is the normalization taken without loss of generality in [ 225 , equations (3.1)–(3.2b), lines 1394–1435 ] and at the start of the proof of [ 228 , Theorem 4, lines 765–770 ] . The gauge here is built from the Perron eigenvector of the adjoint transfer map, giving the trace-preserving orientation. That orientation is what conditions 1 and 2 of [ 228 , Theorem 4, lines 752–758 ] , \(\sum _iA^i(A^i)^\dagger =\mathbb {1}\) and \(\sum _i(A^i)^\dagger \Lambda A^i=\Lambda \) with \(\Lambda {\gt}0\), become after the further gauge by \(\Lambda ^{1/2}\): condition 2 turns into \(\sum _i(B^i)^\dagger B^i=\mathbb {1}\) and condition 1 into the statement that \(\Lambda \) is a positive definite fixed point of the transfer map of \(B\).
Let \(A_j\) be a finite family with positive bond dimensions whose word tuples at some length \(S{\gt}0\) span the full product matrix algebra. This is the simultaneous spanning condition of [ 17 , Section IV.C ] . Let \(T_A=\bigoplus _j\mu _jA_j\), with every \(\mu _j\ne 0\). Suppose that \(h\geq 0\) has range \(R\geq 1\) and that \(\ker H^{\mathrm{open}}_{h,N}=G_N(T_A)\) for all sufficiently large \(N\). There exist primitive trace-preserving tensors \(B_j\) with faithful invariant matrices \(\rho _j\) such that their word tuples still span the full product algebra at length \(S\). Write \(T_B=\bigoplus _j\mu _jB_j\). Then
These equalities hold for every block \(j\) and every length \(N\). Let \(\omega _j\) be the quasi-local state constructed from \(B_j,\rho _j\). For every continuous complex-linear functional \(\varphi \) on the quasi-local algebra,
The original interaction range need not exceed \(S\). This combines the normalization in [ 225 , equations (3.1)–(3.2b) ] with the ground-state classification of [ 225 , Theorems 1.1–1.2 ] , for the original simultaneously spanning family. The passage from more general periodic presentations is separate [ 61 ] .
Suppose \(A_iV=VB_i\) for every letter \(i\). For every virtual matrix \(\sigma \) and every observable \(X\) on \(k\geq 0\) sites,
If additionally \(V^*V=I\), \(\rho =V\sigma V^*\), and \(\operatorname {tr}\sigma \neq 0\), then \(\operatorname {tr}\rho =\operatorname {tr}\sigma \) and \(\eta _{A,\rho }(X)=\eta _{B,\sigma }(X)\). No normalization, stationarity, or positivity is needed for these identities. This concerns the finite insertion formula in [ 225 , Section 3 ] .
Let \(L{\gt}0\) and \(p\leq n\). Under the canonical identification
both the prefix Hamiltonian on \([0,p)\) and every suffix-interval Hamiltonian ending at \(p\) are direct sums over the right-spectator configurations of their operators in the \(n\)-site volume.
If \(D\ge 1\), \(A\) is \(L_0\)-block-injective, and \(L_0{\gt}0\), then
In Nachtergaele’s notation, with \(n=K+L_0\) and \(l=L_0\), this is the common range identity for \(G_{\Lambda _n}\) and \(G_{\Lambda _{n+1}\setminus \Lambda _{n-l}}\) inside \(\Lambda _{n+1}\).
Let \(H_N\) be the canonical range-\(R\) open Hamiltonian, and let \(H_L,H_T\) be its restrictions to a prefix of length \(n\) and a terminal interval of length \(W\), respectively. Then \(H_L+H_T\le 2H_N\). If \(n\le N\) and \(N+R\le n+W+1\), every interaction window lies in one of these intervals, and \(\ker H_L\cap \ker H_T=\ker H_N\). If the finite kernels equal the MPS boundary spaces and the overlap has length at least \(R\), the corresponding extended boundary spaces intersect exactly in \(G_N\).
- MPSTensor.ker_openParentHamiltonianES_le_ker_openPrefix
- MPSTensor.localTermES_eq_zero_of_openSuffixParentHamiltonianES_eq_zero
- MPSTensor.ker_openPrefix_inf_ker_openSuffix_eq_ker_openParentHamiltonianES
- MPSTensor.openPrefix_add_openSuffix_le_two_openParentHamiltonianES
- MPSTensor.leftBoundary_inf_reassocTail_range_eq_groundSpaceES_of_open_kernel
Let \(1\leq R\leq m\), and suppose that every range-\(R\) open parent Hamiltonian on \(R\leq W\leq m+R-1\) sites has norm gap at least \(\gamma \) on its kernel complement. If \(m\gamma {\gt}(R-1)^2\), then
and \(\delta \| v\| \leq \| H^{\mathrm{open}}_{N,R}(A)v\| \) for every \(N\geq R\) and \(v\perp \ker H^{\mathrm{open}}_{N,R}(A)\). This is a consequence of the cyclic finite-range inequality, as derived in [ 56 ] ; it is not the printed periodic nearest-neighbor criterion.
Under the block-family hypotheses of Theorem 12.8.1.33, suppose that at \(x_0\) one \(\gamma {\gt}0\) bounds the open norm gaps on \(G_N(B_{x_0})^\perp \) for every \(N\geq R\). Then some \(\delta {\gt}0\) bounds the open norm gaps on their kernel complements for every \(N\geq R\) and every \(x\) in one neighborhood of \(x_0\).
Let \(A_x^j\) be continuous block tensors with positive physical and bond dimensions. Suppose their length-\(S\) word tuples span the product matrix algebra at every parameter, where \(S{\gt}0\), and put \(B_x=\bigoplus _j\mu _j(x)A_x^j\), with every weight nonzero. Fix \(R\geq S+1\) and a finite upper volume \(M\). If at \(x_0\) the open Hamiltonians on every \(R\leq N\leq M\) have norm gap at least \(\gamma {\gt}0\) on \(G_N(B_{x_0})^\perp \), then every \(\delta {\lt}\gamma \) is a common lower bound on \(G_N(B_x)^\perp \) for those volumes and all \(x\) in some neighborhood of \(x_0\). The same assertion holds for one fixed \(N\geq R\). No continuity of the weights is required. This is the finite-volume continuity step of [ 236 , Appendix A, lines 2575–2578 ] .
Let \(A\) be a periodic tensor of period \(m{\gt}0\) on a nonzero physical space. For \(K,M,r\geq 0\), let \(P\) denote the ground-space projection on \(Km+Mm+r\) sites, let \(P_{\mathrm L}\) project onto the prefix ground space on \(Km+Mm\) sites tensored with the free \(r\) sites, and let \(P_{\mathrm R}\) project onto the terminal ground space on \(Mm+r\) sites tensored with the free \(Km\) sites. Suppose that, for every pair of sequences \(K_M,r_M\), \(\| P-P_{\mathrm L}P_{\mathrm R}\| \to 0\) as \(M\to \infty \). Then there exist \(R{\gt}0\), \(N_0\geq R\), and \(\delta {\gt}0\) such that every \(N\geq N_0\) and every \(v\perp \ker H_{A,R,N}\) satisfy \(\delta \| v\| \leq \| H_{A,R,N}v\| \), where \(H_{A,R,N}\) is the canonical open parent Hamiltonian. At the same range, \(\ker H_{A,R,N}=G_A(N)\) for every \(N\geq R\). The convergence assumption is explicit; this is an auxiliary limit criterion for the projection comparison in [ 225 , Section 6, Lemma commutation(ii) ] .
Let \(R{\gt}0\) and let \(h\) be a positive semidefinite interaction on \(R\) sites. For every \(n\geq 0\), the open sum \(\sum _{a=0}^{n-R}\iota _{a}(h)\) of its nonwrapping translates on \(n\) sites is positive semidefinite; for \(n{\lt}R\) the sum is empty and equal to zero. This is the positivity of the finite-volume Hamiltonians in [ 225 , equation (3.12) ] .
Let \(R{\gt}0\). If the range-\(R\) parent Hamiltonian on \(n\) sites has norm gap \(\kappa {\gt}0\) on its kernel complement, its extension to the prefix of an \((n+q)\)-site chain has the same lower bound. Likewise, if \(R\leq W\leq n\leq N\) and the range-\(R\) Hamiltonian on \(W\) sites has gap at least \(\kappa \), its interaction sum on \([n-W,n)\), acting on the \(N\)-site chain, has gap at least \(\kappa \). The physical coordinate isometry identifies this interval interaction sum with the \(W\)-site Hamiltonian tensored with the identity on the remaining sites. These are the finite-interval gaps used in condition C2 of [ 225 ] .
Let \(m{\gt}0\), \(N\geq R{\gt}0\), and \(P=N+2m\). Put \(A_s=\sum _{q=0}^{m-1}\widetilde h_{s+q}\). Each \(A_s\) is zero or the range-\(R\) open Hamiltonian on an interval \([n-W,n)\), extended by the identity on the remaining sites, with \(R\leq W\leq m+R-1\) and \(W\leq n\leq N\). If every open Hamiltonian on \(R\leq W\leq m+R-1\) sites has norm gap at least \(\gamma {\gt}0\) on its kernel complement, then \(A_s^2\geq \gamma A_s\) for every \(s\).
Every \(\widetilde h_s\) is an orthogonal projection. If \(R\leq e\) and \(e+R\leq P\), then \(\widetilde h_s\widetilde h_{s+e} =\widetilde h_{s+e}\widetilde h_s\) for every \(s\). If \(N\leq P\), \(\sum _s\widetilde h_s=H^{\mathrm{open}}_{N,R}(A)\).
Let \(x\mapsto A_x\) be continuous, \(R\leq N\), and suppose every \(A_x\) is \(R\)-block and \(N\)-block injective. If \(\delta \lVert v\rVert \leq \lVert H^{\mathrm{open}}_{N,R}(A_{x_0})v\rVert \) for all \(v\perp G_N^{\mathrm{ES}}(A_{x_0})\), with \(\delta {\gt}0\), then for every \(\delta '{\lt}\delta \) and every \(x\) near \(x_0\), \(\delta '\lVert v\rVert \leq \lVert H^{\mathrm{open}}_{N,R}(A_x)v\rVert \) for all \(v\perp G_N^{\mathrm{ES}}(A_x)\). This is the continuity step of [ 236 , lines 2575–2578 ] .
Set \(L=l+1\). For every vector \(v\) on the \(N\)-site chain,
Thus the full martingale range satisfies C1 with \(d_{l+1}=1\).
If \(L\leq l\), then
Thus condition C1 of [ 225 ] holds with \(d_l=l-L+1\). This is the source value \(pl-r+1\) in the present convention \(p=1\) and interaction range \(r=L\).
Let \(D\geq 1\), let \(l{\gt}0\), suppose that \(A\) is \(l\)-block injective, and let \(N\geq l+1\). Assume \(0\leq \epsilon _l{\lt}1/\sqrt{l+1}\) and, for every \(0\leq n{\lt}N\),
where \(Q_n\) is the ground-space projection of the interval \([n-l,n+1)\) and \(E_n^{(N)}\) is the fixed-volume martingale difference. Then every \(v\in (G_N^{\mathrm{ES}}(A))^\perp \) satisfies
This intermediate MPS specialization assumes C3 on the full range \(0\leq n{\lt}N\), rather than only above the lower threshold in the source statement. It applies the full-range estimate with \(d_{l+1}=\gamma _{l+1}=1\) and retains the coefficient of Theorem 2.1(i) in [ 225 ] . It is not a formulation of the unrestricted source theorem. Theorem 12.3.7.65 supplies the full-range C3 hypothesis from the physical open-chain estimates.
If \(L{\gt}0\) and \(v\in (\ker H^{\mathrm{open}}_{L,L}(A))^\perp \), then
In particular,
Thus \(\gamma _{l+1}=1\) at the full-window volume \(N=L=l+1\) in [ 225 , Theorem 2.1(i) ] . This neither asserts that the excited subspace is nonzero nor supplies a gap for any \(N{\gt}L\).
If \(A\) is \(L\)-block-injective, then \(\dim G_{L}^{\mathrm{ES}}(A)=D^{2}\). If \(A\) is \(L_{0}\)-block-injective with \(L_{0}{\gt}0\), \(D\geq 1\) and \(N\geq L_{0}+1\), then
The local support spaces of a pure generalized valence-bond-solid state with auxiliary algebra \(M_{k}(\mathbb {C})\) have dimension \(k^{2}\) on long intervals, and they are the kernels of the open-interval Hamiltonian [ 225 , lines 1505–1510 and 1545–1558 ] ; the growth argument on an open chain is [ 228 , lines 1211–1233 ] . The review states that the open-boundary degeneracy is at least the square of the dimension of the irreducible projective representation carried by the boundary [ 17 , line 1174 ] . That dimension is at most \(D\), so the exact count \(D^{2}\) is sharper than the review’s lower bound.
Assume \(D\geq 1\), \(L_0{\gt}0\), and \(N\geq L_0+1\). If \(A\) is \(L_0\)-block-injective, then
This is the canonical MPS instance of the ground-state property in [ 225 , Equation (3.13) ] . The right-hand side is the open boundary-condition space and may therefore be degenerate.
Let \(L\le N\). If \(H^{\mathrm{open}}_{N,L}(A)v=0\), then for every nonwrapping start \(s\) with \(s+L\le N\) and every complementary configuration \(\tau \),
This is the frustration-free extraction in [ 225 , Equation (3.12) ] .
Let \(H\) be the range-\(R\) parent Hamiltonian on an open chain of length \(N\), and let \(H_L\) and \(H_T\) be its interaction sums on a prefix of length \(n\) and on a terminal interval of length \(W\), respectively. Write \(P_L\), \(P_T\), and \(P\) for their orthogonal ground projections. Suppose \(\kappa \geq 0\), and suppose both interval Hamiltonians have a norm gap at least \(\kappa \) on their respective kernel complements. If \(\| P-P_LP_T\| \leq 1/2\), then \((\kappa /16)\| v\| \leq \| Hv\| \) for every \(v\perp \ker H\). The interval interaction sums always satisfy \(H_L\leq H\) and \(H_T\leq H\).
Let \(R{\gt}0\), and let \(h\geq 0\) have kernel \(G_R(A)\). There are \(\kappa ,C{\gt}0\), independent of \(N\), such that
Hence a uniform canonical open-chain gap \(\gamma {\gt}0\) for \(N\geq R\) gives the gap \(\kappa \gamma \) for \(H^{\mathrm{open}}_N(h)\) on its kernel complement. In particular, this applies to every fixed positive parent interaction of a canonical tensor when its normal representatives are simultaneously injective at \(S{\gt}0\) and \(R\geq S+1\).
Suppose \(m{\gt}0\), \(R{\gt}0\), \(Mm\ge R\), and the canonical open kernels equal \(G_N\) for every \(N\ge R\). Cover \(N=Km+Mm+r\) by the aligned prefix of length \((K+M)m\) and the terminal interval of length \(Mm+r\). Assume that, uniformly for \(K\ge K_0\) and \(0\le r{\lt}m\),
If the aligned prefix gaps have a common bound \(\gamma {\gt}0\), the full open Hamiltonians have a common bound \(\min (\gamma ,\eta )/16{\gt}0\), where \(\eta \) is a derived minimum over the terminal residues. For a normalized periodic tensor of period \(m\), the aligned gap is derived from its primitive blocked presentation. Consequently all \(N\ge (K_0+M)m\) have a common positive gap. The displayed projection estimate remains an explicit hypothesis; this criterion does not establish its periodic-tensor decay. It is the finite-Hamiltonian consequence of the three-interval estimate in [ 225 , Section 6, Lemma commutation(ii) ] .
Set \(L=l+1\), let \(0\leq n{\lt}N\), and let \(Q_n\) be the orthogonal projection onto the kernel of \(H^{\mathrm{suffix}}_{N,L;L,n+1}(A)\). Then every vector \(v\) satisfies
Hence C2 holds on the full martingale range with \(\gamma _{l+1}=1\).
Let \(L{\gt}0\) and \(L\leq m\leq N\). The suffix of length \(L\) ending at \(m\) contains one interaction:
Write \(G^{\ker }_{N,L;m}\) for the orthogonal projection onto the kernel of this suffix Hamiltonian. Then
Consequently condition C2 holds in Loewner order with \(\gamma _L=1\):
Equivalently, for every \(v\),
If \(v\in (\ker H^{\mathrm{suffix}}_{N,L;L,m}(A))^\perp \), then
In particular,
In the notation of [ 225 ] , set \(L=l+1\) and \(m=n+1\). The interval is then \(\Lambda _{n+1}\setminus \Lambda _{n-l}=[n-l,n+1)\), so
Condition C2 has the constant
The later condition C3 uses the distinct overlap parameter \(\epsilon _l\).
- MPSTensor.openSuffixParentHamiltonianES_eq_localTermES
- MPSTensor.openSuffixParentHamiltonianES_isSymmetricProjection
- MPSTensor.openSuffixParentHamiltonianES_eq_orthogonal_starProjection_ker
- MPSTensor.openSuffixParentHamiltonianES_eq_id_sub_starProjection_ker
- MPSTensor.openSuffixParentHamiltonianES_C2
- MPSTensor.openSuffixParentHamiltonianES_C2_quadratic_form
- MPSTensor.openSuffixParentHamiltonianES_norm_eq_of_mem_orthogonal_ker
- MPSTensor.openSuffixParentHamiltonianES_unit_gap
Let the original block family have simultaneous injectivity length \(S{\gt}0\) and nonzero weights. For every positive parent interaction of the original weighted tensor at range \(R\geq S+1\), there are trace-preserving primitive representatives, with faithful invariant matrices, preserving every block ground space at every length. Their completed sector states are pure among all quasi-local states and have a common positive literal infinite-volume commutator gap for the original interaction. The original blocks need not be normalized, and the invariant matrices and purity are conclusions. The two free margins may tend to infinity along any filter, as in the preceding theorem.
The preceding finite and infinite-volume conclusions also hold when the original blocks are only assumed to have full simultaneous word span at some positive length \(S\). Their normalization and invariant density matrices are derived internally. No condition \(R\geq S+1\) is required; positivity and the eventual open-chain kernel identity remain the hypotheses on the original interaction.
Let \(d,L{\gt}0\). For every observable \(X\) on an original interval \([a,a+k)\) with \(k{\gt}0\), there exist \(b\in \mathbb Z\), \(N{\gt}0\), and an observable \(Y\) on \(N\) blocked sites such that
Negative interval starts are included. This is the local observable form of the regrouping in [ 225 , lines 825–836 and Section 3 ] .
Each \(E_n\) is an orthogonal projection, and
With Nachtergaele’s endpoint conventions \(G_0=\mathbf1\) and \(G_{N+1}=0\), the differences form a complete family:
These are the assertions following the display labelled \(E_n\) in [ 225 ] .
Let \(Q_n\) be a local ground projection and suppose that
Then every index outside \([n-l,n]\) cancels:
If \(n{\lt}N\), the complete resolution term therefore reduces to
This is the reduction from Enpsi to the first line of Enpsi2 in [ 225 , Theorem 2.1(i) ] .
Let \(\mathcal{H}=\mathbb {C}^d\), and let \(Q\) be an endomorphism of \(\mathcal{H}\otimes \mathcal{H}\). Under the canonical identifications of two- and three-site configurations with pairs and right-associated triples, the matrices of its two adjacent actions are
Equivalently, for all \(a,x,b,a',x',b'\in \{ 0,\ldots ,d{-}1\} \),
This is the homogeneous three-single-site specialization of the factor actions on \(\mathcal{H}_A\otimes \mathcal{H}_X\otimes \mathcal{H}_B\) in [ 13 , Definition D.2 ] ; here \(\mathcal{H}_A=\mathcal{H}_X=\mathcal{H}_B=\mathcal{H}\). No assertion is made for unequal factor spaces.
Let \(A\) be an MPS tensor, and let \(R,W\) be positive integers with \(W\geq 2R\). Suppose that the open range-\(R\) Hamiltonian on \(W\) sites has kernel \(\mathcal G_W(A)\). If the periodic parent Hamiltonians of range \(W\) have a uniform positive gap for all sufficiently large chain lengths, then the same holds at range \(R\).
Let \(h\) be a positive interaction on \(R\) sites whose kernel is the local MPS space \(G_R(A)\), in the sense of [ 17 , Section IV.C ] . Write \(h_R=I-P_{G_R(A)}\) for the canonical parent projection. There are constants \(\kappa ,C{\gt}0\) such that
Let \(0{\lt}\eta {\lt}1\), \(1\le R\le m\), and \(2(R-1)^2\le m\eta \). Suppose the open parent Hamiltonian on \(m+R-1\) sites has gap at least \(1-\eta /2\). Then, for every \(N\ge 2m\) and every \(v\perp \ker H_{R,N}\),
Let \(T\colon E\to F\) and \(S\colon G\to F\) be injective bounded linear maps between real or complex Hilbert spaces, with \(E\) and \(G\) finite-dimensional. Write \(K_T=(T^\dagger T)^{-1}\), \(K_S=(S^\dagger S)^{-1}\), and let \(P_T,P_S\) be the orthogonal projections onto their ranges. For every bounded operator \(X\) on \(F\),
Consequently, along any filter, if these maps are eventually injective, their inverse Gram norms are eventually bounded by constants, and \(\lVert T^\dagger X S\rVert \to 0\), then \(\lVert P_T X P_S\rVert \to 0\). The Hilbert spaces and maps may vary with the index.
Fix \(A\) and \(L\). Suppose \(\delta {\gt}0\) and \(M\in \mathbb N\) satisfy \(\delta \lVert v\rVert \leq \lVert H_{N,L}^{\mathrm{ES}}(A)v\rVert \) for every \(N\geq M\) and every \(v\in (\ker H_{N,L}^{\mathrm{ES}}(A))^\perp \). Then some \(\gamma {\gt}0\) satisfies the inequality with \(\gamma \) in place of \(\delta \) for every \(N\).
For any tensor \(A\), interaction range \(L\), and chain length \(N\), there exists \(\delta _N{\gt}0\) such that \(\delta _N\lVert v\rVert \leq \lVert H_{N,L}^{\mathrm{ES}}(A)v\rVert \) for every \(v\in (\ker H_{N,L}^{\mathrm{ES}}(A))^\perp \).
Let \(A\) and \(B\) have the same bond dimension and satisfy \(G_L(A)=G_L(B)\). Then their canonical parent interactions at length \(L\) agree, hence so do all their translated local terms and their parent Hamiltonians \(H_{N,L}(A)=H_{N,L}(B)\) and \(H_{N,L}^{\mathrm{ES}}(A) =H_{N,L}^{\mathrm{ES}}(B)\) on every periodic chain.
Let \(1\leq R\leq m\), put \(W=m+R-1\), and suppose that \(H^{\mathrm{open}}_{W,R}(A)\) has norm gap \(\gamma \). Thus, for every \(u\in (\ker H^{\mathrm{open}}_{W,R}(A))^\perp \),
If \((R-1)^2{\lt}m\gamma \), define
Then \(\delta {\gt}0\), and for every \(N\geq 2m\) and every \(v\in (\ker H_{N,R}^{\mathrm{ES}}(A))^\perp \),
The coefficient in (308) is the TNLean finite-range derivation documented in docs/paper-gaps/knabe88_finite_range_coefficient.tex. It is not a theorem of the MPS sources cited for the gap definition and the nearest-neighbor Knabe principle.
For a tensor \(A\) and a range \(L{\gt}1\), assume the cyclic-window anticommutator estimate of Theorem 12.8.17. Then there exists \(\gamma {\gt}0\) such that for every \(N\ge 2L\) and every vector \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \) one has \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \). By Lemma 12.3.20, this implies the usual uniform lower bound on every nonzero eigenvalue of the parent Hamiltonian.
Assume \(N{\gt}0\) and \(L\le N\). Then \(\ker H_N(A,L)=G_L^{(N)}(A)\). Equivalently, after the canonical Hilbert-space identification, \(\mathcal K_N^{\mathrm{ES}}(A,L)=e_N^{-1}G_L^{(N)}(A)\), where \(\mathcal K_N^{\mathrm{ES}}(A,L)\) is the zero-energy space of \(H_{N,L}^{\mathrm{ES}}(A)\).
Assume \(N{\gt}0\) and \(L\le N\). If a vector \(\psi \) has zero energy for the finite periodic parent Hamiltonian, \(H_N(A,L)\psi =0\), then every cyclic length-\(L\) restriction of \(\psi \) lies in the local MPS ground space. Equivalently, \(\ker H_N(A,L)\subseteq G_L^{(N)}(A)\).
Let \(h\) be an interaction on \(R\) sites, let \(U\) be a physical matrix, and suppose \([h,U^{\otimes R}]=0\). For every \(N\geq R\), each cyclic extension of \(h\) commutes with \(U^{\otimes N}\). Consequently both the periodic sum and the sum over nonwrapping windows commute with \(U^{\otimes N}\). In configuration coordinates, cyclic extension is the matrix obtained by placing \(h\) on the window and the identity on its complement.
Let \(A\) be a tensor, let \(U,Q\) be physical matrices with \(UQ=QU\), and write \(T_L(V)=V^{\otimes L}\). Then \(T_L(U)\) and \(T_L(Q)\) commute. If \(T_L(U)G_L(A)=G_L(A)\), then \(T_L(U)G_L(QA)=G_L(QA)\). When \(Q\) is invertible, these two invariance statements are equivalent. If \(U\) is unitary and \(G_L(A)\) is invariant, its canonical complementary projection \(h_L(A)\) commutes with \(T_L(U)\). Finally, if \(Q\) is invertible and Hermitian, every interaction \(h\) commuting with \(T_L(U)\) retains this symmetry under the deformation \(T_L(Q^{-1})hT_L(Q^{-1})\).
Let \(P:\mathbb {C}^k\to \mathbb {C}^d\) be a linear map, let \(E\) be the orthogonal projection onto \(\operatorname {ran}P\), and set \(Q=(PP^\dagger )^{1/2}+I-E\). There is a partial isometry \(W\) such that \(Q{\gt}0\), \(P=QW\), and \(WW^\dagger =E\). On the retained physical space this is the polar decomposition of [ 236 , lines 579–584 ] ; the identity extension acts only on the orthogonal complement.
Let \(P:\mathbb {C}^k\to \mathbb {C}^d\), and suppose that unitary matrices \(U\) and \(X\) satisfy \(UP=PX\). Let \(E\) project onto \(\operatorname {ran}P\), and write \(P=QW\) with \(Q=(PP^\dagger )^{1/2}+I-E\) as in Theorem 12.10.20. Then \(U\) commutes with \(PP^\dagger \), \(E\), and \(Q\), and \(UW=WX\). In particular, this applies to the joint physical map of a block family. Physical rotation by a matrix \(L\) replaces that map by \(LP\). These assertions establish the polar step in [ 236 , lines 645–676 ] , with the virtual unitary \(X\) supplied as a hypothesis.
- Matrix.commute_mul_conjTranspose_of_unitary_covariance
- Matrix.commute_leftPolarSupport_of_unitary_covariance
- Matrix.commute_leftPolarPos_of_unitary_covariance
- Matrix.leftPolarIso_covariance_of_unitary_covariance
- MPSTensor.commute_leftPolarPhysicalFactor_of_unitary_covariance
- MPSTensor.blockPhysicalMatrix_rotatePhysical
Let \(A\) be a normalized irreducible tensor of period one with positive bond dimension. There is a positive-definite matrix \(\rho \) such that \(A\) and \(\rho \) form a primitive normalized MPS pair. This is the Perron–Frobenius normalization used in [ 225 , Equations (3.1)–(3.2b) ] .
For a tensor \(A\) and a positive interaction range \(R\), the periodic sum of the canonical parent projection \(h_R=I-P_{G_R(A)}\) is exactly the canonical parent Hamiltonian, at every volume:
Cyclic extension and its periodic sum preserve operator inequalities and commute with scalar multiplication. In particular, for every volume \(N\),
The scalar identity holds for every complex number \(c\).
Let \(1\leq R\leq m\), \(N\geq 2m\), and \(W=m+R-1\). Write \(K_{R,W}\) for the open parent Hamiltonian of interaction length \(R\) on \(W\) sites, and \(H_{R,N}\) for the corresponding periodic Hamiltonian on \(N\) sites. If real constants \(\kappa ,C\) satisfy
where \(h_W\) projects onto \(\mathcal G_W(A)^\perp \), then
Assume \(d\geq 1\). Under the periodic-family, nonzero-weight, positivity, positive-range, and eventual original kernel hypotheses of Theorem 12.9.166, there exist \(L{\gt}0\) and \(\gamma {\gt}0\) such that the same norm gap holds at every original length \(NL\), on the orthogonal complement of its actual kernel. No common blocking length is supplied, and the original family may be empty. These are the aligned finite-chain lengths associated with the regrouping in [ 225 , lines 825–836, Theorem 1.2 ] .
Let \(d\geq 1\), and let \((A_j)\) be a finite family of normalized periodic tensors with periods \(m_j\), and let \(T=\bigoplus _j\mu _jA_j\), with \(\mu _j\neq 0\). Let \(L{\gt}0\) satisfy \(m_j\mid L\) for every \(j\). Suppose \(h\geq 0\) has range \(R{\gt}0\) and \(\ker H_h(n)=\mathcal G_T(n)\) for all sufficiently large \(n\). There is \(\gamma {\gt}0\) such that, for every \(N\geq 0\) and every \(v\perp \ker H_h(NL)\),
This includes short volumes with their actual kernels. The original family may be empty; its sectors need not be inequivalent and no faithful invariant matrices are supplied. This is the explicit periodic-family, block-divisible finite-chain consequence of [ 225 , Theorem 1.2, Section 3, equations (3.12)–(3.14) ] and [ 191 , Lemma 6 ] . The other original residue lengths and general GVBS presentations remain separate [ 61 ] .
Let \(d\geq 1\), and let \((A_j)_{j=0}^{r-1}\) be a finite family of normalized periodic tensors with periods \(m_j\), and let \(T=\bigoplus _j\mu _jA_j\), where \(\mu _j\neq 0\). Let \(h\geq 0\) have interaction range \(R{\gt}0\). Suppose that its open-chain Hamiltonians satisfy \(\ker H_h(N)=\mathcal G_T(N)\) for all sufficiently large \(N\). Then there are \(L{\gt}0\) and a finite pairwise gauge-phase inequivalent primitive family \((B_k)_{k=0}^{g-1}\), of positive bond dimensions with faithful invariant matrices, such that \(\mathcal G_{T^{[L]}}(N)=\mathcal G_{\bigoplus _kB_k}(N)\) for every \(N\geq 0\). Let \(\widetilde\omega _k\) be the corresponding primitive sector states transported from the blocked lattice to the original lattice. For every continuous complex linear functional \(\varphi \),
Here \(\mathcal F_h\) is the zero-energy face of all normalized positive states, and purity is extremality among all states. The original family may be empty and no inequivalence is imposed on it. No translation invariance of \(\varphi \), supplied invariant matrices, or injectivity bound on \(R\) is assumed.
This is the explicit periodic-tensor-family case of [ 225 , Theorems 1.1–1.2, Section 3 ] , using the site grouping at lines 825–836 and [ 191 , Lemma 6 ] . Identification of an arbitrary GVBS presentation with these tensor data and finite support spaces remains separate [ 61 ] .
Let \(A_j\) be a finite family of normalized periodic tensors with periods \(m_j\), and let \(\mu _j\neq 0\). Suppose \(L{\gt}0\) and \(m_j\mid L\) for every \(j\). There are finitely many primitive tensors \(B_k\) of positive bond dimensions, with faithful invariant matrices \(\rho _k\), such that distinct \(B_k\) are not gauge-phase equivalent after any equality of their bond dimensions. For every \(N\geq 0\),
The original family may be empty, and the representative family may consequently be empty. This is the finite-family support consequence of [ 191 , Lemma 6 ] and representative selection in [ 13 , Proposition 2.7 ] .
Every finite weighted family of normalized periodic tensors with nonzero weights admits a positive blocking length and an inequivalent primitive faithful family with identical open-boundary spaces at every blocked length. No original inequivalence or nonempty-family assumption is required. This is the common-period consequence of [ 191 , Lemma 6 ] .
Let \(d{\gt}0\), let \((A_j)\) be a finite, possibly empty, family of normalized periodic tensors of periods \(m_j\), and let \(\mu _j\neq 0\). Put \(T=\bigoplus _j\mu _jA_j\). Let \(h\geq 0\) have arbitrary range \(R{\gt}0\), and suppose that \(\ker H_h(n)=\mathcal G_n(T)\) for every sufficiently large \(n\). There is one \(\gamma {\gt}0\) such that every pure zero-energy state and every centered original local observable satisfy (470), for every filter and every pair of diverging boundary margins. The common blocking length, primitive tensors, faithful invariant matrices, and sector inequivalence are derived from periodicity. Competing states are not assumed translation invariant. For an empty tensor family the zero-energy state face is empty. This concerns explicit periodic tensor families; the general GVBS identification in [ 225 , Theorem 1.2 ] remains separate; see [ 61 ] .
Suppose that \(A\) is periodic of period \(m\) and \(p{\gt}0\). Put \(r=\gcd (m,p)\). There are nonzero cyclic projections \(P_u\) with \(\sum _uP_u=\mathbb {1}_D\) and \(P_uA^i=A^iP_{u+1}\), positive dimensions \(D_\alpha \) with \(\sum _\alpha D_\alpha =D\), and left-canonical tensors \(B_\alpha \) indexed by \(\alpha \in \{ 0,\ldots ,r-1\} \). Their support isometries satisfy
Every \(B_\alpha \) is periodic of period \(m/r\), and for every \(N\ge 0\),
The cyclic compression is that of [ 191 , Lemma 6 ] ; the local-space equality is its supported word contraction.
Let \(A,B\) be normalized irreducible periodic tensors, possibly of different periods and bond dimensions. Suppose that whenever their bond dimensions agree, they are inequivalent up to similarity and phase. For every \(\varepsilon {\gt}0\), all sufficiently large \(n\) satisfy \(|\langle x,y\rangle |\leq \varepsilon \| x\| \| y\| \) for every \(x\in \mathcal G_n(A)\) and \(y\in \mathcal G_n(B)\). The statement concerns boundary spaces of generating tensors and does not assert purity of their associated states.
Let \(A\) be a normalized irreducible periodic tensor. There are \(C{\gt}0\) and \(n_0\) such that every \(n\geq n_0\) and \(x\in \mathcal G_n(A)\) have a virtual boundary \(u\) with \(\Gamma ^A_n(u)=x\) and \(\| u\| \leq C\| x\| \). The bound includes every residue length. This is the boundary normalization step in the mixed-transfer argument of [ 225 , Lemma disjoint ] , applied to periodic generating tensors using [ 191 , Lemma 6 ] .
A normalized irreducible periodic tensor \(A\) of positive period \(m\) admits cyclic projections \(P_a\), positive sector dimensions \(D_a\) summing to \(D\), support isometries \(V_a\), primitive compressed tensors \(B_a\), and faithful invariant matrices \(\rho _a\). They satisfy \(V_aV_a^\dagger =P_a\), \(P_aA_i=A_iP_{a+1}\), \(B_a^i=V_a^\dagger (A^{[m]})^iV_a\), and \((A^{[m]})^iV_a=V_aB_a^i\). The blocked MPS ground space agrees with the joint compressed-sector space at every length, including zero. For each sector, the limiting residual Gram metric is \(Y\mapsto P_{a+r}Y\rho _a/\operatorname{tr}\rho _a\), and the geometric estimate of Theorem 12.9.195 holds uniformly over all original tail lengths \(r\). The primitive sectors, faithful matrices, and row corners are derived from periodicity.
Let \(d\ge 1\), let \((A_j)\) be a finite family of normalized periodic tensors with arbitrary periods, and let \(\mu _j\ne 0\). For \(T=\bigoplus _j\mu _jA_j\), there exists \(R_0{\gt}0\) such that
In particular, the canonical local projection of range \(R_0\) defines a positive interaction with exact open kernels at every \(N\ge R_0\). No inequivalence or independence of the supplied sectors is assumed. The finite family may be empty. This is the finite periodic-tensor presentation of the interaction construction in Section 4 of [ 225 ] ; the passage from general GVBS data to such a presentation remains separate.
Under the preceding hypotheses, there are a positive interaction \(h\) of range \(R{\gt}0\), with exact open kernels for all \(N\ge R\), and a constant \(\gamma {\gt}0\) such that every pure normalized zero-energy state \(\varphi \) and every centered local observable \(X\) on a positive-length interval beginning at \(a\) satisfy
The limit holds along every interval exhaustion with both margins diverging. The constant precedes the state, observable, filter, and margins. Neither translation invariance nor supplied interaction, kernel, separation, or independence data are required. The statement concerns the finite periodic-tensor presentation; the general GVBS presentation step remains separate.
Under the preceding hypotheses, one can choose \(R{\gt}0\) and the canonical positive interaction \(h\) of range \(R\) so that its open kernels are \(\mathcal G_T(N)\) for every \(N\ge R\). There exists \(\delta {\gt}0\) giving a norm gap above the actual open kernel at every original volume, including \(N{\lt}R\). For the same interaction there exists \(\gamma {\gt}0\) such that every pure normalized zero-energy state \(\varphi \) and every centered local observable \(X\) on a positive-length interval satisfy
The limit holds along every exhaustion with both margins diverging. Both constants precede the states, observables, filters, and margins. No translation invariance of a classified state is assumed. The general GVBS presentation step remains separate.
Under the preceding hypotheses, one can choose a positive interaction \(h\) of range \(R{\gt}0\), with exact open kernels for all \(N\ge R\), and one constant \(\gamma {\gt}0\) as follows. For every pure normalized zero-energy state \(\varphi \) of \(h\) and every centered local observable \(X\) on a positive-length interval beginning at \(a\), the literal finite-volume commutator energies converge along every interval exhaustion whose left and right margins diverge:
The constant is chosen before the state, observable, filter, and boundary margins. No translation-invariance assumption is imposed on the state being classified. The middle-space independence assumption is retained in this sufficient criterion.
Under the preceding hypotheses, the same supplied interaction \(h\) has an all-length finite norm gap with a constant \(\delta {\gt}0\) and a constant \(\gamma {\gt}0\) such that every pure normalized zero-energy state \(\varphi \) and every centered local observable \(X\) on a positive-length interval beginning at \(a\) satisfy
The limit holds along every exhaustion with both margins diverging. Both constants precede the states, observables, filters, and margins. No translation invariance of a classified state is assumed. The general GVBS presentation step remains separate.
Let \(d\ge 1\), let \((A_j)\) be a finite family of normalized periodic tensors, let \(\mu _j\ne 0\), and write \(T=\bigoplus _j\mu _jA_j\). Let \(h\ge 0\) be a supplied interaction of range \(R{\gt}0\) whose open Hamiltonians satisfy \(\ker H_N(h)=\mathcal G_T(N)\) for every sufficiently large \(N\). Then there exists \(\delta {\gt}0\) such that every original volume \(N\) and every \(v\perp \ker H_N(h)\) satisfy
The actual kernel is used also on short intervals. No local-kernel identity at range \(R\), lower bound on \(R\) beyond positivity, sector separation, stationary matrix, or finite gap is supplied. The finite family may be empty. The statement concerns finite periodic-tensor presentations; the general GVBS presentation step remains separate. This is the given-interaction conclusion of [ 225 , Theorem 1.2 ] in this setting.
Let \(d\ge 1\), let \((A_j)\) be a finite family of normalized periodic tensors with arbitrary periods, and let \(\mu _j\ne 0\). Assume explicitly that their original middle ground spaces \((\mathcal G_{A_j}(n))_j\) are independent at every sufficiently large length \(n\). For \(T=\bigoplus _j\mu _jA_j\), there exists \(R_0{\gt}0\) such that
In particular, the canonical local projection of range \(R_0\) gives a positive interaction whose open kernel is exactly \(\mathcal G_T(N)\) for every \(N\ge R_0\). The family may be empty. This is an independence criterion: its middle-space hypothesis remains explicit and is not an unrestricted periodic-family existence theorem.
Let \((A_j)\) be a finite family of pairwise inequivalent normalized irreducible periodic tensors, with periods \(m_j\), and let \(L{\gt}0\) be divisible by every \(m_j\). Put \(\Phi _i=\bigoplus _j(A_j)_i\). There is a family of positive-dimensional primitive tensors \((B_s)\) on the \(L\)-site alphabet, indexed by pairs \(s=(j,a)\) with \(a\in \mathbb Z/m_j\mathbb Z\), with faithful stationary matrices \(\rho _s\) and isometries \(V_s\) into the original direct sum, such that
The tensors \(B_s\) are pairwise inequivalent up to similarity and phase. For every \(s\) and every tail length \(r\geq 0\), the matrix
is an orthogonal projection. The family may be empty.
Let \(d\geq 1\), and let \(A_j\) be finitely many periodic tensors with nonzero weights \(\mu _j\), and let \(h\geq 0\) be an interaction of positive range whose open-chain kernels eventually equal the boundary spaces of \(\bigoplus _j\mu _jA_j\). There is \(L{\gt}0\) such that every pure zero-energy state of \(h\) is invariant under translation by every integer multiple of \(L\). No invariance under one-site translation and no minimality of the period is asserted [ 225 , Theorem 1.1 ] .
Let \(d\ge 1\), let \((A_j)\) be a finite family of normalized periodic tensors with arbitrary periods, and let \(\mu _j\ne 0\). For \(T=\bigoplus _j\mu _jA_j\), there exists \(R_0{\gt}0\) such that every \(R\ge R_0\) has both of the following properties:
and there exists \(\delta _R{\gt}0\) such that, for every \(N\ge 0\) and every \(v\perp \ker H_{R,N}^{\mathrm{open}}(T)\),
The finite family may be empty. No inequivalence, invariant matrix, primitive presentation, interaction, or kernel identity is supplied. The norm bound uses the actual kernel also when \(N{\lt}R\). This is the finite periodic-tensor presentation of the gap theorem; the passage from general GVBS data to such a presentation remains separate.
Let \(A\) be a tensor, let \(p,R\geq 1\) be integers, and let \(\gamma {\gt}0\). Suppose that, for every sufficiently large integer \(M\) and every \(u\perp \ker H^{\mathrm{open}}_{pM,R}(A)\), \(\gamma \lVert u\rVert \leq \lVert H^{\mathrm{open}}_{pM,R}(A)u\rVert \). Then there is \(\delta {\gt}0\) such that, for every sufficiently large \(N\) and every \(v\perp \ker H^{\mathrm{ES}}_{N,R}(A)\), \(\delta \lVert v\rVert \leq \lVert H^{\mathrm{ES}}_{N,R}(A)v\rVert \).
Let \(A\) be periodic of period \(m\), and let \(p{\gt}0\) satisfy \(m\mid p\). There is a nonempty finite family \((B_j)_{j=0}^{g-1}\), with positive bond dimensions and faithful invariant matrices \(\rho _j\), such that each \((B_j,\rho _j)\) has normalized primitive fixed-point data, distinct sectors are gauge-phase inequivalent after any equality of their bond dimensions, and
The tensors and matrices are selected from the compressed primitive family. Inequivalence is derived, rather than assumed. This is a finite local-support consequence of the periodic blocking in [ 191 , Lemma 6 ] and the representative selection in [ 13 , Proposition 2.7, Theorem 2.10 ] . The compressed peripheral period uses the correction in [ 86 ] .
Blocking a periodic tensor by its period gives a nonempty finite pairwise gauge-phase inequivalent primitive family with faithful invariant matrices and
Let \(A\) be a normalized periodic tensor. There exist \(R\ge 1\) and \(\delta {\gt}0\) such that the canonical range-\(R\) open-parent Hamiltonian \(H_{R,N}\) satisfies
There is no divisibility condition on \(N\). The interaction range and constant are derived from periodicity alone. This is the normalized periodic tensor specialization of [ 225 , Theorem 1.2 and Section 6 ] ; the general GVBS presentation’s identification with these support spaces remains separate, as recorded in [ 61 ] .
Let \(A\) be a normalized irreducible periodic tensor of period \(m\). There is an orthogonal resolution \(\{ P_u\} _{u\in \mathbb {Z}/m\mathbb {Z}}\) satisfying \(P_uA_i=A_iP_{u+1}\), and a threshold \(n_0\), such that for every \(n\ge n_0\) the length-\(n\) words span precisely \(\{ X:P_uX=XP_{u+n}\text{ for all }u\} \). Consequently,
The lengths here count original sites, with no divisibility requirement. This is the cyclic-sector form of the intersection argument in [ 225 , equations (3.10)–(3.11) ] .
For the same tensor \(A\), there is \(R_0{\gt}0\) such that, for every \(R\ge R_0\) and every \(N\ge R\), the canonical range-\(R\) open parent Hamiltonian has kernel exactly \(G_N(A)\). In particular, there exists a positive interaction \(h\) of some range \(R{\gt}0\) with this kernel identity at every \(N\ge R\). No faithful invariant matrix, cyclic resolution, or short-range intersection is supplied as a hypothesis. This concerns the explicit normalized periodic tensor presentation; it does not assert a presentation theorem for arbitrary GVBS states. It implements the finite-support interaction construction of [ 225 , Lemma existenceinteraction and Theorem 1.1 ] in this tensor setting.
Let \(A\) and \(B\) be periodic tensors. Then at least one of the following alternatives holds:
\(\langle V^{(N)}(A) | V^{(N)}(B) \rangle \to 0\) as \(N\to \infty \);
\(A\) and \(B\) are related by the repeated-block relation, i.e. they differ by a unit-modulus phase and an invertible gauge.
Let \(A\) be a normalized periodic tensor. There exist a range \(R\ge 1\), a positive local matrix \(h\) on \(R\) sites, and constants \(\delta ,\gamma {\gt}0\) with the following properties. The open interaction sum \(H_N\) satisfies
For every pure zero-energy state \(\varphi \) of this same interaction and every observable \(X\) supported on \([a,a+k)\) with \(k{\gt}0\) and \(\varphi (X)=0\), along every filter for which both margins \(\ell _n,r_n\to \infty \), there is a limit
Both constants precede every state, observable and interval exhaustion. The interaction, its range and all support data are derived from periodicity. This is an existence consequence of Theorems 1.1–1.2 of [ 225 ] in the explicit normalized periodic tensor setting. The general GVBS presentation remains separate; see [ 61 ] .
Let \(A\) and \(B\) be normalized periodic tensors. Suppose that there is \(z\ne 0\) such that their periodic vectors satisfy \(|V^{(N)}(B)\rangle =z^N|V^{(N)}(A)\rangle \) for every \(N{\gt}0\). Then their bond dimensions agree, they are gauge and phase equivalent, and \(\mathcal G_A(N)=\mathcal G_B(N)\) for every \(N\ge 0\).
Let \(A\) be a normalized periodic tensor of period \(m\). For arbitrary sequences \(K_M,r_M\in \mathbb N\), put \(N_M=K_Mm+Mm+r_M\). Let \(P_{\mathrm L,M}\) and \(P_{\mathrm R,M}\) be the ground-space projections for the prefix of length \((K_M+M)m\) and the suffix of length \(Mm+r_M\), extended to the full interval. Then
This is the explicit periodic tensor form of the cover estimate used in [ 225 , Lemma commutation (ii) and Section 6 ] . The passage from a general GVBS presentation to this tensor presentation remains separate; see [ 61 ] .
For every normalized periodic tensor \(A\), there exist \(R\ge 1\), a positive local matrix \(h\) on \(R\) sites, and \(\delta {\gt}0\) such that its open-chain interaction sums \(H_N\) satisfy
No local interaction, invariant density, or sector separation is supplied. The scope is the explicit normalized periodic tensor setting of [ 225 , Theorem 1.2, Lemma existenceinteraction, and Section 6 ] .
Let \(A\) be a normalized periodic tensor. There exist \(R{\gt}0\), a positive semidefinite interaction \(h\) on \(R\) sites, and \(\gamma {\gt}0\) such that \(\ker H_N(h)=G_N(A)\) for every \(N\ge R\). If \(\varphi \) is any pure zero-energy state for this interaction, and \(X\) is an observable on an interval of positive length with \(\varphi (X)=0\), then
along every filter for which both interval margins \(\ell ,r\) tend to infinity. The same constant works for all such states, observables, and exhaustions. The physical dimension, interaction, eventual kernel identity, and primitive invariant matrices are derived. This proves the existence and infinite-volume gap conclusions of [ 225 , Theorems 1.1–1.2 ] for an explicit normalized periodic tensor presentation. Identification with a general GVBS presentation, and the all-residue finite-volume gap, remain separate.
If \(A\) is periodic of period \(m\) and \(p\ge 0\), there is a family of nonzero cyclic projections \(P_u\) with \(\sum _uP_u=\mathbb {1}_D\) and \(P_uA^i=A^iP_{u+1}\) for which the dimensions \(D_\alpha \), isometries \(V_\alpha \), and left-canonical tensors \(C_\alpha \) of Theorem 22.4.7 exist, with the same identities and the same MPV at every length, including zero. If \(p{\gt}0\), every resulting block satisfies \(\operatorname{spec}_{\mathrm{per}}(\mathcal{E}_{C_\alpha })=\{ z\in \mathbb {C}:z^{m/\gcd (m,p)}=1\} \) and is periodic of period \(m/\gcd (m,p)\). This is the prescribed-blocking conclusion of [ 191 , Lemma 6 ] .
Let \(A\) be periodic of period \(m\), and let \(p{\gt}0\) satisfy \(m\mid p\). Put \(r=\gcd (m,p)=m\). There are positive bond dimensions \(D_j\) with \(\sum _{j=0}^{r-1}D_j=D\), tensors \(B_j\) on the blocked physical alphabet, and positive definite matrices \(\rho _j\), such that each pair \((B_j,\rho _j)\) has primitive fixed-point data. In particular,
and for every \(N\geq 0\),
No inequivalence assumption is imposed on the sector family. This combines the periodic blocking of [ 191 , Lemma 6 ] with the primitive normalization used in [ 225 , Section 3, equations (3.1)–(3.2b) ] . The peripheral period is the corrected \(m/\gcd (m,p)\) of [ 86 ] .
A periodic tensor of period \(m\) admits \(m\) positive-dimensional primitive sectors after blocking by \(m\). Their bond dimensions sum to \(D\), they have faithful invariant matrices, and
Let \(A\) be a periodic MPS tensor of period \(m\) and positive physical dimension. Let \(h\geq 0\) have interaction range \(R{\gt}0\), and suppose \(\ker H_N(h)=G_N(A)\) for every sufficiently large \(N\). There is a nonempty finite family of pairwise gauge-phase inequivalent primitive tensors \(B_j\), with positive bond dimensions and faithful invariant matrices \(\rho _j\), such that \(G_N(A^{[m]})= G_N(\bigoplus _j B_j)\) at every length \(N\geq 0\). Let \(\omega _j\) denote the primitive state of \((B_j,\rho _j)\) transported to the original lattice by grouping \(m\) consecutive sites. For every continuous complex-linear functional \(\varphi \),
The sector data and their inequivalence are derived from periodicity. The state is not assumed translation invariant, and \(R\) is not subject to an injectivity bound. This is the periodic-tensor case of the ground-state face classification in [ 225 , Theorems 1.1–1.2 and Section 3 ] ; the identification of a general periodic GVBS presentation remains separate, as described in [ 61 ] .
Let the physical dimension be positive and let \(A\) be a periodic MPS tensor. Let \(h\geq 0\) have arbitrary positive range \(R\), and suppose that for all sufficiently large \(N\) the open Hamiltonian has kernel \(G_N(A)\). There is a constant \(\gamma {\gt}0\) common to every pure state \(\varphi \) in the zero-energy face of \(h\) such that, for every positive-length local observable \(X\) with \(\varphi (X)=0\),
Here \(H_{\ell ,r}\) is the original open interaction sum on the interval extending \(\ell \) sites to the left and \(r\) sites to the right of the support of \(X\). The limit exists as a complex number along every filter for which both margins tend to infinity. The same \(\gamma \) applies to every such filter and pair of margins. Both sides are real, and the inequality uses their usual real order.
The primitive sectors and their faithful invariant matrices are derived from periodicity. No translation-invariance assumption is imposed on \(\varphi \). This proves the infinite-volume conclusion for an explicit periodic tensor presentation. The general GVBS presentation of [ 225 , Theorem 1.2 ] requires a separate identification with these finite support spaces; see [ 61 ] .
Let \(A\) be a normalized irreducible tensor of period \(m\) and let \(L{\gt}0\). There exist sector dimensions \(D_a{\gt}0\), tensors \(B_a\) on \(d^L\) spins, and isometries \(V_a\colon \mathbb {C}^{D_a}\to \mathbb {C}^D\), indexed by \(a\in \{ 0,\ldots ,\gcd (m,L)-1\} \), such that \(\sum _aD_a=D\), \(\sum _aV_aV_a^\dagger =I\), and each \(B_a\) is trace-preserving with period \(m/\gcd (m,L)\). Both letter intertwinings hold:
For every \(N,r\geq 0\), including zero, \(\mathcal G_{NL+r}(A)=\sum _a\operatorname {ran} \mathcal F_{A,B_a,V_a}^{N,r}\). This extends the finite support decomposition in [ 191 , Lemma blocking-arbitrary ] to an ungrouped terminal interval. The period formula includes the peripheral-root correction in [ 86 ] .
Let \(A\) be periodic with period \(m\). There are \(m\) primitive normalized compressed tensors \(B_j\), positive virtual dimensions \(D_j\), support isometries \(V_j\), and faithful invariant matrices \(\rho _j\). The tensors \(B_j\) are pairwise gauge-phase inequivalent and satisfy
Let \(F_j(N,r)\) be the correlated rectangular boundary range with coefficient \(\operatorname {tr}(B_j^uV_j^*A^\tau Y)\). Then
for every \(N,r\geq 0\). For every function \(r:\mathbb N\to \mathbb N\),
All primitive normalization and separation data are derived from periodicity; none are assumed separately. This comparison does not assert an overlapping-window defect estimate or a gap at all residue lengths.
Let \(A\) be a normalized periodic tensor of period \(m\). There exist positive-dimensional blocked sectors \(B_j\), indexed by \(j\in \mathbb Z/m\mathbb Z\), and rectangular isometries \(V_j\) such that
and, for every pair of sequences of exterior lengths \(K(M),r(M)\),
The spaces are the joint residual window ranges in the triple configuration coordinates. This establishes their projection comparison from periodicity alone; identifying them with original physical interval spaces is a separate isometric transport. The estimate is the residual-coordinate form of commutation (ii) in [ 225 ] .
Let \((A_j)_{j=0}^{b-1}\) be a finite family of normalized periodic tensors, with arbitrary periods, and let \(\mu _j\ne 0\). There is a subfamily selected by an injective map \(s:\{ 0,\ldots ,g-1\} \to \{ 0,\ldots ,b-1\} \) such that its members are pairwise inequivalent under gauge and phase transformations. Every original member is gauge and phase equivalent to a selected member, after the corresponding bond-dimension identification.
The selected family inherits the original periods and positive bond dimensions. The statement also allows an empty original family. This is an algebraic support identity; it does not assert the joint intersection property or automatic existence of a parent interaction.
Let \(D{\gt}0\) and \(m{\gt}0\). Let \(A\) be a left-canonical irreducible MPS tensor with bond dimension \(D\), and suppose that the peripheral eigenvalues of its transfer map are exactly \(\{ z\in \mathbb {C}:z^m=1\} \). Then \(O_{AA}(mn)\longrightarrow m\).
Let \(x\mapsto A_x\) be a continuous family of one-site injective tensors of nonzero bond dimension, let \(S\) be a compact parameter set, and let \(N\geq 2\). Then there is \(\delta {\gt}0\) such that \(\delta \lVert v\rVert \leq \lVert H_{N,2}(A_x)v\rVert \) for all \(x\in S\) and all \(v\) orthogonal to the kernel of \(H_{N,2}(A_x)\).
Let \(A\) be a periodic tensor of period \(m{\gt}0\), and let \(h\geq 0\) have positive range and eventual kernel \(G_A(n)\). There exists \(\delta {\gt}0\) such that, for every \(N\geq 0\),
The primitive family, its invariant matrices, and its inequivalence are derived from periodicity. Other residue lengths remain separate.
Let \(R=L+Q{\gt}0\) and suppose that the range-\(R\) open parent Hamiltonians on \(K+L\) and \(K+L+Q\) sites have their boundary-condition spaces as kernels. On \(K+L+Q\) sites, let \(P\) project onto the ground space of the first \(K+L\) sites, let \(T\) project onto the ground space of the last \(L+Q\) sites, and let \(F\) project onto the whole-chain ground space. Then
The right-hand side is the three-interval projector error in [ 225 , condition C3\('\) ] .
Let \(B=\bigoplus _{j=1}^r\mu _jA_j\), where every \(\mu _j\) is nonzero and the \(A_j\) are pairwise gauge-phase inequivalent normalized primitive tensors with positive bond dimensions and positive-definite fixed points. Write \(P_I(B)\) for the orthogonal projector onto the ground space on the interval \(I\), extended by the identity on the remaining sites. For every \(\eta {\gt}0\), all sufficiently large \(L\), and every \(K\geq 0\) and \(Q{\gt}0\),
Let \((A_i)_{i\in I}\) be a finite family of tensors, and use the interval spaces and projectors of Theorem 12.7.3. Suppose that, for each \(\varepsilon {\gt}0\) and all sufficiently large \(L\), the following two bounds hold. For distinct \(i,j\), \(x\in G_L(A_i)\), and \(y\in G_L(A_j)\), \(|\langle x,y\rangle |\leq \varepsilon \lVert x\rVert \lVert y\rVert \). For every \(i\), \(K\geq 0\), and \(Q{\gt}0\), \(\lVert Q_iP_i-R_i\rVert \leq \varepsilon \). Then, for every \(\eta {\gt}0\), all sufficiently large \(L\), and every \(K\geq 0\) and \(Q{\gt}0\),
Let \((A_i)\) be a finite family of tensors with the same physical dimension, with bond dimensions allowed to depend on \(i\). Let \(B\) be a nonnegative real matrix with zero diagonal and \(\beta =\lVert B\rVert {\lt}1/2\). Suppose that for \(i\ne j\), \(x\in G_L(A_i)\), and \(y\in G_L(A_j)\), \(|\langle x,y\rangle |\leq B_{ij}\lVert x\rVert \lVert y\rVert \). In the physical space on \(K+L+Q\) sites, put
Let \(P_i,Q_i,R_i\) be the orthogonal projectors onto these spaces, and let \(P,Q,R\) project onto their respective sums. With \(\delta =\beta /(1-\beta )\),
This is the physical interval comparison in the proof of [ 225 , Lemma 6.1(ii) ] , before inserting the individual block error bounds.
Let \(A^j\) and \(B^j\) have the same respective bond dimensions, and let all coefficients \(\mu _j,\nu _j\) be nonzero. If \(G_L(A^j)=G_L(B^j)\) for every \(j\), then the assembled tensors \(A=\bigoplus _j\mu _jA^j\) and \(B=\bigoplus _j\nu _jB^j\) satisfy \(G_L(A)=G_L(B)\). Their canonical periodic parent Hamiltonians of range \(L\) coincide at every volume: \(H^{\mathrm{ES}}_{N,L}(A)=H^{\mathrm{ES}}_{N,L}(B)\).
Let \(B=\bigoplus _{j=1}^r\mu _jA_j\) with \(\mu _j\ne 0\) for every \(j\). For every \(K,L\geq 0\), the extended interval spaces satisfy
For every \(K,L,Q\geq 0\), with the reassociation \(\mathscr A_{K,L,Q}\) of Definition 12.3.7.28, the right interval space of length \(L+Q\) satisfies
Let \(A\) be in canonical form, and let \(B^0,\ldots ,B^{g-1}\) be its distinct normal representatives. Repeated copies and their nonzero coefficients do not change the positive-length local spaces: for every \(L{\gt}0\),
This is the local-space consequence of the canonical decomposition in [ 17 ] .
Assume the ground-space overlap bound of Theorem 12.7.1. Let \(U_A\) be the range of \(\Gamma ^{\mathrm{left}}_{K+L,Q}(A)\) and \(V_A\) the range of the reassociated tail boundary map \(\widetilde\Gamma ^{\mathrm{tail}}_{K;L,Q}(A)\), both in the physical space on \(K+L+Q\) sites. Define \(U_B,V_B\) in the same way. Then \(|\langle x,y\rangle |\leq \varepsilon \lVert x\rVert \lVert y\rVert \) for vectors in any of the pairs of ranges \(U_A\times U_B\), \(V_A\times V_B\), and \(U_A\times V_B\).
Under the hypotheses of Theorem 12.6.3, assume \(\beta {\lt}1/2\). Let \(W_i\subseteq U_i\), and let \(R_i\) be the orthogonal projector onto \(W_i\). Let \(P,Q,R\) project onto \(\sum _i U_i\), \(\sum _i V_i\), and \(\sum _i W_i\), respectively. Then
This is the abstract projector comparison used in [ 225 , Lemma 6.1(ii) ] .
Let \(A,B\) be tensors with the same physical dimension and arbitrary bond dimensions. Suppose \(\varepsilon \geq 0\) and \(|\langle a,b\rangle |\leq \varepsilon \lVert a\rVert \lVert b\rVert \) for every \(a\in G_L(A)\) and \(b\in G_L(B)\). Let \(x,y\) be vectors on three successive intervals of lengths \(K,L,Q\). For each prefix configuration \(u\) and suffix configuration \(v\), define \(x_{u,v}(w)=x(u,w,v)\) and \(y_{u,v}(w)=y(u,w,v)\). If \(x_{u,v}\in G_L(A)\) and \(y_{u,v}\in G_L(B)\) for all \(u,v\), then
No positivity assumption on \(K,L,Q\) is required. This is the finite-sum estimate used to extend the overlap bounds in [ 225 , Lemma 6.1(ii) ] .
Let \((v_i)\) and \((w_i)\) be finite families in a complex inner product space. Let \(B\) be a real matrix with zero diagonal and \(\beta =\lVert B\rVert {\lt}1\). Suppose that, for \(i\ne j\), each of \(|\langle v_i,v_j\rangle |\), \(|\langle w_i,w_j\rangle |\), and \(|\langle v_i,w_j\rangle |\) is bounded by \(B_{ij}\) times the product of the corresponding vector norms. Put \(\delta =\beta /(1-\beta )\). Then
This is the pairing estimate in the proof of [ 225 , Lemma 6.1 ] .
Let \((U_i)\) and \((V_i)\) be finite families of subspaces of a finite-dimensional complex inner product space. Let \(B\) have zero diagonal and \(\beta =\lVert B\rVert {\lt}1\). Suppose that, for \(i\ne j\), \(|\langle u,v\rangle |\leq B_{ij}\lVert u\rVert \lVert v\rVert \) whenever \((u,v)\) lies in \(U_i\times U_j\), \(V_i\times V_j\), or \(U_i\times V_j\). Write \(P_i,Q_i\) for their orthogonal projectors and put \(X_U=\sum _i P_i\), \(X_V=\sum _i Q_i\), \(\delta =\beta /(1-\beta )\). Then
Let \(R{\gt}0\) and suppose that the length-\(n\) open parent Hamiltonian has kernel \(G_n(A)\). On \(n+r\) sites, the kernel of the same interactions restricted to the first \(n\) sites is \(G_n(A)\otimes (\mathbb {C}^d)^{\otimes r}\). A vector belongs to this space if and only if each restriction obtained by fixing the last \(r\) physical coordinates belongs to \(G_n(A)\).
Let \((A_i)_{i\in I}\) be a finite family of normalized primitive tensors with positive bond dimensions and positive-definite fixed points. Suppose that, if the bond dimensions of two distinct members agree, they are not related by a gauge transformation and a phase. Use the interval projectors of Theorem 12.7.3. For every \(\eta {\gt}0\), all sufficiently large \(L\), and every \(K\geq 0\) and \(Q{\gt}0\),
This is the uniform decay consequence of [ 225 , Lemma 6.1(ii) ] for a positive right outer interval.
Let \(d{\gt}0\) and \(B=\bigoplus _j\mu _jA^j\), with nonzero coefficients \(\mu _j\). Suppose that the blocks have positive bond dimensions, are trace-preserving and primitive with positive definite fixed points, and are pairwise inequivalent under conjugation and a nonzero scalar. There is \(L_0{\gt}0\) such that for every \(L_0\le L\le N\),
Let \(d{\gt}0\), \(r\ge 2\), and \(B=\bigoplus _{j=0}^{r-1}\mu _jA^j\), with all coefficients nonzero. Suppose that the blocks have positive bond dimensions, are trace-preserving and primitive with positive definite invariant matrices, and are pairwise inequivalent under conjugation and multiplication by a nonzero scalar. Suppose that each block is block injective at the supplied common length \(L_0{\gt}0\). Then for
the open-chain parent Hamiltonian satisfies \(\ker H^{\mathrm{open}}_{N,R}(B)=G_N(B)\). This is the interaction threshold of [ 228 , Theorem 12 ] in trace-preserving coordinates.
Let \(A^0,\ldots ,A^{r-1}\) be trace-preserving primitive tensors with positive bond dimensions and positive definite fixed points. Suppose that no two distinct blocks are related by conjugation and a nonzero complex scalar. Then there is \(L_0\) such that, for every \(L\ge L_0\), the tuples \((A^{0,w},\ldots ,A^{r-1,w})\), with \(|w|=L\), span \(\prod _jM_{D_j}(\mathbb {C})\).
Let \(A=\bigoplus _j\mu _jA^j\) have nonzero coefficients and normal blocks of positive bond dimensions. Suppose that no two distinct blocks are related by conjugation and a nonzero scalar. There are normalized primitive blocks \(B^j\), with positive definite invariant matrices, which remain pairwise inequivalent and satisfy \(H^{\mathrm{ES}}_{N,L}(A)=H^{\mathrm{ES}}_{N,L}(B)\) for every \(L,N\), where \(B=\bigoplus _j\mu _jB^j\).
Reassociation identifies any subspace of the physical space on \(K+(L+Q)\) sites with the same subspace on \((K+L)+Q\) sites. In particular, when \(L+Q{\gt}0\), the kernel of the single parent interaction on the last \(L+Q\) sites is \((\mathbb {C}^d)^{\otimes K}\otimes G_{L+Q}(A)\) in these common coordinates.
Let \(T\) be an operator on a finite-dimensional physical space and let \((e_s)_s\) be an orthonormal basis for an additional physical space. For \(x=\sum _s x_s\otimes e_s\) and \(y=\sum _s y_s\otimes e_s\),
Consequently \(T\ge 0\) implies \(T\otimes \mathbb {1}\ge 0\), and \(T\le U\) implies \(T\otimes \mathbb {1}\le U\otimes \mathbb {1}\). Scalar multiplication commutes with extension: \((cT)\otimes \mathbb {1}=c(T\otimes \mathbb {1})\). If \(V\) is a unitary identification of physical spaces, then \(VTV^{-1}\le VUV^{-1}\) if and only if \(T\le U\).
Let \((V_i)\) be a finite family of subspaces of a finite-dimensional complex inner product space. Let \(B\) have zero diagonal and \(\beta =\lVert B\rVert {\lt}1/2\), and suppose that \(|\langle u,v\rangle |\leq B_{ij}\lVert u\rVert \lVert v\rVert \) for \(i\ne j\), \(u\in V_i\), and \(v\in V_j\). Write \(P_i\) for the orthogonal projector onto \(V_i\), \(P\) for the projector onto \(\sum _i V_i\), \(X=\sum _i P_i\), and \(\delta =\beta /(1-\beta )\). Then
This is the finite-dimensional projector estimate underlying [ 225 , Lemma 6.1(i) ] .
Let \(0{\lt}R\leq W\leq n\leq N\), and suppose that the range-\(R\) open parent Hamiltonian on \(W\) sites has kernel \(G_W(A)\). On the \(N\)-site chain, the range-\(R\) interaction sum on \([n-W,n)\) and the single parent interaction on the same interval have the same kernel. In particular, for \(W=L+Q\) and a chain of length \(K+L+Q\), the kernel of the terminal range-\(R\) interaction sum is \((\mathbb {C}^d)^{\otimes K}\otimes G_{L+Q}(A)\), the boundary space of condition (C3\('\)) in [ 225 ] .
Let \(d{\gt}0\), and let \(B=\bigoplus _j\mu _j A^j\), where every \(\mu _j\) is nonzero and \(\sum _a(A^{j,a})^\dagger A^{j,a}=\mathbb {1}\) for every \(j\). Suppose that the simultaneous word tuples span the full product matrix algebra at every length \(n\ge L_0\). For \(L_0+1\le L\le N\), the open-chain parent Hamiltonian with interaction range \(L\) satisfies
Suppose that \(\sum _a(A^{j,a})^\dagger A^{j,a}=\mathbb {1}\) for every block \(j\). If the simultaneous length-\(L\) word tuples span the product matrix algebra, the same holds at every length \(n\ge L\).
Assume \(L\leq N\) and let \(E_m=G_m^{(N)}-G_{m+1}^{(N)}\). Then
Consequently, for \(n{\lt}N\) and every vector \(v\),
- LinearMap.IsSymmetric.kernelProjection_commute_of_commute
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- MPSTensor.openPrefixParentHamiltonianES_commute_openSuffixParentHamiltonianES
- MPSTensor.openPrefixGroundProjectionES_commute_openIntervalGroundProjectionES_of_le
- MPSTensor.ker_openPrefixParentHamiltonianES_le_ker_openSuffixParentHamiltonianES
- MPSTensor.openPrefixGroundProjectionES_commute_openIntervalGroundProjectionES_of_lt
- MPSTensor.fixedAmbient_martingaleDifference_commute_openIntervalGroundProjectionES
- MPSTensor.inner_sum_fixedAmbient_martingaleDifference_openIntervalGroundProjectionES_eq_Icc
Let \((A^j)_j\) have simultaneous spanning length \(S{\gt}0\). An invertible change \(M\) of the physical space preserves this spanning property. More generally, a rectangular change with a left inverse preserves the joint word span. If \(x\mapsto M_x\) is continuous and invertible, then on a compact parameter set the weighted block sums of \((M_xA^j)_j\) have a uniform periodic parent gap at every fixed \(R\geq S+1\), for all \(N\geq R\). All block weights are nonzero.
The transfer map \(\mathcal{E}_A\) of an MPS tensor \(A\) is idempotent, \(\mathcal{E}_A^2 = \mathcal{E}_A\), if and only if there exists an isometry \(V \colon \mathbb {C}^d \to \mathbb {C}^d \otimes \mathbb {C}^d\) such that
Let \(A\) have positive bond dimension and let its blocking by \(L\) have a finite isometric resolution into primitive tensors \(B_j\) of positive bond dimensions with faithful stationary matrices \(\rho _j\) and pairwise inequivalent sectors. Suppose the embeddings satisfy
Assume that for every \(j,r\) the matrix
is an orthogonal projection. For \(N=KL+ML+r\), let \(P_N\) project onto \(\mathcal G_A(N)\), let \(P_{\mathrm L}\) project onto \(\mathcal G_A(KL+ML)\otimes \mathcal H_r\), and let \(P_{\mathrm R}\) project onto \(\mathcal H_{KL}\otimes \mathcal G_A(ML+r)\). Along every filter for which \(M\to \infty \), with \(K,r\) arbitrary,
No periodicity assumption on \(A\) is imposed. All sector-resolution and tail-Gram hypotheses are supplied explicitly.
Assume \(D_k \ge 1\) for all \(k\). Every \(\mathbb {C}\)-algebra automorphism \(T\) of \(\prod _{k=0}^{r-1} M_{D_k}(\mathbb {C})\) decomposes as per-block inner automorphisms followed by a block permutation \(\pi \): there exist \(\pi \in S_r\) with \(D_{\pi (k)} = D_k\) and invertible matrices \(X_k \in \mathrm{GL}_{D_k}(\mathbb {C})\) such that, for every tuple \(M = (M_0,\ldots ,M_{r-1})\) and every block index \(k\), \(T(M)_{\pi (k)} = X_k M_k X_k^{-1}\), where \(T(M)_j\) denotes the \(j\)-th block component of \(T(M)\).
Let \(R{\gt}0\) and let \(h\) be a fixed positive operator with kernel \(G_R(A)\), and let \(H_{R,N}(A)\) be the canonical periodic parent Hamiltonian. There are \(\kappa ,C{\gt}0\), independent of \(N\), such that
Let \(A\) be a matrix product tensor and let \(R{\gt}0\). Suppose that the periodic Hamiltonians formed from the orthogonal projection onto \(G_R(A)^\perp \) have a positive gap uniformly in the chain length. Let \(h\) be a fixed positive operator on \(R\) sites with kernel \(G_R(A)\), and let \(H_N(h)\) be the sum of its cyclic translates. Then there is \(\delta {\gt}0\) such that, for every chain length \(N\) and every \(v\perp \ker H_N(h)\),
For \(N{\lt}R\) both periodic Hamiltonians are defined to be zero.
On the retained physical space let \(Q{\gt}0\) and let \((A^j)_j\) be simultaneously one-site injective. Set \(Q_\gamma =\gamma Q+(1-\gamma )I\) for \(0\leq \gamma \leq 1\). Both the canonical two-site parent Hamiltonians of \((Q_\gamma A^j)_j\) and the periodic Hamiltonians obtained from
where \(h_0\) is the initial canonical parent projection, have a positive gap uniform in \(\gamma \) and every \(N\geq 2\). The positive factor is supplied on the retained physical space. This is the deformation in [ 236 , lines 589–594 and 610–625, Appendix A ] .
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Let \((A^j)_j\) be simultaneously one-site injective blocks of positive bond dimensions, let all block weights be nonzero, and let \(Q{\gt}0\) on the retained physical space. Set \(Q_\gamma =\gamma Q+(1-\gamma )I\) for \(\gamma \in [0,1]\). The canonical two-site open parents of \((Q_\gamma A^j)_j\) have a positive gap uniform in \(\gamma \) and every length \(N\geq 2\). The same conclusion holds for the open Hamiltonians with interaction \(h_\gamma =(Q_\gamma ^{-1})^{\otimes 2}h_0(Q_\gamma ^{-1})^{\otimes 2}\), where \(h_0\) is the initial canonical parent projection. In both assertions the gap is on the orthogonal complement of the open Hamiltonian’s kernel. This is the open-chain consequence of the deformation in [ 236 , lines 589–594 and 610–625, Appendix A ] .
Let \(f\) be a positive complex-linear functional on a finite matrix algebra, and let \(H\geq 0\) satisfy \(f(H)=0\). Every left multiple of \(H\) has zero expectation, and
If \(C\geq 0\), \(c\in \mathbb R\), and \(C\leq cH\), then
These are finite-dimensional positive-comparison consequences of the zero-energy argument in [ 225 , Section 6 ] . Normalization of \(f\) is unnecessary.
Let \(f\) be a positive complex-linear functional on a star-ordered complex algebra in which every element admits a decomposition into self-adjoint real and imaginary parts. If \(p=p^*=p^2\) and \(f(p)=f(1)\), then, for every \(X\),
In particular, let \(\varphi \) be a normalized positive norm-one state on the quasi-local algebra. For an interval projection \(p\) with \(\varphi (j_{a,N}(p))=1\), an interval observable \(X\), and \(c\in \mathbb C\),
No translation invariance is assumed.
Let \(\mathcal A\) be a star-ordered complex algebra admitting self-adjoint decomposition, and let \(f:\mathcal A\to \mathbb C\) be positive and complex linear. If \(f(C^*C)=0\), then \(f(YC^*C)=0\) for every \(Y\in \mathcal A\). This is the Cauchy–Schwarz implication used in the zero-energy argument of [ 225 , Section 6 ] .
Let \(f\) be a positive normalized complex-linear functional and \(p=p^*=p^2\). If \(f(H)=0\) and \(\kappa (1-p)\leq H\) for some \(\kappa {\gt}0\), then \(f(p)=1\). In particular, if \(H\geq 0\) is a finite-interval Hamiltonian with \(\ker H=G_N(A)\), then every normalized positive quasi-local state satisfying \(\varphi (j_{a,N}(H))=0\) also satisfies \(\varphi (j_{a,N}(p_A(N)))=1\). No uniform gap is required. This is the finite-dimensional support implication used in [ 225 , Section 3 ] .
Let \(0{\lt}R\leq W\leq N\), let \(h\) be positive, and suppose \(\kappa p_W(A)\leq H_W^{\mathrm{open}}(h)\). Then
If additionally \(\kappa ,\gamma {\gt}0\), both global kernels equal \(G_N(A)\), and \(\gamma \lVert v\rVert \leq \lVert H_{N,W}^{\mathrm{open}}(A)v\rVert \) for \(v\perp G_N(A)\), then
No local parent-kernel equality is assumed for \(h\). This is a finite-window consequence of condition C1 and the comparison argument in Section 6 of [ 225 ] .
Let \(A\) be a normalized irreducible periodic tensor of period \(m\), and let \(L{\gt}0\) be divisible by \(m\). There are positive-dimensional primitive tensors \(B_j\) on the \(L\)-site alphabet, faithful stationary matrices \(\rho _j\), cyclic projections \(P_j\), and isometries \(V_j\) such that
The tensors \(B_j\) are pairwise inequivalent up to similarity and phase. For every tail length \(r\), their literal support projections satisfy \(\sum _{|\tau |=r}A_\tau ^*V_jV_j^*A_\tau =P_{j-r}\).
Restricting the left boundary map to the prefix for a fixed suffix \(\tau \in \{ 0,\ldots ,d{-}1\} ^L\) recovers \(\Gamma _K(Z_\tau )\):
Let \(B=\bigoplus _{j=0}^{r-1}\mu _j A_j\), where \(r\geq 2\), all coefficients are nonzero, and the normalized primitive tensors \(A_j\) are pairwise inequivalent under similarity and phase. Assume positive physical and bond dimensions, positive definite recorded invariant densities, and a common block-injectivity length \(L_0\geq 1\). For every \(R\geq 3(r-1)(L_0+1)+1\), there exist \(\gamma _R{\gt}0\) and \(N_R\) such that
for every \(N\geq N_R\) and every \(v\perp \ker H_{R,N}(B)\). This removes the restriction to the range constructed by the overlap argument, as recorded in [ 131 ] .
Let \(B=\bigoplus _j\mu _jA_j\), where the \(A_j\) are normalized primitive tensors with common positive physical dimension, positive bond dimensions, and positive-definite invariant matrices. Suppose \(\mu _j\ne 0\) and no two blocks are related by similarity and multiplication by any nonzero complex scalar. For every \(0{\lt}\eta {\lt}1\) and every sufficiently large integer \(p\), the periodic range-\(2p\) parent Hamiltonian has gap at least \(1-\eta \) at every sufficiently large chain length.
Let \(B=\bigoplus _{j=0}^{r-1}\mu _j A_j\), where \(r\geq 2\), all coefficients are nonzero, and the normalized primitive tensors \(A_j\) are pairwise inequivalent under similarity and phase. Assume positive physical and bond dimensions, positive definite recorded invariant densities, and a common block-injectivity length \(L_0\geq 1\). For every \(R\geq 3(r-1)(L_0+1)+1\), there exists \(\gamma _R{\gt}0\) such that, for every \(N\in \mathbb N\) and every \(v\perp \ker H_{R,N}(B)\),
Let \(A\) be a normalized primitive tensor with positive bond dimension and a positive-definite invariant matrix. For every \(0{\lt}\eta {\lt}1\), every sufficiently large integer \(l\) has the following property. If \(m\ge l+1\), \(2l^2\le m\eta \), and \(N\ge 2m\), then
for every \(v\perp \ker H_{l+1,N}\). One may take \(m=\max \{ l+1,\lceil 2l^2/\eta \rceil \} \). The threshold for \(l\) depends on the tensor; no dimension-only bound on this threshold is asserted.
Let \(A\) be a normalized primitive tensor with positive bond dimension and a positive-definite invariant matrix. For every \(a{\gt}0\), every sufficiently large integer \(l\) satisfies \(l{\gt}1\), \(l\)-block injectivity, and a number \(\epsilon _l\ge 0\) with \(\epsilon _l\sqrt{l+1}{\lt}a\) such that, for every prefix length \(K\ge 0\),
Here \(G_I\) denotes the projection onto the matrix-product boundary space on the indicated interval, extended by the identity on its complement in the chain of length \(K+l+1\).
Let \(A_0,\ldots ,A_{r-1}\) be normalized primitive tensors with common physical dimension \(d\geq 1\), positive bond dimensions, and positive-definite invariant matrices. Suppose no two distinct blocks are related by a virtual similarity and a phase. Put \(B=\bigoplus _j\mu _jA_j\), with \(\mu _j\ne 0\) for every \(j\). For every \(0{\lt}\eta {\lt}1\), there is \(p_0\) such that, for every \(p\geq p_0\) and every \(N\geq 2p\),
This is a quantitative consequence of [ 225 , Theorem 2.1(ii) and Section 6 ] .
Under the hypotheses of Theorem 12.8.32, for every \(R_*\in \mathbb N\) there are \(p\geq \max \{ 1,R_*\} \) and \(\delta {\gt}0\) such that
for every chain length \(N\) and every \(v\perp \ker H^{\mathrm{open}}_{N,2p}(B)\). For \(N{\lt}2p\) the Hamiltonian is zero and the assertion is vacuous. This is the open-chain form of [ 225 , Theorem 1.2 ] .
Let \(A_0,\ldots ,A_{r-1}\) be normalized primitive tensors with common physical dimension \(d\geq 1\), positive bond dimensions, and positive-definite invariant matrices. Assume that no two distinct blocks are related by a virtual similarity and a phase. Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), with every \(\mu _j\neq 0\). For every \(R_*\in \mathbb N\), there are \(p\geq \max \{ 1,R_*\} \) and \(\gamma {\gt}0\) such that
for every \(M\geq 2\) and every \(v\perp \ker H^{\mathrm{open}}_{pM,2p}(B)\). This follows from the grouped martingale estimate of [ 225 , Theorem 2.1(ii) and Section 6 ] .
Let \(A_0,\ldots ,A_{r-1}\) be normalized primitive tensors with common physical dimension \(d\geq 1\), positive bond dimensions, and positive-definite invariant matrices. Assume that no two blocks are related by a virtual similarity and a phase. Let \(\mu _j\ne 0\) for every \(j\), and set \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\). For every prescribed lower bound \(R_*\), there exist \(R\geq \max \{ 1,R_*\} \), \(\gamma {\gt}0\), and \(N_0\) such that the periodic range-\(R\) parent Hamiltonian satisfies
for every \(N\geq N_0\) and every \(v\perp \ker H_{R,N}\). In particular, the block sum admits a parent interaction with a uniform positive periodic gap.
Let \(B=\bigoplus _j\mu _j A_j\), where the coefficients are nonzero and \((A_j)_j\) is a finite family of normalized primitive tensors of positive bond dimension, pairwise inequivalent under similarity and phase. Assume that each recorded invariant density is positive definite and that the physical dimension is positive. There exists \(R_0\geq 1\) such that, for every \(R\geq R_0\), there exist \(\gamma _R{\gt}0\) and \(N_R\) for which
for every \(N\geq N_R\) and every \(v\perp \ker H_{R,N}(B)\).
Let \(A\) be a normalized primitive tensor with positive-definite invariant matrix \(\rho \), and let \(X\) be a fixed observable with \(\eta _A(X)=0\). Consider a sequence \(N_n=\ell _n+k+r_n\) with both \(\ell _n,r_n\to \infty \). Suppose the operators \(H_n\) are eventually positive, have kernel equal to the open-boundary MPS space, and have a common norm gap \(\gamma \geq 0\) on its orthogonal complement. Let \(\psi _n\) be unit vectors eventually in that ground space. If
then \(\gamma \operatorname {Re}\eta _A(X^*X)\leq e\). The energy limit is an explicit hypothesis here; the passage to the infinite-volume state in [ 225 , Section 6 ] remains separate, as discussed in [ 61 ] .
Let \(A\) be a normalized primitive tensor with positive-definite invariant matrix \(\rho \). Let \(X\) act on \(k\) sites and let \(\ell ,r\to \infty \) along any filter. For any family of unit vectors \(\psi _N\) eventually belonging to the open-boundary MPS space on \(N=\ell +k+r\) sites, \(\| X_{\ell ,r}\psi _N\| ^2\to \operatorname {Re}\eta _A(X^*X)\).
Suppose in addition that \(\rho \) is positive definite. Let \(G_N\) be the open-boundary MPS space and \(P_N\) its orthogonal projection. Then
In particular, centered observables satisfy \(\| P_NX_{\ell ,r}P_N\| \to 0\). For every \(\varepsilon {\gt}0\), eventually \(\| P_NX_{\ell ,r}\psi -\eta _A(X)\psi \| {\lt}\varepsilon \) simultaneously for all \(\psi \in G_N\) with \(\| \psi \| \leq 1\).
With the preceding hypotheses, let \(\psi _N\) be any varying family of vectors eventually belonging to \(G_N\). If \(\| \psi _N\| \leq 1\) and \(\eta _A(X)=0\), then \(\| P_NX_{\ell ,r}\psi _N\| ^2\to 0\). If instead \(\| \psi _N\| =1\), then \(\langle \psi _N,X_{\ell ,r}\psi _N\rangle \to \eta _A(X)\). These assertions concern one primitive sector; the general pure-GVBS infinite-volume statement is discussed in [ 61 ] .
A one-site suffix increment needs no reassociation, so the open-chain projector defect at overlap length \(l\) and prefix length \(K\) equals the whole-increment defect with \(r=K\), \(m=l\), and \(\ell =1\). Consequently, for a primitive tensor with positive-definite fixed point there are \(c{\gt}0\), \(0{\lt}\lambda {\lt}1\), and an injective length \(L_0\geq 1\) such that, whenever \(l\geq L_0\) and \(c\lambda ^l{\lt}1\), for every prefix length \(K\),
This establishes the projector defect bound (132) for primitive matrix product states.
Let \(A\) be primitive with positive-definite fixed point \(\rho \) and nonzero bond dimension. There are constants \(c{\gt}0\) and \(0{\lt}\lambda {\lt}1\) and an injective length \(L_0\geq 1\) such that, whenever \(m\geq L_0\) and \(c\lambda ^m{\lt}1\), the bound (133) holds for every prefix length \(r\) and every \(\ell \geq 1\).
Let \(D{\gt}0\) and suppose \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). If \(\mathcal{E}_A\) is strongly irreducible in the sense of Definition 6.1.1.6, then there is a matrix \(\rho {\gt}0\) such that \((A,\rho )\) is primitive in the sense of Definition 6.6.1.
A finite family of pairwise gauge and phase inequivalent primitive tensors, with faithful invariant matrices, has independent boundary spaces \(G_{B_i}(N)\) for every sufficiently large \(N\). This is the consequence of the overlap estimates in [ 225 , Lemmas disjoint and overlapestimate ] .
Let \(A\) be a primitive MPS tensor whose fixed-point witness \(\rho \) is positive definite, and put \(\mathcal K_\infty =\mathscr R(P_\rho )\). For every \(0{\lt}a{\lt}1\), all sufficiently large \(n\) satisfy
Let \(A\) be a primitive MPS tensor with fixed-point witness \(\rho \). Then there are real constants \(C,r\) with \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that, for every \(n\geq 1\),
Under the same primitive-MPS and positive-definite fixed-point hypotheses,
in operator norm.
Under the same hypotheses, for every \(0{\lt}a{\lt}1\) and all sufficiently large \(n\), the physical boundary map \(\Gamma _n^{\mathrm{ES}}\) is injective and its inverse Gram operator is \(\mathcal K_n^{-1}\), satisfying
For the primitive family of the preceding theorem, one can choose a single \(R{\gt}0\) and \(h\geq 0\) such that the open kernels equal \(G_N(T)\) for every \(N\geq R\), and hence for all sufficiently large \(N\), and
Here \(\omega _j\) is the constructed quasi-local state of \(A_j\). The pure states in \(\mathcal F_h\) are precisely the states \(\omega _j\). Purity is extremality among all normalized positive quasi-local states; no translation invariance of a state being classified is assumed. This is the stated primitive-family case of the existence assertion in [ 225 , Theorem 1.1 and its finite-family consequence ] . More general periodic GVBS presentations remain separate [ 61 ] .
Let \(T=\bigoplus _{j=0}^{b-1}\mu _jA_j\), with nonzero weights, be a finite family of pairwise gauge-phase inequivalent trace-preserving primitive tensors of positive bond dimensions. Suppose each tensor has a faithful invariant matrix. There exist \(R{\gt}0\) and a positive interaction \(h\) on \(R\) sites such that, for every \(N\geq R\),
Neither \(R\) nor a simultaneous injectivity length is supplied. The inequality \(N\geq R\) is stronger than the inclusive-interval bound \(N\geq R+1\) in [ 225 , Theorem 1.1 ] .
Let \(\Gamma _n\) be the Hilbert-space boundary map and let \(\mathcal R\) denote Gram reshuffling. Then
Fix \(A\) and a virtual matrix \(\rho \) with nonzero trace. For each interval length \(k\), the functional \(\eta _A\) is complex linear. If \(\rho \geq 0\), then \(\eta _A(X)\geq 0\) for \(X\geq 0\). If \(E(\rho )=\rho \), then \(\eta _A(I_k)=1\) and adjoining identity factors on the right preserves expectations. If \(\sum _i(A^i)^*A^i=I\), adjoining identity factors on the left preserves expectations. Thus a normalized primitive tensor with its positive invariant matrix defines compatible states on finite intervals, as in [ 225 , Section 3, equations (3.1)–(3.2b) ] .
- MPSTensor.observableInsertionExpectation_one
- MPSTensor.observableInsertionExpectation_nonneg
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- MPSTensor.observableInsertionExpectation_smul
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- MPSTensor.observableInsertionExpectation_one_appendObservable
- MPSTensor.IsPrimitiveMPS.observableInsertionExpectation_one
- MPSTensor.IsPrimitiveMPS.observableInsertionExpectation_nonneg
For any tensor \(A\), positive observable \(X\) on \(k\) sites, and positive virtual matrix \(\rho \), the insertion \(E_X(\rho )\) is positive. This is the positivity of the local GVBS contraction in [ 225 , Section 3, equations (3.1)–(3.2b) ] .
Let \(\rho \geq 0\) satisfy \(E_A(\rho )=\rho \) and \(\operatorname {tr}\rho \neq 0\). For each finite interval, the insertion expectation \(\eta _A\) is a positive continuous linear functional of operator norm one. Consequently
The adjoint identity requires only \(\rho \geq 0\); the norm-one assertion also uses invariance and nonzero trace. Primitivity is unnecessary. This is the finite local state of [ 225 , Section 3, equations (3.1)–(3.2b) ] .
- MPSTensor.observableInsertionExpectationₗ
- MPSTensor.observableInsertionExpectationPositive
- MPSTensor.observableInsertionExpectationPositive_apply
- MPSTensor.norm_observableInsertionExpectationPositive
- MPSTensor.norm_observableInsertionExpectation_le
- MPSTensor.observableInsertionExpectation_conjTranspose
Let \(A\) be a normalized primitive tensor with invariant matrix \(\rho \), and let \(P_\rho (Y)=\operatorname {tr}(Y)\rho /\operatorname {tr}\rho \). For every local observable \(X\), \(P_\rho E_XP_\rho =\eta _A(X)P_\rho \) and \(E^n\to P_\rho \) in operator norm. Along any filter on which \(\ell ,r\to \infty \), \(E^\ell E_XE^r\to \eta _A(X)P_\rho \).
For a finite tensor family, eventual independence of the middle ground spaces and the individual eventual one-step intersection identities imply the eventual joint one-step intersection identity. In particular, this applies to any finite family of pairwise gauge and phase inequivalent normalized primitive tensors with positive bond dimensions and faithful invariant matrices. Thus, writing \(\mathcal G(q)=\sum _j\mathcal G_j(q)\), for all sufficiently large \(n\),
The family may be empty. The primitive assertion concerns the supplied generating tensors, rather than an arbitrary GVBS presentation.
Suppose finitely many pairwise inequivalent primitive tensors \(A_j\) have faithful invariant matrices, and let \(T=\bigoplus _j\mu _j A_j\) with every \(\mu _j\ne 0\). Let \(\omega _j\) be their constructed states. A state \(\varphi \) satisfies
if and only if
Among these supported states, the pure states are precisely the \(\omega _j\). No translation invariance of \(\varphi \) is assumed. This is the primitive-family support classification used in [ 225 , Theorem 1.1 and Section 3 ] ; identification of a general periodic GVBS presentation remains separate.
Let \(A\) be a normalized primitive tensor with positive-definite invariant matrix \(\rho \), let \(h\) act on \(R{\gt}0\) sites, and let \(X\) act on \(k{\gt}0\) sites. Put \(t=R-1\), \(w=2t+k\), \(B=I_t\otimes X\otimes I_t\), and \(Y=B^*[H_w(h),B]\). On chains of length \(N_n=2n+w\), place \(X\) at start \(n+t\). For any eventually unit vectors \(\psi _n\) in the open-boundary MPS space, \(\operatorname {Re}\langle X_n^*[H_{N_n}(h),X_n]\psi _n,\psi _n\rangle \to \operatorname {Re}\eta _A(Y)\).
Let \(A\) be a normalized primitive tensor with positive-definite invariant matrix \(\rho \), and let \(X\) act on \(k{\gt}0\) sites with \(\eta _A(X)=0\). Fix \(R{\gt}0\) and let \(h=I-P_{G_R(A)}\). Suppose that, on the expanding chains \(N_n=2n+2(R-1)+k\), the canonical open Hamiltonians eventually have kernel \(G_{N_n}(A)\) and a common norm gap \(\gamma \geq 0\). Suppose also that a family of unit vectors eventually belongs to these ground spaces. With \(B=I_{R-1}\otimes X\otimes I_{R-1}\),
This is a statement about finite-interval primitive-sector expectations. Their extension to the quasi-local algebra and the GNS Hamiltonian remains separate, as discussed in [ 61 ] .
Let \(A\) be a normalized primitive tensor of bond dimension \(D{\gt}0\) with positive-definite invariant matrix \(\rho \). At each fixed range \(R\geq D^4+1\), there exists \(\gamma {\gt}0\) such that the preceding local commutator inequality holds for every observable on every nonempty finite interval with \(\eta _A(X)=0\). The interaction is \(h=I-P_{G_R(A)}\).
Let \(A\) be a normalized primitive tensor with positive-definite invariant matrix \(\rho \), let \(h\geq 0\) act on \(R{\gt}0\) sites, and let \(X\) act on \(k{\gt}0\) sites with \(\eta _A(X)=0\). Suppose the open interaction sums on chains \(N_n=2n+2(R-1)+k\) eventually have kernel \(G_{N_n}(A)\) and common norm gap \(\gamma \geq 0\). Suppose that a family of unit vectors eventually belongs to these ground spaces. With \(B=I_{R-1}\otimes X\otimes I_{R-1}\),
Let \(A\) be a normalized primitive tensor of bond dimension \(D{\gt}0\) with positive-definite invariant matrix \(\rho \). Let \(h\geq 0\) act on \(R\geq D^4+1\) sites with kernel \(G_R(A)\). There exists \(\gamma {\gt}0\) such that the preceding local commutator bound holds for every observable on every nonempty finite interval with \(\eta _A(X)=0\). This concerns one primitive sector and its finite-interval expectations; the quasi-local, GNS, and pure multiblock conclusions remain separate [ 61 ] .
Let \(T=\bigoplus _{j=0}^{b-1}\mu _jA_j\), with \(\mu _j\ne 0\), be a finite weighted sum of inequivalent trace-preserving primitive tensors with faithful invariant density matrices. Write \(\omega _j\) for their constructed quasi-local states. Let \(h\geq 0\) have range \(R\geq 1\), and suppose \(\ker H_N^{\mathrm{open}}(h)=G_N(T)\) for all sufficiently large \(N\). Then, for every normalized positive state \(\varphi \),
No translation invariance of \(\varphi \) is assumed. This is the ground-state classification associated with [ 225 , Theorems 1.1–1.2 ] for the stated primitive sector family. More general periodic presentations remain separate [ 61 ] .
Let \((B_{j,a})_a\) be finitely many finite families of primitive tensors of positive bond dimensions, with faithful invariant matrices. There is a finite pairwise gauge-phase inequivalent primitive faithful family \((C_b)_b\) such that, for every \(N\geq 0\),
Any of the families may be empty. This is representative selection in [ 13 , Proposition 2.7 ] applied to a finite union of primitive families.
Let \(A\) be a primitive MPS tensor with nonzero bond dimension and a positive-definite fixed point. There exist \(l{\gt}1\) and \(\epsilon _l\geq 0\) such that \(A\) is \(l\)-block injective,
and, for every \(N\geq l+1\) and every \(v\in (G_N^{\mathrm{ES}}(A))^\perp \),
Let \(A\) be a primitive MPS tensor with nonzero bond dimension and a positive-definite fixed point. There exist \(l{\gt}1\), \(\epsilon _l\geq 0\), an integer \(m\geq l+1\), and \(\delta {\gt}0\) such that \(A\) is \(l\)-block injective and
Set
The integer \(m\) may be chosen so that
and
For every \(N\geq 2m\) and every \(v\in (\ker H_{N,l+1}^{\mathrm{ES}}(A))^\perp \),
The existence of a uniform MPS parent-Hamiltonian gap and the nearest-neighbor Knabe principle are stated in [ 228 , lines 1460–1500 ] and [ 17 , lines 2184–2188,2194–2197 ] . Those passages do not state the arbitrary finite-range coefficient (316).
Let \(d\ge 1\), let \((A_j)\) be a finite family of pairwise gauge and phase inequivalent normalized primitive tensors with positive bond dimensions and faithful invariant matrices, and let \(\mu _j\ne 0\). For \(T=\bigoplus _j\mu _jA_j\), there exists \(R_0{\gt}0\) such that
In particular, the canonical local projection of range \(R_0\) is a positive parent interaction with this exact kernel at every \(N\ge R_0\). No simultaneous word span is supplied in this assertion or used in the family intersection argument.
Let \(d,D{\gt}0\), and let \(A\) be a normalized primitive tensor with a positive-definite invariant matrix \(\rho \). Suppose that \(h\geq 0\) has kernel \(G_R(A)\) and \(R\geq D^4+1\). There is a constant \(\gamma {\gt}0\) such that, for every \(a\in \mathbb Z\), \(k\geq 1\), and \(X\in M_{d^k}(\mathbb C)\) with \(\omega _A(j_{a,k}(X))=0\),
where \(t=R-1\), \(w=2t+k\), \(B=I_t\otimes X\otimes I_t\), and \(Y=B^*[H_w(h),B]\). The constant is independent of \(a\), \(k\), and \(X\). This is the local commutator inequality in the constructed primitive quasi-local state, at sufficient range. Identification of all pure multiblock GVBS states and construction of the GNS Hamiltonian remain separate from this consequence of [ 225 , Theorem 1.2 and Section 6 ] .
Let \(d,D{\gt}0\), let \(A\) be a trace-preserving primitive tensor, and let \(\rho {\gt}0\) be its invariant virtual matrix. The completed MPS state \(\omega _A\) is pure among all states of the quasi-local algebra. The states in a proposed convex decomposition need not be translation invariant. This proves the primitive canonical case of the purity assertion in [ 225 , Section 3 ] ; no classification of the multiblock ground-state face is asserted here.
Gauge-equivalent MPS tensors, including a common nonzero scalar factor on every physical matrix, have the same open-boundary MPS space at every length. More generally, normalized normal tensors whose periodic vectors differ by powers of one nonzero scalar have identical open-boundary spaces at every length, even when their bond dimensions are initially allowed to differ. The latter assertion uses the dimension equality and invertible gauge supplied by the fundamental theorem [ 13 , Proposition 2.7, Theorem 2.10 ] .
Let \((A_j)_{j=0}^{r-1}\) be a finite family of normalized primitive tensors of positive bond dimensions, with faithful fixed-point witnesses \(\rho _j\). Let \(\mu _j\neq 0\). There is an injective selection of original indices \(s:\{ 0,\ldots ,g-1\} \to \{ 0,\ldots ,r-1\} \) such that the tensors \(B_k=A_{s(k)}\) retain the supplied primitive and faithful witnesses, and distinct \(B_k\) are not gauge-phase equivalent after any equality of their bond dimensions. Every \(A_j\) is gauge-phase equivalent to one \(B_k\), with the required bond-dimension equality supplied. For every \(N\geq 0\),
The finite family may be empty. This is the exact local-support consequence of selecting phase-class representatives in [ 13 , Proposition 2.7 ] , used for the inequivalent primitive sectors of [ 225 , Section 3 ] .
Let \(A\) be a normalized primitive tensor of positive bond dimension with stationary matrix \(\rho \). If \(\rho \) is positive definite, then blocking by any positive length retains primitivity with the same stationary matrix \(\rho \). A bijective relabelling of the physical letters also preserves the primitive witness with the same \(\rho \); this second assertion requires only the original primitive fixed-point data.
Let \(A\) be a trace-preserving primitive MPS tensor of positive physical and bond dimensions, with positive definite invariant matrix \(\rho \). Let \(\omega _A\) be its constructed quasi-local state. Suppose that \(\varphi \) is any normalized positive norm-one continuous complex-linear state on the quasi-local algebra and that
Then every interval observable has the prescribed insertion expectation:
and consequently \(\varphi =\omega _A\). The competing state need not be translation invariant. This uniqueness statement is a consequence of primitive boundary convergence, as used in [ 225 , Sections 3 and 6 ] ; purity is a separate consequence of uniqueness and inheritance of support in convex decompositions.
Let \(A\) be a primitive MPS tensor with nonzero virtual dimension, and let \(J_K=J^{\mathrm{tail}}_K\) be its length-\(K\) tail virtual map. Then there is a constant \(C{\gt}0\) such that, for every \(K\in \mathbb {N}\),
No positive-definiteness hypothesis on the fixed point is required.
Let \(A\) be a normalized primitive tensor with nonzero bond dimension and a positive-definite fixed point. For every \(\varepsilon {\gt}0\), all sufficiently large \(L\) and every \(K\geq 0\), \(Q{\gt}0\) satisfy
This is the uniform decay of the individual-block errors used in [ 225 , Section 6 ] .
Under the convention that an \(L\)-site interaction vanishes for \(L{\gt}N\), the two-site parent Hamiltonian on a one-site periodic chain satisfies
Let \(P\) and \(Q\) be orthogonal projections on a real or complex Hilbert space. For every vector \(v\),
Suppose \(A\) has invariant-projector closure. Then there are isometries \(V_1,\ldots ,V_r\) with \(V_k^\dagger V_k=\mathbb {1}\), pairwise orthogonal ranges, and \(\sum _kV_kV_k^\dagger =\mathbb {1}\), such that \(A^iV_k=V_kA_k^i\) for the corner tensors \(A_k^i=V_k^\dagger A^iV_k\), every corner tensor \(A_k\) is irreducible, and \(\mathcal{V}(A)=\mathcal{V}(\bigoplus _{k=1}^rA_k)\).
Let \(P_U\), \(P_V\), and \(P_{U\cap V}\) be the orthogonal projections onto \(U\), \(V\), and their intersection. If
then the Friedrichs overlap of \(U\) and \(V\) is at most \(\eta \).
For \(d{\gt}0\), \(L{\gt}0\), \(a\in \mathbb Z\), \(N\geq 0\) and \(X\in M_{(d^L)^N}(\mathbb C)\),
The functional \(T_L\omega \) has the same norm as \(\omega \), is a state exactly when \(\omega \) is a state, and is pure exactly when \(\omega \) is pure. This applies to all state functionals, without an invariance condition under translations.
For every algebraic local observable \(A\), one has \(\omega (j(A))=f_{\mathrm{loc}}(A)\). For every finite region \(\Lambda \) and \(X\in \mathcal A_\Lambda \), one has \(\omega (j_\Lambda (X))=f_\Lambda (X)\).
Every compatible family of finite-region states, with \(d{\gt}0\), has a unique continuous complex-linear extension \(\omega \colon \mathcal A\to \mathbb {C}\) with the prescribed finite-region values. It satisfies \(\omega (I)=1\), \(\lVert \omega \rVert =1\), and, for every \(A\in \mathcal A\), \(|\omega (A)|\leq \lVert A\rVert \) and \(\omega (A^\dagger A)\geq 0\). Compatibility of the finite-region functionals is an explicit hypothesis.
- SpinChain.CompatibleLocalState.quasiLocalFunctional_nonneg_star_mul_self
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Let \(\omega \) be a state, and let \((P_i)_{i\in J}\) be any family of self-adjoint projections with \(\omega (P_i)=1\). Suppose every state \(\phi \) satisfying \(\phi (P_i)=1\) for every \(i\in J\) equals \(\omega \). Then \(\omega \) is pure.
Let \(\phi ,\psi \) be states, let \(0{\lt}t{\lt}1\), and set \(\omega =(1-t)\phi +t\psi \). If \(\omega (X^*X)=0\), then \(\phi (X^*X)=0\). If \(\omega \) is a state, \(P=P^*=P^2\), and \(\omega (P)=1\), then \(\phi (P)=1\).
Let \(j_{a,k}\) denote inclusion of the interval \([a,a+k)\) into the quasi-local algebra. Then, including empty intervals,
For \(k{\gt}0\), the nonwrapping inclusion of \(X\) at position \(b\) in a chain of length \(N\), where \(b+k\leq N\), has quasi-local image \(j_{a+b,k}(X)\) when the chain begins at \(a\).
Under the preceding assumptions, if \(\ell ,r\to \infty \) along any common filter, the quasi-local energy is eventually equal to \(j_{a-t,w}(Y)\) and hence converges to it in norm. Consequently every functional \(\omega \) on the quasi-local algebra satisfies
Continuity is unnecessary here because the observable is eventually constant.
Let \(R,k{\gt}0\), \(t=R-1\), \(w=k+2t\), and let \(h\) be an arbitrary \(R\)-site matrix. Write \(H_N\) for its open-chain sum and \(X_N=I_\ell \otimes X\otimes I_r\), with \(\ell ,r\geq t\) and \(N=\ell +k+r\). Let
Then
Equivalently, putting \(x=j_{a,k}(X)\) and \(\widehat H_N=j_{a-\ell ,N}(H_N)\) gives \(x^*[\widehat H_N,x]=j_{a-t,w}(Y)\). Neither positivity nor a state assumption is required.
For \(d{\gt}0\) and \(a\in \mathbb Z\), coordinate transport and the finite-region inclusion give a star-algebra inclusion \(j_{a,k}:M_{d^k}(\mathbb C)\to \mathcal A\), including \(k=0\). It preserves the matrix operator norm: \(\| j_{a,k}(X)\| =\| X\| \). This is the interval observable convention in [ 225 , Section 3 ] .
Let \(d{\gt}0\), let \(A,B\) be trace-preserving tensors with possibly different bond dimensions, and let \(\rho ,\sigma \geq 0\) be stationary matrices of nonzero trace. If \(\eta _{A,\rho }(X)=\eta _{B,\sigma }(X)\) for every observable at every finite length, then their continuous quasi-local expectation functionals satisfy \(\omega _{A,\rho }=\omega _{B,\sigma }\). Faithfulness and purity are unnecessary. This concerns the generating expectations in [ 225 , Section 3 ] .
Let \(h\) be an \(R\)-site matrix, where \(R{\gt}0\), and let \(H_N\) be its open-chain sum. For every interval position \(a\) and every \(N\geq 0\),
with the sum understood as empty when \(N{\lt}R\). Consequently, any complex-linear functional \(\varphi \) satisfying \(\varphi (j_{a,R}(h))=0\) for every \(a\in \mathbb Z\) also satisfies
for every \(a\) and \(N\). This applies in particular to arbitrary continuous states, without requiring translation invariance.
The set \(\mathcal F_h\) is real convex for every local observable \(h\). If \(h\ge 0\), it is a face of the full state space: whenever \(\omega =(1-t)\phi +t\psi \in \mathcal F_h\), with \(0{\lt}t{\lt}1\) and \(\phi ,\psi \) states, both constituents belong to \(\mathcal F_h\).
Let \(A=\bigoplus _\alpha \mu _\alpha A_\alpha \) have nonzero weights and simultaneous injectivity length \(S{\gt}0\). Let \(h\geq 0\) have \(\ker h=G_R(A)\) at range \(R\geq S+1\). If a normalized positive quasi-local state \(\varphi \) satisfies \(\varphi (j_{a,R}(h))=0\) for every \(a\in \mathbb Z\), then
Consequently the same support constraint holds eventually on the symmetric intervals \([-n,n+1)\). The state need not be translation invariant, and primitive normalization of the original blocks is unnecessary. This is the local support consequence of the zero-energy condition in [ 225 , Theorem 1.1 ] , at the stated sufficient range.
Let \(\varphi \) be a continuous complex-linear functional that is nonnegative on squares, and let \(h\geq 0\) have range \(R{\gt}0\). If \(\varphi \) vanishes on every translate of \(h\), then every stabilized local commutator energy has real, nonnegative expectation. Normalization and translation invariance are not required.
Let \(h\geq 0\) have positive finite interaction range. Suppose that \(\ker H_N(h)=G_N(A)\) for every sufficiently large \(N\). If a normalized positive quasi-local state \(\varphi \) has \(\varphi (j_{a,R}(h))=0\) for every \(a\in \mathbb Z\), then, for every sufficiently large \(n\),
This uses the eventual kernel hypothesis in [ 225 , Theorem 1.2 ] directly. It imposes no local parent-kernel equality and no lower bound on the interaction range in terms of injectivity.
Let \(K,L,Q\in \mathbb {N}\) and let \(Y\colon (\{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} )\to M_{D}(\mathbb {C})\) be a boundary family. For all \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \), \(w\colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \), and \(v\colon \{ 0,\ldots ,Q{-}1\} \to \{ 0,\ldots ,d{-}1\} \),
For residual boundary vectors \(\Gamma _{N,r}^{A,B,V}(Y)\) and \(\Gamma _{N,r}^{A,B,V}(Z)\), one has
This includes zero prefix and tail lengths and requires no normalization or letter intertwining.
Let \(C_rY=(V^\dagger A^\tau Y)_\tau \) and suppose that \(Q_r=\sum _\tau A^{\tau \dagger }VV^\dagger A^\tau \) is an orthogonal projection. Write \(\mathcal Q_rY=Q_rY\). For every prefix length \(N\),
The last identity requires \(\operatorname{tr}\rho \ne 0\). No positivity or injectivity of the prefix tensor is required for these identities.
The common physical map is an isometry, \(V^*V=1\). Moreover, the equal-sector bond embedding and the physical support projector satisfy
and \(P_{2,\mathrm{res}} = 1-q_2\! \left(\bigoplus _j B_j\right)\). If the \(B_j\) form a basis of normal tensors and their direct sum is a renormalization fixed point, the sectorwise Appendix B decompositions and the common residual isometry required above exist.
- MPSTensor.IsResidualIsometryFamily.residualFamilyPhysicalIsometryMatrix_isometry
- MPSTensor.ResidualFamilyAppendixBData.twoSiteBondEmbedding_range
- MPSTensor.ResidualFamilyAppendixBData.twoSiteBondEmbedding_range_eq_directSum_groundSpace
- MPSTensor.ResidualFamilyAppendixBData.twoSiteBondSupportProjection_range
- MPSTensor.ResidualFamilyAppendixBData.twoSiteBondSupportProjection_eq_complement
- MPSTensor.IsBNTCanonicalForm.exists_residualFamilyAppendixBData
Physical reindexing identifies the full window map \(C\) with the residual boundary map on \(K+M\) blocked sites and an original tail of length \(r\). Consequently,
This holds for all lengths, including zero, without injectivity or normalization hypotheses.
Let \(\mathcal V=\bigsqcup _j\{ j\} \times [D_j]\times [D_j]\) be the disjoint union of the within-sector virtual pairs. For \(x=(j,\alpha ,\beta )\) in \(\mathcal V\), set \(u_x(i)=(U_j^i)_{\alpha ,\beta }\). Then \(\sum _i u_x(i)\overline{u_y(i)}=\delta _{x,y}\).
For two inequivalent normalized primitive blocked-prefix tensors of positive bond dimensions with faithful invariant matrices, their correlated residual ranges become uniformly orthogonal as \(N\to \infty \). The estimate is uniform in the original tail length \(r\); the original tensors supplying the tails need no normalization assumption.
Let \(B_j\) be a finite family of pairwise gauge-phase inequivalent normalized primitive blocked-prefix tensors of positive bond dimensions with faithful invariant matrices. Let \(F_j(N,r)\) be their correlated residual ranges. Along any filter for which \(N\to \infty \), and with arbitrary varying \(r\), one has
Assume \(D,E{\gt}0\), \(\rho {\gt}0\), and that \(Q_r\) is an orthogonal projection equal to the exact tail Gram matrix. Let \(\delta _N=\| K_N^B-K_\rho \| \) and \(u=\| R_\rho ^{-1}\| \). If \(u\delta _N\leq a{\lt}1\), then \(F_{N,r}\) is invertible, \(H_{N,r}\) is self-adjoint, and
Without the small-error condition, \(\| \Gamma _{N,r}\| ^2\leq \| R_\rho \| +\delta _N\). These are intermediate estimates for [ 225 , Lemma commutation(ii) ] , rather than an assertion of the three-interval projection estimate.
Assume \(D{\gt}0\), and let the positive-dimensional prefix tensor \(B\) be primitive with faithful invariant matrix \(\rho \). Suppose that every exact tail Gram matrix \(Q_r\) is an orthogonal projection. For all sufficiently large prefix lengths, uniformly over every \(r\geq 0\), the preceding inverse and support equations hold and
Whenever \(u\delta _N\leq \tfrac 12\),
Consequently this inverse error tends to zero along every diverging prefix length, with arbitrary tail lengths. Under the cyclic hypotheses of Theorem 12.9.194, the exact projections are \(Q_r=P_{a+r}\); no restriction on the tail length is needed.
Under the hypotheses of Theorem 12.9.223, define \(D_M=T^*L_1-K_r^{\oplus u}J_1I_rC_r^*G_\infty ^{\oplus \tau }\). Then
This is an exact finite-volume identity. A quantitative bound or a limiting projection estimate requires further hypotheses on the prefix Gram.
Suppose \(B_i\) and \(B_j\) are inequivalent normalized primitive tensors with positive bond dimensions and faithful invariant matrices. For every \(\varepsilon {\gt}0\), all sufficiently large middle lengths \(M\) satisfy, for every \(K,r\geq 0\),
Thus the projection product tends to zero along every diverging middle length, while both outer lengths may vary arbitrarily. The original tensor \(A\) and support matrix \(V_i\) require no normalization or intertwining hypothesis for this conclusion. This is the cross-sector part of the projection comparison in [ 225 , Lemmas disjoint and commutation(ii) ] .
Let \(T_i\) be the right residual window formed from \(A,B_i,V_i\), and let \(L_j\) be the left window formed from \(B_j\), in the coordinates of Definition 12.9.197. Their middle interval has length \(M\), the outer blocked prefix has length \(K\), and the original tail has length \(r\). For rectangular boundary fibers \(x_u\) and square boundary fibers \(y_\tau \),
The two sector bond dimensions may differ, and every length may be zero. If \(\varepsilon \geq 0\) and \(|\langle v,w\rangle |\leq \varepsilon \| v\| \| w\| \) for every \(v\in \mathcal G_M(B_i)\) and \(w\in \mathcal G_M(B_j)\), then
No normalization, injectivity, intertwining, or boundary inverse is assumed.
With the coordinates of Definition 12.9.197, put \((J_2Y)_\tau =V^*A^\tau Y\). Then \(TJ_1=C=L_1J_2\). \(C\) is the residual boundary map at prefix length \(K+M\) and tail length \(r\), transported by the physical configuration bijection. No injectivity, normalization, or intertwining hypothesis is required.
Suppose \(\sum _jV_jV_j^*=1\) and \(V_j^*A^{(L)}_i=(B_j)_iV_j^*\). Let \(U\) be the physical configuration isometry from the original \((K+M)L+r\) sites to the three-window coordinates. Then \(\sum _j\operatorname {ran}C_j(K,M,r)=U G_A((K+M)L+r)\). This includes zero lengths and requires no primitivity or injectivity.
Under the virtual resolution and blocked cointertwining hypotheses of Theorem 12.9.237, let \(U\) be the configuration isometry from the original \(KL+ML+r\) sites to the physical cover coordinates. Then \(\sum _j\operatorname {ran}C_j(K,M,r)=U G_A(KL+ML+r)\). The statement includes zero lengths and requires no primitivity or injectivity.
Let \(B_j\) be a fixed finite family of pairwise inequivalent normalized primitive tensors with positive bond dimensions and faithful invariant matrices. Let \(A\) and the rectangular matrices \(V_j\) be arbitrary. For the full residual window maps \(C_j(K,M,r)\), if \(M\) tends to infinity along an arbitrary filter, then
The prefix \(K\) and tail \(r\) may vary arbitrarily. No normalization or intertwining is required for \(A\) or \(V_j\). This is the full-window sector sum in the projection comparison of [ 225 , Lemma commutation(i) ] .
Let \(\rho \in M_{E}(\mathbb {C})\) have nonzero trace, put \(G_\infty =\operatorname {gramReshuffle}(\mathcal P_\rho )\), and let \(K_r=C_r^*G_\infty ^{\oplus \tau }C_r\). If \(\lVert C_r\rVert \leq 1\), then
The second bound does not require the contractivity assumption. Neither bound requires primitivity or a fixed-point equation. Both coefficients are one and are independent of the spectator lengths.
Use the residual windows \(T,L_1\) at full length \((K+M)L+r\) from Definition 12.9.197. Write \(\Gamma _{M,r}\) for the correlated residual boundary map and \(G_B(n)\) for the ordinary blocked-prefix Gram. Then
Consequently \(\lVert T\rVert \leq \lVert \Gamma _{M,r}\rVert \) and \(\lVert L_1\rVert \leq \lVert \Gamma _B(K+M)\rVert \). These identities require no normalization or injectivity and include zero lengths and empty configuration sets.
Let \(D{\gt}0\), and let \((B_j)_{j\in J}\) be a finite family of positive-dimensional primitive tensors with faithful invariant matrices, pairwise inequivalent up to similarity and phase. Let \(T_j,L_j,C_j\) be the right, left, and full window maps arising from an original tensor and rectangular embeddings \(V_j\). Suppose that each correlated-tail Gram is an orthogonal projection at every tail length. Write \(\mathcal T=\sum _j\operatorname {ran}T_j\), \(\mathcal L=\sum _j\operatorname {ran}L_j\) and \(\mathcal C=\sum _j\operatorname {ran}C_j\). Along any filter with \(M\to \infty \) and arbitrary exterior lengths \(K,r\),
Let \(L_B(K,M,r)\) denote the left residual window of Definition 12.9.197. The coordinate bijection of Definition 12.9.231,
splits the blocked configuration into lengths \(K,M\) and moves the original tail to the last coordinate. If \(U_{K,M,r}\) is the associated unitary permutation, then
All lengths may be zero. No normalization or injectivity is assumed.
Under the isometric virtual resolution and blocked letter intertwining of Theorem 12.9.245, let \(U^{\mathrm{cover}}_{K,M,r}\) be the unitary from original configurations of length \(KL+ML+r\) to the three-window coordinates of Definition 12.9.243. Then
All lengths may be zero. This is the same physical coordinate isometry used for the right and full windows; no additional tensor hypothesis is imposed. The equality supplies the aligned prefix space in [ 225 , Lemma commutation(ii) ] .
Let \(A\) be an MPS tensor and let \(U_{K,M,r}\) be the unitary from original configurations to the three-window coordinates of Definition 12.9.197. Then
Suppose a family \((B_j)_j\) presents the blocked prefix ground space exactly:
Then its joint left-window range is the original aligned prefix ground space, with the tail unconstrained:
In particular, this equality follows for a finite family with an isometric virtual resolution satisfying
All lengths may be zero. No primitivity, normalization, or injectivity is assumed. This is the exact left-window identification used in the three-interval comparison of [ 225 , Lemma commutation(ii) ] .
For every finite sector family \((B_j)_j\) with possibly different bond dimensions,
This comparison includes empty families and requires no normalization. Suppose in addition that the sectors are pairwise inequivalent normalized primitive tensors with positive bond dimensions and faithful invariant matrices. Then for every \(\varepsilon {\gt}0\), all sufficiently large \(M\) satisfy, for every \(K,r\geq 0\),
Consequently the left-window projection-sum error tends to zero along every diverging middle interval, with arbitrary outer-prefix and original-tail lengths. This is the aligned left-sector comparison for [ 225 , Lemma commutation(i) ] .
Let the virtual dimension \(E\) be positive and let \(\rho \in M_{E}(\mathbb {C})\) be positive definite. Put \(G_\infty =\operatorname {gramReshuffle}(\mathcal P_\rho )\) and \(R_\rho Y=Y\rho /\operatorname{tr}\rho \). For the correlated tail map \(C_r\),
Suppose the exact tail Gram matrix is an orthogonal projection \(Q\), so that \(C_r^*C_r=\mathcal Q\) with \(\mathcal QY=QY\). Set \(K_r=C_r^*G_\infty ^{\oplus \tau }C_r\) and \(I_r=\mathcal Q R_\rho ^{-1}\). With the residual window coordinates,
No transfer fixed-point equation, injectivity, or normalization is required for these identities.
In the same coordinates, let \(G_B(M)=\Gamma _B(M)^*\Gamma _B(M)\) be the ordinary prefix Gram. With the Frobenius inner product on virtual matrices,
This identity is exact at every nonnegative length; it asserts no limiting metric or quantitative decay.
Let \(D,E{\gt}0\), and let the blocked tensor \(B\) be primitive with a faithful invariant matrix \(\rho \). Suppose the correlated tail Gram is an orthogonal projection \(Q_r\) for every tail length \(r\). Along any filter with \(M\to \infty \), and with arbitrary \(K,r\),
This is the same-sector mixed-Gram estimate used in commutation (ii), equation (3.15), of [ 225 ] ; the algebraic row-support hypotheses are stated explicitly here.
The fibers are the matrices in Definition 12.9.234, and \(\| \mathcal A\| \le \| C_r\| \), \(\| \mathcal B\| \le \| J_1\| \). Under the hypotheses of Theorem 12.9.223,
These are finite-dimensional estimates underlying the mixed-pairing step of Nachtergaele’s commutation lemma [ 225 ] ; no transfer normalization or injectivity is assumed here.
- MPSTensor.boundaryFamilyEquiv_residualWindowLeftOverlapFactorES_apply
- MPSTensor.boundaryFamilyEquiv_residualWindowRightOverlapFactorES_apply
- MPSTensor.norm_residualWindowLeftOverlapFactorES_le
- MPSTensor.norm_residualWindowRightOverlapFactorES_le
- MPSTensor.residualWindow_centeredMixedGram_eq_overlapFactors
- MPSTensor.norm_residualWindow_centeredMixedGram_le
Let \(D,E{\gt}0\), and let \(B\) be primitive with a faithful invariant matrix \(\rho \). Suppose the exact correlated-tail Gram is an orthogonal projection \(Q_r\) for every \(r\). Along any filter with \(M\to \infty \) and arbitrary exterior lengths \(K,r\),
This is the same-sector part of commutation (ii) in [ 225 ] ; the tail support is retained explicitly.
Let \(\Gamma _{B,V}(M,r)\) be the residual boundary map and let \(U_{M,r}\) be physical reindexing through \(e_{M,r}\). The right window map is the finite direct sum of \(U_{M,r}\Gamma _{B,V}(M,r)\) over the blocked prefix configurations \(u\) of length \(K\). No normalization or intertwining hypothesis is required.
Let \(A\), \(B_j\), and \(V_j\) satisfy \(\sum _jV_jV_j^*=1\) and \(V_j^*A^{(L)}_i=(B_j)_iV_j^*\). Let \(U\) be the configuration isometry from the original \(KL+ML+r\) sites to the physical cover coordinates. Then
Here the terminal boundary map leaves the first \(KL\) sites free and places the original ground space on the last \(ML+r\) sites. No primitivity, injectivity, or isometry of the virtual embeddings is required. This is the right joint-window identification in the projection comparison of [ 225 , Lemma commutation(ii) ] .
Let \((B_j)_j\) be a finite family of pairwise gauge-phase inequivalent primitive tensors with faithful invariant densities, and let \(V_j\) be arbitrary rectangular embeddings. Write \(S_j(K,M,r)\) for the range of the right window map of sector \(j\). As \(M\to \infty \),
uniformly in arbitrary varying prefix lengths \(K\) and original-tail lengths \(r\). The assertion holds along any filter on which \(M\) tends to infinity. Residual-sector projection-sum decay also persists under any varying physical configuration isometry before adding the spectator. This is the right-window comparison in [ 225 , Lemma 6.1(i) ] .
Let \(T,L_1,C\) denote the right, left, and full window maps. Suppose that \(Q_r\) is the exact correlated-tail Gram projection, that the right and full supported Gram inverses satisfy \(G_{M,r}H_{M,r}=Q_r\) and \(G_{K+M,r}H_{K+M,r}=Q_r\), with \(H_{M,r}\) self-adjoint, and that the ordinary Gram \(G_B(K+M)\) is invertible. Then
Here the hats denote the direct sums over the corresponding exterior configurations. This is the supported-boundary cancellation in commutation (ii), equation (3.15), of [ 225 ] .
Let \(B\) have bond dimension \(E\) and satisfy \(\sum _i B_i^*B_i=I\). For every prefix length \(K\geq 0\),
Consequently, the map \(J_1\) of Definition 12.9.197 satisfies \(\| J_1\| \leq \sqrt E\), independently of \(K\). This is the prefix bound in the overlapping-window argument of [ 225 , Lemma 6.1(ii) ] .
If \(A\) has nonzero virtual dimension, is a normal tensor in the sense of [ 13 ] , and is a renormalization fixed point, then for every chain length \(N{\gt}2\) the two-site local terms commute. This theorem records the commutation part of the nearest-neighbor source condition, not the ground-space spanning part.
If \(A\) has nonzero virtual dimension, is a normal tensor in the sense of [ 13 ] , and is a renormalization fixed point, then for every chain length \(N{\gt}2\) the MPS vector \(V^{(N)}(A)\) satisfies the commutation and zero-energy equations for nearest-neighbor local terms:
The zero-energy equation (6) is the parent-Hamiltonian frustration-freeness statement.
Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and let \(B=\bigoplus _{j=1}^g A_j\) contain one copy of every basis tensor. If \(B\) is a renormalization fixed point, then for every \(N{\gt}2\) its translated two-site parent interactions commute and
Hence \(B\) satisfies the all-chain nearest-neighbor commuting parent-Hamiltonian ground-space condition, establishing the multiplicity-one distinct-sector forward implication of [ 13 , Theorem 3.10 ] .
Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and set \(A=\bigoplus _{j=1}^g A_j\). Then
This statement concerns the multiplicity-one, unit-weight direct sum; it does not include repeated copies or arbitrary raw weights.
Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and let \(B=\bigoplus _{j=1}^g A_j\) contain one copy of every basis tensor. If \(B\) is a renormalization fixed point, then for every \(N{\gt}2\) its translated two-site parent interactions commute pairwise. Thus \(B\) satisfies the nearest-neighbor commuting parent-Hamiltonian condition.
Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and let \(B=\bigoplus _{j=1}^g A_j\) contain one copy of every basis tensor. If \(B\) is a renormalization fixed point, then for every \(N{\gt}2\),
This establishes the multiplicity-one distinct-sector case of [ 13 , Theorem 3.10 ] .
Let \(A\) have positive bond dimension and be a normal renormalization fixed-point tensor in the sense of [ 13 ] . If the canonical-form tensor consists of the single normal sector \(A\), then for every \(N{\gt}2\), \(\ker H_2^{(N)}(A) =\operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \). Its translated two-site interactions also commute and annihilate \(V^{(N)}(A)\). Hence the singleton family satisfies the all-chain nearest-neighbor commuting parent-Hamiltonian ground-space condition.
Let \(T:E\to F\) be a complex linear map from a normed space to an inner-product space, and let \((W_i)_{i\in I}\) be a finite family of subspaces of \(F\). Suppose \(C_i\geq 0\) and
Suppose also that each \(x_i\in W_i\) admits a preimage \(u_i\) with \(Tu_i=x_i\) and \(\| u_i\| \leq C_i\| x_i\| \). Every \(x\in \sum _iW_i\) then admits a preimage \(u\) satisfying
This is a consequence of the supplied lower Gram bound, of the form used in the overlap estimate in [ 225 , Section 4 ] .
Let \(T\) be the weighted sum of tensors \(A_\alpha \), with nonzero weights. Suppose \(h\geq 0\) acts on \(R{\gt}0\) sites and has kernel \(G_R(T)\). For a \(k\)-site matrix \(X\) with \(k{\gt}0\), put \(t=R-1\), \(w=k+2t\), and
where \(H_w\) is the open-chain sum of \(h\). For every block \(\alpha \) and every virtual matrix \(\rho _\alpha \geq 0\),
If \(A_\alpha \) is trace preserving and \(\rho _\alpha \) is an invariant matrix of nonzero trace, the corresponding constructed quasi-local state satisfies \(\omega _\alpha (j_{a,w}(Y))\geq 0\) for every interval position \(a\). No simultaneous injectivity bound is imposed.
Under the preceding asymptotic overlap hypothesis, let \(W_n\subseteq E_n\) have orthogonal projection \(Q_n\), and let the operators \(X_n\) be eventually uniformly bounded. If \(\| P_{j,n}X_nQ_n\| \to 0\) for every sector \(j\), then \(\| P_nX_nQ_n\| \to 0\). The input space \(W_n\) may be arbitrary and need not be a member of the finite family.
Under the same sector-state assumptions, suppose that \(\ker H_N=\mathcal G_N\) for every sufficiently large \(N\), where \(\mathcal G_N\) is the joint open-boundary MPS space. Then \(\omega _\alpha (j_{a,R}(h))=0\) for every \(a\) and every sector \(\alpha \). No injectivity condition at the interaction range is required. This is the sector-membership consequence of the finite-volume kernel hypothesis in [ 225 , Theorem 1.2 ] .
Let \(T=\bigoplus _j\mu _jA_j\) with every \(\mu _j\neq 0\), and let \(A_\alpha \) be a trace-preserving block with positive invariant density \(\rho _\alpha \) of nonzero trace, and let \(\omega _\alpha \) be its constructed quasi-local state. Let \(h\) act on \(R{\gt}0\) sites. Suppose that, for some \(N\geq R\), the joint MPS space \(G_N(T)\) is contained in \(\ker H_N\), where \(H_N\) is the open-chain sum of \(h\). Then \(\omega _\alpha (j_{a,R}(h))=0\) for every interval position \(a\). Positivity of \(h\) is unnecessary.
Let \(J\) be finite, let \(G_j(n)\) be subspaces of finite-dimensional complex Hilbert spaces, and let \(P_j(n)\) and \(P(n)\) be the projections onto \(G_j(n)\) and \(\sum _jG_j(n)\), respectively. Fix \(c_j\in \mathbb {C}\). Suppose \(\| \sum _jP_j(n)-P(n)\| \to 0\), the operators \(X_n\) are eventually uniformly bounded, and \(\| P(n)X_nP_j(n)-c_jP_j(n)\| \to 0\) for every \(j\). Then \(\| P(n)X_nP(n)-\sum _jc_jP_j(n)\| \to 0\) along the same filter.
Let the subspaces in the preceding theorem vary along an arbitrary filter, allowing the finite-dimensional ambient Hilbert spaces to vary. Suppose the three projection-sum errors tend to zero, and suppose \(\| P_{S_i}P_{T_i}-P_{C_i}\| \to 0\) for each \(i\), while \(\| P_{S_i}P_{T_j}\| \to 0\) for each \(i\ne j\). Then \(\| P_UP_V-P_W\| \to 0\). The finite index set is fixed and may be empty.
Let \(G_{j,n}\) be a finite family of subspaces of finite-dimensional complex Hilbert spaces \(E_n\), along an arbitrary filter. Suppose that for every \(\varepsilon {\gt}0\), eventually every pair \(i\neq j\) satisfies \(|\langle x,y\rangle |\leq \varepsilon \| x\| \| y\| \) for \(x\in G_{i,n}\) and \(y\in G_{j,n}\). Write \(P_{j,n}\) for their orthogonal projections and \(P_n\) for the projection onto their joint span. Then \(\| \sum _jP_{j,n}-P_n\| \to 0\).
Let \((A_j)\) be a finite family of pairwise gauge-phase inequivalent tensors, of bond dimensions \(D_j\). If \(i\neq j\) and \(D_j=D_i\), then the tensor \(A_j\), transported by this bond-space identification, is not gauge-phase equivalent to \(A_i\). No normalization, positive dimension, or nonempty-family assumption is required. This is the separation condition of [ 13 , Proposition 2.7 ] written in the bond-space orientation used by the sector support and gap arguments.
Let \(T=\bigoplus _j\mu _j A_j\), where every \(\mu _j\ne 0\). If \(A_j\) is trace preserving and has a nonzero positive invariant matrix, its constructed state \(\omega _j\) satisfies
for every \(a\in \mathbb Z\) and \(N\geq 0\). Primitivity is unnecessary.
Let \(H\) be a linear operator on a finite-dimensional complex Hilbert space, let \(W=\ker H\), and suppose \(W\subseteq U\cap V\). Write \(P_W,P_U,P_V\) for the orthogonal projections onto these subspaces. Suppose \(\kappa \geq 0\), \(c{\lt}1\), \(\| P_W-P_UP_V\| \leq c\), and \(\kappa ((I-P_U)+(I-P_V))\leq H\) in the order of self-adjoint operators. Then \(\kappa (1-c)\| v\| \leq \| Hv\| \) for every \(v\perp W\). This is a two-subspace consequence of the ground-projection estimate in [ 225 , Section 6, the three-interval commutation lemma, part (ii) ] .
For the preceding tensor, the periodic range-two parent Hamiltonian has a nonzero eigenvector of eigenvalue \(\varepsilon _N=1-\cos (2\pi /N){\gt}0\) for every \(N\geq 2\). These eigenvalues tend to zero. Consequently there are no \(\delta {\gt}0\) and \(N_0\) such that
Thus the unrestricted assertion in [ 17 , Section IV.C, lines 2183–2187 ] is false. The sufficient-range theorems remain valid, since this example has \(R=S=2\). See [ 27 ] .
- SpinChain.oneMagnon_edge_sum
- ZMod.cycleLaplacian_stdAddChar
- ZMod.cycleFourierGap_pos
- ZMod.tendsto_cycleFourierGap_zero
- MPSTensor.shortRangeHeisenbergMagnon
- MPSTensor.shortRangeHeisenbergMagnon_ne_zero
- MPSTensor.parentHamiltonianES_shortRangeHeisenbergMagnon
- MPSTensor.not_exists_eventual_uniform_gap_shortRangeHeisenbergTensor_two
Let \(V:\mathbb C^D\to \mathbb C^d\otimes \mathbb C^D\) be a supplied map, let \((A_i)_{ab}=V_{(i,a),b}\), and set \(F_X(B)=V^*(X\otimes B)V\). For every matrix \(\rho \) and every finite list \((X_1,\ldots ,X_n)\),
Here \(E\) is the physical observable transfer with forward matrix words. The equality includes \(n=0\). It holds in particular for an isometry, whose slices satisfy \(\sum _iA_i^*A_i=I\); neither normalization, positivity, nor stationarity is needed for the trace identity itself. This is the explicit order relation for the supplied representation in [ 225 , equations (3.5)–(3.6) ] , and makes no assertion that arbitrary GVBS boundary data have this representation.
Let \(p\geq 1\), \(M\geq 2\), and \(pM\leq N\). Sample the canonical range-\(2p\) interactions at starts \(0,p,\ldots ,p(M-2)\), and write \(H_{\mathrm{sp}}\) for their sum on \(N\) sites. Let \(U\) be the kernel of the full range-\(2p\) interaction on the prefix of length \(pM\). Suppose the grouped martingale projections satisfy \(\| Q_nE_n\| \leq \epsilon {\lt}1/\sqrt2\), with \(\epsilon \geq 0\). Then
The grouped projector estimate remains valid after adjoining free sites on the right. If \(N=pM+q\) with \(q{\gt}0\), adjoining the terminal range-\(2p\) parent projection \(T\) gives \(H_{\mathrm{sp}}+T\leq H^{\mathrm{open}}_{N,2p}\). This is a consequence of [ 225 , Theorem 2.1(ii) ] .
- MPSTensor.sparseGroupedOpenParentHamiltonianES
- MPSTensor.grouped_martingaleDifference_norm_le_of_projector_defect_of_le
- MPSTensor.sparseGroupedOpenParentHamiltonianES_energy_bound_of_grouped_c3
- MPSTensor.sparseGroupedOpenParentHamiltonianES_isPositive
- MPSTensor.sparseGroupedOpenParentHamiltonianES_projection_bound_of_grouped_c3
- MPSTensor.sparseGroupedOpenParentHamiltonianES_add_terminal_le
Let \(p\geq 1\), \(M\geq 2\), \(0{\lt}q{\lt}p\), and \(N=pM+q\). Assume that every range-\(2p\) open Hamiltonian at length at least \(2p\) has its boundary-condition space as kernel. Suppose that the three-interval ground-projection defect is at most \(0\leq \epsilon {\lt}1/\sqrt2\) whenever the overlap has at least \(p\) sites, uniformly in both outer intervals, with the right interval nonempty. Then
Let \(K,L\in \mathbb {N}\) and \(X\in M_{D}(\mathbb {C})\). Let \(\varphi \) be a complex function of the prefix configurations \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) and \(\psi \) a complex function of the suffix configurations \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \). Then the vector \((u,\tau )\mapsto \varphi (u)\, \Gamma _{K+L}(X)(u,\tau )\) belongs to the range of \(\Gamma ^{\mathrm{tail}}_{K,L}\), and the vector \((u,\tau )\mapsto \psi (\tau )\, \Gamma _{K+L}(X)(u,\tau )\) belongs to the range of \(\Gamma ^{\mathrm{left}}_{K,L}\).
The tail Gram is the direct sum over prefix configurations of copies of \(\mathcal K_L\), and the left Gram is the direct sum over suffix configurations of copies of \(\mathcal K_K\). Whenever the corresponding local boundary map is injective, the inverse Gram is the direct sum of the local inverse Grams. These identities include empty spectator index types and assert no norm estimate.
Equip each virtual matrix fiber with the unweighted Hilbert–Schmidt inner product and flatten the finite family of fibers into a Euclidean space. The resulting continuous tail and left boundary maps have ranges \(\mathcal V_{K,L}(A)\) and, at suffix length one, \(\mathcal U_K(A)\). If \(J^{\mathrm{tail}}_K(X)_u=XA^u\) and \(J^{\mathrm{left}}_L(X)_\tau =A^\tau X\), then
Let \(q\in \mathbb N\), let \(X=(X_u)_{u\in \{ 0,\ldots ,d{-}1\} ^K}\), and let \(Z=(Z_j)_{j\in \{ 0,\ldots ,d{-}1\} ^q}\) be virtual boundary-matrix families. After the canonical identification of \(K+(l+q)\) with \((K+l)+q\), the mixed Gram for the tail window of length \(l+q\) and the left window with a \(q\)-site increment is
where \(U\) is Frobenius vectorization and \(\mathcal K_l=\Gamma _l^\dagger \Gamma _l\). The multiplication order is fixed by \(\operatorname{tr}(A^{\tau ,j}X_u)=\operatorname{tr}(A^\tau (A^jX_u))\) and \(\operatorname{tr}(A^{u,\tau }Z_j)=\operatorname{tr}(A^\tau (Z_jA^u))\). The one-site identity is the specialization \(q=1\). This identity expresses the algebraic form of condition (C3\('\)) in [ 225 , Equation (2.5) ] .
Specialize Theorem 12.3.5.22 to \(q=1\). Let \(X=(X_u)_{u\in \{ 0,\ldots ,d{-}1\} ^K}\) and \(Z=(Z_j)_{j\in \{ 0,\ldots ,d{-}1\} ^1}\) be the resulting boundary families. If \(\rho \in M_{D}(\mathbb {C})\) has nonzero trace and \(\mathcal K_\infty =\mathscr R(P_\rho )\), then
Here \(\mathcal K_\infty U(Z_jA^u) =U((\operatorname{tr}\rho )^{-1}Z_jA^u\rho )\), so the limiting multiplication is on the right and is generally not the identity.
Let \(A,B\) have the same physical alphabet and suppose \(V^\dagger V=I\) and \(A^iV=VB^i\) for every letter. Then \(\Gamma _L^A(VXV^\dagger )=\Gamma _L^B(X)\) for every \(L\), and \(G_L(B)\subseteq G_L(A)\). If also \(VV^\dagger A^i=A^i\), their periodic vectors agree at every positive length. Finally, if invertible matrices \(X,Y\) satisfy \(XV=VY\) and \(\sum _jU_{ij}A^j=XA^iX^{-1}\), then \(\sum _jU_{ij}B^j=YB^iY^{-1}\) [ 236 , Section II.F.2 ] .
- MPSTensor.groundSpaceMap_eq_of_isometric_bond_intertwiner
- MPSTensor.groundSpace_le_of_isometric_bond_intertwiner
- MPSTensor.trace_evalWord_eq_of_supported_isometric_bond_intertwiner
- MPSTensor.mpv_eq_of_supported_isometric_bond_intertwiner
- MPSTensor.rotatePhysical_gauge_covariance_of_isometric_bond_intertwiner
Let \(A\) satisfy \(\sum _i A_i^*A_i=I\), and let \(\rho \geq 0\) have nonzero trace and satisfy \(\sum _i A_i\rho A_i^*=\rho \). There exist a tensor \(B\) of positive bond dimension and a faithful stationary matrix \(\sigma {\gt}0\) such that
In particular, the physical dimension is positive. There also exist a range \(R{\gt}0\), a positive interaction \(h\), and constants \(\delta ,\gamma {\gt}0\) such that
The same interaction satisfies the finite-volume estimate
Here \(\mathcal F_h\) denotes the normalized positive states with zero expectation of every translate of \(h\). For every pure \(\varphi \in \mathcal F_h\), every local observable \(X\) on a positive-length interval \([a,a+k)\) with \(\varphi (X)=0\), and every interval exhaustion whose left and right margins both diverge,
The interaction and constant are chosen before the pure state, observable, filter, and margins. The generated state may be mixed; neither its purity nor translation invariance of the competing states is required. The finite kernels are \(G_B(N)\); equality with \(G_A(N)\) is not asserted.
This is a consequence for supplied stationary generating data in [ 225 , Section 3 ] . Their construction from arbitrary GVBS boundary limits remains separate, as documented in [ 61 ] .
Let \(A\) satisfy \(\sum _i A_i^*A_i=I\), and let \(\rho \geq 0\) have nonzero trace and satisfy \(\sum _i A_i\rho A_i^*=\rho \). There exist a tensor \(B\) of positive bond dimension and a faithful stationary matrix \(\sigma {\gt}0\) such that
In particular, the physical dimension is positive. There also exist a range \(R{\gt}0\), a positive interaction \(h\), and \(\gamma {\gt}0\) such that
Here \(\mathcal F_h\) denotes the normalized positive states with zero expectation of every translate of \(h\). For every pure \(\varphi \in \mathcal F_h\), every local observable \(X\) on a positive-length interval \([a,a+k)\) with \(\varphi (X)=0\), and every interval exhaustion whose left and right margins both diverge,
The interaction and constant are chosen before the pure state, observable, filter, and margins. The generated state may be mixed; neither its purity nor translation invariance of the competing states is required. The finite kernels are \(G_B(N)\); equality with \(G_A(N)\) is not asserted.
This is a consequence for supplied stationary generating data in [ 225 , Section 3 ] . Their construction from arbitrary GVBS boundary limits remains separate, as documented in [ 61 ] .
Let \(A\) be an MPS tensor, let \(\rho \geq 0\) have nonzero trace and satisfy \(\sum _iA_i\rho A_i^*=\rho \), and let \(P\) be its support projection. The matrix \(\rho \) need not be faithful. For every \(k\geq 0\) and every \(k\)-site matrix \(X\),
If moreover \(\sum _iA_i^*A_i=I\), there is a density matrix \(\sigma _k\) with \(\eta _{A,\rho }(X)=\operatorname {tr}(\sigma _kX)\) for every \(X\) and \(\operatorname {ran}\sigma _k=G_A^P(k)\). This describes the local support spaces of normalized stationary generating data, as in [ 225 , Section 3 and the proof of the overlap lemma in Section 4 ] , without the faithfulness that the source obtains from minimality. Passing from an arbitrary GVBS boundary limit to normalized stationary data remains separate [ 61 ] .
Let \(A\) satisfy \(\sum _iA_i^*A_i=I\), and let \(\rho \geq 0\) have nonzero trace with \(E_A(\rho )=\rho \). There exist a positive dimension \(E\), an isometry \(V\), a normalized tensor \(B\), and a faithful stationary matrix \(\sigma {\gt}0\) such that
Let \(\varphi \) be the quasi-local state defined by the original generators \(A,\rho \). For every \(k\geq 0\), there is a matrix \(\tau _k\geq 0\) of trace one such that, for every translated interval \([a,a+k)\) and every matrix observable \(X\) on that interval,
The physical dimension is derived from the compressed normalization. No faithfulness of \(\rho \) or primitivity of the generating tensor is required. This concerns supplied stationary generating data in [ 225 , Section 3 ] and its finite support identification in Section 4; constructing such data from an arbitrary GVBS boundary limit remains separate [ 61 ] .
Let \(A\) satisfy \(\sum _i(A^i)^*A^i=I\), let \(\rho \geq 0\) be a fixed point of its transfer map with \(\operatorname {tr}\rho \neq 0\), and let \(h\geq 0\) be an interaction of positive range whose open-chain kernels equal the MPS boundary spaces \(G_N(A)\) for all sufficiently large \(N\). Then the quasi-local state generated by \(A\) and \(\rho \) has zero local energy for \(h\). No primitivity, faithfulness of \(\rho \), or purity is assumed [ 225 , Theorems 1.1–1.2 ] .
Let \(A\) satisfy \(\sum _iA_i^*A_i=I\), and let \(\rho \geq 0\) have nonzero trace with \(E_A(\rho )=\rho \). There exist a positive dimension \(E\), an isometry \(V\), a tensor \(B\), and a faithful matrix \(\sigma {\gt}0\) such that
\(\eta _{A,\rho }(X)=\eta _{B,\sigma }(X)\) for every observable on every finite interval, including length zero. All conclusions except \(\sum _iB_i^*B_i=I\) hold without the normalization of \(A\). This is a restriction of supplied stationary generating data to their support, as in [ 244 ] , restriction to full-rank fixed points, and the normalized generating representation of [ 225 , Section 3 ] . Constructing generating data from an arbitrary GVBS boundary limit remains separate [ 61 ] .
Let \(A\) satisfy \(\sum _iA_i^*A_i=I\), and let \(\rho \geq 0\) have nonzero trace with \(E_A(\rho )=\rho \). There exist a positive dimension \(E\), an isometry \(V\), a normalized tensor \(B\), and a faithful stationary matrix \(\sigma {\gt}0\) with
The physical dimension is positive, and the actual continuous functionals agree: \(\omega _{A,\rho }=\omega _{B,\sigma }\). No purity or minimality is assumed. This is a consequence for supplied stationary generating data in [ 225 , Section 3 ] . Their construction from arbitrary GVBS boundary limits remains separate [ 61 ] .
Let \((V_i)_{i\in I}\) be finitely many subspaces of a complex Hilbert space, and let \(B\) be a real matrix with zero diagonal and \(\| B\| {\lt}1\). Suppose that, for \(i\ne j\), \(x\in V_i\), and \(y\in V_j\),
Then the family \((V_i)\) is independent: a vanishing sum of vectors, one from each subspace, has every component zero.
Let \(v_1,\ldots ,v_m\) lie in a complex inner product space, and let \(B\in M_m(\mathbb {R})\) have zero diagonal. Suppose that \(|\langle v_i,v_j\rangle |\leq B_{ij}\lVert v_i\rVert \lVert v_j\rVert \) whenever \(i\neq j\). With the Euclidean operator norm on \(B\),
This is the lower Gram estimate in the proof of [ 225 , Lemma 9 ] .
Let \(U,V_1,\ldots ,V_m\) be subspaces of a complex inner product space. Let \(a\in \mathbb {R}^m\) and \(B\in M_m(\mathbb {R})\), where \(B\) has zero diagonal and \(\lVert B\rVert {\lt}1\). Suppose that \(|\langle u,v\rangle |\leq B_{ij}\lVert u\rVert \lVert v\rVert \) for \(i\neq j\), \(u\in V_i\), and \(v\in V_j\). Suppose also that \(|\langle x,u\rangle |\leq a_i\lVert x\rVert \lVert u\rVert \) for every \(i\), \(x\in U\), and \(u\in V_i\). Then every \(x\in U\) and \(y\in V_1+\cdots +V_m\) satisfy
This is [ 225 , Lemma 9 ] , expressed as a bound on inner products.
Let \(U,V_1,\ldots ,V_m\) be subspaces of a complex inner product space, and let \(\varepsilon \geq 0\) satisfy \(\varepsilon m\leq 1\). Suppose that every pair of vectors in distinct \(V_i\) satisfies \(|\langle u,v\rangle |\leq \varepsilon \lVert u\rVert \lVert v\rVert \), and that the same inequality holds for vectors in \(U\) and \(V_i\), for every \(i\). Set
Then \(\eta \leq 1\), and every \(x\in U\) and \(y\in V_1+\cdots +V_m\) satisfy \(|\langle x,y\rangle |\leq \eta \lVert x\rVert \lVert y\rVert \). This is the uniform bound in [ 225 , Lemma 9 ] .
Let \(I\) be a finite index set and let \(A=(A_{ij})_{i,j\in I}\) be a complex matrix. Then
For the preceding slices,
Thus an isometry gives trace-preserving normalized tensor matrices. The insertion identity holds for any supplied \(V\), without an isometry assumption. No dimension, stationarity, or faithfulness assumption is needed. This identifies the one-site coordinates of [ 225 , equation (3.6) ] .
Identify configurations on one site with their physical letter. For any tensor \(A\), its one-site physical observable transfer is
For the slices of \(V\), this is trace-dual to the preceding Heisenberg insertion:
The matrices \(X,B,\rho \) are arbitrary. These identities neither construct a supplied isometry from arbitrary GVBS data nor identify multi-site expectations with a particular ordering of source sites.
Let \(D{\gt}0\) and let \(V:\mathbb C^D\to \mathbb C^d\otimes \mathbb C^D\) be an isometry, with tensor slices \(A_i\). Then \(\sum _iA_i^*A_i=I\), and the transfer map of \(A\) has a positive fixed point \(\rho \) of trace one. There are a faithful normalized tensor \(B\) with a positive definite fixed point that generates the same quasi-local state, a range \(R{\gt}0\), and a positive interaction \(h\) on \(R\) sites whose open-chain kernels equal \(G_N(B)\) for \(N\geq R\), such that the generated state has zero energy, the open Hamiltonians of \(h\) have one positive gap at every length, and one constant \(\gamma {\gt}0\) bounds the commutator energy below in every pure zero-energy state, for every centered observable on an interval of positive length and every interval exhaustion whose left and right margins both diverge [ 225 , equation (3.6), Theorems 1.1–1.2, and Section 6 ] . The isometry is supplied; its construction from a general presentation is not asserted [ 61 ] .
Let \(T:E\to \mathcal H\) be a continuous linear map from a finite-dimensional Hilbert space to a complete Hilbert space. Suppose \(Q=Q^*\), \(TQ=T\), and \((T^*T)H=Q\). Then \(P_{\operatorname {ran}T}=THT^*\).
More generally, let \(T:E\to \mathcal H\), \(L:F\to \mathcal H\), and \(C:G\to \mathcal H\) have finite-dimensional domains and a common factorization \(C=TJ_1=LJ_2\). For \(S\in \{ T,L,C\} \), suppose \(Q_S=Q_S^*\), \(SQ_S=S\), and \((S^*S)H_S=Q_S\). If also \(H_T=H_T^*\), then
This is the supported-boundary algebra for the interval projector comparison in [ 225 , Lemma 6.1(ii) ] .
Let \(T:E\to F\), \(L:G\to F\), and \(C:W\to F\) be bounded maps, with finite-dimensional virtual spaces and a complete physical Hilbert space. Suppose
Then
For a family along any filter, assume this factorization eventually holds, the product \(\| T\| \| H_T\| \| H_L\| \| L\| \) is eventually bounded, and \(\| R\| \to 0\). Then \(\| P_TP_L-P_C\| \to 0\). The spaces may vary with the index. This implication supplies the norm comparison in [ 225 , Lemma commutation (ii) ] ; the factorization, inverse bounds, and residual decay are explicit hypotheses here.
For every prefix \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) and suffix \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \),
The range of the tail boundary map is exactly the tail ground space:
For every \(K,L\in \mathbb {N}\), transport of the tail boundary map range by the inverse canonical \(\ell ^2\) isomorphism identifies it with the open-chain tail ground submodule:
Concretely, a vector \(v\) in the \((K+L)\)-site Euclidean space belongs to \(\mathcal V_{K,L}(A)\) exactly when the corresponding function satisfies the tail ground condition (Definition 11.5.5.6).
Restricting the tail boundary map to the suffix for a fixed prefix \(u\in \{ 0,\ldots ,d{-}1\} ^K\) recovers \(\Gamma _L(Y_u)\):
Under the same hypotheses, there is an integer \(p{\gt}0\), measured in sites of \(A\), such that \(A\) is \(p\)-block injective and
For bounded linear operators \(A,a:G\to W\), \(B,b:F\to G\), and \(C,c:E\to F\) between normed spaces,
No finite-dimensional or completeness hypothesis is required.
Let the normed spaces and operators in Lemma 12.9.241 vary along an arbitrary filter. Suppose the three individual differences tend to zero in norm, and the products \(\| B\| \| C\| \), \(\| a\| \| C\| \), and \(\| a\| \| b\| \) have a common eventual upper bound. Then \(\| ABC-abc\| \to 0\).
Fix a tensor \(A\) with physical dimension \(d{\gt}0\) and a coefficient \(c\in \mathbb R\). For every chain length \(N\) and site \(i\in \{ 0,\ldots ,N{-}1\} \), set \(h_i^{(N)}:=h_{i,\mathrm{ES}}^{(N,2)}(A)\), the Euclidean range-two parent term of this same tensor \(A\) at \(i\). Suppose that, for every three-site vector \(w\),
Then, for every \(N\geq 3\), every site \(i\in \{ 0,\ldots ,N{-}1\} \), and every \(N\)-site vector \(v\), the forward pair satisfies
and the backward pair satisfies
All site indices are taken modulo \(N\), so these conclusions include pairs and three-site windows crossing the periodic cut.
Suppose the first two length-two parent interactions commute on the three-site chain: \(h_0^{(3)}h_1^{(3)} =h_1^{(3)}h_0^{(3)}\). Then, for every \(N{\gt}2\) and every \(i\in \mathbb Z/N\mathbb Z\), the two adjacent translates commute: \(h_i^{(N)}h_{i+1}^{(N)} =h_{i+1}^{(N)}h_i^{(N)}\). For each exterior configuration \(\tau \), let \(R_{i,\tau }^{(3)}\) restrict to the cyclic three-site window beginning at \(i\). It intertwines the two \(N\)-site terms with the two three-site terms:
Thus the same local \(A\! X\! B\) calculation is placed at every cyclic translate through the two restriction identities above. The hypothesis \(N{\gt}2\) is essential to this transport: at \(N=2\) the two cyclic windows contain the same sites in opposite orders. This case is not part of [ 13 , Theorem 3.10 ] .
If \(A\) is injective, then \(G_3(A) = \ker (q_2(A)_{AX}) \cap \ker (q_2(A)_{XB})\). More generally, on \(L+1\) sites, simultaneous annihilation by the two adjacent length-\(L\) parent interactions is equivalent to membership in \(G_{L+1}(A)\) whenever \(A\) is injective and \(L{\gt}1\).
Let \(A\) be an irreducible MPS tensor with \(D {\gt} 0\) and some \(A^i \neq 0\). Then there exist a positive real \(r\), a positive definite matrix \(\sigma \), and a tensor \(B\) gauge-equivalent to \(r^{-1/2}A\) such that
and \(\sum _i (B^i)^\dagger B^i = \mathbb {1}\).
This corresponds to the spectral rescaling and trace-preserving gauge construction of [ 13 , Appendix A ] .
For a nonempty closed chain, the trace of the word \(A^{\sigma _0}\cdots A^{\sigma _{L-1}}\) is the sum over cyclic virtual-index configurations:
For \(P\geq 0\), \(\widehat P_{AX}P=P=P\widehat P_{AX}\).
If \(P=P^\dagger \) and \(A\) and \(B\) are decorrelated, then
Let \(P\), \(P_{AX}\), and \(P_{XB}\) be orthogonal projectors, and write their lifts to the tripartite space as \(\widehat P_{AX}=P_{AX}\otimes \mathbb {1}_B\) and \(\widehat P_{XB}=\mathbb {1}_A\otimes P_{XB}\). Suppose that the lifted projectors commute, \(\widehat P_{AX}\widehat P_{XB} =\widehat P_{XB}\widehat P_{AX}\). Then
This is the passage from [ 13 , Definition D.2, lines 2205–2218 ] to the projector identity at lines 2279–2289.
For matrix units on \(A\) and \(B\),
Let \(P=P^\dagger =P^2\). Then \(P\) is the ground-space projector of a parent commuting Hamiltonian on \(AX\) and \(XB\) if and only if regions \(A\) and \(B\) satisfy complex-linear decorrelation.
Let \(P=P^\dagger =P^2\). Then \(P\) is the ground-space projector of a parent commuting Hamiltonian on \(AX\) and \(XB\) if and only if regions \(A\) and \(B\) are decorrelated when tested on Hermitian observables. This is [ 13 , Proposition D.3, lines 2221–2289 ] .
If \(P=P^\dagger =P^2\) and \(A\) and \(B\) are decorrelated relative to \(P\), then
For \(P\geq 0\), \(\widehat P_{XB}P=P=P\widehat P_{XB}\).
Suppose that the canonical length-two parent terms of \(A\) commute on a three-site chain. Then \(Q_A\) is Hermitian, and its natural placements on the first and last pairs of a three-fold tensor product commute:
Under the preceding hypothesis, there are positive integers \(d_q,m_q\), a unitary identification
and Hermitian matrices \(R_q\) on \(\mathcal H\otimes \mathcal H_{q,l}\) and \(S_q\) on \(\mathcal H_{q,r}\otimes \mathcal H\) such that, after conjugating the middle site by this unitary, the interaction decomposes as
on the first pair and
on the last pair. This is the spatial decomposition of [ 5 , Lemma 2.1 ] for the canonical parent interaction.
Let \(X\) be a nonempty set, let \(\mathcal F\) be a filter on an index set \(I\), and let \(e:I\times X\to \mathbb R\). Suppose \(e(i,x_i)\to 0\) along \(\mathcal F\) for every function \(i\mapsto x_i\) from \(I\) to \(X\). Then, for every \(\varepsilon {\gt}0\), eventually \(e(i,x)\leq \varepsilon \) simultaneously for every \(x\in X\). No topology or countability assumption on \(X\) is required, and \(e\) need not be nonnegative. This elementary implication applies to the uniform projection estimates in [ 225 , Lemma commutation(ii) ] .
Let \(A\) be primitive with positive-definite fixed point \(\rho \) and nonzero bond dimension. There is an integer \(p{\gt}0\), measured in sites of \(A\), such that \(A\) is \(p\)-block injective and, for every prefix length \(K\),
Let \(A\) be injective with \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), let \(\Lambda \) be a positive definite matrix with \(\mathcal{E}_A^\dagger (\Lambda )=\Lambda \), and let \(u\) be a unitary with \(\sum _j u_{ij}A^j=\zeta \, XA^iX^{-1}\) for all \(i\), where \(X\in \mathrm{GL}_{D}(\mathbb {C})\) and \(\zeta \neq 0\). Then \(X=c\, W\) for a nonzero scalar \(c\) and a unitary \(W\) with \(W\Lambda W^\dagger =\Lambda \). In particular \(X\Lambda =\Lambda X\).
Let \(A_{ij}=t_{ij}E_{ij}\), where \(t\) is a real matrix. If every entry of \(t\) is nonzero, then the letters of \(A\) span the full matrix algebra. If the squared entries in every column sum to one, then
Writing \(T_{abcef}=t_{ab}t_{bc}t_{ce}t_{ef}\), one has
For every configuration \(\sigma =(i_0j_0,i_1j_1,i_2j_2,i_3j_3)\) and every real function \(u\) on the vertices,
- MPSTensor.FNWDimensionConstant.pathConfig_eq_iff
- MPSTensor.FNWDimensionConstant.inner_pathVector
- MPSTensor.FNWDimensionConstant.groundSpaceMap_pathConfig
- MPSTensor.FNWDimensionConstant.inner_pathVector_groundSpaceMap
- MPSTensor.FNWDimensionConstant.pathVector_first_mark_apply
- MPSTensor.FNWDimensionConstant.pathVector_last_mark_apply
Let \(\rho \in M_D(\mathbb {C})\) be positive definite. The \(\rho \)-weighted inner product on \(M_D(\mathbb {C})\) is
This is the auxiliary inner product in [ 194 , Equation (5.6) ] .
Let \((P_i)_{i\in I}\) be a finite family of positive operators on a finite-dimensional complex inner product space, with real weights \(0{\lt}w_i\leq 1\), and let \(\gamma \geq 0\). If
for every \(v\in \bigl(\ker \sum _{i\in I}w_iP_i\bigr)^\perp \), then
for every \(v\in \bigl(\ker \sum _{i\in I}P_i\bigr)^\perp \).
Let \((P_i)_{i\in I}\) be a finite family of positive operators on an inner product space. A positive operator \(P\) annihilates a vector \(v\) as soon as \(\operatorname {Re}\langle Pv,v\rangle =0\), and consequently, for real weights \(w_i{\gt}0\),
equivalently \((\sum _{i\in I}w_iP_i)v=0\) exactly when \(P_iv=0\) for every \(i\in I\). For the empty family both sums are zero and all three kernels are the full space. This is the analytic kernel step of the blocked-sum comparison in Lemma block-obc of Rozmán–Molnár–Schuch 1 , in which the blocked parent Hamiltonian is a weighted sum \(\sum _i\gamma _ih_i\) of positive local terms with \(\gamma _i{\gt}0\).
- LinearMap.IsPositive.apply_eq_zero_of_re_inner_eq_zero
- WeightedPositiveKernel.re_inner_weighted_sum_apply
- WeightedPositiveKernel.apply_weighted_sum_eq_zero_iff
- WeightedPositiveKernel.apply_sum_eq_zero_iff
- WeightedPositiveKernel.ker_weighted_sum_eq_iInf
- WeightedPositiveKernel.ker_sum_eq_iInf
- WeightedPositiveKernel.ker_weighted_sum_eq_sum
Let \((P_i)_{i\in I}\) be a finite family of positive operators on an inner product space, with real weights \((w_i)_{i\in I}\). If \(w_i\geq 0\) for every \(i\), then \(\sum _{i\in I}w_iP_i\) is positive. If \(w_i\leq 1\) for every \(i\), then
in Loewner order. This is the comparison step of the blocked-sum argument in the source cited in Theorem 12.8.11, where the coefficients satisfy \(0{\lt}\gamma _i\leq 1\).
If \(A\) is left-canonical, then the length-\(Q\) left virtual map \(J_Q^{\mathrm{left}}\) preserves the Hilbert–Schmidt norm. In particular,
With the notation of Theorem 12.3.5.22,
Let \(\rho \in M_{D}(\mathbb {C})\) have nonzero trace and set \(\mathcal K_\infty =\mathscr R(P_\rho )\). Then
The overlap factors satisfy
Let \(T=\widetilde\Gamma ^{\mathrm{tail}}_{K;L,Q}\), \(S=\Gamma ^{\mathrm{left}}_{K+L,Q}\), and \(C=\Gamma _{K+L+Q}^{\mathrm{ES}}\). Assume separately that \(T\), \(S\), and \(C\) are injective. Since \(TJ_K^{\mathrm{tail}}=C=SJ_Q^{\mathrm{left}}\), the difference \(P_TP_S-P_C\) equals the exact inverse-Gram residual obtained from these two factorizations.
If \(A\) is left-canonical, then for every \(Q\in \mathbb {N}\),
where \(\Omega _Q=\{ 0,\ldots ,({-}1\} d^Q)\) is the set of physical words of length \(Q\).
Let \((A_j)_{j=1}^g\) and \((B_j)_{j=1}^g\) be tensor families with the same physical dimension and the same bond dimension \(D_j\) in each labelled block. Suppose that for every \(j\) there is an arbitrary invertible matrix \(X_j\in \mathrm{GL}_{D_j}(\mathbb {C})\) such that, for every physical index \(i\),
Then, at every fixed length \(L\),
Both directions hold for general invertible gauges without unitarity constraints, allowing span certificates in left-canonical form to be transported to unital gauges.
For every \(n\),
Consequently, if \(L{\gt}0\) and \(A\) is \(L\)-block injective, then \(A\) is \(m\)-block injective for every \(m\ge L\). This combines the block-injectivity discussion in [ 13 , Section II ] with [ 13 , Appendix C.3, Lemma L ] .