Dependencies

By (46), \(\operatorname{tr}(YA^w)\neq 0 \iff k\equiv 1\pmod2\), matching the odd fermion parity of (40); by (43), \(\operatorname{tr}(A^w)\neq 0 \iff k\equiv 0\pmod2\). Consequently, \(\operatorname{tr}(YA^w)\operatorname{tr}(A^w)=0\) for every word \(w\).

If every site carries the same local vector \(\sum _i a^i|i\rangle \), the one-site tensor with \(1\times 1\) matrices \(a^i\) generates the product vector \(\bigl(\sum _i a^i|i\rangle \bigr)^{\otimes N}\) both as an open-boundary state with trivial boundary vectors and as a matrix product vector with periodic boundary conditions.

The length-\(2\) blocked AKLT tensor groups two physical sites into one, with blocked physical dimension \(9 = 3^2\) and the same bond dimension \(D = 2\).

The length-\(2\) blocked on-site action of \(\mathbb {Z}_2 \times \mathbb {Z}_2\) on the \(9\)-dimensional blocked physical space is the Kronecker square \(g \longmapsto P_g \otimes P_g\) of the single-site spin-\(1\) \(\pi \)-rotation representation \(P_g\) that implements the symmetry of Theorem 15.3.11. Identifying the blocked physical index with the pair of single-site indices, the entry of \(P_g \otimes P_g\) at \((i_1 i_2, j_1 j_2)\) is the product \((P_g)_{i_1 j_1}(P_g)_{i_2 j_2}\) over the two grouped sites. This is the on-site action that makes the blocked twist agree with blocking the single-site twist.

The Cartesian presentation of the AKLT tensor has \(d = 3\), \(D = 2\), and \(A^x = \sigma _x\), \(A^y = \sigma _y\), \(A^z = \sigma _z\), the three Pauli matrices. This is the rotationally natural form used in [ 17 ] (around line 1159); it is the AKLT state in the Cartesian basis of the spin-\(1\) site, the counterpart of the \(|m\rangle \)-basis form of Definition 15.3.7.

The AKLT factor system on \(\mathbb {Z}_2 \times \mathbb {Z}_2\) is \(\omega (g,h) = (-1)^{(g_1 + g_2)\, h_1}\) (the value \(-1\) when \(g_1 + g_2 = 1\) and \(h_1 = 1\), and \(1\) otherwise). It is the factor system of the explicit projective representation with virtual gauges \(i\sigma _y\) and \(\sigma _z\); the diagonal term \(g_1 h_1\) relative to the cluster case reflects \((i\sigma _y)^2 = -\mathbb {1}\).

On \(N\) spin-\(1\) sites with open boundaries, put

There is no interaction joining the last site to the first. This is the polynomial of  [ 228 , local TeX lines 340–349 ] with the open boundaries of  [ 17 , lines 1171–1174 ] .

Let \(S^x,S^y,S^z\) be the spin-\(1\) operators in the basis \(|0\rangle =|m=0\rangle \), \(|1\rangle =|m=+1\rangle \), \(|2\rangle =|m=-1\rangle \), with \(S^z=\operatorname{diag}(0,1,-1)\), \(S^+|0\rangle =\sqrt2|1\rangle \), \(S^+|2\rangle =\sqrt2|0\rangle \), \(S^x=\tfrac 12(S^++S^-)\) and \(S^y=\tfrac 1{2i}(S^+-S^-)\). This is the labelling in which the three matrices of the AKLT tensor of Definition 15.3.7 are proportional to \(\sigma ^z,\sqrt2\sigma ^+,-\sqrt2\sigma ^-\) on the indices \(0,1,2\). For two sites \(j,k\) of a chain let \(\vec S_j\cdot \vec S_k=\sum _\alpha S^\alpha _jS^\alpha _k\). On a periodic chain of \(N\) spin-\(1\) sites, with indices modulo \(N\), the AKLT Hamiltonian is

[ 2 , 3 ] ; see also [ 228 , local TeX lines 340–349 ] , and, up to an additive constant, [ 236 , lines 2126–2130 ] . Its two-site term is \(h=\vec S_0\cdot \vec S_1+\tfrac 13(\vec S_0\cdot \vec S_1)^2\).

The basis of [ 17 , Appendix A, “The AKLT state” ] is

The second basis replaces \(|+\rangle \) with \(-\frac{i}{\sqrt2}(|-1\rangle +|+1\rangle )\) and keeps \(|-\rangle \) and \(|0\rangle \).

Set \(d = 3\) (spin-1) and \(D = 2\). The AKLT tensor is the family \(\{ A^0, A^1, A^2\} \subset M_{2}(\mathbb {C})\) defined via the Pauli matrices by

Here \(\sigma _z = |0\rangle \! \langle 0| - |1\rangle \! \langle 1|\), so equivalently \(A^1 = \frac{1}{\sqrt{3}} \sigma _+\) and \(A^2 = -\frac{1}{\sqrt{3}} \sigma _-\) for \(\sigma _+ = \sqrt{2} |0\rangle \! \langle 1|\) and \(\sigma _- = \sqrt{2} |1\rangle \! \langle 0|\). Equivalently, the matrices are the Clebsch–Gordan coefficients coupling spin-\(1\) and spin-\(\tfrac 12\) to spin-\(\tfrac 12\). This is the representative of the tensor of Definition 15.3.1 used below (Lemma 15.3.8).

Place a spin-\(\tfrac 12\) singlet on every bond, encoded by

and project the two spin-\(\tfrac 12\) particles at each site onto their joint spin-\(1\) subspace. With the physical states labelled by \(S_z=0,\pm 1\), the resulting tensor with \(d=3\) and \(D=2\) is

The three left factors are the spin-\(1\) states \(|\uparrow \uparrow \rangle \), \((|\uparrow \downarrow \rangle +|\downarrow \uparrow \rangle )/\sqrt2\), and \(|\downarrow \downarrow \rangle \) written as coefficient matrices. This is the tensor of [ 17 , Appendix A, “The AKLT state” ] , for the state of  [ 2 ] .

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 \(A^{[L]}\).

A finite block family has no gauge-phase-equivalent distinct blocks when, for any two distinct indices with the same bond dimension, the corresponding blocks are not gauge-phase equivalent.

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.

On a periodic chain of \(N\) sites, the stabilizer at site \(i\) is the operator

with site indices modulo \(N\). For \(N\ge 3\) the sites \(i\), \(i+1\), \(i+2\) are distinct and \(K_i = \sigma ^z_i\sigma ^x_{i+1}\sigma ^z_{i+2}\); the results below assume \(N\ge 3\).

The cluster-state factor system on \(\mathbb {Z}_2 \times \mathbb {Z}_2\) is \(\omega (g,h) = (-1)^{g_2\, h_1}\) (the value \(-1\) when \(g_2 = 1\) and \(h_1 = 1\), and \(1\) otherwise). It is the factor system of the explicit projective representation with virtual gauges \(\sigma _z\) and \(\sigma _x\), both squaring to the identity, so there is no diagonal term.

The three-site stabilizer of [ 228 , local TeX lines 374–387 ] is \(K = \sigma ^z_1\sigma ^x_2\sigma ^z_3\). In the computational basis, with \(\sigma ^z|s\rangle = (-1)^s|s\rangle \) and \(\sigma ^x|s\rangle = |s+1\rangle \), it acts on a three-site vector \(v\) by

The results below are proved for the transpose of the tensor of Definition 15.7.1, the family

and transferred back by Lemma 15.7.5.

Set \(d = D = 2\). The cluster-state tensor of  [ 17 , Appendix A, “The cluster state” ] is the family \(\{ A^0, A^1\} \subset M_{2}(\mathbb {C})\) given by

Equivalently, \((A^s)_{\alpha \beta } = \delta _{\alpha s}\, (-1)^{s\beta }/\sqrt{2}\).

The commutator phase of a \(2\)-cochain \(\omega \) at a pair \(g,h\) is \(\beta _\omega (g,h):=\omega (g,h) \omega (h,g)^{-1}\), defined for any \(\mathbb {C}^{\times }\)-valued \(2\)-cochain.

For \(a\in \{ 0,1\} \) and \(N\ge 0\),

For every \(\lambda _2\in \mathbb {C}\), define the associated correlation-length quantity by

For a nonzero subleading eigenvalue in the physical range \(0{\lt}|\lambda _2|{\lt}1\), this quantity is positive. If all subleading spectral values vanish and correlations therefore vanish after finitely many transfer steps, the total definition gives the limiting value \(\xi =0\).

Set \(d=D=2\) and

The first and second virtual diagonal entries define the bond-dimension-one product tensors for \(|0\rangle \) and \(|+\rangle \), respectively.

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,

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 \((i_1,\ldots ,i_L)\).

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

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.

The tensor with physical index \((i,j)\in G\times G\) and bond dimension \(|G|\) is

[ 17 , Section III.A, “Virtual symmetries” ] . Only \(g=\alpha i^{-1}\) contributes, so \(A^{ij}_{\alpha \beta }=|G|^{-1}\) if \(\beta =\alpha i^{-1}j\) and \(0\) otherwise.

The (unnormalized) GHZ state on \(N\) sites of a \(d\)-level system is

Set \(d = D = 2\). The GHZ tensor is the family \(\{ A^0, A^1\} \subset M_{2}(\mathbb {C})\) defined by

For a system of \(N\) sites, the resulting MPV is the GHZ state

Set \(D = d\). The GHZ tensor on a \(d\)-level system is the family \(\{ A^i\} _{i=0}^{d-1}\subset M_{d}(\mathbb {C})\) with entries

that is, \(A^i=|i\rangle \! \langle i|\)  [ 17 , Appendix A, “The GHZ state” ] .

The virtual \(\mathbb {Z}_2\) representation on the bond space is the homomorphism \(\mathbb {Z}_2 \to M_{2}(\mathbb {C})\) sending the identity to \(I\) and the generator to \(\sigma _z = \operatorname{diag}(1, -1)\).

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,

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 \(L\).

The local ground space is the image \(G_L(A):=\operatorname{range}(\Gamma _L)\subseteq (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\).

For boundary matrices \(X,Y\in M_{D}(\mathbb {C})\), set explicitly

The virtual-boundary nondecay predicate holds for \(u\) and \(\Lambda \) if there exist \(X,Y\) and a positive constant \(c{\gt}0\) such that \(c\le \| R_L(u;X,Y)\| \) for every \(L\). Thus some virtual-boundary sequence does not decay to zero.

This virtual-boundary condition asserts the existence of boundary matrices with non-vanishing transfer overlap under the twist \(u\). The relationship with physical endpoint operators (Definition 10.3.1.7) is analyzed in [ 183 ] .

A tensor \(A\) is injective if its matrices span the full matrix algebra: \(\operatorname{span}_{\mathbb {C}}\{ A^i : i = 0,\ldots ,d{-}1\} = M_{D}(\mathbb {C})\).

A scalar 2-cochain \(\omega \colon G \times G \to \mathbb {C}^{\times }\) is a 2-cocycle if it satisfies, for all \(g,h,k \in G\),

A tensor \(A\) satisfies the virtual-insertion zero-correlation-length condition if its transfer map is idempotent (\(\mathcal{E}_A^2=\mathcal{E}_A\)) and it satisfies virtual-insertion distance independence (Definition 17.5.11).

Set \(d=D=2\). The Kitaev-chain tensor is

with the twist \(Y=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\) at the boundary [ 17 , Appendix A, “The Kitaev chain” ] , [ 9 ] .

For a word \(w=(i_1,\ldots ,i_N)\) the twisted periodic amplitude is \(\operatorname{tr}(YA^{i_1}\cdots A^{i_N})\), the coefficient of \(w\) in (40).

A tensor \(A\) is \(L\)-block injective if

An injective tensor is \(1\)-block injective.

The tensor \(C^i=B^i\, (Y\oplus 1)\in M_{3}(\mathbb {C})\), with \(B^i\) as in (8), places the singlet (7) on the two spin levels \(0,1\) of the bond and carries the empty level \(2\) through. Explicitly \(C^0=E_{02}-E_{21}\) and \(C^1=E_{12}+E_{20}\).

Let \(S^\alpha =\sigma ^\alpha /2\) for \(\alpha =x,y,z\) be the spin-\(\tfrac 12\) operators, and for two sites \(j,k\) of a chain let \(\mathbf S_j\cdot \mathbf S_k=\sum _\alpha S^\alpha _jS^\alpha _k\). On a periodic chain of \(N\) sites, with indices modulo \(N\),

[ 17 , Appendix A, “The Majumdar-Ghosh model” ] . The three-site term is \(h=\tfrac 12(\mathbf S_0\cdot \mathbf S_1+\mathbf S_1\cdot \mathbf S_2 +\mathbf S_0\cdot \mathbf S_2)\) on the sites \(0,1,2\) of a window.

Let \(f(a,b)\) be a pair amplitude on two sites. For a chain of \(N\) sites with configurations \(\sigma =(\sigma _0,\ldots ,\sigma _{N-1})\) put

For even \(N\) these are the coverings \((1,2)(3,4)\cdots (N-1,N)\) and \((2,3)(4,5)\cdots (N,1)\) of the periodic chain by the pair state \(\sum _{ab}f(a,b)|ab\rangle \). The normalized singlet amplitude is \(s(a,b)=Y_{ab}/\sqrt2\). For a word \(w\), the pair product is the product of \(f\) over the consecutive pairs \((w_0,w_1),(w_2,w_3),\ldots \), and zero if \(|w|\) is odd.

Set \(d=2\). Let \(E_{ab}\) denote the matrix units of \(M_{3}(\mathbb {C})\), and let

The review prints the tensor \(A=\bigl[|0\rangle {\bigl[(02|+(20|\bigr]}+|1\rangle {\bigl[(12|+(21|\bigr]}\bigr]\otimes Y\) [ 17 , Appendix A, “The Majumdar-Ghosh model” ] . Read literally, with \((ab|\) the bra \(E_{ab}\) of the pair of virtual indices and \(\otimes \) the Kronecker product, it is the tensor \(\widetilde A^i=B^i\otimes Y\in M_{6}(\mathbb {C})\) on the bond \(\mathbb {C}^3\otimes \mathbb {C}^2\cong \mathbb {C}^6\), with \(Y\) as in (7).

Set \(d = 2\) (spin-\(\tfrac 12\)) and \(D = 3\). The Majumdar–Ghosh tensor is the family \(\{ A^0, A^1\} \subset M_{3}(\mathbb {C})\) defined by

The level \(0\) is the closed link between two dimers, the levels \(1,2\) carry the spin of an open singlet partner, and the relative sign \(-\tfrac 1{\sqrt2}\) is the antisymmetry of the singlet.

A (translation-invariant, PBC) MPS tensor with physical dimension \(d\) and bond dimension \(D\) is a collection of matrices \(\{ A^i\} _{i=0}^{d-1}\), where \(A^i \in M_{D}(\mathbb {C})\), indexed by a physical index \(i \in \{ 0, \ldots , d{-}1\} \). Such a tensor defines an MPV family. Diagrammatically,

The black node denotes the tensor \(A\), the horizontal legs are virtual, and the upper leg is the physical index \(i\).

The transfer map \(\mathcal{E}_A\) of an MPS tensor \(A\) satisfies transfer map idempotence when \(\mathcal{E}_A\circ \mathcal{E}_A=\mathcal{E}_A\).

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

of \(N\) copies of the local tensor \(A\), with the outer virtual legs closed by the trace.

The MPV overlap between two tensors \(A\) (of bond dimension \(D_1\)) and \(B\) (of bond dimension \(D_2\)) at system size \(N\) is

Diagrammatically, the overlap contracts the MPV ring of \(A\) against the conjugate ring of \(B\) along their shared physical indices, both virtual rings closed by the trace:

This pairing is linear in the first MPV coefficient and conjugate-linear in the second. By Lemma 2.8.4, it is the complex conjugate of the Hilbert-space inner product from Definition 2.8.2.

For a finite family of blocks, put \(j \sim k\) when there is a nonzero scalar factor \(\zeta \) such that \(|V^{(N)}(B_k)\rangle =\zeta ^N|V^{(N)}(B_j)\rangle \) for every positive length \(N\). The associated tuple enumerates the finite quotient, chooses one representative per class, enumerates that representative first in its class, records the scalar relating the representative to each member, and proves that distinct representatives are not MPV-phase equivalent. The pairs \((j,q)\), with \(j\) a phase class and \(q\) a copy in that class, are in bijection with the original block indices.

The subspace spanned by the MPS vector \(V^{(N)}(A)(\sigma )=\operatorname{tr}(A^{\sigma _0}\cdots A^{\sigma _{N-1}})\).

A factor system \(\omega \) has a non-trivial class when it is not cohomologous to the constant cocycle \(1\).

A tensor \(A\) is normal [ 17 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

An MPS tensor \(A\) has on-site symmetry under a representation \(U:G\to \mathrm{GL}_d(\mathbb {C})\) if \(\mathcal{V}(A)=\mathcal{V}(\widetilde A_g)\) for every \(g\in G\). This equality of periodic MPV families provides the condition for constructing virtual symmetry gauges.

For a tensor \(A\) with bond dimension \(D\), a left boundary covector \(v_L\in \mathbb {C}^D\), a right boundary vector \(v_R\in \mathbb {C}^D\), and a word \(w=(i_1,\ldots ,i_k)\), the open-boundary contraction is the bilinear pairing

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) ] .

The open-boundary state on \(N\) sites assigns to each configuration \(\sigma =(\sigma _1,\ldots ,\sigma _N)\) the amplitude \((v_L| A^{\sigma _1}\cdots A^{\sigma _N}\, |v_R)\).

The chain Hamiltonian is

Denote by \(H_{N,L}^{\mathrm{ES}}(A)\) the conjugate of \(H_N(A,L)\) by the canonical identification between the coefficient space and \(\ell ^2(\{ 0,\ldots ,d-1\} ^N)\).

The parent interaction at range \(L\) is the canonical orthogonal projector specified in (14).

Let \(g\mapsto U_g\) be a representation of a finite group \(G\) on the virtual system of a tensor \(A\), and let \(\Pi _U\) be the projector onto the \(U_g\)-invariant subspace. The tensor is \(G\)-injective if \(\mathcal{P}(A)U_g=\mathcal{P}(A)\) for every \(g\) and \(\mathcal{P}(A)\) is injective on the invariant subspace [ 235 , Section “Two dimensions: \(G\)-injective PEPS” ] . It is \(G\)-isometric if moreover \(\mathcal{P}(A)\) restricted to the invariant subspace is a multiple of an isometry, \(\langle \mathcal{P}(A)x,\mathcal{P}(A)y\rangle =c\, \langle x,y\rangle \) for invariant \(x,y\) and one constant \(c{\gt}0\) [ 235 , Section “Isometric PEPS” ] . When the virtual action factors as \(U_g=U_g^{\ell }\otimes U_g^r\otimes U_g^t\otimes U_g^b\), the invariance of a four-leg tensor reads

\(=\)

The four factors include the contragredient actions required by the orientations of the virtual bonds.

A left-canonical irreducible MPS tensor \(A\) of bond dimension \(D\) is \(m\)-periodic (for some \(m\ge 1\)) if the peripheral spectrum of its transfer map \(\mathcal{E}_A\) is exactly the \(m\)-th roots of unity \(\{ e^{2\pi i k/m}:0\le k{\lt}m\} \). This is the normalized form-II convention used later in the source. A \(1\)-periodic tensor is exactly an irreducible tensor with primitive transfer map (the corresponding definition in the companion quantum-channel volume  [ 216 , “Primitive map” ] ).

A tensor \(A\) of bond dimension \(D\) is a translation-invariant canonical representation when its bond indices split into finitely many blocks of positive size, and there are weights \(1\ge \lambda _j{\gt}0\) and tensors \(A^j\) on the blocks with

such that every block satisfies the three conditions of Theorem 7.5.6: \(\sum _iA_i^j(A_i^j)^\dagger =\mathbb {1}\), there is a diagonal positive definite \(\Lambda ^j\) with \(\sum _i(A_i^j)^\dagger \Lambda ^jA_i^j=\Lambda ^j\), and the identity is the only fixed point of \(X\mapsto \sum _iA_i^jX(A_i^j)^\dagger \) up to scalars. This is the block form of [ 228 , Theorem 4 ] , which lists the blocks consecutively.

An MPS tensor \(A\) is a renormalization fixed point if there exists an isometry \(V\colon \mathbb {C}^d\to \mathbb {C}^{d^2}\) with coefficients \(V_{(i_1,i_2),j}\) satisfying \(V^\dagger V=\mathbb {1}\) such that for all physical indices \(i_1,i_2\in \{ 0,\ldots ,d-1\} \),

This expresses the physical two-site blocking relation \(AA=A\) of [ 13 , Definition 3.2 ] .

For local vectors \(|\phi ^s\rangle =\sum _{i}a^{i,[s]}|i\rangle \in \mathbb {C}^d\), \(s=1,\ldots ,N\), the product vector \(|\psi \rangle =|\phi ^1\rangle \otimes \cdots \otimes |\phi ^N\rangle \) has coefficient \(a^{i_1,[1]}\cdots a^{i_N,[N]}\) on \(|i_1,\ldots ,i_N\rangle \). Its tensor has bond dimension \(D=1\): at site \(s\) the matrix of physical index \(i\) is the \(1\times 1\) matrix \(a^{i,[s]}\). The product vector \(\bigotimes _{s\in I}|\phi ^s\rangle \) is defined in the same way for any finite set of sites \(I\).

Set \(A^0=\operatorname {diag}(q,1)\) and \(A^1=|0\rangle \langle 1|\). Write \(\Omega _N=|0\cdots 0\rangle \) and \(\Psi _N(q)=\sum _{k=0}^{N-1}q^k|0\cdots 1_k\cdots 0\rangle \).

Fixing the first physical index \(i\) gives the state \(\sigma \mapsto \psi (i,\sigma _1,\ldots ,\sigma _L)\).

Fixing the last physical index \(j\) gives the state \(\sigma \mapsto \psi (\sigma _1,\ldots ,\sigma _L,j)\).

Given a matrix \(M \in \mathbb {C}^{m\times d}\) and an MPS tensor \(A\) with physical dimension \(d\), the physical-index rotation is the tensor with physical dimension \(m\) given by \((A_M)^i := \sum _j M_{ij}\, A^j\) for \(i=0,\ldots ,m-1\). When the matrix is a unitary \(u\), the rotation is written \(A_u\).

Let \(A^{[1]},\ldots ,A^{[N]}\) be tensors with bond dimension \(D\), one for each site, and let \(v_L,v_R\in \mathbb {C}^D\). The open-boundary contraction of the configuration \((\sigma _1,\ldots ,\sigma _N)\) is

When all sites carry the same tensor this is the open-boundary contraction of Definition 15.5.2.

Let \(O_1,O_2\) act on two disjoint contiguous regions of positive lengths \(L_1,L_2\geq 1\) in a periodic chain, with complementary gap lengths \(n_1,n_2\geq 0\). Physical correlations are independent of distance if, for all \(m_1,m_2\geq 0\) satisfying \(n_1+n_2=m_1+m_2\),

This allows either region to be translated along the chain without crossing the other, including adjacent regions (\(n_i=0\)) [ 13 , Definition 3.3 ] .

Let \(\rho \) be a projective representation of \(G\) on \(\mathbb {C}^D\) with factor system \(\omega \), let \(\phi \) be a character of \(G\), and put \(W_g=\rho (g^{-1})\). The SPT fixed-point tensor has physical space \(\mathbb {C}^D\otimes \mathbb {C}^D\) and letters \(A^{(a,b)}=D^{-1/2}|a\rangle \langle b|\), and \(G\) acts on the physical space by \(U(g)=\phi (g)\, W_g^{T}\otimes W_g^{-1}\). The projective representation need not be unitary; when it is, \(W_g^{-1}=W_g^{\dagger }\), so with \(X_g=W_g^{\dagger }\) the conjugation \(W_gA^iW_g^{-1}\) is the source’s \(X_g^{\dagger }A^iX_g\).

The transfer map associated to a tensor \(A\) is the linear map \(\mathcal{E}_A : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) defined by

Diagrammatically, the transfer map is the double-layer contraction

in which the upper node denotes \(A\), the lower node denotes \(A^\dagger \), and the physical index is summed over between them.

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

For \(a,b\in \{ 0,1\} \), the vector \(|ab\rangle \in (\{ 0,1\} ^{2}\to \mathbb {C})\) is the indicator of the configuration \((\sigma _1,\sigma _2)=(a,b)\).

For \(N\geq 0\), the W-state amplitude is the indicator of the single-excitation configurations:

A tensor \(B\) with physical dimension \(2\) and bond dimension \(D\) is a periodic representation of \(|W_N\rangle \) when \(\operatorname {tr}(B^{\sigma _1}\cdots B^{\sigma _N})=|W_N\rangle (\sigma )\) for every configuration \(\sigma \) of \(N\) sites.

Following [ 17 , Appendix A, “The W state” ] , set \(d=D=2\). The W tensor is the site-independent family

with left boundary covector \((l|=(0|\) and right boundary vector \(|r)=|1)\). Here \(A^1\) is the single raising operator: it sends \(|1)\mapsto |0)\) and annihilates \(|0)\), and squares to zero.

For \(N\geq 0\) and \(m\geq 0\), let \(u^{(N)}_m\) be the indicator of the configurations of \(N\) sites with exactly \(m\) excitations. In particular \(u^{(N)}_1\) is the W amplitude of Definition 15.5.4.

The word span at length \(n\) is the subspace

This is  [ 233 , Section II ] .

The commutator phase of the AKLT factor system on the two generators is \(-1\).

With the physical labels \(0,1,2\) of Definition 15.3.7 read as \(S_z=0,+1,-1\), the tensor \(A\) of Definition 15.3.1 and the tensor \(T\) of Definition 15.3.7 satisfy \(A^{s}=\sqrt{3/2}\, \sigma _z T^{s}\sigma _z^{-1}\) for every \(s\).

The matrices of (3) are \(A^{0}=\sigma _z/\sqrt2\), \(A^{+1}=-|0\rangle \! \langle 1|\), and \(A^{-1}=|1\rangle \! \langle 0|\).

The commutator phase of the cluster-state factor system on the two generators is \(-1\).

The tensor of Definition 15.7.1 is the transpose of the tensor of Definition 15.7.4, and the two are gauge equivalent: with \(H = \bigl(\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\bigr)\), the review tensor is \(H A^s H^{-1}\), since \(H|\pm \rangle = \sqrt{2}\, |0\rangle \) and \(\sqrt{2}\, |1\rangle \) respectively. Hence both tensors generate the same matrix product vectors at every length, and their length-\(2\) blockings are gauge equivalent by the same matrix.

In the valence-bond description of  [ 242 ] , neighbouring virtual qubits share the bond \(|00\rangle + |01\rangle + |10\rangle - |11\rangle \) and each site maps both of its virtual qubits to the physical one by \(|0\rangle \langle 00| + |1\rangle \langle 11|\), so that the coefficient of \(|s\rangle \) on a ring of \(N \geq 1\) sites is \(\prod _{j=1}^{N} H_{s_j s_{j+1}}\) with \(H = \bigl(\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\bigr)\) and indices modulo \(N\). Let \(B^s = |s\rangle \! \langle s| H\). Then for every \(N \geq 1\) and every \(s \in \{ 0,1\} ^N\),

so \(B\) is a matrix product description of the valence-bond state, and \(B^s = \sqrt{2}\, A^s\).

If \(x\geq 0\), then the complex coercion of its real square root satisfies \(((\sqrt{x}:\mathbb R):\mathbb C)^2=x\).

If \(|\lambda _2| {\lt} 1\) and \(\lambda _2 \ne 0\), then \(\xi {\gt} 0\).

The GHZ tensor on a \(d\)-level system with \(d=2\) is the tensor of Definition 15.2.7.

If \(\psi \in G_{L+1}(A)\), i.e. if \(\psi (\sigma )=\operatorname{tr}(A^\sigma X)\) for some \(X\), then, for each \(j\), the state obtained by fixing the last index to \(j\) lies in \(G_L(A)\) with witness \(A^jX\).

If \(\psi \in G_{L+1}(A)\) with \(\psi (\sigma )=\operatorname{tr}(A^\sigma X)\), then, for each \(i\), the state obtained by fixing the first index to \(i\) lies in \(G_L(A)\) with witness \(XA^i\).

Let \(A\) be a tensor of \(D \times D\) matrices over physical dimension \(d\), \(L\)-block injective with \(0 {\lt} L\). Then \(A\) is \(m\)-block injective for every \(m \ge L\). This is the chain form of the statement that any region of at least the blocking size is injective, derived in the closing section of [ 220 ] (“Normal MPS: alternative proof”) from its union lemma.

If the word \(w\) contains the letter \(1\) exactly \(k\) times, then \(A^{i_1}\cdots A^{i_N}=Y^k\).

Let \(X=\operatorname {diag}(1,i)\), which is invertible with inverse \(\operatorname {diag}(1,-i)\), and let \(\varphi _0=1\), \(\varphi _1=i\). Then \(A^jX=\sqrt2\, \varphi _j\, XE^j\) for \(j=0,1\): up to the on-site phase \(\operatorname {diag}(1,i)\) on the physical index and the factor \(\sqrt2\), the Kitaev-chain tensor is gauge equivalent to the even-parity tensor.

For every word, the product \(M=A^{i_1}\cdots A^{i_N}\) satisfies

Of the four ways to close the virtual indices at the two ends of an open chain, the two diagonal closures give the same amplitudes and the two off-diagonal closures differ by a sign  [ 9 ] .

\(A^0=\mathbb 1\), \(A^1=Y\), \((A^1)^2=Y^2=-\mathbb 1\), and \(Y\) commutes with \(A^0\) and \(A^1\) [ 9 ] .

Every vector of \(\mathcal G_{K}(A)\) equals \(\Gamma _{K}(X)\) for a matrix \(X\) with \(GX=(-1)^{K}XG\). For \(K\ge 3\), a matrix \(X\) with \(GX=(-1)^{K}XG\) and \(\Gamma _{K}(X)=0\) is zero.

For \(M\ge 3\), \((\mathcal G_{M}(A)\otimes \mathbb {C}^2)\cap (\mathbb {C}^2\otimes \mathcal G_{M}(A)) =\mathcal G_{M+1}(A)\). Consequently, for \(N\ge 3\) every vector of \(\mathcal G_{N,3}(A)\) lies in \(\mathcal G_{N}(A)\).

If \(0{\lt}N\) and \(L\le N\), then \(V^{(N)}(A)\in \mathcal G_{N,L}(A)\).

Let \(\zeta A\) denote the tensor with matrices \(\{ \zeta A^i\} _{i=0}^{d-1}\). Then, for every \(N\) and every configuration \(\sigma = (i_1, \ldots , i_N)\), \(V^{(N)}(\zeta A)_\sigma = \zeta ^{N} V^{(N)}(A)_\sigma \).

If \(A\) has on-site symmetry under \(U\), then so does \(cA\) for every \(c\in \mathbb {C}\), and so does every tensor gauge equivalent to \(A\).

For every \(A\), \(L\), and \(N\), the operator \(H^{\mathrm{open}}_{N,L}(A)\) is positive.

If \(\psi \in \mathcal G_{N,L}(A)\), then \(T_s\psi \in \mathcal G_{N,L}(A)\) for every cyclic translation \(s\).

The Hilbert-space local interaction \(P_L^{\mathrm{ES}}(A)\) annihilates exactly the Hilbert-space local ground space: \(P_L^{\mathrm{ES}}(A)v=0\) if and only if \(v\in G_L^{\mathrm{ES}}(A)\).

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.

Let \(\sum _iA_iA_i^\dagger =\mathbb {1}\), let the dual map \(X\mapsto \sum _iA_i^\dagger XA_i\) have a positive definite fixed point, and let every fixed point of \(X\mapsto \sum _iA_iXA_i^\dagger \) be a multiple of the identity. Then no orthogonal projection other than \(0\) and \(\mathbb {1}\) is invariant under every \(A_i\).

Let \(A\) be in the canonical form of Definition 23.3.6 with blocks \(A^1,\ldots ,A^b\) of sizes \(D_1,\ldots ,D_b\), each \(L_0\)-block injective for a common \(L_0{\gt}0\), no two related by a gauge transformation and a phase. For every \(n\geq \max (L_0,3(b-1)(L_0+1))\) the tuples \((A^1_{w},\ldots ,A^b_{w})\) over words \(w\) of length \(n\) span \(\bigoplus _{j}M_{D_j}(\mathbb {C})\). The same holds for any family of blocks \(A^1,\ldots ,A^b\), not assumed to come from a canonical form, provided each block satisfies \(\sum _iA^j_i(A^j_i)^\dagger =\mathbb {1}\), its dual map \(X\mapsto \sum _i(A^j_i)^\dagger XA^j_i\) has a positive definite fixed point, and no orthogonal projection other than \(0\) and \(\mathbb {1}\) is invariant under every \(A^j_i\).

Let \(A^1,\ldots ,A^b\) be tensors of bond dimensions \(D_1,\ldots ,D_b\) whose word tuples span \(\bigoplus _jM_{D_j}(\mathbb {C})\) at lengths \(R\) and \(R'\), and let \(c_1,\ldots ,c_b\) be nonzero. If for every word \(\tau \) of length \(R'\) the function \(\sigma \mapsto \sum _j c_j\, \operatorname{tr}\bigl(A^j_{\sigma }A^j_{\tau }\bigr)\) of words \(\sigma \) of length \(R\) lies in a subspace \(V\), then \(\sum _jD_j^2\leq \dim V\).

For every physical word \(\sigma \),

The standard boundary contraction \(\langle 0|A^{\sigma _1}\cdots A^{\sigma _N}|1\rangle \) is \(\Psi _N(q)(\sigma )\).

On a ring of at least two sites, the cyclic successor is a permutation without fixed points. In cyclic-group coordinates it is addition by one, and its inverse is subtraction by one. Exchanging two entries of an extracted window exchanges the corresponding physical sites when that window is inserted back into the configuration.

At real hopping \(\mu \), the canonical two-site parent interaction is

Equivalently, with occupation projections \(n_L,n_R\), it is \((\mu ^2n_L+n_R-\mu T)/(1+\mu ^2)\), where \(T\) exchanges \(01\) and \(10\) and annihilates \(00\) and \(11\).

If \(A^{[s]}=A\) for every site \(s\), the site-dependent open-boundary contraction of \((\sigma _1,\ldots ,\sigma _N)\) is the amplitude of the open-boundary state of \(A\) at that configuration.

Each Hilbert-space local term \(h_{i,\mathrm{ES}}\) is positive, hence the full Hilbert-space parent Hamiltonian \(H_{N,\mathrm{ES}}\) is positive.

If \(R\in \mathrm{GL}_{D}(\mathbb {C})\) satisfies \((\widetilde A_g)^i=R\, A^i\, R^{-1}\) for all \(i\), then \((\widetilde{(A^{[L]})}^{\, U^{[L]}}_g)^I=R\, (A^{[L]})^I\, R^{-1}\) for every blocked index \(I\).

Split \(N=R+R'\) sites into a first piece of \(R\) sites and a second piece of \(R'\) sites. For configurations \(\sigma \) of the first piece and \(\tau \) of the second,

Hence for every \(\tau \) the function \(\sigma \mapsto \langle \sigma \tau | W_N \rangle \) lies in the span of \(u^{(R)}_0\) and \(u^{(R)}_1\), and the reduced state of \(|W_N\rangle \) on the first piece has rank at most two.

Let \(B\) be a tensor of bond dimension \(D\ge 1\) whose exact word spans equal \(M_{D}(\mathbb {C})\) at all sufficiently large lengths. If \(S_1(B)\) contains a matrix with a nonzero eigenvalue, then \(S_{D^2}(B)=M_{D}(\mathbb {C})\). This is the nonzero-eigenvalue bound of [ 244 , Theorem 6.9(3) ] , in its word-span form.

Although the cluster-state tensor is not injective at length \(1\), the length-\(2\) blocked tensor is injective. Using \(\langle 0 | \pm \rangle = \tfrac {1}{\sqrt{2}}\), \(\langle 1 | + \rangle = \tfrac {1}{\sqrt{2}}\), and \(\langle 1 | - \rangle = -\tfrac {1}{\sqrt{2}}\), the four products \(A^{ij} = A^i A^j\) are

Since \(\{ |+\rangle ,|-\rangle \} \) and \(\{ \langle 0|,\langle 1|\} \) are bases, these four rank-one matrices are linearly independent and span \(M_{2}(\mathbb {C})\).

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.

The length-\(2\) blocked AKLT tensor is injective: its nine matrices span \(M_{2}(\mathbb {C})\).

The length-\(2\) blocked AKLT tensor is on-site symmetric under \(\mathbb {Z}_2 \times \mathbb {Z}_2\), with each group element acting by the Kronecker square \(P_g \otimes P_g\) of the corresponding single-site spin-\(1\) \(\pi \)-rotation \(P_g\) (Definition 15.3.19).

For \(N\ge 3\), the periodic two-site chain ground space of the AKLT tensor is spanned by the AKLT state:

Thus the two-site parent Hamiltonian has a unique ground state although the AKLT tensor becomes injective only after blocking \(L_0 = 2\) sites [ 17 , line 2095 ] . On two periodic sites the two-site chain ground space is \(\mathcal G_2(A_{\mathrm{AKLT}})\) itself, so the condition \(N\ge 3\) is necessary.

The AKLT correlation length, computed from the subleading eigenvalue \(\lambda _2 = -\tfrac 13\), equals

It is finite and positive.

Let \(\Lambda \) be a positive definite fixed point of the adjoint transfer map of the length-\(2\) blocked AKLT tensor. Every eigenspace of \(\Lambda \) has even dimension.

The two virtual matrices that implement the symmetry of Theorem 15.3.11, namely \(i\sigma _y\) and \(\sigma _z\), anticommute.

For the AKLT tensor \(A\), the three-site intersection property holds:

where \(\mathcal G_{[a,b]}\) denotes the local ground space \(\mathcal G_{b-a+1}(A)\) placed on sites \(a,\ldots ,b\). This is the identity \(\mathcal G_{1,2}\cap \mathcal G_{2,3} = \mathcal G_{1,2,3}\) that [ 17 , line 2095 ] asks to check by hand; it is not an instance of the general intersection property, since the AKLT tensor is not injective on one site.

For the AKLT tensor \(A\) of Definition 15.3.7, a two-site vector \(v\) lies in \(\mathcal G_2(A)\) if and only if

The AKLT factor system (Theorem 15.3.15), built from the anticommuting gauges \(i\sigma _y\) and \(\sigma _z\) (Theorem 15.3.12), is not a coboundary: its commutator phase on the two generators is \(-1\). Its class in \(H^2(\mathbb {Z}_2 \times \mathbb {Z}_2, \mathrm{U}(1)) \cong \mathbb {Z}_2\) is therefore non-trivial, so the AKLT chain realizes a non-trivial SPT phase rather than a trivial product state.

The transfer map of the AKLT tensor has a unique eigenvalue of maximal modulus (a simple spectral-radius eigenvalue); in particular the AKLT tensor is normal. Concretely, the length-\(2\) blocked tensor is injective.

The AKLT tensor is not injective at a single site: \(\operatorname{span}_{\mathbb {C}}\{ A^0, A^1, A^2\} \) is the three-dimensional space of traceless matrices \(\mathfrak {sl}(2,\mathbb {C})\), which is a proper subspace of \(M_{2}(\mathbb {C})\).

The AKLT factor system is a genuine \(2\)-cocycle, being the factor system of an explicit projective representation, so it represents an element of \(H^2(\mathbb {Z}_2 \times \mathbb {Z}_2, \mathrm{U}(1))\).

For \(N\ge 2\), the kernel of the open-chain two-site projector parent Hamiltonian of the AKLT tensor is the open-boundary MPS space,

and it is four-dimensional. The count is for this open-chain projector parent Hamiltonian. The passage [ 17 , lines 1171–1174 ] states a fourfold degeneracy for the AKLT chain with open boundary conditions, whose displayed Hamiltonian is the polynomial spin interaction. Summing the local identity \(P^{(2)}=\frac12\bigl(h_{\mathrm{poly}}+\frac23 I\bigr)\) over the \(N-1\) open bonds gives \(H_{\mathrm{proj}}=\frac12\bigl(H_{\mathrm{poly}}+\frac{2(N-1)}{3}I\bigr)\). The projector kernel is the ground eigenspace of the polynomial Hamiltonian, equivalently the kernel of this shifted operator, and not the kernel of the polynomial Hamiltonian itself. For \(N=2\) the polynomial interaction has eigenvalues \(-\frac23\) and \(\frac43\), so its kernel is trivial, while the projector kernel is four-dimensional. The identification of the projector kernel with the open-boundary matrix-product space uses the boundary vectors of that passage together with the three-site intersection check of [ 17 , line 2095 ] , which the review prints for the periodic chain. As a consequence of Theorem 12.3.7.19 at \(L_{0}=2\), and not as a statement of the review, the three-site open-chain parent Hamiltonian also has a four-dimensional kernel for \(N\ge 3\). The spin-\(0\) statement of the same passage is proved in Theorem 15.10.24.

For \(N\ge 2\), the ground energy is \(E_0=-2(N-1)/3\), and

The degeneracy is the open-boundary statement of  [ 17 , lines 1171–1174 ] ; the explicit energy is the consequence of the polynomial normalization used here.

For \(N\ge 2\), the open AKLT ground space contains one spin-\(0\) state and a spin-\(1\) triplet. In the boundary parametrization \(\Gamma _N(X)(\sigma )=\operatorname {tr}(A^\sigma X)\),

The singlet line is \(\Gamma _N(\mathbb {C}I)\), and the triplet is \(\Gamma _N(\{ X:\operatorname {tr}(X)=0\} )\). Thus the ground-space intersections with the squared-spin eigenspaces for \(0\) and \(2\) have dimensions \(1\) and \(3\), respectively. The singlet count is stated in  [ 17 , lines 1171–1174 ] ; the triplet description follows from the displayed action.

Each of the three Pauli matrices \(\sigma _x\), \(\sigma _y\), \(\sigma _z\) is an eigenvector of the AKLT transfer map with eigenvalue \(-\tfrac 13\):

For \(N\ge 2\) the Hamiltonian (57) satisfies

where \(H^{(2)}_{N,\mathrm{AKLT}}\) is the two-site parent Hamiltonian of the AKLT tensor. Hence every eigenvalue of \(H_{\mathrm{AKLT},N}\) is real and at least \(-\tfrac {2N}{3}\). For \(N\ge 3\) the eigenspace for \(-\tfrac {2N}{3}\) is \(\mathbb {C}\, V^{(N)}(A_{\mathrm{AKLT}})\), so the AKLT state is the unique ground state of \(H_{\mathrm{AKLT},N}\) [ 2 ] , [ 228 , local TeX lines 340–349 ] . The ground energy \(-\tfrac {2N}{3}\) is derived here from the identity above; the local source states only the ground-state property, for no specified boundary condition.

The basis (4) is orthonormal, and so is the basis obtained by replacing \(|+\rangle \) with \(-\frac{i}{\sqrt2}(|-1\rangle +|+1\rangle )\): the matrices whose columns are these vectors are unitary.

The tensor of Definition 15.3.1 has the same local ground spaces, and a periodic matrix product state spanning the same line, as the representative of Definition 15.3.7. Hence on a periodic chain of \(N\geq 3\) sites its two-site parent Hamiltonian has the AKLT state as its unique ground state.

Let \(U\) be the unitary whose columns are the vectors of (4). The tensor (3) expressed in this basis, that is, the tensor \(B^{a}=\sum _{s}\overline{U_{sa}}\, A^{s}\) obtained by rotating the physical index by \(U^{\dagger }\), is

This is the form \(\sigma _i/\sqrt2\) printed in [ 17 , Appendix A, “The AKLT state” ] up to the virtual gauge \(\sigma _y\) and the scalar \(-1\): \(B^{a}=\sigma _y\big(-\sigma _a/\sqrt2\big)\sigma _y^{-1}\) for \(a=-,+,0\) with \(\sigma _-=\sigma _x\), \(\sigma _+=\sigma _y\), \(\sigma _0=\sigma _z\). Hence on \(N\) sites the state coefficients of \(B\) are \((-1)^N\) times those of the printed form, and the two describe the same state up to a global sign. In the basis with \(|+\rangle =-\frac{i}{\sqrt2}(|-1\rangle +|+1\rangle )\) the tensor becomes exactly \(\sigma _i/\sqrt2\). The printed identity \(A^{+}=\sigma _y/\sqrt2\) in the basis (4) holds only up to this gauge and sign, as recorded in [ 151 ] .

The three coefficient matrices \(P^{s}\) with \(A^{s}=P^{s}Y\) are symmetric, \((P^{s})^{T}=P^{s}\), so they are states of the symmetric (spin-\(1\)) subspace of two spin-\(\tfrac 12\) particles, and they are orthonormal: \(\operatorname{tr}\big((P^{s})^{\dagger }P^{t}\big)=\delta _{st}\).

For every \(g\in \mathbb {Z}_2\times \mathbb {Z}_2\), the twist of the tensor \(A\) of Definition 15.3.1 by \(g\) satisfies \(\sum _j (P_g)_{ij}A^{j}\, X_g=X_g\, A^{i}\), where \(X_g\) is the projective representation generated by \(i\sigma _y\) and \(\sigma _z\). Its factor system is the non-trivial class of \(H^2(\mathbb {Z}_2\times \mathbb {Z}_2,\mathrm{U}(1))\) (Theorem 15.3.16).

The tensor of Definition 15.3.1 is normal and on-site symmetric under the \(\mathbb {Z}_2\times \mathbb {Z}_2\) action of Theorem 15.3.11. Its Pauli form \(\sigma _i/\sqrt2\) of Theorem 15.3.6 is on-site symmetric under the spin-\(1\) representation of \(\mathrm{SO}(3)\).

The Cartesian AKLT tensor is on-site symmetric under the spin-\(1\) (defining) representation of \(\mathrm{SO}(3)\) on the physical index.

On two spin-\(1\) sites,

where \(P^{(2)}\) is the orthogonal projector onto the subspace of total spin \(2\), the eigenspace of \(\vec S_{\mathrm{tot}}^{2}\) for \(6=2(2+1)\), with \(\vec S_{\mathrm{tot}}=\vec S_0+\vec S_1\). This projector is the two-site parent interaction of the AKLT tensor, the orthogonal projector onto \(\mathcal G_2(A_{\mathrm{AKLT}})^{\perp }\). Writing the Hamiltonian as a sum of such spin-\(2\) projectors is the form of  [ 2 ] .

For every element \(g\) of \(\mathbb {Z}_2 \times \mathbb {Z}_2\), the length-\(2\) blocked AKLT tensor has virtual-boundary nondecay in the sense of Definition 10.3.2.10 for the twist by \(g\), including the identity group element, with the maximally mixed boundary state \(\Lambda = \tfrac {1}{2}\mathbb {1}\) as its stationary boundary state.

The value \(1\) is the leading eigenvalue of the AKLT transfer map, with the identity \(\mathbb {1}\) as eigenvector, and \(-\tfrac 13\) is a subleading eigenvalue, with \(\sigma _z\) as eigenvector. The subleading modulus is therefore \(\tfrac 13\).

Let \(R=e^{i\pi S_y}\) be the spin-\(1\) rotation, which sends \(|0\rangle \mapsto -|0\rangle \) and exchanges \(|\pm 1\rangle \), and let \(X=i\sigma _y\). The AKLT tensor satisfies \(\overline{\sum _jR_{ij}A^j}=X^\dagger A^iX\) and \((A^i)^{T}=-X^\dagger A^iX\) for every \(i\), and \(X\bar X=-\mathbb {1}\) [ 17 , Section III.A ] .

Every unitary gauge implementing the time-reversal or the reflection symmetry of the AKLT tensor has index \(X\bar X=-\mathbb {1}\).

Each single-site spin-\(1\) \(\pi \)-rotation matrix \(P_g\) of the \(\mathbb {Z}_2 \times \mathbb {Z}_2\) action is unitary: \(P_g P_g^\dagger = \mathbb {1}\).

The AKLT tensor is on-site symmetric under the \(\mathbb {Z}_2 \times \mathbb {Z}_2\) subgroup of \(\mathrm{SO}(3)\) generated by two commuting \(\pi \)-rotations about orthogonal axes, acting on the spin-\(1\) physical index.

If \(A\) is normal, \(D \ge 1\), and injective after blocking \(L_0{\gt}0\) sites, with \(N \ge 2\), \(L_0+1\le N\), and \(L_0 {\lt} L \le N\), then \(\mathcal G_{N,L}(A) = \operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \).

The length-\(2\) blocked cluster transfer map satisfies \(\mathcal{E}_A^2 = \mathcal{E}_A\). In closed form, it sends every \(X\) to the scalar \((X_{00} + X_{11})/2\) times the identity, so its image is one-dimensional.

For every \(N \geq 1\) and every \(s \in \{ 0,1\} ^N\),

with indices modulo \(N\), where \(\mathrm{CZ}_{a,b}\) multiplies the coefficient of \(|s\rangle \) by \((-1)^{s_a s_b}\). Thus the periodic matrix product vector of the tensor of Definition 15.7.1 is exactly the normalized cluster state obtained by acting with controlled-\(Z\) gates between nearest neighbours of a ring on \(|+\rangle ^{\otimes N}\)  [ 17 , Appendix A, “The cluster state” ] , [ 230 ] . For \(N \geq 3\) the \(N\) gates act on the distinct bonds of the ring. For \(N = 2\) the two gates act on the same pair and cancel, giving \(|+\rangle ^{\otimes 2}\); for \(N = 1\) the single gate \(\mathrm{CZ}_{1,1}\) is the Pauli \(Z\), giving \(|-\rangle \).

Let \(\Lambda \) be a positive definite fixed point of the adjoint transfer map of the length-\(2\) blocked cluster-state tensor. Every eigenspace of \(\Lambda \) has even dimension.

The virtual matrices implementing the two generators of Theorem 15.7.10, namely \(V_1 = \sigma _z\) and \(V_2 = \sigma _x\), anticommute (\(V_1 V_2 = -V_2 V_1\)).

For the cluster tensor \(A\) of Definition 15.7.4,

the \(+1\) eigenspace of \(K = \sigma ^z_1\sigma ^x_2\sigma ^z_3\).

The cluster-state factor system (Theorem 15.7.14), built from the anticommuting gauges \(\sigma _z\) and \(\sigma _x\) (Theorem 15.7.11), is not a coboundary: its commutator phase on the two generators is \(-1\). Its class in \(H^2(\mathbb {Z}_2 \times \mathbb {Z}_2, \mathrm{U}(1)) \cong \mathbb {Z}_2\) is therefore non-trivial, so the cluster state realizes a non-trivial SPT phase.

Although the cluster-state tensor is not injective at a single site, its length-\(2\) blocking is injective: the four length-\(2\) products span \(M_{2}(\mathbb {C})\). In particular, the cluster-state tensor is normal.

The cluster-state tensor is not injective: \(\operatorname{span}_{\mathbb {C}}\{ A^0, A^1\} = \operatorname{span}_{\mathbb {C}}\{ |+\rangle \! \langle 0|, |-\rangle \! \langle 1|\} \) has dimension \(2\), not \(\dim M_{2}(\mathbb {C}) = 4\).

The cluster-state factor system is a genuine \(2\)-cocycle, being the factor system of an explicit projective representation, so it represents an element of \(H^2(\mathbb {Z}_2 \times \mathbb {Z}_2, \mathrm{U}(1))\).

The tensor of Definition 15.7.1 is not injective, its length-\(2\) blocking is injective, and so it is normal. The length-\(2\) blocked tensor is on-site symmetric under the \(\mathbb {Z}_2 \times \mathbb {Z}_2\) action of Theorem 15.7.10; the two generators are implemented on the bond space by \(\sigma _x\) and \(\sigma _z\), the conjugates by \(H\) of the gauges of Theorem 15.7.11, and these form a projective representation with the factor system of Definition 15.7.12, whose class is non-trivial (Theorem 15.7.15). The blocked tensor has string order for every group element with the maximally mixed boundary state, and has zero correlation length.

Let \(A_{\mathrm{src}}\) be the tensor with the two matrices displayed in [ 228 , local TeX lines 378–387 ] , the source’s physical labels \(1,2\) written as \(0,1\):

Then

the \(-1\) eigenspace of \(K = \sigma ^z_1\sigma ^x_2\sigma ^z_3\).

Let \(N\ge 3\) and put \(U=(\sigma ^z)^{\otimes N}\) and \(H=\sum _i K_i\). Then

The common \(-1\) eigenspace of the \(K_i\) and the ground eigenspace of \(H\) are both \(\mathbb {C}|V^{(N)}(A_{\mathrm{src}})\rangle \). More precisely, with the usual inner product \(\langle v,w\rangle =\sum _s\overline{v_s}w_s\), the bound below holds for every vector \(v\):

Thus the ground energy is \(-N\), and the normalized ground state is unique up to phase. This proves the periodic-chain claim for the matrices in [ 228 ] , for odd as well as even \(N\ge 3\).

For \(N\ge 3\), the periodic three-site ground space of the source tensor is the common \(-1\) eigenspace of the translated stabilizers:

Let \(B\) be the SPT fixed-point tensor of the projective representation of \(\mathbb {Z}_2\times \mathbb {Z}_2\) by \(\sigma _z\) and \(\sigma _x\). The length-\(2\) blocked cluster-state tensor satisfies \(A^i=\sum _jV_{ij}B^j\) for a unitary \(V\) on the four-dimensional physical space. This unitary intertwines the two symmetries: \(u(g)\, V=V\, U(g)\) for every \(g\), where \(u\) is the on-site \(\mathbb {Z}_2\times \mathbb {Z}_2\) action on the blocked cluster state and \(U\) is the physical action of the fixed point. Every virtual projective representation of \(B\) has a non-trivial class [ 17 , Section III.A ] .

For \(N\ge 3\), the common \(+1\) eigenspace of the translated stabilizers is the periodic three-site chain ground space of the cluster tensor:

For \(N\ge 3\),

and this common eigenspace is one-dimensional.

For every element \(g\) of \(\mathbb {Z}_2 \times \mathbb {Z}_2\), the length-\(2\) blocked cluster-state tensor has virtual-boundary nondecay in the sense of Definition 10.3.2.10 for the twist by \(g\), including the identity group element, with the maximally mixed boundary state \(\Lambda = \tfrac {1}{2}\mathbb {1}\) as its stationary boundary state.

Let \(u\) be the on-site \(\mathbb {Z}_2\times \mathbb {Z}_2\) action of Theorem 15.7.10 and let \(X\) be the projective representation with factor system of Definition 15.7.12 whose generators act by \(\sigma _z\) and \(\sigma _x\). For every \(g\) and every letter \(i\) of the length-\(2\) blocked cluster-state tensor \(A\),

The length-\(2\) blocked cluster-state tensor is on-site symmetric under \(\mathbb {Z}_2 \times \mathbb {Z}_2\). The two generators act on the \(4\)-dimensional blocked physical space \((\mathbb {C}^2)^{\otimes 2}\) by \(\sigma _x \otimes \mathbb {1}\) and \(\mathbb {1}\otimes \sigma _x\), which commute and each square to the identity, defining a linear representation of \(\mathbb {Z}_2 \times \mathbb {Z}_2\).

The length-\(2\) blocked cluster tensor has zero correlation length: its idempotent transfer map (Theorem 15.7.20) gives correlations independent of the separation. Like the GHZ state, and unlike the AKLT state, the cluster state is a renormalization fixed point with zero correlation length.

For \(N\ge 2\) and \(a\in \{ 0,1\} \), one has \(|a\rangle ^{\otimes N}\in \mathcal G_{N,2}(A_{\mathrm{GHZ}})\).

Let \(A_0\) and \(A_+\) be the bond-dimension-one product tensors obtained from the first and second virtual diagonal entries of \(A\). Their formal MPV overlap at length \(n\) is

The tensor in Definition 15.9.1 satisfies physical correlation independence for arbitrary disjoint physical regions, including adjacent regions.

For \(N{\gt}0\) and \(\boldsymbol i=(i_1,\ldots ,i_N)\in \{ 0,1\} ^N\),

Hence the generated MPV is \(|0\rangle ^{\otimes N}+|+\rangle ^{\otimes N}\), where \(|+\rangle =(|0\rangle +|1\rangle )/\sqrt2\).

The transfer map is not idempotent, and the tensor has no physical blocking isometry. Thus it does not satisfy the pure-state renormalization fixed-point equation.

If \(A\) is periodic with period \(m\), then the cyclic-sector decomposition of \(A^{(m)}\) may be chosen with rectangular support isometries \(V_k\) such that

In particular, the same isometries realize the corner-letter identity \(\varphi _k(C_k^i)=P_k(A^{(m)})^iP_k\).

Let \(A\) be any tensor. If

then \(\Gamma _{L_0}(XA^j)=\Gamma _{L_0}(A^jY)\) for every physical letter \(j\).

If \(A\) and \(B\) are gauge equivalent, then they generate the same MPV family.

For \(N\ge 2\), the common nearest-neighbour cyclic ground space of the GHZ tensor satisfies

This is the cyclic-window form of the two-fold degeneracy in [ 17 , line 2205 ] .

For every \(N\), the matrix product vector of the tensor of Definition 15.2.1 on \(N\) sites is the GHZ state \(\sum _{i=0}^{d-1}|i,\ldots ,i\rangle \)  [ 17 , Appendix A, “The GHZ state” ] . For \(N=0\) both sides equal \(d\).

For \(d\ge 2\) the GHZ tensor on a \(d\)-level system is not injective.

Let the symmetric group on \(\{ 0,\ldots ,d-1\} \) act on \(\mathbb {C}^d\) by permutation matrices. The GHZ tensor on a \(d\)-level system is on-site symmetric under this action. In particular, for \(d\ge 1\) it is on-site symmetric under the cyclic group \(\mathbb {Z}_d\) acting by the cyclic shift \(|i\rangle \mapsto |i+1 \bmod d\rangle \).

The transfer map of the GHZ tensor on a \(d\)-level system extracts the diagonal, \(\mathcal{E}_A(X)_{\alpha \beta }=\delta _{\alpha \beta }X_{\alpha \alpha }\). In particular \(\mathcal{E}_A^2=\mathcal{E}_A\), so the tensor is a renormalization fixed point.

Let \(G\) act on the two virtual legs by \(L_g\otimes \bar L_g\). The map \(\mathcal P(A)=\sum _{ij}\sum _{\alpha \beta }A^{ij}_{\alpha \beta } |ij\rangle (\alpha \beta |\) is invariant under this action and injective on its invariant subspace, so the tensor is \(G\)-injective [ 17 , Section III.A, “Virtual symmetries” ] , [ 235 , Section “Two dimensions: \(G\)-injective PEPS” ] .

The transfer map of the \(G\)-injective tensor is \(\mathcal{E}_A(X)_{\alpha \beta }=|G|^{-1}\sum _{k\in G}X_{\alpha k,\beta k}\). It is idempotent, so every nonzero eigenvalue of the transfer map equals \(1\) [ 17 , Section III.A, “Virtual symmetries” ] .

Let \(G\) act on the physical index by \(L_g\otimes \bar L_g\), that is \((i,j)\mapsto (gi,gj)\). Each matrix \(A^{ij}\) is unchanged by this action, so the tensor is on-site symmetric under the regular representation [ 17 , Section III.A, “Virtual symmetries” ] .

For \(G=\mathbb {Z}_2\) the tensor is \(A^{ij}=\tfrac 12\sigma _x^{i+j}\). Let \(H=\bigl(\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\bigr)\). After the Fourier transform \(\tfrac 12 H\otimes H\) on the physical index \((i,j)\), the resulting tensor is gauge equivalent, through \(H\) on the bond, to the GHZ tensor on two-level systems blocked over two sites [ 17 , Section III.A, “Virtual symmetries” ] .

For the GHZ tensor \(A\) of Definition 15.2.7,

The GHZ tensor is a renormalization fixed point.

The GHZ tensor is not injective: \(\operatorname{span}_{\mathbb {C}}\{ A^0, A^1\} \) is the two-dimensional space of diagonal matrices in \(M_{2}(\mathbb {C})\), which does not equal \(M_{2}(\mathbb {C})\).

The order-parameter operator \(\sigma _z\) is a fixed point of the GHZ transfer map, \(\mathcal{E}_A(\sigma _z) = \sigma _z\).

The transfer map of the GHZ tensor satisfies \(\mathcal{E}_A^2 = \mathcal{E}_A\).

The matrix \(\sigma _z\) commutes with every GHZ matrix \(A^i\). A non-scalar matrix commuting with all \(A^i\) is the hallmark of a non-injective (\(\mathbb {Z}_2\)-injective) tensor: the on-site symmetry of Theorem 15.2.14 swaps the two one-dimensional injective blocks \(A^0 \leftrightarrow A^1\), while \(\sigma _z\) acts on each block by a phase.

Let \(\mathbb {Z}_2 = \{ 1, \sigma _x\} \) act on \(\mathbb {C}^2\) by the Pauli-\(X\) matrix \(\sigma _x\). The GHZ tensor is on-site symmetric under this action: for each \(g \in \mathbb {Z}_2\), the twisted tensor \(\widetilde A_g\) defines the same MPV family as \(A\), i.e. \(\mathcal{V}(A)=\mathcal{V}(\widetilde A_g)\).

The GHZ tensor has zero correlation length: its idempotent transfer map (Theorem 15.2.12) gives correlations independent of the separation. This contrasts with the AKLT state (Theorem 15.3.1.4), whose subleading eigenvalue \(-\tfrac 13\) produces a finite correlation length \(1/\log 3\).

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 \(\psi \in G_{K+1}(A)\).

If \(A\) is \(L_0\)-block injective, then \(\Gamma _{L_0}\) is injective.

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 \(A\) be an injective tensor, symmetric under a unitary representation \(U\). Assume \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), \(\Lambda \) is positive definite, \(\operatorname{tr}(\Lambda )=1\), and \(\mathcal{E}_A^\dagger (\Lambda )=\Lambda \). Then the virtual-boundary nondecay predicate holds for every group element \(g\).

Let the word \(w\) contain the letter \(1\) exactly \(k\) times. Then

Neither amplitude depends on the length \(N\) except through \(k\).

Read as an ordinary matrix product state, the Kitaev-chain tensor is reducible: the vector \((1,i)\) is a common eigenvector of \(A^0\) and \(A^1\), with eigenvalues \(1\) and \(i\), and the orthogonal projection onto the line it spans is a nontrivial projection invariant under both matrices. The tensor is not injective, and it is not normal: no blocking length makes the products span the full matrix algebra [ 17 , Section III.A.4 ] .

For \(c\in \mathbb C\) let \(a_c\) be the tensor of bond dimension one with \(a_c^0=1\) and \(a_c^1=c\); it is injective. Write \(a_\pm =a_{\pm i}\). For every word \(w\),

so the ordinary contraction of the twisted state is a sum of two injective matrix product states [ 17 , Section III.A.4 ] .

For every even \(N{\gt}0\), \(|V^{(N)}(C)\rangle =P^{\mathrm{even}}_Y+P^{\mathrm{odd}}_Y\), the sum of the two nearest-neighbour coverings by the singlet \(|10\rangle -|01\rangle \). This is the Majumdar–Ghosh superposition of [ 17 , Appendix A, “The Majumdar-Ghosh model” ] , with the tensor read as in [ 112 ] .

For every even \(N{\gt}0\), \(|V^{(N)}(C)\rangle =\sqrt2^{\, N/2}\, |V^{(N)}(A)\rangle \).

For even \(N{\gt}0\) and \(L\le N\), both coverings \(P^{\mathrm{even}}_s\) and \(P^{\mathrm{odd}}_s\) lie in \(\mathcal G_{N,L}(A)\).

For every even \(N{\gt}0\), \(|V^{(N)}(A)\rangle =P^{\mathrm{even}}_s+P^{\mathrm{odd}}_s\), the sum of the two nearest-neighbour coverings by the normalized singlet \((|10\rangle -|01\rangle )/\sqrt2\). For odd \(N\) the periodic vector vanishes.

On an even periodic chain of length \(N = 2m\ge 4\), let \(H_{N,\mathrm{MG}}^{(3)}\) be the range-three parent Hamiltonian of the Majumdar–Ghosh tensor, the periodic sum of translates of the local projector with kernel \(\mathcal G_{3}(A_{\mathrm{MG}})\). Then \(H_{N,\mathrm{MG}}^{(3)}\) has a two-dimensional ground space \(\mathcal G_{N,3}(A_{\mathrm{MG}})\), spanned by the two nearest-neighbour singlet coverings \((1,2)(3,4)\cdots (N-1,N)\) and \((2,3)(4,5)\cdots (N,1)\).

On an odd periodic chain of length \(N\ge 5\), the space \(\mathcal G_{N,3}(A_{\mathrm{MG}})\) is zero.

For \(N\ge 3\), every \(\psi \in \mathcal G_{N,3}(A)\) satisfies \(H_N\psi =-\tfrac {3N}{8}\psi \). In particular, for even \(N\ge 4\) each of the two singlet coverings, and for \(N\ge 3\) the periodic vector \(|V^{(N)}(A)\rangle \), is an eigenvector of (13) with eigenvalue \(-\tfrac {3N}{8}\).

The Majumdar–Ghosh tensor is isometric: \((A^0)^\dagger A^0 + (A^1)^\dagger A^1 = \mathbb {1}\).

The three-site local ground space \(\mathcal G_3(A)\) equals the eigenspace of \(h\) for the eigenvalue \(-\tfrac 38\).

The Majumdar–Ghosh tensor is not injective: every element of \(\operatorname{span}_{\mathbb {C}}\{ A^0, A^1\} \) has vanishing \((2,2)\) entry, so the span is a proper subspace of \(M_{3}(\mathbb {C})\).

The Majumdar–Ghosh tensor is not normal: no blocking length makes the products of that length span the full matrix algebra \(M_{3}(\mathbb {C})\). This non-normality is the algebraic signature expected from the two-periodic dimer-covering structure.

For every word \(w\), \(\operatorname{tr}\bigl(\widetilde A^{w}\bigr)=\operatorname{tr}\bigl(B^{w}\bigr)\, \operatorname{tr}\bigl(Y^{|w|}\bigr)\).

For \(N\ge 3\), the Hamiltonian (13) satisfies

so \(H_{N}\ge -\tfrac {3N}{8}\), and the eigenspace of \(H_{N}\) for \(-\tfrac {3N}{8}\) is the kernel of \(H_{N,\mathrm{MG}}^{(3)}\). For even \(N\ge 4\), \(-\tfrac {3N}{8}\) is the smallest eigenvalue of \(H_{N}\), and its eigenspace is spanned by the two singlet coverings \((1,2)(3,4)\cdots (N-1,N)\) and \((2,3)(4,5)\cdots (N,1)\). This is the statement of [ 17 , Appendix A, “The Majumdar-Ghosh model” ] that the Majumdar–Ghosh state is the ground state of \(H_{N}\) and a superposition of the two coverings.

For odd \(N\ge 5\), \(-\tfrac {3N}{8}\) is not an eigenvalue of \(H_{N}\), so every eigenvalue of \(H_{N}\) is strictly larger than \(-\tfrac {3N}{8}\).

Let \(N=2m{\gt}0\) and \(\delta (a,b)=\delta _{ab}\). Then

the sum of the two coverings by the pair state \(|00\rangle +|11\rangle \), and \(|V^{(N)}(\widetilde A)\rangle \) is not a scalar multiple of \(|V^{(N)}(A)\rangle \).

The operator \(\tfrac 43\bigl(h+\tfrac 38\bigr)=\tfrac 13(P_{01}+P_{12}+P_{02})\) is the orthogonal projector onto \(\mathcal G_{3}(A)^{\perp }\), the three-site parent interaction of the Majumdar–Ghosh tensor.

On three spin-\(\tfrac 12\) sites let \(S^\alpha =S^\alpha _0+S^\alpha _1+S^\alpha _2\) and \(\mathbf S^{2}=\sum _\alpha (S^\alpha )^{2}\). Then \(\mathbf S^{2}=\tfrac 94+4h=\tfrac 34+3Q\) with \(Q=\tfrac 43\bigl(h+\tfrac 38\bigr)\) the projector of Theorem 15.4.26. Hence \(\mathcal G_{3}(A)\) is the eigenspace of \(\mathbf S^{2}\) for \(\tfrac 34\), the subspace of total spin \(\tfrac 12\), and \(Q\) is the orthogonal projector \(P^{3/2}\) onto the eigenspace of \(\mathbf S^{2}\) for \(\tfrac {15}4\), the subspace of total spin \(\tfrac 32\).

The transfer map of the Majumdar–Ghosh tensor sends the identity to the diagonal matrix \(\mathcal{E}_A(\mathbb {1}) = \operatorname{diag}(2, \tfrac 12, \tfrac 12)\).

Let \(K\) be a tensor with a bond level \(z\) such that \(K^i_{xy}=0\) whenever \(x\neq z\) and \(y\neq z\), and \(K^i_{zz}=0\) for every \(i\). Put \(f(a,b)=(K^aK^b)_{zz}\). Then \((K^w)_{zz}\) is the pair product of \(w\). For even \(N{\gt}0\),

and for odd \(N\) the periodic vector vanishes.

If \(A\) is \(L_0\)-block-injective with \(L_0{\gt}0\), \(D\ge 1\), and \(N\ge L_0+1\), then \(V^{(N)}(A)\ne 0\) in the \(N\)-site space.

If a commuting pair \(g,h\) has \(\beta _\omega (g,h)\neq 1\), then \(\omega \) has a non-trivial class.

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 \(G_N(A)\).

For every \(A\), \(L\), and \(N\),

If \(L{\gt}0\), then

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.

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) ] .

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 \(H_{N,L}^{\mathrm{ES}}(A)\).

If \(D \ge 1\), \(A\) is injective after blocking \(L_0{\gt}0\) sites, and \(L_0 {\lt} L \le N\), then the parent Hamiltonian of interaction range \(L\) has a unique periodic ground state. These inequalities imply \(N\ge 2\) and \(N\ge L_0+1\), extending the finite-size range \(N\ge 2L_0\) of [ 228 , Theorem 10 ] .

If a tensor of bond dimension \(D\) is periodic with period \(m\), then \(m\leq D\).

This is a normalized form of the state-vector conclusion in [ 228 , Theorem 5 ] . Let \(A\) be an irreducible left-canonical tensor whose peripheral transfer spectrum is the set of \(p\)-th roots of unity. If \(p\mid N\), there are orthogonal cyclic projections \((Q_u)_{u\in \mathbb Z/p\mathbb Z}\) such that \(\sum _uQ_u=\mathbb {1}\), \(Q_uA^i=A^iQ_{u+1}\), and \(Q_{u+N}=Q_u\). For every ring length \(N\geq 0\), its coefficients satisfy the boundary-aware decomposition

For \(N{\gt}0\), this becomes the source’s decomposition into the \(p\) explicit component chains:

Each \(B_u\) is a \(p\)-periodic MPS of bond dimension \(D\) with local matrices given by (1). At \(N=0\), (6) reduces to the boundary traces \(\operatorname{tr}(Q_u)\) from (4). If \(p\nmid N\), then \(|V^{(N)}(A)\rangle =0\).

Every MPS tensor admits a decomposition satisfying Definition 7.5.1.

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

Every product state is a matrix product state with bond dimension one [ 17 , Appendix A, “Product states” ] : for the tensor of Definition 15.1.1, with \(A^{[s]\, i}=a^{i,[s]}\) and trivial boundary vectors \(v_L=v_R=1\), the site-dependent open-boundary contraction of every configuration \((i_1,\ldots ,i_N)\) is the coefficient of the product vector,

so that \(|\psi \rangle =\sum _{i_1,\ldots ,i_N}a^{i_1,[1]}\cdots a^{i_N,[N]}\, |i_1,\ldots ,i_N\rangle \) is the matrix product state of this tensor.

At \(q=1\), the tensor and boundary vector are the W-state tensor and unnormalized W vector. Already on the two-site ring, the W vector is a nonvacuum zero-energy state. For \(|q|{\lt}1\), amplitudes decay as \(|q|^k\) from the left edge. For \(|q|{\gt}1\), the amplitude \(k\) sites from the right edge relative to the last-site amplitude equals \(q^{-k}\) and tends to zero.

Set \(f_j=\exp (2\pi \mathrm{i}j/N)\) and \(\psi _N=W(f)\). For every \(N\ge 2\), \(\psi _N\ne 0\) and

The vector \(\psi _N\) is orthogonal to the whole kernel of \(H_1\), including the uniform W state, rather than only to the vacuum.

There are no \(\delta {\gt}0\) and \(N_0\) such that every ring of length \(N\ge N_0\) satisfies \(\delta \| v\| \le \| H_1v\| \) for all \(v\perp \ker H_1\). This is thermodynamic gaplessness in the finite-periodic-volume criterion; no zero gap above the enlarged kernel of a fixed finite ring is asserted.

The uniform one-particle vector \(W_N=\sum _j|0\cdots 1_j\cdots 0\rangle \) satisfies \(H_\mu W_N=c(\mu )W_N\) for every real \(\mu \) and \(N\ge 2\). For \(\mu {\gt}0\) and \(\mu \ne 1\), \(c(\mu ){\gt}0\) is the largest possible norm-gap constant in (35), hence the exact gap above the unique periodic vacuum. At \(\mu =1\), the W vector is instead another zero-energy state; this does not assert that the finite-ring gap above the enlarged kernel is zero.

For site amplitudes \(f_j\), write \(W(f)=\sum _j f_j|0\cdots 1_j\cdots 0\rangle \). On the one-particle sector, the actual critical local term is one half of identity minus exchange of its two sites. Thus for \(N\ge 2\),

The indices are cyclic, with both oriented edges retained for \(N=2\). In particular, the constant-amplitude W vector is a zero-energy state.

The open nearest-neighbour parent Hamiltonian has kernel \(\mathcal G_N\) and ground-state dimension two for every \(N\ge 2\).

The full boundary matrix-product space is \(\mathcal G_N=\operatorname {span}\{ \Omega _N,\Psi _N(q)\} \). It has dimension two for every \(N\ge 1\).

On every ring of \(N\ge 2\) sites, the actual PVBS parent Hamiltonian satisfies

The sum defining \(H_\mu \) contains both oriented windows when \(N=2\).

If \(N\ge 2\) and \(q^N\ne 1\), the periodic parent Hamiltonian has kernel \(\operatorname {span}\{ \Omega _N\} \) and ground-state dimension one. In particular, this conclusion holds at every such length if \(|q|\ne 1\).

For \(\mu \ge 0\), set \(c(\mu )=(\mu -1)^2/(1+\mu ^2)\). If \(v\) has zero vacuum amplitude, then \(\langle v,H_\mu v\rangle \ge c(\mu )\| v\| ^2\). Consequently, for every \(N\ge 2\) and \(v\perp \ker H_\mu \),

Let \(A\) be normal with \(D\ge 1\) and let \(X\) be unitary with \((A^i)^{T}=\zeta \, X^\dagger A^iX\) for all \(i\). Then \(X\bar X=\pm \mathbb {1}\), and the sign is the same for every unitary \(X\) with this property [ 17 , Section III.A ] .

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 ] .

For distinct sites \(j\neq k\), \(\mathbf S_j\cdot \mathbf S_k=\tfrac 12P_{jk}-\tfrac 14\), where \(P_{jk}\) exchanges the spins at \(j\) and \(k\).

In the setting of Theorem 10.4.10.2, let \(u,v\) be unitaries with virtual gauges \(X,Y\), that is, \(\sum _j u_{ij}A^j=\zeta \, XA^iX^{-1}\) and \(\sum _j v_{ij}A^j=\eta \, YA^iY^{-1}\) with \(\zeta ,\eta \neq 0\). If \(XY=-YX\), then every eigenspace of \(\Lambda \) has even dimension. This sharpens Theorem 10.4.10.4 when the projective representation contains an anticommuting pair.

The map \(g\mapsto U(g)\) is a representation of \(G\). For every \(g\) and \(i\),

Thus the tensor is on-site symmetric up to \(\phi \), with the vector transformation (71). For \(D\geq 1\), there is a virtual projective representation realizing (72), and every virtual projective representation realizing that relation with the same character \(\phi \) has a factor system cohomologous to \(\omega \). If \(\rho \) is unitary and \(|\phi (g)|=1\) for every \(g\), then \(U\) is a unitary representation [ 17 , Section III.A ] .

Let \(D\geq 1\). The transfer map of the SPT fixed-point tensor is \(X\mapsto D^{-1}\operatorname{tr}(X)\, \mathbb {1}\), an idempotent of rank one, and the tensor is injective, hence normal, so its letters are linearly independent [ 17 , Section III.A ] .

Every rotation \(R \in \mathrm{SO}(3)\) is realized by conjugating the Pauli vector by some \(U \in \mathrm{SU}(2)\): there is \(U \in \mathrm{SU}(2)\) with \((\tfrac 12\operatorname{tr}(\sigma _i U\sigma _j U^{-1}))_{ij} = R\). Equivalently, the adjoint double cover \(\mathrm{SU}(2) \to \mathrm{SO}(3)\) is surjective.

Let \(A\) be normal with \(D\ge 1\), let \(P\) satisfy \(P\bar P=\mathbb {1}\), and let \(X\) be unitary with \(\sum _jP_{ij}\bar A^j=\zeta \, X^\dagger A^iX\) for all \(i\). Then \(X\bar X=\pm \mathbb {1}\), and the sign is the same for every unitary \(X\) with this property [ 17 , Section III.A ] .

If \(\mathcal{E}_A\circ \mathcal{E}_A=\mathcal{E}_A\), then every nonzero eigenvalue of \(\mathcal{E}_A\) equals \(1\), the form in which [ 17 , Section III.A ] states the renormalization fixed-point property.

Let \(G\in M_{D}(\mathbb {C})\) and \(c\in \mathbb {C}\) satisfy \(GA^i=c\, A^iG\) for every \(i\). Then for \(0{\lt}N\) and \(L\le N\) the vector \(\sigma \mapsto \operatorname{tr}(A^{\sigma _0}\cdots A^{\sigma _{N-1}}G)\) lies in \(\mathcal G_{N,L}(A)\).

Let \(B^1,\ldots ,B^b\) be binary tensors of positive bond dimensions, each unital and irreducible, with a positive definite fixed point of its dual transfer map. Suppose no two blocks differ by a gauge transformation and a phase, and each is \(L_0\)-block injective for a common \(L_0{\gt}0\). If \(\sum _jc_j|V^{(N)}(B^j)\rangle =c|W_N\rangle \) with every \(c_j\ne 0\) and \(c\ne 0\), then, writing \(F=\max \{ L_0,3(b-1)(L_0+1)\} \),

The same conclusion holds for canonical-form coefficients \(c_j=\lambda _j^N\). Here \(b-1\) means zero when \(b=0\). The threshold in (25) is at most \(2F\); for \(L_0\ge 3\) it is \(L_0+F\).

Let \(A\) be a translation-invariant tensor of bond dimension \(D\) in the canonical form of Definition 23.3.6, with \(b\) blocks, no two of which are related by a gauge transformation and a phase. Suppose that for some \(N\) and some \(c\neq 0\) the periodic vector of \(A\) on \(N\) sites is \(c|W_N\rangle \).

1. If every block is \(L_0\)-block injective for a common \(L_0{\gt}0\), then \(N{\lt}2\max \bigl(L_0,\, 3(b-1)(L_0+1)\bigr)\).

2. If every block is \(L\)-block injective for some \(L{\gt}0\) and its matrix \(A_0^j\) for the letter \(0\) is invertible, then \(N{\lt}2\max \bigl(D^2+1,\, 3(D-1)(D^2+2)\bigr)\), so \(D=\Omega (N^{1/3})\).

3. If the dual transfer channel of every block is primitive, then \(N{\lt}2\max \bigl((D^2+1)^2,\, 3(D-1)((D^2+1)^2+1)\bigr)\).

The first item holds more generally for \(\sum _jc_j|\phi _{A^j}\rangle =c|W_N\rangle \) with arbitrary nonzero coefficients \(c_j\) in place of \(\lambda _j^N\), for blocks satisfying the three conditions of the canonical form. The first item is the inequality of the proof in [ 228 ] , conditional on the canonical form with condition C1, and with the threshold enlarged by \(L_0\) so that it also covers a single block. The distinctness hypothesis is stricter than the source’s pairwise-different block states. The reduction of an arbitrary tensor to this form holds when \(N\geq 2\) has no divisor between \(2\) and \(D\) (Theorem 15.5.24); the review’s asymptotic bound \(D^3\log D=\Omega (N)\) at every length is outside the scope of these results  [ 46 ] .

For every \(N\), the open-boundary state of the W tensor with boundary vectors \((0|\) and \(|1)\) equals the W state:

Let \(A\) be any translation-invariant tensor of bond dimension \(D\). Let \(N\geq 2\), and suppose no integer \(m\) with \(2\leq m\leq D\) divides \(N\). If the periodic vector of \(A\) on \(N\) sites is \(c|W_N\rangle \) with \(c\neq 0\), then

The admissible lengths include every \(N\) whose prime divisors exceed \(D\), such as \(N=p^2\) for a prime \(p{\gt}D\).

Let \(A\) be a binary tensor of bond dimension \(D\) in canonical form, with no two blocks related by a gauge transformation and a phase. Suppose each block \(B^j\) is injective at some positive length and there are \(X_j\in \operatorname{span}_{\mathbb {C}}\{ B_i^j\} \), \(\mu _j\ne 0\), and \(\phi _j\ne 0\) satisfying \(X_j\phi _j=\mu _j\phi _j\). If \(V^{(N)}(A)=cW_N\) with \(c\ne 0\), then, for \(L=\max \{ 1,D^2\} \),

For positive \(D\), \(L=D^2\), so this gives \(D=\Omega (N^{1/3})\). The maximum with one includes \(D=N=0\).

Let \(B^1,\ldots ,B^b\) have bond dimensions \(D_1,\ldots ,D_b\) and suppose their word tuples of length \(R'\) span \(\bigoplus _jM_{D_j}(\mathbb {C})\). Let every \(c_j\ne 0\). If each function \(\sigma \mapsto \sum _jc_j\operatorname{tr}(B^j_\sigma B^j_\tau )\) lies in a subspace \(V\) of functions on words of length \(R\), then every function \(\sigma \mapsto \sum _j\operatorname{tr}(X_jB^j_\sigma )\) also lies in \(V\). If the specified block \(B^j\) is \(R\)-block injective, then \(D_j^2\le \dim V\).

For \(N\geq 2\), no tensor of bond dimension \(1\) is a periodic representation of \(|W_N\rangle \).

Let \(B\) be a tensor with physical dimension \(2\) and bond dimension \(D\). If \(B\) is a periodic representation of \(|W_k\rangle \) for every \(1\leq k\leq D\), then \((B^0)^D=0\), and \(B\) is a periodic representation of \(|W_N\rangle \) for no \(N{\gt}D\). Consequently there is a length \(1\leq N\leq D+1\) for which \(B\) is not a periodic representation of \(|W_N\rangle \), and no single tensor of fixed bond dimension is a periodic representation of \(|W_N\rangle \) for every \(N\geq 1\).

Closing the tensor of Definition 15.5.1 with a trace gives

In particular the trace contraction vanishes on every single-excitation configuration, and for \(N\geq 1\) it is not \(|W_N\rangle \).

Let \(A\) be any translation-invariant tensor of bond dimension \(D\) and let \(N\) be prime. If its periodic vector is \(c|W_N\rangle \) with \(c\neq 0\), then the bound (30) holds. In particular, \(D=\Omega (N^{1/5})\) along prime lengths.

On three sites, each of the three single-excitation configurations \(|100\rangle \), \(|010\rangle \), \(|001\rangle \) receives W-state amplitude \(1\), while the vacuum \(|000\rangle \) and the doubly-excited \(|110\rangle \) receive amplitude \(0\).

Let \(r\leq 2\), let \(x_0,\ldots ,x_{r-1}\) and \(y_0,\ldots ,y_{r-1}\) be complex numbers, and let \(c\neq 0\). Suppose \(N\geq 2\), and \(N\geq 3\) when \(r=2\). Then the equalities

for all \(1\leq m\leq \min (3,N)\) cannot all hold. In particular a sum of at most two translation-invariant product states, with one-site amplitudes \(x_k\) and \(y_k\) for the letters \(0\) and \(1\), is not a nonzero multiple of \(|W_N\rangle \): its amplitude on the configuration \(1^m0^{N-m}\) is the left-hand side of (23). This supplies the step that [ 228 , lines 2208–2210 ] calls trivial, after the rank count has reduced the canonical form to at most two blocks of size one by one.

Let \(A\) be a normalized MPS tensor with \(D{\gt}0\), \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), that is primitive in the sense of Definition 6.1.1.3. Then

where \(\iota (A)=\min \{ n\ge 1:S_n(A)=M_{D}(\mathbb {C})\} \) is the full-Kraus-rank index. This is  [ 233 , Theorem 1, case (1) ] .

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 ] .

A tensor satisfies the virtual-insertion zero-correlation-length condition (Definition 17.5.18) if and only if its transfer map is idempotent: