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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 16.2.4. 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 [ CPGSV21 ] (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 16.2.1.

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 \((i\sigma _y)^2 = -\mathbb {1}\).

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-\(\tfrac 12\).

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

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 [ CPGSV21 , 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.

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.

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

where \(|\pm \rangle = \tfrac {1}{\sqrt{2}}(|0\rangle \pm |1\rangle )\). The review  [ CPGSV21 ] writes the reflected convention \(A^0 = |0\rangle \! \langle +|\), \(A^1 = |1\rangle \! \langle -|\); the transpose taken here is the same state read in the opposite direction and carries the same SPT order.

The commutator phase of a \(2\)-cochain \(\omega \) at a pair \(g,h\) is \(\beta _\omega (g,h):=\omega (g,h) \omega (h,g)^{-1}\). It is defined for any \(\mathbb {C}^{\times }\)-valued \(2\)-cochain; the cocycle condition is not needed.

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

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

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 \(\sigma _z = \operatorname{diag}(1, -1)\).

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.

Virtual-boundary nondecay is an existence statement over the virtual matrices \(X,Y\), for a fixed twist \(u\); no claim is made for any particular boundary choice. Definition 11.3.6 records the physical-endpoint string-order notion from the source, which quantifies existentially over \(u,x,y\) and requires a positive limit rather than a lower bound uniform in the length. That paper also notes that the correlator for a particular endpoint choice can vanish even when the physical symmetry is present.

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 \(g,h,k \in G\),

This auxiliary one-block predicate is the conjunction of \(\mathcal{E}_A^2=\mathcal{E}_A\) and virtual-insertion distance independence. It is not the physical zero-correlation-length condition of Definition 18.2.6.

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 three bond levels are the two spin-\(\tfrac 12\) values of an open singlet partner together with the closed inter-dimer link; the relative sign \(-\tfrac 1{\sqrt2}\) is the antisymmetry of the singlet. The equality between the matrix product vectors of these matrices and the even-ring dimer superposition, with normalization and cyclic convention fixed, is a separate verification, not part of this definition.

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 of \(A\) satisfies the idempotence criterion 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 2.7.4, it is the complex conjugate of the Hilbert-space inner product from Definition 2.7.2.

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

A tensor \(A\) is normal [ CPGSV21 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

The predicate called on-site symmetry requires \(\mathcal{V}(A)=\mathcal{V}(\widetilde A_g)\) for every \(g\in G\), where \(U:G\to \mathrm{GL}_d(\mathbb {C})\) acts on the physical index. This is equality of periodic MPV families and is the hypothesis used to construct virtual local gauges. It is not, by itself, the source notion that every reduced density operator of an infinite physical state is invariant under \(U(g)^{\otimes N}\).

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

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

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

This is the relation \(AA=A\) in [ CPGSV16 , Definition 3.2 ] .

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 when, for every \(m_1,m_2\geq 0\) satisfying \(n_1+n_2=m_1+m_2\), one has

Thus either region may be translated without crossing the other. Zero gaps, corresponding to adjacent regions, are included. This is [ CPGSV16 , Definition 3.3 and lines 490–496 ] .

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.

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:

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.

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

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

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 \(\xi {\gt} 0\).

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 \(M_{2}(\mathbb {C})\).

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 16.2.12).

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

It is finite and positive.

The two virtual matrices that implement the symmetry of Theorem 16.2.4, namely \(i\sigma _y\) and \(\sigma _z\), anticommute.

The AKLT factor system (Theorem 16.2.8), built from the anticommuting gauges \(i\sigma _y\) and \(\sigma _z\) (Theorem 16.2.5), 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 \(H^2(\mathbb {Z}_2 \times \mathbb {Z}_2, \mathrm{U}(1))\).

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

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

For every element \(g\) of \(\mathbb {Z}_2 \times \mathbb {Z}_2\), the length-\(2\) blocked AKLT tensor has string order in the sense of Definition 11.3.11 for the twist by \(g\), 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 \(\tfrac 13\).

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.

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.

The virtual matrices implementing the two generators of Theorem 16.5.5, namely \(V_1 = \sigma _z\) and \(V_2 = \sigma _x\), anticommute (\(V_1 V_2 = -V_2 V_1\)).

The cluster-state factor system (Theorem 16.5.9), built from the anticommuting gauges \(\sigma _z\) and \(\sigma _x\) (Theorem 16.5.6), 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 \(H^2(\mathbb {Z}_2 \times \mathbb {Z}_2, \mathrm{U}(1))\).

For every element \(g\) of \(\mathbb {Z}_2 \times \mathbb {Z}_2\), the length-\(2\) blocked cluster-state tensor has string order in the sense of Definition 11.3.11 for the twist by \(g\), with the maximally mixed boundary state \(\Lambda = \tfrac {1}{2}\mathbb {1}\) as its stationary boundary state.

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 16.5.12) 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 16.6.1 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.

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 [ CPGSV21 , line 2205 ] .

For the GHZ tensor \(A\) of Definition 16.1.1,

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 16.1.7 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 16.1.5) gives correlations independent of the separation. This contrasts with the AKLT state (Theorem 16.2.1.4), whose subleading eigenvalue \(-\tfrac 13\) produces a finite correlation length \(1/\log 3\).

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 \(g\).

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, spanned by the two nearest-neighbour singlet coverings \((1,2)(3,4)\cdots (N-1,N)\) and \((2,3)(4,5)\cdots (N,1)\). This two-fold degeneracy reflects the spontaneous breaking of the one-site translation symmetry of the spin-\(\tfrac 12\) chain \(H=\sum \boldsymbol {S}_i\cdot \boldsymbol {S}_{i+1} +\tfrac 12\sum \boldsymbol {S}_i\cdot \boldsymbol {S}_{i+2}\) of [ CPGSV21 ] .

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

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.

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

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

The transfer map of an MPS tensor is idempotent if and only if the tensor has a physical blocking isometry.

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.

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

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 \(0\).

The auxiliary conjunction \((\mathcal{E}_A^2=\mathcal{E}_A\ \text{and virtual-insertion distance independence})\) is equivalent to \(\mathcal{E}_A^2=\mathcal{E}_A\). This is a logical simplification of the auxiliary convention, not the physical ZCL theorem of [ CPGSV16 , Theorem 3.8 ] .