15 Concrete Examples
Concrete matrix product state tensors illustrate injectivity, canonical forms, symmetries, and parent Hamiltonians [ 17 , Section III ] . Each example is specified by its physical dimension \(d\), bond dimension \(D\), and matrix family \(\{ A^i\} \). The cases formalized here comprise foundational models from the quantum information and many-body literature: product states, the Greenberger–Horne–Zeilinger (GHZ) state [ 202 , 201 ] , the Affleck–Kennedy–Lieb–Tasaki (AKLT) spin-\(1\) valence-bond state [ 2 , 3 ] , the Majumdar–Ghosh dimer ground state [ 218 , 217 ] , the W state [ 192 ] , the one-dimensional cluster state [ 7 , 230 ] , the Kitaev (Majorana) chain, of which only the ungraded contraction is formalized [ 209 ] , and the correlation-independent non-fixed-point tensor of Cirac et al. [ 14 ] , followed by their parent-Hamiltonian constructions [ 194 , 228 ] .
15.1 Product states
Every product state is a matrix product state with bond dimension one [ 17 , Appendix A, “Product states” ] . For site-dependent local tensors, the matrix product is closed by boundary vectors.
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 \(I\).
The boundary vectors are the scalars \(v_L=v_R=1\). Theorem 15.1.4 identifies the resulting contraction with the product vector; Corollary 15.1.5 gives the translation-invariant specialization and its periodic closure.
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.
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.
Both sides are \(v_L^{\mathsf T}A^{\sigma _1}\cdots A^{\sigma _N}v_R\).
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.
A product of \(1\times 1\) matrices is the \(1\times 1\) matrix of the product of their entries; induct on \(N\).
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.
15.2 The GHZ state
The Greenberger–Horne–Zeilinger (GHZ) state [ 202 , 201 ] is the prototypical non-injective, renormalization-fixed-point MPS [ 17 , Section III.A ] . On a \(d\)-level system it is an MPS with bond dimension \(d\) [ 17 , Appendix A, “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 indices are closed by the ordinary trace. The resulting periodic vector is identified in Theorem 15.2.3. Theorems 15.2.4–15.2.6 give the idempotent transfer map, non-injectivity for \(d\ge 2\), and on-site permutation symmetry. For the qubit tensor, the virtual symmetry and zero correlation length are Theorems 15.2.16 and 15.2.19; its parent ground space is Theorem 15.10.3.
The (unnormalized) GHZ state on \(N\) sites of a \(d\)-level system is
A product \(A^{\sigma _1}\cdots A^{\sigma _N}\) is the diagonal matrix whose \(x\)-th entry is \(1\) if \(\sigma _1=\cdots =\sigma _N=x\) and \(0\) otherwise; its trace counts the \(x\) with this property.
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.
Since \(A^i=|i\rangle \! \langle i|\), the term \(A^iX(A^i)^\dagger \) is \(X_{ii}|i\rangle \! \langle i|\). Extracting the diagonal is idempotent.
For \(d\ge 2\) the GHZ tensor on a \(d\)-level system is not injective.
Every \(A^i\) has vanishing \((0,1)\) entry, while the all-ones matrix does not.
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 twisted tensor of a permutation \(\pi \) is the relabelling \(i\mapsto A^{\pi ^{-1}(i)}\), so its matrix product vector is the GHZ state evaluated on the relabelled configuration, which equals the GHZ state.
For \(d=2\) the tensor of Definition 15.2.1 is the two-level GHZ tensor of Definition 15.2.7, which is the representative used in the rest of this section.
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 GHZ tensor on a \(d\)-level system with \(d=2\) is the tensor of Definition 15.2.7.
Both tensors have \(A^i=|i\rangle \! \langle i|\) for \(i\in \{ 0,1\} \).
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 \(a\in \{ 0,1\} \) and \(N\ge 0\),
The transfer map of the GHZ tensor satisfies \(\mathcal{E}_A^2 = \mathcal{E}_A\).
The transfer map acts by \(\mathcal{E}_A(X)_{ij} = \delta _{ij} X_{ij}\) (extraction of the diagonal). This operation is idempotent.
The GHZ tensor is a renormalization fixed point.
The claim follows from Theorem 15.2.11.
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 \(M_{2}(\mathbb {C})\).
Every element of \(\operatorname{span}_{\mathbb {C}}\{ A^0, A^1\} \) has zero off-diagonal entries, so the matrix with all entries equal to \(1\) is not in the span.
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 identity element is trivial. For the generator \(g = \sigma _x\), the twisted tensor satisfies \(\widetilde A_g^i = A^{1-i}\), i.e. the two matrices are swapped. Conjugation by \(\sigma _x\) implements the same swap, giving gauge equivalence and hence the same MPV family.
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 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.
Both \(A^0\) and \(A^1\) are diagonal, so each commutes with the diagonal matrix \(\sigma _z\).
The order-parameter operator \(\sigma _z\) is a fixed point of the GHZ transfer map, \(\mathcal{E}_A(\sigma _z) = \sigma _z\).
Since \(\mathcal{E}_A(X) = \sum _i A^i X (A^i)^{\dagger }\) and both \(A^i\) are diagonal, the claim is the entrywise computation \(\mathcal{E}_A(\sigma _z) = \sigma _z\).
A primitive (injective) tensor has a one-dimensional transfer-map fixed-point space, spanned by a positive matrix that equals the identity only in a normalized gauge. That the GHZ transfer map fixes the additional non-scalar operator \(\sigma _z\) is the transfer-matrix signature of the extra non-decaying mode behind its long-range ferromagnetic correlations.
The claim follows from the idempotence of the GHZ transfer map and the equivalence of zero correlation length with transfer-map idempotence (Theorem 17.5.20).
15.2.1 The \(G\)-injective fixed-point tensor
For a finite group \(G\), the review constructs a \(G\)-injective MPS that is a renormalization fixed point and generalizes the GHZ state to arbitrary groups [ 17 , Section III.A, “Virtual symmetries” ] . Write \(L_g\) for the left regular representation, \(L_g|k\rangle =|gk\rangle \).
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 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” ] .
The term \(A^{ij}X(A^{ij})^\dagger \) has entries \(|G|^{-2}X_{\alpha i^{-1}j,\beta i^{-1}j}\); summing over \(j\) and then \(i\) gives the formula. Applying it twice averages over \(k\) and then over \(k'\), and the substitution \(k'\mapsto kk'\) returns the single average.
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 condition \(\beta =\alpha i^{-1}j\) is unchanged by \((\alpha ,\beta )\mapsto (h\alpha ,h\beta )\). An invariant vector satisfies \(x_{\alpha \beta }=x_{1,\alpha ^{-1}\beta }\), and the component of \(\mathcal P(A)x\) at \((1,w)\) is \(|G|^{-1}\sum _\alpha x_{\alpha ,\alpha w} =x_{1,w}\), so \(\mathcal P(A)x=0\) forces \(x=0\).
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” ] .
\((gi)^{-1}(gj)=i^{-1}j\).
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” ] .
The transformed tensor at \((a,b)\) is \(\tfrac 14\sum _{ij}(-1)^{ai+bj}\sigma _x^{i+j}\), and conjugation by \(H\) turns \(\sigma _x\) into \(\sigma _z\), giving \(\tfrac 14\bigl(1+(-1)^a\sigma _z\bigr)\bigl(1+(-1)^b\sigma _z\bigr) =|a\rangle \! \langle a|\, |b\rangle \! \langle b|\).
15.3 The AKLT state
The Affleck–Kennedy–Lieb–Tasaki (AKLT) state [ 2 , 3 ] is the canonical example of a normal (block-injective) MPS with non-trivial symmetry-protected topological (SPT) order [ 229 , 236 , 17 ] .
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 ] .
On a ring the virtual indices are closed by the trace; on an open chain the two end spins remain as boundary data. The physical-basis and scalar-gauge relations to the normalized representative are Theorem 15.3.6 and Lemma 15.3.8. Theorems 15.3.31 and 15.3.32 give normality, on-site symmetry and the projective virtual action. For the normalized representative, the correlation length is \(1/\log 3\) (Theorem 15.3.1.4). The periodic parent-Hamiltonian statement is Theorem 15.3.33, and the open-boundary ground space is described in Theorem 15.10.18.
The matrices of (3) are \(A^{0}=\sigma _z/\sqrt2\), \(A^{+1}=-|0\rangle \! \langle 1|\), and \(A^{-1}=|1\rangle \! \langle 0|\).
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: \(\operatorname{tr}\big((P^{s})^{\dagger }P^{t}\big)=\delta _{st}\).
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 \(|0\rangle \).
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 vectors \(|\pm \rangle \) are supported on \(|\pm 1\rangle \) and \(|0\rangle \) is orthogonal to both; the inner product of \(|+\rangle \) and \(|-\rangle \) is \(\frac{-i}{2}(1-1)=0\), and each vector has norm \(1\). A phase on \(|+\rangle \) preserves all of these relations.
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 ] .
Since \(\langle +|\pm 1\rangle =-i/\sqrt2\), one has \(B^{+}=-\frac{i}{\sqrt2}(A^{-1}+A^{+1})=-\frac{i}{\sqrt2}Y=-\sigma _y/\sqrt2\), while \(B^{-}=(A^{-1}-A^{+1})/\sqrt2=\sigma _x/\sqrt2\); conjugating the phase of \(|+\rangle \) reverses the sign of \(B^{+}\). Conjugation by \(\sigma _y\) fixes \(\sigma _y\) and reverses the signs of \(\sigma _x\) and \(\sigma _z\), which gives the gauge relation; a gauge leaves the state unchanged and the scalar \(-1\) multiplies it by \((-1)^N\).
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).
Conjugation by \(\sigma _z\) fixes \(\sigma _z\) and reverses the sign of \(|0\rangle \! \langle 1|\) and \(|1\rangle \! \langle 0|\); compare with Lemma 15.3.2 entrywise.
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 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 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 two virtual matrices that implement the symmetry of Theorem 15.3.11, namely \(i\sigma _y\) and \(\sigma _z\), anticommute.
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 \((i\sigma _y)^2 = -\mathbb {1}\).
The commutator phase of the AKLT factor system on the two generators is \(-1\).
The AKLT factor system is a genuine \(2\)-cocycle, being the factor system of an explicit projective representation, so it represents an element of \(H^2(\mathbb {Z}_2 \times \mathbb {Z}_2, \mathrm{U}(1))\).
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 subgroup \(\mathbb {Z}_2 \times \mathbb {Z}_2\) is the minimal subgroup of \(\mathrm{SO}(3)\) admitting a non-trivial \(2\)-cocycle.
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 length-\(2\) blocked AKLT tensor is injective: its nine matrices span \(M_{2}(\mathbb {C})\).
This is the blocked-tensor form of the normality of the single-site tensor (Theorem 15.3.10): the single-site tensor is \(2\)-block injective, which is exactly injectivity of its length-\(2\) blocking.
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).
Writing \(A\) for the single-site tensor and \(A^{[2]}\) for its length-\(2\) blocking, twisting the blocked tensor by \(P_g \otimes P_g\) equals blocking the twisted single-site tensor, \((A^{[2]})^{P_g \otimes P_g} = (A^{P_g})^{[2]}\), where the superscript denotes the twist. Since the single-site tensor is symmetric, \(A^{P_g}\) has the same matrix product vector family as \(A\) (Theorem 15.3.11), and blocking preserves the matrix product vector family, so \((A^{[2]})^{P_g \otimes P_g}\) has the same matrix product vector family as \(A^{[2]}\).
Each single-site spin-\(1\) \(\pi \)-rotation matrix \(P_g\) of the \(\mathbb {Z}_2 \times \mathbb {Z}_2\) action is unitary: \(P_g P_g^\dagger = \mathbb {1}\).
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 ] .
Both identities are checked entry by entry on the three matrices of the AKLT tensor, which are real.
Every unitary gauge implementing the time-reversal or the reflection symmetry of the AKLT tensor has index \(X\bar X=-\mathbb {1}\).
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 blocked tensor is injective and satisfies \(\mathcal{E}(\mathbb {1})=\mathbb {1}\). The two generators of \(\mathbb {Z}_2\times \mathbb {Z}_2\) act by unitary Kronecker squares with virtual gauges \(i\sigma _y\) and \(\sigma _z\), which carry over from the single site by Lemma 10.1.9 and anticommute. Apply Theorem 10.4.10.5.
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 blocked tensor is injective (Theorem 15.3.20) and on-site symmetric under \(\mathbb {Z}_2 \times \mathbb {Z}_2\) (Theorem 15.3.21), and each group element acts by a real symmetric involutive matrix, hence unitarily. The maximally mixed state \(\Lambda = \tfrac {1}{2}\mathbb {1}\) is positive definite and has trace \(1\). The single-site AKLT matrices satisfy \(\sum _i A^i(A^i)^\dagger = \mathbb {1}\) and \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\). These identities propagate to the length-\(2\) blocked tensor \(A^{[2]}\), whose matrices are the products \(A^{i_1}A^{i_2}\). In particular, the blocked transfer map is unital:
Identity (5) sums the right-canonical relation \(\sum _i A^i(A^i)^\dagger = \mathbb {1}\) over the inner letter and then the outer; (6) sums the left-canonical relation \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\) over the outer letter and then the inner, using \(\Lambda = \tfrac {1}{2}\mathbb {1}\). Applying Theorem 10.4.35 yields virtual-boundary nondecay for every \(g\).
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.
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.
Factor \(R = R_z(\alpha )R_x(\beta )R_z(\gamma )\) by its \(ZXZ\) Euler decomposition and map each coordinate rotation to an explicit \(\mathrm{SU}(2)\) generator:
The Cartesian AKLT tensor is on-site symmetric under the spin-\(1\) (defining) representation of \(\mathrm{SO}(3)\) on the physical index.
The bond index carries the projective spin-\(\tfrac 12\) representation: the gauge for a rotation is either of its two \(\mathrm{SU}(2)\) preimages, so the assignment is a representation only up to sign and factors through the double cover \(\mathrm{SU}(2) \to \mathrm{SO}(3)\). Its class is the non-trivial element of \(H^2(\mathrm{SO}(3),\mathrm{U}(1)) \cong \mathbb {Z}_2\), witnessing the SPT phase.
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).
With the physical states ordered \(S_z=0,+1,-1\), the two generators act by
and the virtual matrices are
For \(g=(1,0)\) the twist is \(\widetilde A_g^{0}=-A^{0}\), \(\widetilde A_g^{+1}=A^{-1}\), \(\widetilde A_g^{-1}=A^{+1}\), and with the matrices of Lemma 15.3.2,
since \(\sigma _y\) and \(\sigma _z\) anticommute; the letter \(-1\) is checked in the same way. For \(g=(0,1)\) the twist multiplies \(A^{\pm 1}\) by \(-1\) and fixes \(A^{0}\), which is conjugation by \(\sigma _z\). The case \(g=(1,1)\) is the composite of the two, and \(g=1\) is trivial.
A virtual gauge and a nonzero rescaling leave the local ground spaces unchanged and multiply the periodic state by \((3/2)^{N/2}\); apply Theorem 15.10.17.
15.3.1 Correlation length
The AKLT state is the canonical example of a finite correlation length, originally evaluated by Affleck, Kennedy, Lieb, and Tasaki [ 2 , 3 ] and Fannes, Nachtergaele, and Werner [ 194 ] . Its transfer map has leading eigenvalue \(1\) and a single subleading eigenvalue \(-\tfrac 13\) of multiplicity three; the three Pauli matrices are the subleading eigenvectors and the identity is the leading one. The subleading modulus \(\tfrac 13\) gives the correlation length \(\xi = 1/\log 3\), and connected correlations decay as \((\tfrac 13)^n\).
If \(x\geq 0\), then the complex coercion of its real square root satisfies \(((\sqrt{x}:\mathbb R):\mathbb C)^2=x\).
Apply the real identity \((\sqrt{x})^2=x\) and commute the real embedding with powers.
Each of the three Pauli matrices \(\sigma _x\), \(\sigma _y\), \(\sigma _z\) is an eigenvector of the AKLT transfer map with eigenvalue \(-\tfrac 13\):
Each identity is an entrywise computation from the AKLT matrices, using \((\sqrt{3})^2 = 3\) and \((\sqrt{2})^2 = 2\) to clear the prefactors.
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 \(\tfrac 13\).
The AKLT correlation length, computed from the subleading eigenvalue \(\lambda _2 = -\tfrac 13\), equals
It is finite and positive.
The subleading modulus is \(|{-\tfrac 13}| = \tfrac 13\), so rewriting the correlation length at \(\lambda _2 = -\tfrac 13\) gives \(\xi = -1/\log (\tfrac 13) = 1/\log 3\). Since \(0 {\lt} \tfrac 13 {\lt} 1\), positivity is the specialization of Lemma 14.2.7 to \(\lambda _2 = -\tfrac 13\).
15.4 The Majumdar–Ghosh state
The Majumdar–Ghosh state is the exact dimer ground state of the spin-\(\tfrac 12\) antiferromagnetic Heisenberg chain with next-nearest-neighbour coupling at the Majumdar–Ghosh point \(J_2 = J_1 / 2\) [ 218 , 217 ] : \(H=\sum \boldsymbol {S}_i\cdot \boldsymbol {S}_{i+1} +\tfrac 12\sum \boldsymbol {S}_i\cdot \boldsymbol {S}_{i+2}\). On an even periodic chain it is the symmetric superposition of the two nearest-neighbour singlet coverings \((1,2)(3,4)\cdots \) and \((2,3)(4,5)\cdots (N,1)\), forming the one-dimensional prototype of a resonating valence-bond state. The review [ 17 , Appendix A, “The Majumdar-Ghosh model” ] writes it as a matrix product state with a tensor built from the singlet matrix
whose entry \(Y_{ab}\) is the coefficient of \(|ab\rangle \) in the singlet \(|10\rangle -|01\rangle \).
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).
The periodic coefficients are traces of products of these matrices. The literal reading above and the singlet-bond reading have different periodic vectors; their relation and the source ambiguity are recorded in [ 112 ] and Theorem 15.4.10. The singlet-covering state is identified in Theorem 15.4.8, and its bond-dimension-three tensor is left-canonical, non-injective and non-normal (Theorems 15.4.11, 15.4.13 and 15.4.14). Its two-dimensional parent ground space on even rings of at least four sites is Theorem 15.4.24; the corresponding Hamiltonian comparison is Theorem 15.4.28.
For every word \(w\), \(\operatorname{tr}\bigl(\widetilde A^{w}\bigr)=\operatorname{tr}\bigl(B^{w}\bigr)\, \operatorname{tr}\bigl(Y^{|w|}\bigr)\).
By induction on \(w\) and the mixed-product rule \((B^i\otimes Y)(M\otimes Y^n)=B^iM\otimes Y^{n+1}\), the word evaluation is \(\widetilde A^{w}=B^{w}\otimes Y^{|w|}\) up to the identification of the bond with \(\mathbb {C}^6\), which preserves traces. The trace of a Kronecker product is the product of the traces.
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.
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.
The row \(z\) of \(K^aK^b\) is \(f(a,b)\) times the \(z\)-th unit row, since a path leaving \(z\) must enter a level \(x\neq z\) and then return to \(z\). Induction on the word in steps of two letters gives the entry \((K^w)_{zz}\), and a word of length one has \((K^a)_{zz}=0\). For a word \(a\, w\),
because the rows \(x\neq z\) of \(K^a\) are supported in column \(z\) and \(K^a_{zz}=0\). The two terms are the \((z,z)\) entries for \(a\, w\) and for its rotation \(w\, a\), which are the two pair coverings of the configuration. Pair products of words of odd length vanish.
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 ] .
The level \(z=2\) satisfies the hypotheses of Theorem 15.4.4 for \(C\), and \((C^aC^b)_{22}=Y_{ab}\) by direct multiplication.
The representative used below has the same structure, with the distinguished level \(0\) and the normalized singlet as pair amplitude.
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.
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.
The level \(z=0\) satisfies the hypotheses of Theorem 15.4.4 for \(A\): rows \(1,2\) of \(A^0\) and \(A^1\) are supported in column \(0\), and \(A^i_{00}=0\). Direct multiplication gives \((A^aA^b)_{00}=Y_{ab}/\sqrt2\).
For every even \(N{\gt}0\), \(|V^{(N)}(C)\rangle =\sqrt2^{\, N/2}\, |V^{(N)}(A)\rangle \).
Both vectors are sums of the two pair coverings, with amplitudes \(Y_{ab}\) and \(Y_{ab}/\sqrt2\); each covering is a product of \(N/2\) amplitudes.
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 level \(z=2\) satisfies the hypotheses of Theorem 15.4.4 for \(\{ B^0,B^1\} \), with \((B^aB^b)_{22}=\delta _{ab}\), so Theorem 15.4.4 gives \(\operatorname{tr}(B^{\sigma })=P^{\mathrm{even}}_\delta (\sigma )+P^{\mathrm{odd}}_\delta (\sigma )\). Since \(Y^2=-I_2\), \(\operatorname{tr}(Y^{2m})=2(-1)^m\), and Theorem 15.4.2 gives (12). On the configuration with all spins \(0\) the right side of (12) equals \(4(-1)^m\neq 0\), while Theorem 15.4.8 gives \(V^{(N)}(A)_{0\cdots 0}=0\) because \(s(0,0)=0\).
The Majumdar–Ghosh tensor is isometric: \((A^0)^\dagger A^0 + (A^1)^\dagger A^1 = \mathbb {1}\).
Direct multiplication of the two displayed matrices gives \((A^0)^\dagger A^0=\operatorname{diag}(\tfrac 12,1,0)\) and \((A^1)^\dagger A^1=\operatorname{diag}(\tfrac 12,0,1)\).
The transfer map of the Majumdar–Ghosh tensor sends the identity to the diagonal matrix \(\mathcal{E}_A(\mathbb {1}) = \operatorname{diag}(2, \tfrac 12, \tfrac 12)\).
Substitute the two matrices in \(\mathcal{E}_A(\mathbb {1})=A^0(A^0)^\dagger +A^1(A^1)^\dagger \) and multiply.
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 matrix unit \(E_{22}\) has nonzero \((2,2)\) entry, while every linear combination of \(A^0\) and \(A^1\) has zero \((2,2)\) entry.
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 bond carries a two-periodic grading with complementary support sets
Left multiplication with a generator exchanges these sets. Let \(E_{ij}\) denote the \((i,j)\) matrix unit. Then, for every blocking length \(N\),
Hence no fixed-length word span is all of \(M_{3}(\mathbb {C})\).
Read literally, the valence-bond formula of [ 17 ] specifies the six-dimensional tensor of Definition 15.4.1, which by Theorem 15.4.10 gives coverings by \(|00\rangle +|11\rangle \) rather than by singlets. Placing the singlet on the spin levels of a three-level bond (Definition 15.4.5) gives the Majumdar–Ghosh superposition; see [ 112 ] .
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.
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 \(k\).
The completeness relation \(\sum _\alpha S^\alpha _{pa}S^\alpha _{qb} =\tfrac 12\delta _{pb}\delta _{qa}-\tfrac 14\delta _{pa}\delta _{qb}\) is checked on the sixteen index values.
The three-site local ground space \(\mathcal G_3(A)\) equals the eigenspace of \(h\) for the eigenvalue \(-\tfrac 38\).
By Theorem 15.4.17, \(h=\tfrac 14(P_{01}+P_{12}+P_{02})-\tfrac 38\), so the eigenspace consists of the vectors \(v\) with \(v(\sigma P_{01})+v(\sigma P_{12})+v(\sigma P_{02})=0\) for every \(\sigma \), that is,
Every \(\Gamma _3(X)\) satisfies (14) by expansion. Conversely, a vector satisfying these conditions is \(\Gamma _3(X)\) for the matrix \(X\) with \(X_{01}=-2v_{110}\), \(X_{02}=-2v_{001}\), \(X_{10}=\sqrt2\, v_{100}\), \(X_{20}=-\sqrt2\, v_{011}\) and all other entries zero.
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 \(\mathcal G_{N,L}(A)\).
For the window starting at site \(i\), split the configuration as \(u\, v\) with \(|u|=i\), so that \(v\, u\) begins with the window. Then \(\operatorname{tr}(A^uA^vG)=\operatorname{tr}(A^vGA^u)=c^{i}\operatorname{tr}(A^vA^uG)\), and \(A^{vu}G\) is the window word followed by the boundary matrix \(c^i A^{\mathrm{out}}G\), where \(A^{\mathrm{out}}\) is the product over the sites outside the window.
For even \(N{\gt}0\) and \(L\le N\), both coverings \(P^{\mathrm{even}}_s\) and \(P^{\mathrm{odd}}_s\) lie in \(\mathcal G_{N,L}(A)\).
The grading \(G=\operatorname{diag}(1,-1,-1)\) satisfies \(GA^i=-A^iG\). For even \(N\), \(\operatorname{tr}(A^{\sigma }G)=2(A^{\sigma })_{00}-\operatorname{tr}(A^{\sigma }) =P^{\mathrm{even}}_s(\sigma )-P^{\mathrm{odd}}_s(\sigma )\) by Theorem 15.4.4 and Theorem 15.4.8. The two coverings are half the sum and half the difference of this vector and \(|V^{(N)}(A)\rangle \), both of which lie in \(\mathcal G_{N,L}(A)\) by Theorem 15.4.19.
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 \(-\tfrac {3N}{8}\).
Each window of three sites \(i,i+1,i+2\) of \(\psi \) lies in \(\mathcal G_3(A)\), so by Theorem 15.4.18 the three transposed copies of \(\psi \) in that window sum to zero. Summing over \(i\) and shifting the index of the middle transposition gives \(2\sum _iP_{i,i+1}\psi +\sum _iP_{i,i+2}\psi =0\). Theorem 15.4.17 then gives \(H_N\psi =\tfrac 14\bigl(2\sum _iP_{i,i+1}\psi +\sum _iP_{i,i+2}\psi \bigr) -\tfrac {3N}{8}\psi =-\tfrac {3N}{8}\psi \).
Because the Majumdar–Ghosh tensor is non-normal (Theorem 15.4.14), the intersection property of Theorem 11.5.4.3 does not apply to it. The bond grading \(G=\operatorname{diag}(1,-1,-1)\), which satisfies \(GA^i=-A^iG\), takes the place of normality in the following two theorems.
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.
Since \(G^2=1\) and \(GA^i=-A^iG\), the identity \(\Gamma _{K}(GXG)=(-1)^{K}\Gamma _{K}(X)\) holds, so \(X\) may be replaced by \(\tfrac 12\bigl(X+(-1)^{K}GXG\bigr)\). For \(K=3\) the matrix \(X\) is supported on the entries \((0,1)\), \((0,2)\), \((1,0)\), \((2,0)\), and the coordinates of \(\Gamma _{3}(X)\) at \(001\), \(100\), \(011\), \(110\) are \(-\tfrac 12X_{02}\), \(\tfrac 1{\sqrt2}X_{10}\), \(-\tfrac 1{\sqrt2}X_{20}\) and \(-\tfrac 12X_{01}\). For the step from \(K\) to \(K+1\), fixing the last site to \(a\) gives \(\Gamma _{K}(A^aX)=0\), where \(A^aX\) has parity \((-1)^{K}\); hence \(A^aX=0\) for both \(a\), and \(X=\sum _a(A^a)^\dagger A^aX=0\) by Theorem 15.4.11.
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 \(\mathcal G_{N}(A)\).
Let \(\psi \) lie in the intersection. By Theorem 15.4.22, its restriction with first letter \(i\) is \(\Gamma _{M}(Y_i)\) and its restriction with last letter \(j\) is \(\Gamma _{M}(Z_j)\), with \(Y_i\) and \(Z_j\) of parity \((-1)^{M}\). Comparing on the middle \(M-1\) sites gives \(\Gamma _{M-1}(A^jY_i)=\Gamma _{M-1}(Z_jA^i)\) (Theorem 11.5.4.1), and both matrices have parity \((-1)^{M-1}\). For \(M\ge 4\) the second statement of Theorem 15.4.22 gives \(A^jY_i=Z_jA^i\); with \(X=\sum _j(A^j)^\dagger Z_j\) the left-canonical equation \(\sum _j(A^j)^\dagger A^j=1\) and Theorem 11.5.4.2 give \(Y_i=XA^i\), hence \(\psi =\Gamma _{M+1}(X)\). For \(M=3\), the conditions (14) on the four three-site restrictions with a fixed first or last letter give \(\psi =\Gamma _{4}(X)\) for the matrix \(X\) with \(X_{00}=-2\psi _{0110}\), \(X_{11}=-2\psi _{1100}\), \(X_{12}=2\psi _{0100}\), \(X_{21}=2\psi _{1011}\), \(X_{22}=-2\psi _{0011}\) and all other entries zero, as a check of the sixteen coordinates shows. The second statement follows by enlarging a window of three sites one site at a time.
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 \((2,3)(4,5)\cdots (N,1)\).
The two coverings lie in \(\mathcal G_{N,3}(A)\) by Theorem 15.4.20. Conversely, let \(\psi \in \mathcal G_{N,3}(A)\). By Theorem 15.4.23, \(\psi =\Gamma _{N}(X)\) and its translate by one site, which lies in \(\mathcal G_{N,3}(A)\) by Lemma 13.3.1.38, is \(\Gamma _{N}(Y)\), with \(X\) and \(Y\) commuting with \(G\). Comparing the two cuts on the remaining \(N-1\) sites (Theorem 11.11.1.10) gives \(\Gamma _{N-1}(XA^a)=\Gamma _{N-1}(A^aY)\), so \(XA^a=A^aY\) by Theorem 15.4.22. The entries of these two equations force \(X=\operatorname{diag}(x,y,y)=\tfrac {x+y}2\, 1+\tfrac {x-y}2\, G\), and \(\Gamma _{N}(1)=P^{\mathrm{even}}_s+P^{\mathrm{odd}}_s\), \(\Gamma _{N}(G)=P^{\mathrm{even}}_s-P^{\mathrm{odd}}_s\) as in the proof of Theorem 15.4.20. The two coverings are linearly independent because \(\Gamma _{N}\) is injective on the matrices commuting with \(G\). The kernel of \(H_{N,\mathrm{MG}}^{(3)}\) is \(\mathcal G_{N,3}(A)\) by Theorem 12.3.17.
On an odd periodic chain of length \(N\ge 5\), the space \(\mathcal G_{N,3}(A_{\mathrm{MG}})\) is zero.
As in the proof of Theorem 15.4.24, a vector \(\Gamma _{N}(X)\) of \(\mathcal G_{N,3}(A)\) and its translates by one and two sites give \(XA^a=A^aY\) and \(YA^a=A^aZ\), now with \(X,Y,Z\) anticommuting with \(G\), so that \(X_{00}=X_{11}=X_{12}=X_{21}=X_{22}=0\). The entries of \(XA^a=A^aY\) give
and the same entries of \(YA^a=A^aZ\) give \(Y_{10}=Y_{20}=0\). Hence \(X_{01}=X_{02}=0\), and \(X=0\).
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.
Write \(Q=\tfrac 13(P_{01}+P_{12}+P_{02})\), which equals \(\tfrac 43(h+\tfrac 38)\) by Theorem 15.4.17. Each transposition is a permutation of the configurations, hence self-adjoint, so \(Q\) is self-adjoint. On three spin-\(\tfrac 12\) sites the antisymmetric part vanishes, so \(1+C+C^{2}=P_{01}+P_{12}+P_{02}\) for the cyclic shift \(C\), and \((P_{01}+P_{12}+P_{02})^{2}=3+3(C+C^{2})=3(P_{01}+P_{12}+P_{02})\); this is checked on the eight configurations. Hence \(Q^{2}=Q\). The kernel of \(Q\) is the eigenspace of \(h\) for \(-\tfrac 38\), which is \(\mathcal G_{3}(A)\) by Theorem 15.4.18. A self-adjoint idempotent is the orthogonal projector onto the orthogonal complement of its kernel.
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 \(\tfrac 32\).
Expanding the squares, \(\mathbf S^{2}=\sum _{j,k}\mathbf S_j\cdot \mathbf S_k\), where \(\sum _\alpha (S^\alpha )^{2}=\tfrac 34\) on each site and \(\mathbf S_j\cdot \mathbf S_k=\tfrac 12P_{jk}-\tfrac 14\) for \(j\neq k\) by Theorem 15.4.17. On three sites this gives \(\mathbf S^{2}=\tfrac 94-\tfrac 32+(P_{01}+P_{12}+P_{02})=\tfrac 34+3Q\). Since \(Q\) is an orthogonal projector, \(\mathbf S^{2}v=\tfrac 34v\) exactly when \(Qv=0\), and \(\mathbf S^{2}v=\tfrac {15}4v\) exactly when \(Qv=v\); the kernel of \(Q\) is \(\mathcal G_{3}(A)\) and its range is \(\mathcal G_{3}(A)^{\perp }\).
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.
Theorem 15.4.17 gives \(H_{N}=\tfrac 12\sum _{i}P_{i,i+1}+\tfrac 14\sum _{i}P_{i,i+2}-\tfrac {3N}{8}\). The cyclic shift of the summation index gives \(\sum _{i}P_{i+1,i+2}=\sum _{i}P_{i,i+1}\), so \(H_{N}=\tfrac 14\sum _{i}\bigl(P_{i,i+1}+P_{i+1,i+2}+P_{i,i+2}\bigr) -\tfrac {3N}{8}\). By Theorem 15.4.26, \(H_{N,\mathrm{MG}}^{(3)} =\tfrac 13\sum _{i}\bigl(P_{i,i+1}+P_{i+1,i+2}+P_{i,i+2}\bigr)\). Comparing the two sums gives (15). The parent Hamiltonian is positive by Lemma 12.3.8, so a nonzero \(v\) with \(H_{N}v=\mu v\) satisfies \(0\le \langle v,(H_{N}+\tfrac {3N}{8})v\rangle =(\mu +\tfrac {3N}{8})\lVert v\rVert ^{2}\), and hence \(\mu +\tfrac {3N}{8}\ge 0\). The eigenspace for \(-\tfrac {3N}{8}\) is the kernel of the right side of (15), which is the span of the two coverings by Theorem 15.4.24. The coverings are nonzero, so \(-\tfrac {3N}{8}\) is an eigenvalue.
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 \(-\tfrac {3N}{8}\).
By Theorem 15.4.28, on an even ring the ground space of \(H_{N}\) is two-dimensional. The one-site translation exchanges the two singlet coverings, so this degeneracy reflects the spontaneous breaking of the one-site translation symmetry of the chain.
15.5 The W state
The W state on \(N\) sites is the single-excitation symmetric state
the equal-weight superposition of all configurations with exactly one excitation [ 192 ] . Following [ 17 , Appendix A, “The W state” ] , it is presented with open boundary conditions, where the matrix product is closed by a left and a right boundary vector rather than by a trace. The tensor of Definition 15.5.1 is the one printed there, in the same basis and with the same boundary vectors, so the results below are stated for it directly.
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.
Theorem 15.5.5 proves the open-boundary state identity. Closing the same matrices by a trace instead gives the vacuum term of Theorem 15.5.7. The single-length periodic bond-dimension question is distinct: the review’s general bound \(D^3\log D=\Omega (N)\) remains open, with the conditional and restricted-length results summarized in Remark 15.5.26 and [ 46 ] .
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 \((v_L| A^{\sigma _1}\cdots A^{\sigma _N}\, |v_R)\).
For \(N\geq 0\), the W-state amplitude is the indicator of the single-excitation configurations:
For every \(N\), the open-boundary state of the W tensor with boundary vectors \((0|\) and \(|1)\) equals the W state:
Because \(A^0\) is the identity and \(A^1\) is the raising operator with \(A^1A^1=0\), the ordered product \(A^{\sigma _1}\cdots A^{\sigma _N}\) depends only on the number of excitations of \(\sigma \): it is the identity with no excitation, the raising operator with one excitation, and zero with two or more. Reading off the \((0,1)\) matrix element against the boundary vectors therefore gives \(1\) on the single-excitation configurations and \(0\) elsewhere, which is the W-state amplitude.
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 \(0\).
Direct instances of Theorem 15.5.5: the excitation count is one in the first three cases and zero or two in the others.
The representation of Definition 15.5.1 is not translationally invariant, because its boundary is not periodic [ 17 , Appendix A, “The W state” ] . The following computation illustrates this remark by closing the same tensor with a trace.
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 \(|W_N\rangle \).
The ordered product is the identity, the raising operator, or zero according to the number of excitations, as in the proof of Theorem 15.5.5. Their traces are \(2\), \(0\), and \(0\). For \(N\geq 1\) the configuration \(|10\cdots 0\rangle \) has W-state amplitude \(1\) and trace contraction \(0\).
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.
For \(N\geq 2\), no tensor of bond dimension \(1\) is a periodic representation of \(|W_N\rangle \).
Write \(B^0=b_0\) and \(B^1=b_1\) as scalars. The vacuum amplitude gives \(b_0^N=0\), so \(b_0=0\). The configuration \(|10\cdots 0\rangle \) then has amplitude \(b_1b_0^{N-1}=0\), whereas its W-state amplitude is \(1\).
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 \(N\geq 1\).
The vacuum amplitudes give \(\operatorname {tr}((B^0)^k)=0\) for \(1\leq k\leq D\). By the Newton–Girard identities the characteristic polynomial of \(B^0\) is \(x^D\), and the Cayley–Hamilton theorem gives \((B^0)^D=0\). For \(N{\gt}D\) the configuration \(|10\cdots 0\rangle \) then has amplitude \(\operatorname {tr}(B^1(B^0)^{N-1})=0\), whereas its W-state amplitude is \(1\). If \(B\) represented every \(|W_N\rangle \) with \(1\leq N\leq D+1\), the case \(N=D+1\) would contradict this.
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.
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.
The word \(\sigma \tau \) has exactly one excitation if and only if one of the two pieces has one excitation and the other has none, which is (20). Fixing \(\tau \), the right-hand side is \(u^{(R')}_0(\tau )\, u^{(R)}_1+u^{(R')}_1(\tau )\, u^{(R)}_0\).
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 \(A^j_i\).
For one block this is \(L_0\)-block injectivity, which persists at all larger lengths. For several blocks it is the direct-sum lemma of [ 228 ] for distinct blocks with condition C1; the length \(L_0\) in the threshold covers the single-block case.
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 \(\sum _jD_j^2\leq \dim V\).
The linear map sending \((X_j)_j\in \bigoplus _jM_{D_j}(\mathbb {C})\) to \(\sigma \mapsto \sum _jc_j\operatorname{tr}(A^j_\sigma X_j)\) is injective, because the word tuples at length \(R\) span and the trace pairing is nondegenerate. Its image lies in \(V\), because the word tuples at length \(R'\) span as well.
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.
For \(r=0\) the case \(m=1\) gives \(c=0\). For \(r=1\), combining the cases \(m=1\) and \(m=2\) gives \(c\, y_0=0\), so \(y_0=0\) and again \(c=0\). For \(r=2\) both sequences \(m\mapsto y_k^mx_k^{N-m}\) are annihilated by the recurrence with characteristic polynomial \((x_0S-y_0)(x_1S-y_1)\); applied to the cases \(m=1,2,3\) it gives \(c\, y_0y_1=0\). Hence one of the \(y_k\) vanishes, and the remaining single product state is excluded as for \(r=1\).
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 \(c|W_N\rangle \).
If every block is \(L_0\)-block injective for a common \(L_0{\gt}0\), then \(N{\lt}2\max \bigl(L_0,\, 3(b-1)(L_0+1)\bigr)\).
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 \(D=\Omega (N^{1/3})\).
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 ] .
Cut the chain into two pieces of at least \(n_0=\max (L_0,3(b-1)(L_0+1))\) sites. By Lemmas 15.5.13 and 15.5.14, the reduced state of the first piece has rank at least \(\sum _j D_j^2\), while Lemma 15.5.12 bounds it by two. Hence there are at most two blocks, each of size one by one, with scalar letters \(A^j_0=x_j\) and \(A^j_1=y_j\), so that the amplitude of \(1^m0^{N-m}\) is \(\sum _jc_j\, y_j^mx_j^{N-m}\). Choosing \(z_j\) with \(z_j^N=c_j\) absorbs each coefficient into the one-site amplitudes, \(c_j\, y_j^mx_j^{N-m}=(z_jy_j)^m(z_jx_j)^{N-m}\). By Theorem 15.5.15, a sum of at most two translation-invariant product states is not a nonzero multiple of \(|W_N\rangle \) for \(N\geq 3\), a contradiction when \(N\geq 2n_0\). For the second item, an invertible letter gives condition C1 at \(L_0=D^2+1\) by the fixed-length Wielandt theorem; for the third, the general quantum Wielandt bound gives \(L_0\leq (D^2+1)^2\).
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 an arbitrary tuple \((X_j)_j\), simultaneous spanning gives coefficients \(a_\tau \) such that \(\sum _\tau a_\tau c_jB^j_\tau =X_j\). Hence
To obtain the dimension bound, choose \(X_k=0\) for \(k\ne j\). The map \(X_j\mapsto (\operatorname{tr}(X_jB^j_\sigma ))_\sigma \) is injective: if all its values vanish, \(R\)-block injectivity gives \(\operatorname{tr}(X_jY)=0\) for every matrix \(Y\), and matrix units then give \(X_j=0\). Its image lies in \(V\) by (24), so \(D_j^2\le \dim V\).
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 \(L_0+F\).
Suppose \(N\ge \max \{ L_0+F,12(b-1)\} \) and cut at \(L_0\). The word tuples on the other side, of length \(N-L_0\ge F\), span the product algebra by Lemma 15.5.13. Individual injectivity on the first side and Theorem 15.5.17 give \(D_j^2\le 2\), since the W-state cut rank is at most two. Thus every \(D_j=1\).
For scalar letters \(b_i^j\), unitality reads \(\sum _i|b_i^j|^2=1\). A nonzero letter spans the one-dimensional matrix algebra, so the same blocks are injective at length one. Reapply Theorem 15.5.16 with this length to obtain
a contradiction. Finally, \(L_0+F\le 2F\) and \(12(b-1)\le 2F\). If \(L_0\ge 3\) and \(b\ge 2\), then \(3(b-1)(L_0+1)\ge 12(b-1)\); if \(b\le 1\), the latter term is zero.
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.
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 \(D=N=0\).
If a tensor of bond dimension \(D\) is periodic with period \(m\), then \(m\leq D\).
The cyclic sectors have positive dimensions \(D_j\) and support isometries \(V_j:\mathbb {C}^{D_j}\to \mathbb {C}^D\), with \(V_j^\dagger V_j=\mathbb {1}_{D_j}\), \(V_jV_j^\dagger =P_j\), and \(\sum _jP_j=\mathbb {1}_D\). Taking traces gives
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 \(A_i\).
If \((\mathbb {1}-P)A_iP=0\) for an orthogonal projection \(P\), then \(P-\sum _iA_iPA_i^\dagger =\sum _iB_iB_i^\dagger \) with \(B_i=PA_i(\mathbb {1}-P)\) is positive, and its pairing with the positive definite dual fixed point vanishes. Hence \(P\) is a fixed point, a multiple of the identity, so \(P\) is \(0\) or \(\mathbb {1}\).
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.
Rescale by the inverse square root of the spectral radius of the transfer map and gauge by its positive definite eigenvector; the peripheral spectrum of the resulting trace-preserving irreducible channel is the group of \(m\)-th roots of unity. Period one means that \(1\) is the only peripheral eigenvalue, which is primitivity.
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 \(p{\gt}D\).
Write the periodic vector as \(\sum _j\lambda _j^N|\phi _{A^j}\rangle \) with unital irreducible blocks \(A^j\). Each block is irreducible by Lemma 15.5.22, and, rescaled and gauged, periodic by Lemma 15.5.23, with a period \(m_j\leq D_j\leq D\) by Theorem 15.5.21. If \(m_j\neq 1\), then \(2\leq m_j\leq D\), so the divisor assumption gives \(m_j\nmid N\). The block state vanishes by Theorem 7.4. If \(m_j=1\) the block is primitive, and the general quantum Wielandt bound of Theorem 6.4.8 gives injectivity after \(L_0=(D^2+1)^2\) sites. Group the blocks into the phase classes of Definition 8.6.1.7: the \(q\)-th member \(A^{(j,q)}\) of class \(j\) satisfies \(|\phi _{A^{(j,q)}}\rangle =\xi _{j,q}^N|\phi _{A^{(j,0)}}\rangle \), with \(A^{(j,0)}\) the class representative. Then
and no two representatives are related by a gauge transformation and a phase. Dropping the classes with \(c_j=0\) or period other than one leaves at most \(D\) blocks. The generalized first item of Theorem 15.5.16 then gives (30).
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.
If \(N\leq D\), the bound is immediate. Otherwise a divisor \(m{\gt}1\) of the prime \(N\) equals \(N{\gt}D\). Thus no \(m\) with \(2\leq m\leq D\) divides \(N\), and Theorem 15.5.24 applies.
Theorem 15.5.10 concerns one tensor representing the W state on a range of lengths. The review states the stronger single-length bound: a translation-invariant representation of \(|W_N\rangle \) with bond dimension \(D\) requires \(D^3\log D=\Omega (N)\), hence \(D=\Omega (N^{1/(3+\delta )})\) for every \(\delta {\gt}0\), obtained by combining the representation theory of translation-invariant matrix product states with a quantitative quantum Wielandt inequality [ 17 , Appendix A, “The W state” ] . This single-length bound is outside the scope of the results here: Theorem 15.5.16 proves the conditional single-length bound of its source, and Theorem 15.5.25 establishes the bound \(D=\Omega (N^{1/5})\) at prime lengths. Theorem 15.5.24 also covers lengths with no divisor between \(2\) and \(D\) [ 46 ] .
15.6 Product Vacua with Boundary States
The one-species PVBS chain has a product vacuum and a geometric boundary particle [ 4 , Section II, equations (1)–(8) ] . We use left-to-right matrix products and write \(q=\mu e^{i\theta }\) for the hopping parameter and phase. The review’s real parameter is \(q=1+\lambda \) [ 17 , Section IV.C.4 ] . Its printed right-localization threshold and unconditional periodic-uniqueness claim require the corrections explained in [ 43 ] . The following finite-chain results allow all complex \(q\); \(q=0\) is an algebraic extension of the original positive-hopping family. No spectral-gap assertion is included.
Set \(A^0=\operatorname {diag}(q,1)\) and \(A^1=|0\rangle \langle 1|\). Write \(\Omega _N=|0\cdots 0\rangle \) and \(\Psi _N(q)=\sum _{k=0}^{N-1}q^k|0\cdots 1_k\cdots 0\rangle \).
For every physical word \(\sigma \),
The standard boundary contraction \(\langle 0|A^{\sigma _1}\cdots A^{\sigma _N}|1\rangle \) is \(\Psi _N(q)(\sigma )\).
Multiplication by \(A^0\) contributes \(q\) before the unique particle; multiplication by \(A^1\) creates it. Two particles give the zero matrix. Induction on the word length gives (32).
The full boundary matrix-product space is \(\mathcal G_N=\operatorname {span}\{ \Omega _N,\Psi _N(q)\} \). It has dimension two for every \(N\ge 1\).
Taking the trace against a boundary matrix \(X\) gives \((q^NX_{00}+X_{11})\Omega _N+X_{10}\Psi _N(q)\). The entries \(X_{11}\) and \(X_{10}\) realize arbitrary coefficients. Evaluation on the vacuum and on a particle at the first site proves linear independence.
The open nearest-neighbour parent Hamiltonian has kernel \(\mathcal G_N\) and ground-state dimension two for every \(N\ge 2\).
A two-site vector satisfies the local constraint precisely when \(a_{11}=0\) and \(a_{01}=q a_{10}\). The left and right restrictions of a putative longer ground vector lie in the shorter vacuum-particle spaces. Their common two-site constraints eliminate the two-particle coefficient and relate the remaining particle coefficients by \(q\). This proves the intersection property from length two onward. Iterating it identifies all nonwrapping local constraints with \(\mathcal G_N\). The existing positive-parent kernel theorem then identifies the actual Hamiltonian kernel.
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 \(|q|\ne 1\).
Every periodic ground vector satisfies the nonwrapping constraints, hence is \(a\Omega _N+b\Psi _N(q)\). A cyclic translate is also a ground vector. Comparing its particle amplitudes at the first two sites yields \(b=bq^N\), forcing \(b=0\). The vacuum satisfies every cyclic constraint.
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.
Substitution gives the W tensor at \(q=1\) and verifies the two periodic constraints. Evaluation on a one-particle configuration separates the W vector from the vacuum span. The localization statements follow from the exact geometric coefficients and convergence of powers of a scalar of modulus less than one.
15.6.1 Exact Periodic Gap in the One-Species, Zero-Phase Model
For positive real hopping \(\mu \), the one-particle dispersion in [ 4 , Section II, after equation (8) ] is \(1-2\mu \cos (k)/(1+\mu ^2)\) at zero phase. The general multi-species thermodynamic gap formula stated there is a conjecture. The finite-ring result below is a proved one-species specialization, with an explicit operator argument; its scope is recorded in [ 127 ] .
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.
Transport addition through the equivalence between finite site indices and the cyclic group. For the exchange identity, express each transposition as two function updates and commute those updates with insertion of the cyclic window.
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 \(11\).
The image of the displayed operator is orthogonal to the two-site vacuum-particle space, and subtracting its image from any vector leaves a vector in that space. Uniqueness of orthogonal projection identifies the displayed operator with the actual parent interaction.
On every ring of \(N\ge 2\) sites, the actual PVBS parent Hamiltonian satisfies
The sum defining \(H_\mu \) contains both oriented windows when \(N=2\).
The difference between \(h_\mu \) and \(2\mu h_1/(1+\mu ^2)\) is the diagonal field \([(\mu ^2-\mu )n_L+(1-\mu )n_R]/(1+\mu ^2)\). Summing cyclically gives the same total occupation at the left and right ends of the windows, yielding (34).
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 \(v\perp \ker H_\mu \),
The critical parent \(H_1\) is positive. Every nonvacuum configuration contains at least one particle, so \(\langle v,\mathcal N v\rangle \ge \| v\| ^2\) when the vacuum coefficient vanishes. Apply (34). The vacuum belongs to the actual kernel, so kernel-orthogonal vectors meet this condition. The norm bound follows from Cauchy–Schwarz.
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.
The uniform one-particle vector is invariant under exchange of two sites and has no adjacent double occupation. Every critical local term therefore annihilates it. It has particle number one, so (34) gives its eigenvalue \(c(\mu )\). For noncritical hopping, this is positive, and symmetry of the Hamiltonian makes the eigenvector orthogonal to its kernel. Applying any proposed norm-gap bound to this nonzero vector gives the matching upper bound \(c(\mu )\).
15.6.2 Critical One-Particle Dynamics and the Thermodynamic Gap
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.
A one-particle configuration has no doubly occupied bond. The \(|11\rangle \langle 11|\) part of the local projector therefore vanishes, leaving the singlet exchange term. Sum the exchange differences using the existing one-magnon permutation identity. Constant coefficients have zero cycle Laplacian.
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.
The existing cyclic Fourier identity gives \(Lf=\eta _N f\). The particle amplitude at the zero site is one, so \(\psi _N\) is nonzero. The eigenvalue is positive for \(N\ge 2\). For every ground vector \(w\), symmetry gives \(0=\langle H_1w,\psi _N\rangle =\eta _N\langle w,\psi _N\rangle \), proving the full-kernel orthogonality.
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 Fourier energies \(\eta _N\) tend to zero. Given a proposed \(\delta {\gt}0\) and threshold \(N_0\), choose \(N\ge \max \{ N_0,2\} \) with \(0{\lt}\eta _N{\lt}\delta \). Applying the proposed inequality to the nonzero kernel-orthogonal vector \(\psi _N\) forces \(\delta \le \eta _N\), a contradiction. This proves the critical specialization of the one-particle dispersion following equation (8) in [ 4 ] ; the broader multi-species scope remains as recorded in [ 127 ] .
15.7 The cluster state
The cluster state [ 7 , 230 ] is a universal resource for measurement-based quantum computation; its one-dimensional chain provides another canonical example of a non-trivial SPT phase [ 236 , 17 ] . We follow the presentation of [ 17 , Appendix A, “The cluster state” ] . Throughout, \(|\pm \rangle = \tfrac {1}{\sqrt{2}}(|0\rangle \pm |1\rangle )\).
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, \((A^s)_{\alpha \beta } = \delta _{\alpha s}\, (-1)^{s\beta }/\sqrt{2}\).
Close the virtual indices periodically by the trace. Theorem 15.7.2 identifies this vector with the controlled-\(Z\) circuit state, including its short-ring convention; Lemma 15.7.5 relates the two tensor representatives. Theorem 15.7.6 gives normality, the \(\mathbb {Z}_2\times \mathbb {Z}_2\) symmetry and its virtual action, string order under the stated boundary restriction, and zero correlation length after blocking two sites. The nontrivial projective class is Theorem 15.7.15; the periodic stabilizer ground state for \(N\ge 3\) is Theorem 15.10.10, using the gauge-equivalent representative.
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 \(|-\rangle \).
Since \(A^s = |s\rangle \! \langle h_s|\) with \(h_0 = +\) and \(h_1 = -\), the trace of the product is \(\prod _j \langle h_{s_j} | s_{j+1} \rangle \) with indices modulo \(N\), and \(\langle h_a | b \rangle = (-1)^{ab}/\sqrt{2}\). On the other side, every coefficient of \(|+\rangle ^{\otimes N}\) equals \(2^{-N/2}\), and the gates multiply the coefficient of \(|s\rangle \) by \(\prod _j (-1)^{s_j s_{j+1}}\).
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 \(B^s = \sqrt{2}\, A^s\).
The entries are \((B^s)_{\alpha \beta } = \delta _{\alpha s} H_{s\beta }\), so in the expansion of the trace as a sum over cyclic index paths only the path \(\alpha _j = s_j\) survives, and it contributes \(\prod _j H_{s_j s_{j+1}}\). The identity \(B^s = \sqrt{2}\, A^s\) is read off from \(|s\rangle \! \langle s|H = \sqrt{2}\, |s\rangle \! \langle h_s|\) with \(h_0 = +\) and \(h_1 = -\).
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.
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.
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.
Each property is invariant under the gauge equivalence of Lemma 15.7.5 and holds for the transposed representative by Theorems 15.7.7, 15.7.8, 15.7.10, 15.7.19 and 15.7.21. The twisted blocked tensors and their virtual action are the conjugates by \(H\) of those of the transposed representative, so the intertwining relations carry over. For the string order, the blocked transfer channel of the transposed representative is unital and its adjoint fixes the maximally mixed state; since \(H\) is real symmetric with \(H^2 = 2\, \mathbb {1}\), conjugation by \(H\) carries both properties to the blocked tensor of Definition 15.7.1.
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 \(\dim M_{2}(\mathbb {C}) = 4\).
Both \(A^0\) and \(A^1\) are rank-one matrices with linearly independent column spaces, so they are linearly independent. Their span is two-dimensional, which is strictly less than \(\dim M_{2}(\mathbb {C}) = 4\).
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.
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 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 virtual matrices implementing the two generators of Theorem 15.7.10, namely \(V_1 = \sigma _z\) and \(V_2 = \sigma _x\), anticommute (\(V_1 V_2 = -V_2 V_1\)).
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 commutator phase of the cluster-state factor system on the two generators is \(-1\).
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 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.
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 \(A\),
For \(g\) the identity both sides equal \(A^i\). For each of the other three elements, with \(X_g\) equal to \(\sigma _z\), \(\sigma _x\) or \(\sigma _z\sigma _x\), a direct computation on the four blocked letters gives \(\sum _ju(g)_{ij}A^j=X_gA^iX_g^{-1}\); multiplying on the right by \(X_g\) gives (39).
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 ] .
Take \(V_{i,(a,b)}=\sqrt2\, (A^i)_{ab}\); the four blocked letters are orthogonal with \(\operatorname{tr}(A^iA^{j\dagger })=\tfrac 12\delta _{ij}\), so \(V\) is unitary. Both tensors realize the symmetry \(g\) with the same virtual gauge, so \(\sum _j(u(g)V)_{ij}B^j=\sum _j(VU(g))_{ij}B^j\), and the letters \(B^j\) are linearly independent. The class statement follows from Theorem 10.4.7.4 and the non-triviality of the cluster factor system.
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 blocked tensor is injective and satisfies \(\mathcal{E}(\mathbb {1})=\mathbb {1}\); the two generators of \(\mathbb {Z}_2\times \mathbb {Z}_2\) act by unitary permutation matrices with anticommuting virtual gauges \(\sigma _z\) and \(\sigma _x\). Apply Theorem 10.4.10.5.
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.
The blocked tensor is injective (Remark 15.7.9) and on-site symmetric under \(\mathbb {Z}_2 \times \mathbb {Z}_2\) (Theorem 15.7.10), and each group element acts by a real symmetric involutive permutation matrix, hence unitarily. The maximally mixed state \(\Lambda = \tfrac {1}{2}\mathbb {1}\) is positive definite and has trace \(1\). The four blocked matrices in (38) satisfy \(\sum _i A^i (A^i)^\dagger = \mathbb {1}\) and \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\), so the transfer map is unital and \(\Lambda \) is a fixed point of the adjoint transfer map. Applying Theorem 10.4.35 yields virtual-boundary nondecay for every \(g\).
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 single-site cluster transfer map is not idempotent, but the four blocked matrices give the closed form \(\mathcal{E}_A(X) = \tfrac {1}{2}(X_{00} + X_{11}) \mathbb {1}\), which is manifestly idempotent.
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.
The claim follows from the idempotence of the blocked cluster transfer map and the equivalence of zero correlation length with transfer-map idempotence (Theorem 17.5.20).
15.8 The Kitaev chain
The Kitaev chain, or Majorana chain [ 209 ] , is a chain of spinless fermions. In the limit where the Majorana modes pair up only across neighbouring sites, its ground state is the non-trivial fixed point of the topological phase, and it is described by a graded (fermionic) matrix product state with bond dimension two and a twist at the boundary [ 17 , Appendix A, “The Kitaev chain” ] , [ 9 ] .
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 ] .
On a ring of \(N\) sites the state is
where \(\otimes _g\) is the graded tensor product; the state has odd fermion parity [ 17 , Section II.B.4 ] .
Scope.
The results below concern the ordinary contraction of the matrices, which gives the coefficients of (40) in the ordered graded product basis; the graded contraction rules and the graded notion of injectivity are outside the scope of these results [ 70 ] .
The boundary twist enters as \(\operatorname{tr}(YA^{i_1}\cdots A^{i_N})\). Theorem 15.8.5 and Corollary 15.8.6 compute both twisted and untwisted coefficients and their parity supports. Lemma 15.8.7 gives the open closures. Theorems 15.8.8 and 15.8.9 describe non-normality and the two injective sectors of the ordinary matrix tensor. These are not statements of graded injectivity; that distinction is maintained in [ 70 ] .
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).
\(A^0=\mathbb 1\), \(A^1=Y\), \((A^1)^2=Y^2=-\mathbb 1\), and \(Y\) commutes with \(A^0\) and \(A^1\) [ 9 ] .
Direct multiplication of \(2\times 2\) matrices.
If the word \(w\) contains the letter \(1\) exactly \(k\) times, then \(A^{i_1}\cdots A^{i_N}=Y^k\).
Induction on the word, using \(A^0=\mathbb 1\) and \(A^1=Y\).
Let the word \(w\) contain the letter \(1\) exactly \(k\) times. Then
Neither amplitude depends on the length \(N\) except through \(k\).
By Lemma 15.8.4 the products are \(Y^k\) and \(Y^{k+1}\). Since \(Y^2=-\mathbb 1\), an even power \(Y^{2m}\) is \((-1)^m\mathbb 1\), of trace \(2(-1)^m\), and an odd power \(Y^{2m+1}\) is \((-1)^mY\), of trace zero.
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 ] .
Both \(\mathbb 1\) and \(Y\) have this form, and so does every power of \(Y\).
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 ] .
The line spanned by \((1,i)\) is invariant under \(A^0=\mathbb 1\) and under \(A^1=Y\). The linear functional \(M\mapsto M_{00}-M_{11}\) vanishes on every product by (47) (Lemma 15.8.7), while the matrix unit \(E_{00}=|0\rangle \! \langle 0|\) satisfies \((E_{00})_{00}-(E_{00})_{11}=1\neq 0\), so \(\operatorname {span}\{ A^w : |w|=N\} \subsetneq M_{2}(\mathbb {C})\) for every \(N\).
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 ] .
With \(k\) the number of letters \(1\) in \(w\), the right-hand side is \(i^{k+1}+(-i)^{k+1}\), which is the closed form of the twisted amplitude.
The Kitaev-chain tensor is related to the even-parity tensor \(E^0=\tfrac 1{\sqrt2}\mathbb 1\), \(E^1=\tfrac 1{\sqrt2}\sigma ^x\), whose periodic contraction is the equal superposition of even-parity configurations.
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.
\(X\sigma ^xX^{-1}=\sigma ^y\) and \(i\sigma ^y=Y\).
15.9 Correlation independence without a renormalization fixed point
Example 3.4 of [ 13 , 14 ] demonstrates physical correlation independence without satisfying the renormalization fixed-point condition.
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.
Close the virtual indices by the trace. Theorem 15.9.2 identifies the resulting periodic vector. Physical correlation independence is Theorem 15.9.3, while Theorem 15.9.4 proves that the tensor is not a renormalization fixed point. The blocked overlap calculation is Theorem 15.9.5.
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 \(|+\rangle =(|0\rangle +|1\rangle )/\sqrt2\).
Both matrices are diagonal. The first diagonal entry contributes \(1\) only to the all-zero word, while the second contributes \((1/\sqrt2)^N\) to every word.
The tensor in Definition 15.9.1 satisfies physical correlation independence for arbitrary disjoint physical regions, including adjacent regions.
Every word matrix is diagonal. Consequently every observable-inserted transfer operator and the ordinary transfer operator act entrywise on bond matrices, so they commute. The two-region expectation therefore depends on the two complementary gap lengths only through their sum.
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.
Let \(E_{01}=|0\rangle \! \langle 1|\). The transfer map obeys
Hence \(\mathcal{E}_A^2\ne \mathcal{E}_A\). The equivalence between transfer idempotence and a physical blocking isometry gives the second conclusion.
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
Only the all-zero configuration contributes, with amplitude \((1/\sqrt2)^n\).
15.10 Parent Hamiltonian statements for the examples
Following the parent-Hamiltonian construction of [ 17 , lines 1988–2000 ] reviewed in Chapter 11, originally introduced for finitely correlated states by Fannes, Nachtergaele, and Werner [ 194 ] and developed for matrix product states by Perez-Garcia et al. [ 228 ] , the range-\(L\) local parent space is
Then \(\ker h_L = \mathcal G_L(A)\), the range-\(L\) parent Hamiltonian is \(H_N^{(L)} = \sum _i (h_L)_i\), and \(\operatorname{GS}(H)=\ker (H-E_0(H)\mathbb {1})\) denotes the ground eigenspace.
For the GHZ tensor \(A\) of Definition 15.2.7,
By definition, \(\Gamma _2(X)(\sigma _1,\sigma _2) =\operatorname{tr}(A^{\sigma _1}A^{\sigma _2}X)\). For the GHZ matrices \(A^0=\operatorname{diag}(1,0)\) and \(A^1=\operatorname{diag}(0,1)\), whenever \(\sigma _1\ne \sigma _2\), one has \(A^{\sigma _1}A^{\sigma _2}=0\). Hence \(\Gamma _2(X)=X_{00}|00\rangle +X_{11}|11\rangle \), and the image of \(\Gamma _2\) is \(\operatorname{span}_{\mathbb {C}}\{ |00\rangle ,|11\rangle \} \).
For \(N\ge 2\) and \(a\in \{ 0,1\} \), one has \(|a\rangle ^{\otimes N}\in \mathcal G_{N,2}(A_{\mathrm{GHZ}})\).
For a cyclic nearest-neighbour window and outside assignment \(\tau \), set \(R_{i,\tau }:=\left.|a\rangle ^{\otimes N}\right|_{[i,i+1],\tau }\). For \(b,c\in \{ 0,1\} \) with \(b\ne c\), the completed configuration is not constant, hence \(R_{i,\tau }(b,c)=0\). Therefore
so \(|a\rangle ^{\otimes N}\in \mathcal G_{N,2}(A_{\mathrm{GHZ}})\).
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 ] .
Let \(\psi \in \mathcal G_{N,2}(A_{\mathrm{GHZ}})\). For every cyclic nearest-neighbour window and outside assignment \(\tau \), one has
Taking the outside assignment from the word \(\sigma \) gives \(\left.\psi \right|_{[i,i+1],\sigma } (\sigma _i,\sigma _{i+1})=\psi (\sigma )\). Hence a mixed nearest-neighbour word has zero coefficient: \(\sigma _i\ne \sigma _{i+1}\Longrightarrow \psi (\sigma )=0\). A nonconstant binary cyclic word has a mixed edge: \(\sigma \notin \{ (0,\ldots ,0),(1,\ldots ,1)\} \Longrightarrow \exists i,\ \sigma _i\ne \sigma _{i+1}\). Therefore
Conversely, Theorem 15.10.2 gives \(|0\rangle ^{\otimes N},|1\rangle ^{\otimes N} \in \mathcal G_{N,2}(A_{\mathrm{GHZ}})\), and hence their span lies in \(\mathcal G_{N,2}(A_{\mathrm{GHZ}})\).
For the GHZ tensor of Definition 15.2.7,
The periodic nearest-neighbour parent Hamiltonian therefore satisfies
using the cyclic-window equality \(\mathcal G_{N,2}(A_{\mathrm{GHZ}}) =\operatorname{span}\{ |0\rangle ^{\otimes N},|1\rangle ^{\otimes N}\} \) from Theorem 15.10.3. This is the ferromagnetic Ising parent Hamiltonian [ 17 , lines 1194–1196 and line 2205 ] ; see also [ 196 , Section 3 ] for the displayed two-site space and interaction.
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
For the cluster tensor \(A\) of Definition 15.7.4,
the \(+1\) eigenspace of \(K = \sigma ^z_1\sigma ^x_2\sigma ^z_3\).
Write \(q = 1/\sqrt2\). The matrices \(A^s = |\pm \rangle \! \langle s|\) have entries \(A^s_{ij} = q(-1)^{is}\delta _{sj}\), so
The sign \((-1)^{s_1s_2+s_2s_3}\) changes by \((-1)^{s_1+s_3}\) when \(s_2\) is replaced by \(s_2+1\), so \(K\Gamma _3(X) = \Gamma _3(X)\) by (49). Conversely, if \(Kv = v\), then \(v(s_1,1,s_3) = (-1)^{s_1+s_3}v(s_1,0,s_3)\), and the matrix \(X\) with rows \(X_{s0} = v(0,0,s) + v(1,0,s)\), \(X_{s1} = v(0,0,s) - v(1,0,s)\) satisfies \(\Gamma _3(X) = 2q^3\, v\) by (49); hence \(v = \Gamma _3(X/(2q^3)) \in \mathcal G_3(A)\).
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 \(K = \sigma ^z_1\sigma ^x_2\sigma ^z_3\).
With \(A_{\mathrm{src}}^t = |1-t\rangle \langle r_t|\), \(r_0 = (1,1)\), \(r_1 = (1,-1)\), one has \(\langle r_t | u \rangle = (-1)^{tu}\) and
Replacing \(t_2\) by \(t_2+1\) changes the exponent by \(1+t_1+t_3\) modulo \(2\), so \(K\Gamma _3(X) = -\Gamma _3(X)\). Conversely, if \(Kv = -v\), then \(v(t_1,1,t_3) = -(-1)^{t_1+t_3}v(t_1,0,t_3)\), and the matrix \(X\) with \(X_{0,1-t} = (-1)^t\bigl(v(t,0,0)+v(t,0,1)\bigr)\), \(X_{1,1-t} = (-1)^t\bigl(v(t,0,0)-v(t,0,1)\bigr)\) satisfies \(\Gamma _3(X) = 2v\) by (50); hence \(v = \Gamma _3(X/2)\in \mathcal G_3(A_{\mathrm{src}})\).
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 \(N\ge 3\).
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 a cyclic three-site window at \(i\) and an outside assignment \(\tau \), the restriction \(\psi |_{[i,i+2],\tau }\) satisfies \(K\bigl(\psi |_{[i,i+2],\tau }\bigr) = (K_i\psi )|_{[i,i+2],\tau }\), since \(K_i\) reads \(\sigma _i\), \(\sigma _{i+2}\) and flips \(\sigma _{i+1}\) inside the window. Thus \(K_i\psi = \psi \) implies that every restriction is fixed by \(K\); conversely, if every restriction is fixed by \(K\), then evaluating at the window extracted from \(\sigma \) with outside assignment \(\sigma \) itself gives \((K_i\psi )(\sigma ) = \psi (\sigma )\). By Theorem 15.10.6, a restriction is fixed by \(K\) exactly when it lies in \(\mathcal G_3(A_{\mathrm{cl}})\).
For \(N\ge 3\),
and this common eigenspace is one-dimensional.
By Theorem 15.10.9 the common eigenspace is \(\mathcal G_{N,3}(A_{\mathrm{cl}})\). The cluster tensor is injective after blocking \(L_0 = 2\) sites (Theorem 15.7.8), so Theorem 11.11.2.3 with \(L = 3 \le N\) gives \(\mathcal G_{N,3}(A_{\mathrm{cl}}) = \mathbb {C}\, V^{(N)}(A_{\mathrm{cl}})\), and Theorem 11.11.2.4 gives \(\dim \mathcal G_{N,3}(A_{\mathrm{cl}}) = 1\).
For \(N\ge 3\), the periodic three-site ground space of the source tensor is the common \(-1\) eigenspace of the translated stabilizers:
The three-site calculation in Theorem 15.10.7 identifies \(\mathcal G_3(A_{\mathrm{src}})\) with the \(-1\) eigenspace of \(K=ZXZ\). The periodic ground space imposes this local condition at every position around the ring. Each translated operator \(K_i\) acts only on its three sites, so we can apply the local calculation with the outside spins fixed.
Fix a site \(i\) and fix the spins outside \([i,i+2]\) to a configuration \(\tau \). The remaining coefficients form a three-site vector \(\psi |_{[i,i+2],\tau }\). The operator \(K_i\) acts only on these three sites: it flips the middle spin and multiplies by the two endpoint signs. Since \(N\ge 3\), these sites are distinct, and therefore
It follows that \(K_i\psi =-\psi \) if and only if \(K\bigl(\psi |_{[i,i+2],\tau }\bigr) =-\psi |_{[i,i+2],\tau }\) for every \(\tau \). For the reverse implication, every coefficient of \(K_i\psi +\psi \) occurs in one of these three-site vectors, so all its coefficients vanish.
Substituting the local ground-space identity, we obtain \(K_i\psi =-\psi \) for every \(i\) exactly when every three-site restriction belongs to \(\mathcal G_3(A_{\mathrm{src}})\), which is the defining condition for \(\psi \in \mathcal G_{N,3}(A_{\mathrm{src}})\).
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 \(N\ge 3\).
Applying \(\sigma ^z\) at every site changes every stabilizer eigenvalue from \(+1\) to \(-1\). We first relate the vector associated with the normalized cluster tensor to the vector defined by the source matrices.
Put
Direct multiplication gives, for \(t=0,1\),
For a configuration \(s=(s_0,\ldots ,s_{N-1})\), adjacent inverse factors cancel in the ordered product:
Taking the trace in (52) and using \(\operatorname {tr}(GBG^{-1})=\operatorname {tr}(B)\) gives
The parity factor is precisely the action of \(U\) on the coefficient at \(s\). This proves (51); nonvanishing follows from the nonzero cluster vector, since \((\sqrt2)^N\ne 0\) and \(U^2=\mathbb {1}\).
Conjugating by \(\sigma ^z\) fixes \(\sigma ^z\) and sends \(\sigma ^x\) to \(-\sigma ^x\). The three sites in each \(K_i\) are distinct, so \(UK_iU=-K_i\). If \(K_iv=v\) for every \(i\), then \(K_i(Uv)=-Uv\) for every \(i\). Conversely, if \(K_iw=-w\) for every \(i\), then \(K_i(Uw)=Uw\); applying \(U\) again returns \(w\). Thus \(U\) maps the common \(+1\) eigenspace bijectively onto the common \(-1\) eigenspace. By Theorem 15.10.10 and (51),
This space is one-dimensional.
Each \(K_i\) changes coefficient signs and permutes the computational basis, so it preserves the norm. Expanding the squares gives
Equality holds exactly when every \((\mathbb {1}+K_i)v\) vanishes, that is, when \(v\in \mathbb {C}|V^{(N)}(A_{\mathrm{src}})\rangle \). Conversely, summing the equations \(K_i|V^{(N)}(A_{\mathrm{src}})\rangle =-|V^{(N)}(A_{\mathrm{src}})\rangle \) gives \(H|V^{(N)}(A_{\mathrm{src}})\rangle =-N|V^{(N)}(A_{\mathrm{src}})\rangle \). Together with (53), this identifies \(-N\) as the attained ground energy and its eigenspace as \(\mathbb {C}|V^{(N)}(A_{\mathrm{src}})\rangle \).
The cluster tensor of Definition 15.7.4 is the reflected convention of the tensor displayed in [ 17 , lines 2364–2369 ] . By (49) and its reflected counterpart, the three-site parent space of either convention is the \(+1\) eigenspace of \(\sigma ^z_1\sigma ^x_2\sigma ^z_3\). Since each \(K_i\) is a self-adjoint involution, \(\tfrac 12(\mathbb {1}-K_i)\) is the orthogonal projector onto its \(-1\) eigenspace, and Theorem 15.10.10 says that the cluster state is the unique ground state of \(\sum _i \tfrac 12(\mathbb {1}- \sigma ^z_i\sigma ^x_{i+1}\sigma ^z_{i+2})\), with ground energy \(0\).
The older review states instead that cluster states are unique ground states of \(\sum _i \sigma ^z_i\sigma ^x_{i+1}\sigma ^z_{i+2}\) with the sign as printed [ 228 , local TeX lines 374–387 ] . Both statements are correct for their own tensors: on the matrices \(A_{\mathrm{src}}\) displayed there every stabilizer has eigenvalue \(-1\) (Theorem 15.10.7), while on \(A_{\mathrm{cl}}\) it has eigenvalue \(+1\). The two tensors are related by \(A_{\mathrm{src}}^t = \sqrt2\, \sigma ^x A^t\) for the tensor \(A\) of [ 17 , lines 2364–2369 ] , and since \(\langle \pm |\sigma ^x = \pm \langle \pm |\) the states differ by \(\sigma ^z\) on every site, which conjugates each \(K_i\) to \(-K_i\) and exchanges the two Hamiltonians \(\sum _i K_i\) and \(-\sum _i K_i = \sum _i(\mathbb {1}-K_i) - N\mathbb {1}\). Theorem 15.10.12 proves this on-site relation for the matrix product vectors and identifies the ground energy and eigenspace for the printed matrices; see [ 50 ] .
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
Write \(A^0 = c\, \sigma ^z\), \(A^1 = b\, \sigma ^+\), \(A^2 = -b\, \sigma ^-\) with \(c = 1/\sqrt3\), \(b = \sqrt2/\sqrt3\), so that \(b^2 = 2c^2\). The products are \(A^0A^0 = c^2\mathbb {1}\), \(A^0A^1 = -A^1A^0 = cb\, |0\rangle \! \langle 1|\), \(A^0A^2 = -A^2A^0 = cb\, |1\rangle \! \langle 0|\), \(A^1A^2 = -b^2|0\rangle \! \langle 0|\), \(A^2A^1 = -b^2|1\rangle \! \langle 1|\), and \(A^1A^1 = A^2A^2 = 0\). Hence \(\Gamma _2(X)(a,b) = \operatorname{tr}(A^aA^bX)\) has coordinates
in the order \(00,01,02,10,11,12,20,21,22\), which satisfies (54) because \(2c^2 = b^2\). Conversely, given \(v\) satisfying (54), the matrix \(X\) with \(X_{00} = -c^2v_{12}\), \(X_{01} = cb\, v_{02}\), \(X_{10} = cb\, v_{01}\), \(X_{11} = -c^2v_{21}\) satisfies \(\Gamma _2(X) = c^2b^2\, v\) by (55), so \(v = \Gamma _2(X/(c^2b^2))\).
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.
The inclusion \(\supseteq \) is the restriction property of local ground spaces (Lemmas 11.5.2.1 and 11.5.2.2). For \(\subseteq \), let \(\psi \) be a three-site vector whose restrictions \(\psi (i,\cdot ,\cdot )\) and \(\psi (\cdot ,\cdot ,k)\) all lie in \(\mathcal G_2(A)\), so that each satisfies (54). With \(c\), \(b\) as in the proof of Theorem 15.10.14, the products \(A^aA^bA^c\) give
Set \(X_{00} = c\, \psi _{102}\), \(X_{01} = -b\, \psi _{002}\), \(X_{10} = b\, \psi _{001}\), \(X_{11} = c\, \psi _{210}\). Then \(\Gamma _3(X) = c^2b^2\, \psi \) coordinate by coordinate: the twelve coordinates absent from (56) vanish for \(\psi \) by the constraints \(v_{11} = v_{22} = 0\) on the six restrictions, together with \(\psi _{101} = -\psi _{110} = 0\) and \(\psi _{202} = -\psi _{220} = 0\); the sign relations \(\psi _{010} = -\psi _{001} = -\psi _{100}\), \(\psi _{020} = -\psi _{002} = -\psi _{200}\), \(\psi _{012} = -\psi _{102} = \psi _{120}\), \(\psi _{201} = -\psi _{210}\) come from \(v_{01}+v_{10} = 0\) and \(v_{02}+v_{20} = 0\); and the remaining coordinates use \(2v_{00}+v_{12}+v_{21} = 0\) with \(b^2 = 2c^2\): \(2\psi _{000} = \psi _{102}-\psi _{210}\), \(\psi _{021} = \psi _{210}\), \(\psi _{121} = -2\psi _{001}\), \(\psi _{212} = -2\psi _{002}\). Hence \(\psi = \Gamma _3(X/(c^2b^2)) \in \mathcal G_3(A)\).
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.
Let \(\psi \in \mathcal G_{N,2}(A_{\mathrm{AKLT}})\). Every cyclic three-site window restricts on either side to a cyclic two-site window, so both restrictions of \(\psi |_{[i,i+2],\tau }\) lie in \(\mathcal G_2(A_{\mathrm{AKLT}})\), and Theorem 15.10.16 places \(\psi |_{[i,i+2],\tau }\) in \(\mathcal G_3(A_{\mathrm{AKLT}})\). Hence \(\mathcal G_{N,2}(A_{\mathrm{AKLT}})\subseteq \mathcal G_{N,3}(A_{\mathrm{AKLT}})\). The AKLT tensor is injective after blocking \(L_0 = 2\) sites (Theorem 15.3.10), so Theorem 11.11.2.3 with \(L = 3\le N\) gives \(\mathcal G_{N,3}(A_{\mathrm{AKLT}}) = \mathbb {C}\, V^{(N)}(A_{\mathrm{AKLT}})\). The reverse inclusion holds because the periodic MPS vector satisfies every cyclic two-site constraint (Lemma 11.9.8). Finally \(V^{(N)}(A_{\mathrm{AKLT}})\ne 0\) by Theorem 11.9.9 with \(L_0 = 2 {\lt} N\), so \(\dim \mathcal G_{N,2}(A_{\mathrm{AKLT}}) = 1\).
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 \(N=2\) the Hamiltonian is the projector onto the orthogonal complement of \(\mathcal G_{2}(A_{\mathrm{AKLT}})\). For \(N\ge 3\), a kernel vector satisfies every nonwrapping two-site constraint. Each nonwrapping three-site window restricts on either side to a nonwrapping two-site window, so Theorem 15.10.16 places every three-site restriction in \(\mathcal G_{3}(A_{\mathrm{AKLT}})\); this is the local check of [ 17 , line 2095 ] , used here on nonwrapping windows. The AKLT tensor is injective after blocking \(L_{0}=2\) sites (Theorem 15.3.10). For \(N\ge 3\), with \(s\) ranging over starts satisfying \(s+3\le N\) and \(\tau \) over complementary configurations,
The last step is the open-chain intersection theorem. The reverse inclusion holds for every tensor. The span description is Theorem 12.3.7.18, and Theorem 12.3.7.19 with \(D=2\) gives dimension \(D^{2}=4\).
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 \(h=\vec S_0\cdot \vec S_1+\tfrac 13(\vec S_0\cdot \vec S_1)^2\).
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 ] .
In the basis of Definition 15.10.19, \(\vec S_0\cdot \vec S_1=S^z\otimes S^z+\tfrac 12(S^+\otimes S^-+S^-\otimes S^+)\) has integer entries: it fixes \(|11\rangle \) and \(|22\rangle \), exchanges \(|01\rangle \leftrightarrow |10\rangle \) and \(|02\rangle \leftrightarrow |20\rangle \), and maps \(|00\rangle \mapsto |12\rangle +|21\rangle \), \(|12\rangle \mapsto |00\rangle -|12\rangle \), \(|21\rangle \mapsto |00\rangle -|21\rangle \). A computation on the nine coordinates shows that the right side \(Q\) of (58) is \(\sum _kw_kw_k^{\mathsf T}/\lVert w_k\rVert ^{2}\) for the five vectors \(|11\rangle \), \(|22\rangle \), \(|01\rangle +|10\rangle \), \(|02\rangle +|20\rangle \), \(|12\rangle +2|00\rangle +|21\rangle \). Hence \(Q\) is real symmetric and idempotent, and its kernel is cut out by the five constraints (54), which is \(\mathcal G_2(A_{\mathrm{AKLT}})\) by Theorem 15.10.14. A self-adjoint idempotent is the orthogonal projector onto the orthogonal complement of its kernel. The same computation gives \((\vec S_0\cdot \vec S_1)Q=Q\), so a vector is fixed by \(\vec S_0\cdot \vec S_1\) exactly when it is fixed by \(Q\). Since \(\sum _\alpha (S^\alpha )^{2}=2\) on each site, \(\vec S_{\mathrm{tot}}^{2}=4+2\, \vec S_0\cdot \vec S_1\), so the eigenspace of \(\vec S_{\mathrm{tot}}^{2}\) for \(6\) is the set of vectors fixed by \(\vec S_0\cdot \vec S_1\), which is the range of \(Q\).
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.
Restricting a chain vector to the window \(i,i+1\) turns \(\vec S_i\cdot \vec S_{i+1}\) into \(\vec S_0\cdot \vec S_1\), so the local term of the two-site parent Hamiltonian at \(i\) is \(\tfrac 12\bigl(h_i+\tfrac 23\bigr)\) by Theorem 15.10.20. Summing over the \(N\) bonds gives the identity. The parent Hamiltonian is positive by Lemma 12.3.8, so a nonzero \(v\) with \(H_{\mathrm{AKLT},N}v=\mu v\) satisfies \(\mu +\tfrac {2N}{3}\ge 0\). The eigenspace for \(-\tfrac {2N}{3}\) is the kernel of \(H^{(2)}_{N,\mathrm{AKLT}}\), which is \(\mathcal G_{N,2}(A_{\mathrm{AKLT}})\) by Theorem 12.3.17 and \(\mathbb {C}\, V^{(N)}(A_{\mathrm{AKLT}})\), a line, by Theorem 15.10.17.
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 ] .
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.
Sum the local identity \(h+\frac23I=2P^{(2)}\) over the \(N-1\) nonwrapping bonds. The sum of projectors is positive and its kernel is the four-dimensional open matrix-product space, so the lower energy bound is attained precisely on that space.
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.
In the chosen bond basis let \(J^x=\sigma ^x/2\), \(J^y=-\sigma ^y/2\) and \(J^z=-\sigma ^z/2\). The local identity \(\sum _a(S^\alpha )_{ia}A^a=A^iJ^\alpha -J^\alpha A^i\) telescopes along the open chain. Cyclicity of the trace then gives \(S^\alpha _{\mathrm{tot}}\Gamma _N(X)=\Gamma _N([J^\alpha ,X])\). The Pauli identities yield \(\sum _\alpha [J^\alpha ,{[J^\alpha ,X]}]=2X-\operatorname {tr}(X)I\). Block injectivity for \(N\ge 2\) makes \(\Gamma _N\) injective. The scalar matrices have dimension one and the traceless matrices dimension three; the preceding theorem identifies their images with the ground sectors of the physical polynomial Hamiltonian.