17 Concrete Examples
This chapter collects concrete MPS tensors from [ CPGSV21 , Section III ] . Each example is specified by its physical dimension \(d\), bond dimension \(D\), and explicit matrix family \(\{ A^i\} \). Key structural properties (injectivity, RFP status, symmetry) are stated for each tensor.
17.1 The GHZ state
The Greenberger–Horne–Zeilinger (GHZ) state is the prototypical non-injective, renormalization-fixed-point MPS.
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
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 17.1.4.
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 17.1.7 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 24.5.13).
17.2 The AKLT state
The Affleck–Kennedy–Lieb–Tasaki (AKLT) state is the canonical example of a normal (block-injective) MPS with non-trivial symmetry-protected topological (SPT) order.
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-\(\tfrac 12\).
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 17.2.4, 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 17.2.8), built from the anticommuting gauges \(i\sigma _y\) and \(\sigma _z\) (Theorem 17.2.5), is not a coboundary: its commutator phase on the two generators is \(-1\). Its class in \(H^2(\mathbb {Z}_2 \times \mathbb {Z}_2, \mathrm{U}(1)) \cong \mathbb {Z}_2\) is therefore non-trivial, so the AKLT chain realizes a non-trivial SPT phase rather than a trivial product state.
This \(\mathbb {Z}_2 \times \mathbb {Z}_2\) is the smallest subgroup of \(\mathrm{SO}(3)\) that still carries a non-trivial \(2\)-cocycle. That minimality, and the full continuous classification over \(\mathrm{SO}(3)\), are not used in this discrete-subgroup example (Remark 17.2.20).
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 17.2.4. Identifying the blocked physical index with the pair of single-site indices, the entry of \(P_g \otimes P_g\) at \((i_1 i_2, j_1 j_2)\) is the product \((P_g)_{i_1 j_1}(P_g)_{i_2 j_2}\) over the two grouped sites. This is the on-site action that makes the blocked twist agree with blocking the single-site twist.
The 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 17.2.3): 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 17.2.12).
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 17.2.4), 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: \(P_g P_g^\dagger = \mathbb {1}\).
For every element \(g\) of \(\mathbb {Z}_2 \times \mathbb {Z}_2\), the length-\(2\) blocked AKLT tensor has string order in the sense of Definition 12.3.11 for the twist by \(g\), with the maximally mixed boundary state \(\Lambda = \tfrac {1}{2}\mathbb {1}\) as its stationary boundary state.
The blocked tensor is injective (Theorem 17.2.13) and on-site symmetric under \(\mathbb {Z}_2 \times \mathbb {Z}_2\) (Theorem 17.2.14), 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:
Moreover, \(\Lambda = \tfrac {1}{2}\mathbb {1}\) is a fixed point of the adjoint transfer map:
The first identity sums the right-canonical relation \(\sum _i A^i(A^i)^\dagger = \mathbb {1}\) over the inner letter and then the outer; the second sums the left-canonical relation \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\) over the outer letter and then the inner, using \(\Lambda = \tfrac {1}{2}\mathbb {1}\). These are exactly the hypotheses of Theorem 12.4.4, which then yields string order for every \(g\). Like the cluster state, the AKLT state is an explicit witness of that criterion, sharing the same virtual-boundary caveat on the string order parameter.
The Cartesian presentation of the AKLT tensor has \(d = 3\), \(D = 2\), and \(A^x = \sigma _x\), \(A^y = \sigma _y\), \(A^z = \sigma _z\), the three Pauli matrices. This is the rotationally natural form used in [ CPGSV21 ] (around line 1159); it is the AKLT state in the Cartesian basis of the spin-\(1\) site, the counterpart of the \(|m\rangle \)-basis form of Definition 17.2.1.
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.
17.2.1 Correlation length
The AKLT state is the canonical example of a finite correlation length. 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 \((\tfrac 13)^n\).
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 16.2.5 to \(\lambda _2 = -\tfrac 13\).
17.3 The Majumdar–Ghosh state
The Majumdar–Ghosh state is the exact dimer ground state of the spin-\(\tfrac 12\) chain \(H=\sum \boldsymbol {S}_i\cdot \boldsymbol {S}_{i+1} +\tfrac 12\sum \boldsymbol {S}_i\cdot \boldsymbol {S}_{i+2}\). On an even periodic chain it is the symmetric superposition of the two nearest-neighbour singlet coverings \((1,2)(3,4)\cdots \) and \((2,3)(4,5)\cdots (N,1)\), the one-dimensional analog of a resonating valence-bond state.
Set \(d = 2\) (spin-\(\tfrac 12\)) and \(D = 3\). The Majumdar–Ghosh tensor is the family \(\{ A^0, A^1\} \subset M_{3}(\mathbb {C})\) defined by
The three bond levels are the two spin-\(\tfrac 12\) values of an open singlet partner together with the closed inter-dimer link; the relative sign \(-\tfrac 1{\sqrt2}\) is the antisymmetry of the singlet. The equality between the matrix product vectors of these matrices and the even-ring dimer superposition, with normalization and cyclic convention fixed, is a separate verification, not part of this definition.
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 \(M_{3}(\mathbb {C})\).
The review writes the tensor in the valence-bond shorthand \(A = [|0\rangle ((02|+(20|) + |1\rangle ((12|+(21|)]\otimes Y\) with the singlet \(Y = \left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)\). Read literally as \((ab| = |a\rangle \! \langle b|\) on a separate three-level factor tensored with the two-level singlet space, that shorthand would describe a six-dimensional bond. Its two-site contractions do not have the singlet-pair support pattern: same-letter traces can be nonzero while opposite-letter traces vanish. The shorthand is therefore treated here as a schematic of the valence-bond construction: contracting the singlet \(Y\) leaves a matrix product operator on a three-dimensional bond, with the antisymmetry of \(Y\) supplying the relative sign. The contracted tensor is the \(D = 3\) representative in Definition 17.3.1. The source-notation gap and the missing matrix-product-vector identification are recorded in docs/paper-gaps/rmp_majumdar_ghosh_tensor_gap.tex.
The remaining target for the Majumdar–Ghosh example is the ground-space description of its parent Hamiltonian. Because the tensor is non-normal (Theorem 17.3.5) with a two-periodic bond grading, its periodic ground space is governed by the block-diagonal boundary-condition machinery for non-normal tensors (Section 15.3) rather than by the injective single-state result. Thus the relevant target is the following range-three periodic ground-space statement.
On an even periodic chain of length \(N = 2m\ge 4\), let \(H_{N,\mathrm{MG}}^{(3)}\) be the range-three parent Hamiltonian of the Majumdar–Ghosh tensor, the periodic sum of translates of the local projector with kernel \(\mathcal G_3(A_{\mathrm{MG}})\). Then \(H_{N,\mathrm{MG}}^{(3)}\) has a two-dimensional ground space, spanned by the two nearest-neighbour singlet coverings \((1,2)(3,4)\cdots (N-1,N)\) and \((2,3)(4,5)\cdots (N,1)\). This two-fold degeneracy reflects the spontaneous breaking of the one-site translation symmetry of the spin-\(\tfrac 12\) chain \(H=\sum \boldsymbol {S}_i\cdot \boldsymbol {S}_{i+1} +\tfrac 12\sum \boldsymbol {S}_i\cdot \boldsymbol {S}_{i+2}\) of [ CPGSV21 ] .
17.4 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. Unlike the preceding examples it has no translation-invariant representation with bond dimension bounded independently of \(N\); 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.
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 \((v_L| A^{\sigma _1}\cdots A^{\sigma _N}\, |v_R)\).
Set \(d=D=2\). The W tensor is the site-independent family
with left boundary covector \((l|=(0|\) and right boundary vector \(|r)=|1)\). Here \(A^1\) is the single raising operator: it sends \(|1)\mapsto |0)\) and annihilates \(|0)\), and squares to zero.
For \(N\geq 0\), 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 17.4.5: the excitation count is one in the first three cases and zero or two in the others.
The open-boundary representation above is not translation invariant: the chain is closed by boundary vectors, not by a trace. The review observes that there is no translation-invariant representation whose bond dimension is bounded independently of \(N\): any translation-invariant matrix product state equal to \(|W_N\rangle \) must have bond dimension growing with \(N\), satisfying a bound of the form \(D^3\log D=\Omega (N)\). The review obtains this by combining the matrix product state representation theory of [ PGVWC07 ] with the quantitative quantum Wielandt inequality. That asymptotic lower bound is not used in the present example.
17.5 The cluster state
The cluster state is a universal resource for measurement-based quantum computation and provides another example of a non-trivial SPT phase.
Set \(d = D = 2\). The cluster-state tensor is the family \(\{ A^0, A^1\} \subset M_{2}(\mathbb {C})\) defined by
where \(|\pm \rangle = \tfrac {1}{\sqrt{2}}(|0\rangle \pm |1\rangle )\). The review [ CPGSV21 ] writes the reflected convention \(A^0 = |0\rangle \! \langle +|\), \(A^1 = |1\rangle \! \langle -|\); the transpose taken here is the same state read in the opposite direction and carries the same SPT order.
The 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 17.5.5, namely \(V_1 = \sigma _z\) and \(V_2 = \sigma _x\), anticommute (\(V_1 V_2 = -V_2 V_1\)).
The cluster-state factor system 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 17.5.9), built from the anticommuting gauges \(\sigma _z\) and \(\sigma _x\) (Theorem 17.5.6), is not a coboundary: its commutator phase on the two generators is \(-1\). Its class in \(H^2(\mathbb {Z}_2 \times \mathbb {Z}_2, \mathrm{U}(1)) \cong \mathbb {Z}_2\) is therefore non-trivial, so the cluster state realizes a non-trivial SPT phase.
For every element \(g\) of \(\mathbb {Z}_2 \times \mathbb {Z}_2\), the length-\(2\) blocked cluster-state tensor has string order in the sense of Definition 12.3.11 for the twist by \(g\), with the maximally mixed boundary state \(\Lambda = \tfrac {1}{2}\mathbb {1}\) as its stationary boundary state.
The blocked tensor is injective (Remark 17.5.4) and on-site symmetric under \(\mathbb {Z}_2 \times \mathbb {Z}_2\) (Theorem 17.5.5), 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 (??) 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. These are exactly the hypotheses of Theorem 12.4.4, which then yields string order for every \(g\). The cluster state is thus the first explicit witness of that criterion.
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 17.5.12) gives correlations independent of the separation. Like the GHZ state, and unlike the AKLT state, the cluster state is a renormalization fixed point with zero correlation length.
The claim follows from the idempotence of the blocked cluster transfer map and the equivalence of zero correlation length with transfer-map idempotence (Theorem 24.5.13).
17.6 Parent Hamiltonian statements for the examples
The parent-Hamiltonian construction of [ CPGSV21 , lines 1988–2000 ] , reviewed in Chapter 13, is recorded here in the form
Thus
We write \(\operatorname{GS}(H)=\ker (H-E_0(H)\mathbb {1})\) for the ground eigenspace.
For the GHZ tensor \(A\) of Definition 17.1.1,
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 [ CPGSV21 , 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 17.6.2 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 17.1.1,
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 17.6.3. This is the ferromagnetic Ising parent Hamiltonian [ CPGSV21 , lines 1194–1196 and line 2205 ] ; see also [ FGSW\(^{+}\)15 , Section 3 ] for the displayed two-site space and interaction.
The one-dimensional cluster tensor in Definition 17.5.1 is the reflected convention of the tensor displayed in [ CPGSV21 , lines 2364–2369 ] . The sourced stabilizer is \(K_i^{ZXZ}=\sigma ^z_i\sigma ^x_{i+1}\sigma ^z_{i+2}\). The corresponding positive parent interaction is
[ PGVWC07 , local TeX lines 374–387 ] . Thus an \(XZX\) formula should be used for this tensor only after establishing an equivalence such as
For the AKLT tensor of Definition 17.2.1, let \(\mathcal G_{[a,b]}\) denote the local ground space on sites \(a,\ldots ,b\). Then \(L_0=2\), and the three-site intersection property is
This is the dimensionally explicit form of the identity in [ CPGSV21 , line 2095 ] . The displayed local spin-chain Hamiltonian is
up to an additive constant [ PGVWC07 , local TeX lines 340–349 ] ; [ SPGC11 , lines 2126–2130 ] . The spin-\(2\) projector form is related to the polynomial Hamiltonian by the spin-addition identity
for two spin-\(1\) particles. For open boundary conditions and \(N\geq 2\),
[ CPGSV21 , lines 1169–1174 ] .