25 Examples of PEPS on the torus
The simplest PEPS place one four-leg tensor at every site of a square lattice on the torus. This chapter fixes that setting and computes the states of the standard examples of [ 17 , Appendix A ] : the GHZ state, the cluster state, the CZX model, the AKLT state, the resonating valence bond state, and the PEPS whose coefficients are the Boltzmann weights of classical models. The toric code and the quantum double models are treated with the theory of \(G\)-injective tensors in Chapter 26.
25.1 Torus PEPS of a four-leg tensor
Two-dimensional PEPS on the discrete torus are defined by a four-leg tensor \(A^i_{\alpha \beta \gamma \delta }\) placed at every site of a square lattice, with virtual indices ordered top, right, down, left [ 17 , Appendix A, “Two dimensions: PEPS” ] . On a torus, the four legs of site \(v=(x,y)\) correspond to the edges from \(v\) to \((x,y+1)\) (top), from \(v\) to \((x+1,y)\) (right), from \((x,y-1)\) to \(v\) (down), and from \((x-1,y)\) to \(v\) (left).
Let \(a^i_{\alpha \beta \gamma \delta }\) be a tensor with physical dimension \(d\) and bond dimension \(D\). Its torus PEPS on the torus of width \(w\) and height \(h\) has bond dimension \(D\) on every edge and, at every site \(v\), the local tensor \(a\) with its four virtual indices read on the top, right, down and left legs of \(v\) [ 17 , Appendix A, “Two dimensions: PEPS” ] . The right leg of \(v\) is the left leg of \(v+(1,0)\), and the top leg of \(v\) is the down leg of \(v+(0,1)\).
If a function \(f\) on the sites of the torus satisfies \(f(x+1,y)=f(x,y)\) and \(f(x,y+1)=f(x,y)\) for all \((x,y)\), then \(f\) is constant.
Induction on the representatives \(0,\ldots ,w-1\) of the first coordinate and \(0,\ldots ,h-1\) of the second.
Let \(w,h\ge 3\). The right edges and the up edges of the sites are pairwise distinct and exhaust the edges of the torus, and the coefficient of the torus PEPS of \(a\) is
the sum running over all assignments of bond indices \(h_{(x,y)}\) to the horizontal edges from \((x,y)\) to \((x+1,y)\) and \(v_{(x,y)}\) to the vertical edges from \((x,y)\) to \((x,y+1)\).
Every edge is horizontal or vertical, and every horizontal edge is the right edge of one of its endpoints and every vertical edge the up edge of one of its endpoints. For \(w\ge 3\) the right edges of distinct sites are distinct: otherwise \(p=q+(1,0)\) and \(q=p+(1,0)\), so \(2=0\) in the horizontal coordinate. The same holds vertically for \(h\ge 3\). A horizontal edge is never vertical. Reindexing the sum over edge labellings along this bijection gives the formula.
On a torus of width two the right and left neighbours of a site coincide and the two edges joining them are one edge of the lattice graph, so the four legs of a site are not four distinct bonds. The contraction formula, and the cluster, CZX, AKLT, RVB and classical-model statements below, therefore require \(w,h\ge 3\); the source constructions are local and apply to every torus [ 173 ] .
A virtual string on a torus PEPS acts on the bonds it crosses, diagonally in the bond basis: a bond carrying the index \(\alpha \) contributes an element \(X_\alpha \) of a monoid \(M\), and the contributions are multiplied in the order in which the string crosses the bonds. For a matrix algebra \(M\) this is a matrix product operator on the virtual level.
Let \(a\) be a four-leg tensor and \(\alpha \mapsto X_\alpha \) a map from the bond indices to a monoid \(M\) with \(X_tX_r=X_lX_b\) whenever \(a^{s}_{trbl}\neq 0\). Fix a rectangle of \(m\) columns and \(n\) rows of sites of the torus. Consider the string from its upper left to its lower right corner that crosses the left legs of the leftmost column, from top to bottom, and then the down legs of the bottom row, from left to right, and the string that crosses the top legs of the top row and then the right legs of the rightmost column. On every labelling of the bonds whose term in the contraction of the torus PEPS of \(a\) is nonzero the two strings have the same product, and so, for a semiring \(M\) with a \(\mathbb {C}\)-module structure, the contracted network with the first string inserted equals the contracted network with the second string inserted. This is the pulling-through equation of a matrix product operator symmetry [ 17 , Section “MPO and PEPO” ] for strings that are diagonal in the bond basis.
Write \(P_{ij}\) for the contribution of the left leg of the site in column \(i\) and row \(j\) of the rectangle, counted from its upper left site, and \(Q_{ij}\) for that of its top leg. The right leg of that site is the left leg of the site in column \(i+1\), and its down leg is the top leg of the site in row \(j+1\), so the hypothesis at a site with nonzero entry reads \(Q_{ij}P_{i+1,j}=P_{ij}Q_{i,j+1}\). Induction on \(n\) gives \(P_{i0}\cdots P_{i,n-1}\, Q_{in}=Q_{i0}\, P_{i+1,0}\cdots P_{i+1,n-1}\) for one column, and induction on \(m\), moving one column at a time, gives the identity for the rectangle. A nonzero term of the contraction has a nonzero entry at every site, and the terms that vanish contribute zero to both sides.
25.2 The GHZ state
The two-dimensional GHZ state is a superposition of the \(d\) constant spin configurations [ 17 , Appendix A, “The GHZ state” ] . All bonds are contracted periodically. Separate horizontal and vertical assignments of bond indices retain the self-loops and parallel bonds at small lattice sizes. The simple-graph representation requires \(w,h\ge 2\) [ 173 ] .
The GHZ tensor has \(D=d\) and \(A^i_{\alpha \beta \gamma \delta }=\delta _{i=\alpha =\beta =\gamma =\delta }\) [ 17 , Appendix A, “The GHZ state” ] .
On every torus with \(w,h\ge 1\), the GHZ tensor generates the unnormalized state \(\sum _{i=0}^{d-1}|i,i,\ldots ,i\rangle \) [ 17 , Appendix A, “The GHZ state” ] . With horizontal and vertical assignments of bond indices \(h_{(x,y)},v_{(x,y)}\) as in Lemma 25.1.3, its coefficient is
For \(w,h\ge 2\), the simple-graph GHZ PEPS has the same coefficients.
A virtual configuration contributes only if every leg of every site carries the physical index of that site. The right leg of \(v\) is the left leg of \(v+(1,0)\) and the top leg of \(v\) is the down leg of \(v+(0,1)\), so \(\sigma \) is invariant under both unit shifts and is constant by Lemma 25.1.2. Every edge is then labelled by that constant, so exactly one virtual configuration contributes when \(\sigma \) is constant and none otherwise.
For \(d\ge 2\), the four-leg GHZ tensor is not injective. For \(d\ge 2\) and \(w,h\ge 2\), the simple-graph GHZ PEPS is not vertex-injective.
The four-leg basis vector with labels \((0,1,0,0)\) has zero image. In the simple graph, give the right bond at one vertex label \(1\) and every other incident bond label \(0\). The top and right bonds are distinct, so no physical label can agree with both. This virtual basis vector is mapped to zero.
Every permutation \(\pi \) of the \(d\) labels acts on the GHZ site tensor by
For every positive \(w,h\), the contraction in (1) is unchanged when \(\sigma \) is replaced by \(\pi \circ \sigma \). For \(w,h\ge 2\), the simple-graph GHZ PEPS therefore satisfies \(\operatorname{Coeff}(\pi \circ \sigma )=\operatorname{Coeff}(\sigma )\). In particular, for \(d\ge 1\) the on-site \(\mathbb {Z}_d\) shift is a symmetry, and its virtual action is the same shift on every bond.
A permutation preserves equality of the five labels in the site tensor. Reindexing the sum over constant configurations in Theorem 25.2.2 proves the state identity. Take \(\pi (i)=i+g\) for the cyclic shift.
25.3 The cluster state
The cluster tensor is \(A=|0\rangle (00{+}{+}|+|1\rangle (11{-}{-}|\) with \((\pm |=[(0|\pm (1|]/\sqrt{2}\): the top and right legs copy the physical index \(s\), and the down and left legs carry \((-1)^{sb}/\sqrt{2}\) at bond index \(b\) [ 17 , Appendix A, “The cluster state” ] .
On a torus with \(w,h\ge 3\) and \(N=wh\) sites the cluster PEPS has coefficient
By Lemma 25.1.3 the top and right legs force every bond index to equal the physical index of the site below or to the left of it. The remaining factor at \(v\) is \(\tfrac 12(-1)^{\sigma _v\sigma _{v-(0,1)}+\sigma _v\sigma _{v-(1,0)}}\); shifting the product over \(v\) by one site in each direction gives the formula.
Let a finite set of bonds have distinct endpoints \(a(e),b(e)\) in a finite vertex set \(V\). Parallel bonds are counted with multiplicity. The product of controlled-\(Z\) gates has diagonal coefficient \(g(\sigma )=\prod _e(-1)^{\sigma _{a(e)}\sigma _{b(e)}}\). The stabilizer \(K_v\) flips the qubit at \(v\) and applies \(Z\) at the opposite endpoint of every incident bond. Thus
On the square torus, take the horizontal and vertical positive bonds; the normalized circuit state has coefficients \(2^{-N/2}g(\sigma )\) [ 17 , Appendix A, “The cluster state” ] .
For the loop-free bond family of Definition 25.3.3, \(K_v\psi =\psi \) for every \(v\) if and only if \(\psi =cg\) for some \(c\in \mathbb {C}\).
Flipping \(v\) multiplies \(g\) by the product of the neighbouring \(Z\) signs:
Since \(s_v(\sigma )^2=1\), this gives \(K_vg=g\). Conversely, \(K_v\psi =\psi \) reads \(\psi (\sigma )=s_v(\sigma )\psi (\sigma \mathbin {\oplus }\mathbf{1}_v)\), so
for every site \(v\). Changing the sites one at a time reduces every configuration to the all-zero configuration. Hence \(g(\sigma )\psi (\sigma )=\psi (0)\) and \(\psi (\sigma )=\psi (0)g(\sigma )\).
On a torus with \(w,h\ge 3\) and \(N=wh\), the printed cluster PEPS is \(2^{-N/2}\) times the normalized controlled-\(Z\) circuit state and is nonzero. Its span is precisely the common \(+1\) eigenspace of
25.4 The CZX model
In the CZX model of [ 11 ] every site of the square lattice carries four qubits, one at each corner, and the ground state on a closed surface is the product over plaquettes of the states \(|0000\rangle +|1111\rangle \) on the four qubits around each plaquette. Blocking the four qubits of a site, [ 17 , Appendix A, “The CZX model” ] writes this state as a PEPS of bond dimension four.
Label the four qubits of a site \(|ijkl\rangle \), with \(i\) at the top left, \(j\) at the top right, \(k\) at the bottom right and \(l\) at the bottom left, and label the four basis vectors of a bond by pairs of qubits. The CZX tensor is
so each leg carries the pair of corner qubits on its side, listed clockwise around the site [ 17 , Appendix A, “The CZX model” ] . The CZX PEPS places this tensor at every site with its down and left labels read in reverse, so that each bond identifies the pair \((a,b)\) at one end with the pair \((b,a)\) at the other.
The reversal corrects the printed formula. The review contracts every bond with equal labels at its two ends [ 17 , Section “MPS and PEPS” ] . Contracted in this way, the printed tensor sets the top-right qubit of a site equal to the bottom-left qubits of its right and upper neighbours, and the resulting state is a product of GHZ states on diagonal loops that wind around the torus, not the plaquette state. Two neighbouring sites list the qubits of their common side in opposite orders, and the plaquette state is obtained by identifying the pair \((a,b)\) at one end of a bond with \((b,a)\) at the other [ 163 ] .
On a torus with \(w,h\ge 3\) the CZX PEPS has coefficient
where \((i_v,j_v,k_v,l_v)\) are the four qubits of \(\sigma (v)\) and the factor at \(v\) is the coefficient of \(|0000\rangle +|1111\rangle \) on the four qubits around the plaquette at the upper right corner of \(v\) [ 17 , Appendix A, “The CZX model” ] , [ 11 ] .
By Lemma 25.1.3 the top and right legs force the vertical bond above \(v\) to carry \((i_v,j_v)\) and the horizontal bond to its right to carry \((j_v,k_v)\). The down and left legs then require \((i_{v-(0,1)},j_{v-(0,1)})=(l_v,k_v)\) and \((j_{v-(1,0)},k_{v-(1,0)})=(i_v,l_v)\) at every site, and these conditions, shifted by one site, are equivalent to the plaquette conditions.
The on-site \(\mathbb {Z}_2\) symmetry of the CZX model is generated by \(U_{CZX}=U_XU_{CZ}\), where \(U_X=X_1X_2X_3X_4\) flips the four qubits of a site and \(U_{CZ}=CZ_{12}CZ_{23}CZ_{34}CZ_{41}\) applies controlled-\(Z\) gates to its four pairs of neighbouring qubits [ 11 ] . On the state \(|ijkl\rangle \) it acts by \(U_{CZX}|ijkl\rangle =(-1)^{ij+jk+kl+li}|\bar\imath \bar\jmath \bar k\bar l\rangle \).
The operator \(U_{CZX}=U_XU_{CZ}\) on the sixteen states of a site, and its product \(U_{CZX}^{\otimes V}\) over the sites \(V\) of the lattice, whose entry at \((\sigma ',\sigma )\) is \(\prod _v(U_{CZX})_{\sigma '_v\sigma _v}\) [ 11 ] .
Flipping every qubit preserves the plaquette constraints of Theorem 25.4.2. On a configuration satisfying them, let \(P_v\) be the value of the plaquette at the upper right corner of \(v\). The sign of \(U_{CZX}\) at \(v\) is \((-1)^{e_v}\) with \(e_v=P_{v-(1,0)}P_v+P_vP_{v-(0,1)}+P_{v-(0,1)}P_{v-(1,1)} +P_{v-(1,1)}P_{v-(1,0)}\). Shifting the summation variable, the sums over \(v\) of the third and fourth terms equal those of the first and second, so \(\sum _ve_v\) is even: each controlled-\(Z\) gate between two neighbouring plaquettes appears twice.
A virtual leg carries two qubits \((a,b)\). The operator \(V=(X\otimes X)\, CZ\) acts on a leg by \(V|ab\rangle =(-1)^{ab}|\bar a\bar b\rangle \).
Applying \(U_{CZX}\) to the physical index of the CZX tensor equals applying \(V\) to each of its four virtual legs [ 17 , Section III.B, “Non-injective PEPS: SPT phases” ] : writing \(A^s_{trbl}\) for the entry of the CZX tensor with physical index \(s\) and top, right, bottom and left legs \(t,r,b,l\),
Both sides are supported on the flipped legs of the flipped qubits, and the sign \((-1)^{ij+jk+kl+li}\) of \(U_{CZX}\) is the product of the signs of \(V\) on the four legs \((i,j)\), \((j,k)\), \((k,l)\), \((l,i)\).
On the boundary of a region, neighbouring boundary legs carry the same qubit of their common plaquette, one effective spin per boundary plaquette [ 11 ] .
For \(N{\gt}0\) and effective spins \(c\in \{ 0,1\} ^N\), the closed chain of \(N\) boundary legs carrying \(c\) has the pair \((c_m,c_{m+1})\) on leg \(m\), indices modulo \(N\). The map \(\iota \) from \(N\) effective spins into the legs is
For every \(N{\gt}0\), the chain of boundary legs determines \(c\), so \(\iota \) is an isometry, \(\iota ^\dagger \iota =\mathbb {1}\). It satisfies \(V^{\otimes N}\iota =\iota \, U_N\) and \(\iota ^\dagger V^{\otimes N}\iota =U_N\), where \(U_N|c\rangle =(-1)^{\sum _mc_mc_{m+1}}|\bar c\rangle \) is the CZX matrix product unitary of Theorem 31.3.1.2; in terms of the review’s operator \(O(A)\) with the printed matrices, \(V^{\otimes N}\iota =(-1)^N\iota \, O(A)\) [ 17 , Appendix A, “The CZX MPU” ] , [ 11 ] . For actual positive proper bounded rectangles, the physical region image and this cyclic boundary action are identified in Section 25.5, together with the actual reduced-density support and rank, preserved by trace normalization. Nonzero spectral values, entropy, isometric normalization of the physical boundary map, and nonrectangular regions remain outside that result [ 164 ] .
The review states that the CZX model lies in the non-trivial symmetry-protected sector of its on-site \(\mathbb {Z}_2\) symmetry [ 17 , Appendix A, “The CZX model” ] . This classification is not formalized for the two-dimensional state. Its boundary symmetry is a matrix product unitary; the three-cocycle attached to a group of matrix product unitaries is constructed in Theorems 19.5.1.6 and 19.5.1.7, and for the CZX boundary representation its class is computed and shown to be non-trivial in Theorem 31.3.3.8.
25.5 The physical boundary symmetry of a CZX rectangle
Let \(R=\lbrack x_0,x_0+w)\times \lbrack y_0,y_0+h)\) be a coordinate rectangle in a \(W\times H\) torus. Assume \(W,H\geq 3\), \(w,h{\gt}0\), \(w{\lt}W\), \(h{\lt}H\), and \(x_0+w\leq W\), \(y_0+h\leq H\). Write \(P=2w+2h\). The following construction uses the actual open-region map \(A_R\), summing only internal bonds and leaving crossing bonds free. It identifies the plaquette spins in the boundary discussion of [ 11 ] , source lines 330–345, with the image of this physical region map.
Number the crossing bonds clockwise, first across the top, then down the right, across the bottom, and up the left. The corresponding first plaquette corners are \((i,h)\), \((w,h-j)\), \((w-k,0)\), and \((0,\ell )\) on the four sides. Reverse the native pair order on the bottom and left.
This numbering is a bijection from \(\{ 0,\ldots ,P-1\} \) to the actual crossing bonds. Its cyclic successor is ordinary addition in \(\mathbb Z/P\mathbb Z\). Every bond has the specified endpoint in \(R\), the perimeter corners are distinct in the torus, and \(P\) is even. The assertions include rectangles meeting a coordinate seam.
The four native edge families are disjoint and injective. The existing crossing-bond cardinality theorem gives surjectivity. Proper side lengths make the closed corner grid injective modulo each torus period.
For any finite region, a physical configuration determines at most one nonzero incident-bond assignment. Its open-region coefficient is one precisely when this assignment exists and has the specified boundary; otherwise it is zero. Different boundary columns have disjoint physical supports, so the region Gram matrix is diagonal in the native boundary basis. No labels on edges wholly outside the region are summed.
The printed CZX tensor fixes all four incident labels from the four physical qubits. Every incident edge touches a region vertex, so this determines the entire assignment. The contraction sum has at most one nonzero summand, whose value is one.
For a native boundary configuration \(\mu \), let \(f_R\mu \) flip both bits on each crossing bond and let \(\phi _R(\mu )=\prod _{e\in \partial R}(-1)^{a_eb_e}\), where \(\mu _e=(a_e,b_e)\).
Let \(U_R\) apply the CZX on-site symmetry at each site of \(R\). Then
This identity holds for the actual open contraction of every finite region. Reversal of a native bond pair preserves its phase and commutes with its simultaneous bit flip.
Apply the one-site pull-through identity inside the finite contraction. Reindex vertex–edge incidences by edge–endpoint incidences. Each internal bond contributes the square of its controlled phase, hence one; each crossing bond contributes its phase once. Reindex the incident-bond sum by simultaneous bit flips.
Let \(E_R\mu \) be the clockwise label sequence, with bottom and left pairs reversed. This is a bijective change of coordinates. For \(c\in \{ 0,1\} ^P\), set \(\iota (c)_i=(c_i,c_{i+1})\), define \(\mu (c)=E_R^{-1}\iota (c)\), and let \(F_R|c\rangle =A_R|\mu (c)\rangle \).
If a coefficient of \(A_R|\mu \rangle \) is nonzero, then \(E_R\mu =\iota (c)\) for some \(c\in \{ 0,1\} ^P\).
Extract the nonzero incident-bond assignment. On each straight side, the shared internal bond equates the adjacent physical qubits. At each corner, the equality is within one site tensor. These eight cases give the cyclic adjacent-pair constraint, including the final wraparound.
Given bits \(q\) at torus plaquette corners, assign site \((x,y)\) the qubits \((q_{x,y+1},q_{x+1,y+1},q_{x+1,y},q_{x,y})\) and give each native bond the corresponding adjacent pair.
Each local tensor and each actual region witness coefficient equals one. For every \(c\) there is a physical configuration \(\sigma _c\) with \(F_R(\sigma _c,c')=\delta _{c,c'}\). In particular, \(F_R\) is injective.
Extend arbitrary boundary bits from the distinct perimeter corners to torus plaquette bits, assigning zero elsewhere. The resulting physical configuration has coefficient one in the selected column and zero in all other columns by disjointness.
The image of the actual region map is exactly the span of the columns of \(F_R\), and its dimension is \(2^P\).
Every nonzero native boundary column is an effective column; every effective column is a native one. The selectors prove independence, and there are \(2^P\) effective spin configurations.
On the faithful physical boundary coordinates,
Reorder crossing bonds clockwise. Pair reversal preserves the controlled phase and commutes with bit flip, giving one controlled phase for each adjacent pair of effective spins.
The review’s printed CZX matrix product operator \(O(A)\) satisfies \(U_RF_R=F_RO(A)\) on the actual rectangle image.
The printed matrices give \(D_PX^{\otimes P}=(-1)^PX^{\otimes P}D_P\). The rectangle perimeter \(P\) is even, so this sign is one. The normalized bra-ket display is not substituted for those printed matrices.
For every effective boundary column there is a physical configuration on the complement whose closed-state slice is exactly that column. Consequently the actual unnormalized reduced density satisfies
Extend the chosen perimeter bits to global plaquette bits. On the complement, the copied physical configuration has coefficient one for its native crossing labels and zero for all others by disjointness. The actual cut contraction therefore reduces to a single boundary term. Its coefficient-matrix range is the reduced-density range, while the generic support inclusion gives the reverse inclusion.
The trace of \(\rho _R\) is nonzero. Dividing by that trace preserves its range and rank \(2^P\).
A positive semidefinite matrix with zero trace is zero, contradicting the positive rank. Multiplication by a nonzero scalar preserves the column space.
No formula for the nonzero eigenvalues, flatness, or entropy is asserted here. Nor is the unnormalized effective map \(F_R\) asserted to be an isometry. Nonrectangular regions remain outside the perimeter construction.
25.6 The AKLT state
The two-dimensional AKLT state places singlets on the edges of the lattice and projects the four spin-\(\tfrac 12\) at each site onto their symmetric subspace [ 17 , Appendix A, “The AKLT model” ] . Label the basis of four spin-\(\tfrac 12\) by \(s=(s_0,s_1,s_2,s_3)\in \{ 0,1\} ^4\), write \(|s|\) for the number of entries equal to \(1\), and let \(P_{\pi }\) be the permutation of the four factors by \(\pi \in S_4\).
The projector onto the symmetric subspace of \((\mathbb {C}^2)^{\otimes 4}\) is \(\Pi _{\mathrm{sym}}=\frac{1}{24}\sum _{\pi \in S_4}P_{\pi }\).
\(\Pi _{\mathrm{sym}}\) is a Hermitian idempotent commuting with \(U^{\otimes 4}\) for every \(2\times 2\) matrix \(U\). Its entries are \((\Pi _{\mathrm{sym}})_{s,t}=\binom {4}{|s|}^{-1}\) if \(|s|=|t|\) and \(0\) otherwise, and its range, the spin-\(2\) subspace, has dimension five.
\(P_{\pi }P_{\rho }=P_{\rho \pi }\) and \(P_{\pi }^{\dagger }=P_{\pi ^{-1}}\), so the average over \(S_4\) is a Hermitian idempotent, and \(P_{\pi } U^{\otimes 4}=U^{\otimes 4}P_{\pi }\) by reordering the tensor factors. The number of \(\pi \) with \(s=t\circ \pi \) is \(24/\binom {4}{|t|}\) when \(|s|=|t|\) and \(0\) otherwise, by a finite check. The dimension of the range of an idempotent is its trace, \(\sum _{s}\binom {4}{|s|}^{-1}=\sum _{k=0}^{4}\binom 4k\binom 4k^{-1}=5\).
With the singlet matrix \(Y=\left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)\), the AKLT tensor on the square lattice is \(A=\Pi _{\mathrm{sym}}(\mathbb {1}\otimes \mathbb {1}\otimes Y\otimes Y)\), with physical index \(s\in \{ 0,1\} ^4\) (physical dimension \(16\)) and virtual indices ordered top, right, down, left [ 17 , Appendix A, “The AKLT model” ] .
The entries of the AKLT tensor are
and the span of the vectors \(s\mapsto A^s_{\alpha \beta \gamma \delta }\) over all virtual indices is the range of \(\Pi _{\mathrm{sym}}\).
\(Y|0\rangle =|1\rangle \) and \(Y|1\rangle =-|0\rangle \), so the column \((\alpha ,\beta ,\gamma ,\delta )\) of \(\mathbb {1}\otimes \mathbb {1}\otimes Y\otimes Y\) is \((-1)^{\gamma +\delta }\) times the basis vector \((\alpha ,\beta ,1-\gamma ,1-\delta )\). The matrix \(L=\mathbb {1}\otimes \mathbb {1}\otimes Y\otimes Y\) satisfies \(LL^{T}=\mathbb {1}\), so the columns of \(\Pi _{\mathrm{sym}}L\) span the range of \(\Pi _{\mathrm{sym}}\).
Regard \(A\) as the map from the virtual space \((\mathbb {C}^2)^{\otimes 4}\) to the physical space. For every \(U\in SU(2)\),
For an arbitrary \(2\times 2\) matrix \(U\) the same identity holds with \(\bar U\) replaced by \((\operatorname {adj}U)^{T}\).
For a \(2\times 2\) matrix, \(UY=Y(\operatorname {adj}U)^{T}\) by direct computation. Since \(\Pi _{\mathrm{sym}}\) commutes with \(U^{\otimes 4}\), \(U^{\otimes 4}\Pi _{\mathrm{sym}}L=\Pi _{\mathrm{sym}}(U\otimes U\otimes UY\otimes UY)=\Pi _{\mathrm{sym}}L\, (U\otimes U\otimes V\otimes V)\) with \(V=(\operatorname {adj}U)^{T}\). For \(U\in SU(2)\), \(U^{\dagger }U=\mathbb {1}\) and \(U\operatorname {adj}U=(\det U)\mathbb {1}=\mathbb {1}\), so \(\operatorname {adj}U=U^{\dagger }U\operatorname {adj}U=U^{\dagger }\) and \(V=\bar U\).
On a torus with \(w,h\ge 3\) the AKLT PEPS has coefficient
the sum running over the spin-\(\tfrac 12\) labels \(\tau _v\in \{ 0,1\} ^4\) of the four legs of every site. The factor \(Y_{b,a}\) is the coefficient of \(|a\rangle |b\rangle \) in the singlet \(|01\rangle -|10\rangle \), so the state is \(\bigotimes _v\Pi _{\mathrm{sym}}\) applied to a singlet on every edge.
Expand each \(A=\Pi _{\mathrm{sym}}L\) as a sum over the intermediate label \(\tau _v\). The identity factors of \(L\) force the bond index on the top edge of \(v\) to be \(\tau _{v,0}\) and on the right edge to be \(\tau _{v,1}\), and the \(Y\) factors on the down and left legs then give \(Y_{\tau _{v,2},\tau _{v-(0,1),0}}Y_{\tau _{v,3},\tau _{v-(1,0),1}}\); shifting the product by one site gives the formula.
The tensor \(A\) takes values in \((\mathbb {C}^2)^{\otimes 4}\) with physical support on the five-dimensional symmetric subspace, on which the physical action \(U^{\otimes 4}\) acts as the irreducible spin-\(2\) representation [ 17 , Appendix A, “The AKLT model” ] .
25.7 The RVB state
The nearest-neighbour resonating valence bond state is the superposition of all coverings of the lattice by nearest-neighbour singlets. Its PEPS of bond dimension three combines, at each site, the projector
with singlet matrices on the edges; on the square lattice with a translation-invariant orientation of the singlets the tensor is \(A=P(\mathbb {1}\otimes \mathbb {1}\otimes Y\otimes Y)\) [ 17 , Appendix A, “The RVB state” ] . On the bond of dimension three the matrix \(Y\) acts on the spin states \(0,1\) and is extended by \(1\) on the vacuum state \(2\), so that the bond state is \(|01\rangle -|10\rangle +|22\rangle \). The extension by \(0\) would make every entry of \(A\) vanish, since a leg carrying \(Y\) could then never be in the vacuum state [ 168 ] .
The RVB tensor is \(A=P(\mathbb {1}\otimes \mathbb {1}\otimes Y\otimes Y)\) with \(Y=\left(\begin{smallmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{smallmatrix}\right)\), virtual indices ordered top, right, down, left.
\(A^{s}_{urdl}\neq 0\) if and only if exactly one of \(u,r,d,l\) differs from \(2\), that index being \(s\) if it is \(u\) or \(r\) and \(1-s\) if it is \(d\) or \(l\). The nonzero entries equal \(-1\) when the down or left index is \(1\) and \(+1\) otherwise.
\(Y\) maps the bond index \(d\) to \(1-d\) on the spin states, with sign \(-1\) at \(d=1\), and fixes the vacuum. The statement is a finite check over the \(162\) entries.
Let \(w,h\ge 3\). A dimer covering of the torus is a set \(c\) of edges such that every site lies on exactly one edge of \(c\). The RVB PEPS has coefficient
the sum running over all dimer coverings, and each edge of \(c\) written as \((v,v+e)\) with \(e\in \{ (1,0),(0,1)\} \). The factor \(Y_{\sigma _{v+e},\sigma _v}\) is the coefficient of the singlet \(|01\rangle -|10\rangle \) on the two sites, so the state is the superposition of all nearest-neighbour singlet coverings with the translation-invariant orientation.
By Lemma 25.1.3 the coefficient is a sum over bond indices \(h_v\), \(v_v\) on the right and up edges. By Lemma 25.7.2 a configuration contributes only if the edges carrying a non-vacuum index form a dimer covering and the index on each such edge is the physical index of the site below or to the left of it. Conversely every dimer covering determines one such configuration, and its term is the product over sites of the singlet factors on the down and left legs; shifting the product by one site gives the formula.
25.8 PEPS from classical models
Let \(q\ge 1\), let \(h\colon \{ 0,\ldots ,q-1\} ^{2}\to \mathbb {R}\) be a nearest-neighbour interaction and \(\beta \in \mathbb {R}\) an inverse temperature.
On the torus a configuration \(\sigma \) has energy
each nearest-neighbour bond counted once. For symmetric \(h\) this is the sum of \(h\) over unordered bonds.
Let \(M=\sum _{i,j}e^{-\beta h(i,j)/2}|i)(j|\). The classical-model tensor is
the top and right legs copy the physical index \(s\), and the down and left legs carry \(M_{sb}\) and \(M_{sl}\) at bond indices \(b\) and \(l\) [ 17 , Appendix A, “PEPS from classical models” ] . The classical-model PEPS places this tensor at every site of the torus.
On a torus of width and height at least three the classical-model PEPS has coefficient \(\operatorname{Coeff}(\sigma )=e^{-\beta H(\sigma )/2}\), so \(|\operatorname{Coeff}(\sigma )|^{2}=e^{-\beta H(\sigma )}\) is the Gibbs weight of \(\sigma \) [ 173 ] .
By Lemma 25.1.3 the top and right legs force every bond index to equal the physical index of the site below or to the left of it. The remaining factor at \(v\) is \(M_{\sigma _{v}\sigma _{v-(0,1)}}M_{\sigma _{v}\sigma _{v-(1,0)}}\), one factor \(e^{-\beta h/2}\) for each bond at \(v\) read towards its lower and left neighbours, and the product over \(v\) is \(e^{-\beta H(\sigma )/2}\). The coefficient is real, so its squared modulus is \(e^{-\beta H(\sigma )}\).
No bond carries a product of two copies of \(M\): the entry \(e^{-\beta h/2}\) is not a square root of a transfer matrix, and no symmetry of \(h\) is used.
Let the torus have width and height at least three and let \(O=\sum _{\sigma }O(\sigma )|\sigma \rangle \langle \sigma |\) be a diagonal observable. For the classical-model PEPS state \(|\psi \rangle =\sum _{\sigma }\operatorname{Coeff}(\sigma )|\sigma \rangle \),
that is, \(\langle \psi |O|\psi \rangle /\langle \psi | \psi \rangle \) is the expectation value of \(O\) in the Gibbs state \(e^{-\beta H}/Z\) [ 17 , Appendix A, “PEPS from classical models” ] .
Replace \(\overline{\operatorname{Coeff}(\sigma )}\operatorname{Coeff}(\sigma )\) by \(e^{-\beta H(\sigma )}\) in both sums, by Theorem 25.8.3.
In particular, classical correlation functions are expectation values of the PEPS. The review adds that at the critical inverse temperature \(\beta _{c}\) the PEPS has correlations decaying as a power law, so its parent Hamiltonian must be gapless [ 17 , Appendix A, “PEPS from classical models” ] , since a gapped local Hamiltonian has exponentially decaying correlations [ 204 , 226 ] .