Dependencies

If all three blocks are \(G\)-injective and all physical alphabets are finite, every vector in \(\mathcal L\cap \mathcal R\) has a unique boundary operator \(X\) commuting with \(\rho (G)\). Consequently

Under the preceding hypotheses, each vector in \(\mathcal K_n\) is produced by a unique invariant virtual boundary operator. Moreover

If \(A\) is \(G\)-injective, then the tensor obtained by blocking \(L\ge 1\) sites, \((i_1,\dots ,i_L)\mapsto A^{i_1}\cdots A^{i_L}\), is \(G\)-injective.

Let \(R_g\) be the permutation matrix of \(h\mapsto hg\). The color shift of the dual tensor is the action of \(R_g\) on the incoming and \(R_g^{-1}\) on the outgoing legs. Hence, in the torus network of the dual tensor with colors in \(G\) on the bonds, a string of \(R_g\)’s along the left and bottom sides of a rectangle of \(m{\lt}w\) columns and \(n{\lt}h\) rows of sites gives the same network as the string along its top and right sides [ 235 , Section “Two dimensions: \(G\)-injective PEPS”, eq. (2d:move-strings) ] .

Suppose \(G\) is nontrivial and both \(I,J\) are nonempty. The projector \(|1,\ldots ,1\rangle \langle 1,\ldots ,1|\) on the selected legs, extended by the complementary identity, does not preserve the global invariant space. Thus the full matrix algebra on a nonempty proper subset of the original boundary legs does not restrict to that space.

Under the regular native torus hypotheses of Theorem 26.7.20, let \(u,w\) be two actual bond assignments. If their holonomies around \(p_v\) have distinct conjugacy classes, then their globally contracted coefficient matrices satisfy

Thus every physical coefficient column of one is orthogonal to every coefficient column of the other, without a normalization hypothesis. This is a consequence of the local class measurement in [ 235 , Theorem 6.15 ] .

Given a tensor \(A\) and a blocking length \(L\), the \(L\)-blocked tensor is the tensor with physical index set \(\{ 0,\ldots ,d{-}1\} ^L\) (hence physical dimension \(d^L\)) and the same bond dimension \(D\), whose matrices are indexed by words \((i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\):

Blocking coarse-grains \(L\) neighbouring sites into one tensor: the inner virtual bonds are contracted, the \(L\) physical legs merge into a single composite index, and the two outer bonds remain as the bond of \(A^{[L]}\).

For \(N{\gt}0\) and \(L\le N\), define the periodic parent-Hamiltonian ground space

Here the condition \(\psi |_{[i,i+L-1]}\in G_L(A)\) means that, for every choice of physical indices outside the window, the restriction of \(\psi \) to that window lies in \(G_L(A)\). The one-index notation \(G_L(A)\) denotes the local ground space \(\mathcal G_L\) of [ 17 , Section IV.C ] . The two-index notation \(\mathcal G_{N,L}(A)\) used here denotes the periodic intersection of those local constraints over all translated length-\(L\) windows on the \(N\)-site ring. When \(N=0\) or \(L{\gt}N\), set \(\mathcal G_{N,L}(A):=\top \) by convention.

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

The nearest-neighbor commuting condition is the length-\(2\) specialization of the preceding commutation equation. For every \(i,j\in \{ 0,\ldots ,N{-}1\} \), one has \(h_i h_j=h_j h_i\). The source definition [ 13 , Definition 3.9 ] also requires the corresponding parent Hamiltonian to have \(L=2\) and to have the prescribed ground-space spanning property for \(N{\gt}2\).

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

Work on the half-edge space of the whole graph. Restriction to \(R\) retains every half-edge at its vertices, including dangling boundary legs. Let \(\rho _v(g)\) multiply all labels at \(v\) by \(g\) on the left; \(\rho _v(gh)=\rho _v(g)\rho _v(h)\). Let \(A_v\) average these permutation operators, and set \(B_e=B_{V,e}\). Let \(D_R\) be the diagonal indicator of flatness of the restriction to \(R\). Index the three families by \(i\in V\sqcup E\sqcup \mathcal P(V)\), writing \(C_v=A_v\), \(C_e=B_e\), and \(C_R=D_R\). Set

Let \(\mathcal S_R^0\) be the range of the canonical regular open-region matrix. Define \(\widehat{\mathcal S}_R^0\) by requiring every complementary slice \(x_\tau \) to lie in \(\mathcal S_R^0\). Extend a regional gauge by \(1\) outside \(R\), and retain an edge multiplier only on edges with both ends in \(R\). These actions preserve the exterior coordinates: \(L_k(\alpha ,\tau )=(L_k\alpha ,\tau )\) and \(T_r(\alpha ,\tau )=(T_r\alpha ,\tau )\) for internal \(r\). Restriction of \((\alpha ,\tau )\) to \(R\) recovers \(\alpha \). If \(r=1\) on boundary edges, discarding its exterior values leaves its regional action unchanged.

For finitely many pairs of dimensions \((m_k,d_k)\), the finite group \(G=\prod _kG_{m_k}\) has the unitary representation \(U(g)=\bigoplus _k g_k\otimes I_{d_k}\) on \(\bigoplus _k\mathbb C^{m_k}\otimes \mathbb C^{d_k}\).

For a rooted spanning tree of a connected region, permute its accessible boundary registers by \(y_b\mapsto y_{\tau b}\) and retain all internal reference, normalized vertex and cycle registers. This is the boundary exchange in the two independent column operations of [ 235 ] , lines 2489–2507.

Equip \(\mathbb {C}^I\) with \(\langle x,y\rangle =\sum _i\overline{x_i}y_i\). For a subspace \(S\subseteq \mathbb {C}^I\), let \(\Pi _S\) be its orthogonal projector in the coordinate basis. Then \(\Pi _S^2=\Pi _S=\Pi _S^*\), \(\operatorname {ran}\Pi _S=S\), and \(\Pi _Sx=x\) exactly when \(x\in S\).

Let \(G\) be a finite group and let \(R\) be a finite region of an ordered graph. Fix a rooted spanning tree \(T\) of the induced graph on \(R\) and a non-tree internal edge \(e\). Write the exposed coordinates of a physical half-edge configuration \(\alpha \) as \((y,a,r,z)\). For a conjugacy class \(C\) of \(G\), define the diagonal operator

This is the chosen-cycle form of the conjugacy-class measurement in the proof of [ 235 , Theorem 6.15 ] .

For the occurring irreducible characters, with dimensions \(d_\chi \) and multiplicities \(m_\chi \), define

These are the reciprocal coefficients of the weighted trace operator in [ 235 , Lemma 4.4 ] .

Let \(\chi _U\) be the character of \(U\), and let \(\chi _1,\dots ,\chi _r\) be the distinct characters of the irreducible subrepresentations of \(U\), with \(d_i=\chi _i(1)\) and multiplicities \(m_i=\frac{1}{|G|}\sum _g\chi _U(g)\chi _i(g^{-1})\). The isotypic projectors are \(P_i=\frac{d_i}{|G|}\sum _g\chi _i(g^{-1})U_g\), and

[ 235 , Section “MPS: the \(G\)-injective case” ] .

Native horizontal and vertical bond labels retain their direction and endpoint polarity. The four incidence labels at a site are ordered top, right, down, left. Around a rectangular sweep of side lengths \(W,H\), with the embedding \(\kappa \) of Definition 26.9.222, take the translated collar and swept support

requiring \(W+1{\lt}w\) and \(H+1{\lt}h\).

For native group labels \(u=(u_h,u_v)\), define the clockwise holonomy at \(z=(x,y)\) by

Curl is additive. For the labels \(u=(g^{n_h},g^{n_v})\), \(H_u(z)=g^{(\operatorname {curl}n)(z)}\).

For finite \(G\), \(w,h\geq 3\), and a homogeneous tensor \(A\), use the left-regular matrices \(U_g=L_g\). For a physical region \(S\), let \(M_{u,S}(\sigma _S,\sigma _{S^c})\) be the actual native contraction with these physical indices fixed. It agrees entrywise with the cut matrix of the ordered-edge regular tensor network. Write \(S_z\) for the four original physical sites of plaquette \(z\).

Sum \((1,0),(-1,0),(0,1),(0,-1)\) over the east, west, north and south steps of \(p\) to obtain \(d(p)\in \mathbb Z^2\). Concatenation adds displacements, and reversal negates them. For \(p:a\to b\), reducing the two coordinates of \(d(p)\) modulo \(w\) and \(h\) gives \(b-a\in X\).

For \(R\subseteq X\), let \(p\sim _R q\) be the relation generated by removal of an adjacent step and its reverse and by the square move

allowed when \((x+1,y+1)\in R\). The relation is closed under reflexivity, symmetry, transitivity, reversal and concatenation. The endpoints remain fixed. For \(f:X\to \mathbb Z\), define

This describes the elementary deformations used in [ 235 , proof of Lemma 6.14 ] .

Let \(G\) be a group, \(g\in G\), and \(U:G\to \operatorname {GL}(V)\) a representation on a finite-dimensional virtual space with a fixed basis. Assign \(u_h^p(v)=g^{n_h(p)(v)}\) and \(u_v^p(v)=g^{n_v(p)(v)}\) to the two native bond families, and put the actual matrices \(U_{u_h^p(v)}\), \(U_{u_v^p(v)}\) on those bonds. For a swept set \(R\), one may instead retain these matrices only on bonds incident to \(R\) and prescribe arbitrary fixed matrices \(E_e\) on every bond whose two endpoints lie outside \(R\). Denote the actual closed PEPS coefficient, with site tensors \((A_v)\) and physical configuration \(\sigma \), by \(\Psi _p^E(\sigma )\).

Let \(X=\mathbb Z_w\times \mathbb Z_h\), where \(w,h{\gt}0\). A plaquette is labelled by its lower-left primal vertex. A finite dual path is a sequence of east, west, north and south unit steps, retaining the step labels at periodic seams and when parallel bonds have the same endpoints. Write \(n(p)=(n_h(p),n_v(p))\) for its integer crossing counts on rightward and downward native bonds. With \(\delta _s(v)=\mathbf1_{\{ v=s\} }\), the positive steps contribute

Opposite steps negate these counts; concatenation adds them. Thus \(n(p^{-1})=-n(p)\), with multiplicities retained. In the orientation of [ 235 , Definition 6.13 ] , an eastward dual step inserts \(U_g\) on a downward bond and a southward step inserts \(U_{g^{-1}}\) on a rightward bond.

Choose \((x_0,y_0)\in X\) and \(W,H\geq 0\). Lifted vertices are pairs \((i,j)\in \mathbb N^2\). The horizontal steps join \((i,j)\) and \((i+1,j)\) for \(i{\lt}W\), \(j\leq H\); the vertical steps join \((i,j)\) and \((i,j+1)\) for \(i\leq W\), \(j{\lt}H\). Both orientations retain their direction labels under \(\kappa (i,j)=(x_0+i,y_0+j)\) modulo the periods. Vertices outside the rectangle are isolated. The rooted comb path \(C_{i,j}\) consists of \(i\) east steps on the bottom row followed by \(j\) north steps. Its vertical segment is the column at \(i\). The swept primal set is \(R=\{ \kappa (i+1,j+1):i{\lt}W,\ j{\lt}H\} \). An embedded patch also requires \(W{\lt}w\) and \(H{\lt}h\).

Let \(D\) be a nonempty finite set and \(\phi \in \mathbb C^A\). Define \(J_\phi :\mathbb C^D\to \mathbb C^{D\times A}\) by \((J_\phi )_{(d,a),s}=\mathbf1_{\{ d=s\} }\phi (a)\), and set

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

\(=\)

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

A strip consists of an ordered list of tensor blocks \(A_1,\ldots ,A_n\) on a common finite-dimensional virtual space \(\mathcal V\). Each block has its own finite physical alphabet \(I_r\), and a physical configuration is \((i_1,\ldots ,i_n)\). Its full boundary map is

The common adjacent-pair space \(\mathcal K_n\) consists of all physical vectors such that, after fixing every physical index outside a pair \(r,r+1\), the remaining two-block vector belongs to \(\operatorname {ran}\mathcal P(A_rA_{r+1})\). Equivalently, impose the first-pair condition on every remaining configuration and then impose the same adjacent conditions on every first-index-fixed tail. For \(n=2\) there is just one pair condition; for \(n\le 1\) there are no conditions. These are the actual local parent conditions used in the intersection iteration of [ 235 , Theorem 4.11 ] .

Let \(A^i,B^j,C^k\) be operator-valued tensors on a common finite-dimensional virtual space \(\mathcal V\), with independent physical alphabets. Write

Define \(\mathcal L\) to consist of three-block vectors whose every fixed-\(k\) slice belongs to \(\mathcal G_{AB}\), and \(\mathcal R\) to consist of those whose every fixed-\(i\) slice belongs to \(\mathcal G_{BC}\). Thus

These are the local boundary spaces with the unused physical factor unrestricted, as in [ 235 , Theorem 4.8 ] .

Let \(G\) and \(H\) be finite groups acting linearly on complex vector spaces \(\mathcal V\) and \(\mathcal W\), respectively. Let \(T:\mathcal V\to \mathcal P\) be \(G\)-injective and let \(S:\mathcal W\to \mathcal P\) be \(H\)-injective, with \(\operatorname {im}T=\operatorname {im}S\). The virtual comparison is the linear isomorphism

Both restrictions are isomorphisms onto the common physical range.

Let \(\Gamma \) be a finite simple graph with an ordered vertex set \(\mathcal V\), let \(X\) be a finite virtual alphabet, and let \(P_v\) be the physical alphabet at vertex \(v\). For local coefficients \(a_v:X^{E(v)}\times P_v\to \mathbb C\), define the actual contracted coefficient

Physical alphabets may depend on the vertex; no auxiliary tensor structure or assumed coefficient identity is needed.

Let \(\Gamma \) be a finite simple graph with ordered endpoints, let \(X\) be a finite virtual alphabet, and let \(P_v\) be a physical alphabet at each vertex. For site coefficients \(A_v\) and matrices \(K_e\), define

Here \(\zeta |_v\) reads the head or tail label at the actual incidence. Thus each bond matrix has row at the head and column at the tail. This is an auxiliary inserted-bond extension of the closed contraction calculation in [ 235 , Section 7 ] .

Orient every edge from its smaller endpoint \(t(e)\) to its larger endpoint \(h(e)\). The site coordinates \(\sigma _v:E(v)\to X\) are equivalently the bond pairs \(\beta _e=(\sigma _{h(e)}(e),\sigma _{t(e)}(e))\). This bijection induces a linear equivalence of coefficient spaces and its corresponding Hilbert-space coordinate isometry.

For a finite vertex set and matrices \(F_v:\mathbb C^{P_v}\to \mathbb C^{Q_v}\), define the product physical operation by

The physical alphabets may vary independently with the vertex.

Let \(G\) be finite and \(U\) a matrix representation on \(\mathbb C^X\). The local averaging site is \(|G|^{-1}\sum _g\) of the product of \(U(g)\) at head legs and \(U(g^{-1})^{\mathsf T}\) at tail legs. Inserting a matrix \(W\) on every virtual leg means summing the original site coefficient against the product of its oriented endpoint entries. Apply this insertion to the averaging site to obtain the dressed tensor; taking \(W=\Theta \) gives the fourth-root-weighted tensor. These are the finite-graph local contractions underlying [ 235 , Section 7 ] , with normalized site averages.

For a matrix product tensor \(A\), let \(\Gamma _{n}^{\dagger }\psi =\sum _{\tau }\psi (\tau )\, (A^{\tau })^{\dagger }\), the adjoint of \(\Gamma _{n}\) for the Hilbert–Schmidt inner product, and let \(P_{n}=\Gamma _{n}\Gamma _{n}^{\dagger }\), the operator on \(n\) sites with matrix entries \(\operatorname {tr}[A^{\sigma }(A^{\tau })^{\dagger }]\). For \(n=2\) this is the operator of [ 235 , Lemma 6.11 ] .

Let \(A\subset \mathbb Z^2\) be finite, and let \(Q_a\) be the closed unit square centered at \(a\). The contact graph has vertex set \(A\); distinct \(a,b\) are adjacent when \(|a_1-b_1|\leq 1\) and \(|a_2-b_2|\leq 1\). Thus shared corners, as well as shared sides, are retained.

Index each oriented lattice edge by its starting corner \(a-(1/2,1/2)\), with horizontal edges pointing east and vertical edges pointing north. For a finitely supported signed vertical flow \(V:\mathbb Z^2\to \mathbb Z\), define its ray potential at the cell center \(q\) by

The sum is finite because \(V\) is finitely supported.

Let \(A\subset \mathbb Z^2\) be finite, with occupied closed-cell union \(P_A\). Its exterior collar is

The exterior band consists of the unoccupied integer centers \(q\) such that \(\| q-a\| _\infty \leq 1\) for some \(a\in A\). Join two such centers by an edge precisely when they are horizontal or vertical neighbors and one common occupied center \(a\in A\) satisfies both \(\| q-a\| _\infty \leq 1\) and \(\| r-a\| _\infty \leq 1\). The common-center requirement controls the whole segment, rather than its endpoints alone.

Every actual joining bond is a crossing bond of each of the two disjoint regions, with the same underlying native edge.

Let the boundary pairs be \(N=(N_0,N_1)\), \(E=(E_0,E_1)\), \(S=(S_0,S_1)\) and \(W=(W_0,W_1)\). Contract the four internal bonds of the \(2\times 2\) block around an A plaquette:

The boundary transformation \(C\) applies \((a,b)\mapsto (a,a+b)\) on each of the four pairs. This is the CNOT transformation in the blocking construction of [ 235 , Section 7 ] .

Let \(\mathsf B\) be the matrix of the actual four-site contraction, with physical rows and eight-bit boundary columns. Let \(\mathsf K\) have entries \(\mathsf K_{\sigma ,p}=\delta _{\sigma ,F(p)}\). Define the inclusion \(E\) by fixing the four second boundary registers at zero:

All indices take values in \(\mathbb Z_2\). Write the virtual legs in the order top, right, bottom, left. The two checkerboard orientations of the elementary tensor are

These are the tensor of [ 235 , Section 7, “Kitaev’s code state” ] and its rotated orientation. For four colors define

An open block has one pair of boundary labels on each side, ordered north, east, south, west. Each pair is read clockwise along its side. Its four physical spins are ordered clockwise from the upper-right site.

Let \(C\) be the four pair CNOTs. Their native boundary action is \(C_v=\mathcal E_v^{-1}C\mathcal E_v\). Let \(\mathsf A_v\) be the actual open tensor map, with the four original native spins as its rows and all native crossing assignments as its columns. Let \(\mathsf K_v\) be the color-difference tensor with the same physical rows, and let \(E_v\) include the four colors into the native boundary registers by setting the four second registers to zero.

For the locally oriented plaquette at \(v=(x,y)\), enumerate its crossing bonds by a side \(i\in \{ N,E,S,W\} \) and a place \(b\in \{ 0,1\} \) in that side. The places follow clockwise boundary order. This is a bijective enumeration of the actual crossing edges. It gives an equivalence \(\mathcal E_v\) from native boundary configurations to four binary pairs. These are the boundary registers of the blocking construction in [ 235 , Section 7, “Kitaev’s code state” ] .

Let the two torus periods be at least three. At an arbitrary site \(v=(x,y)\), take the actual four-site plaquette with corners \((x,y)\), \((x+1,y)\), \((x+1,y+1)\) and \((x,y+1)\). Order the physical sites clockwise from the upper right. Place \(T_1\) at the upper-right and lower-left corners, and \(T_0\) at the other two corners, with virtual legs in the order top, right, bottom, left. These are the local orientations in the blocking diagram of [ 235 , Section 7, “Kitaev’s code state” ] .

Read an actual crossing-bond assignment as four boundary pairs \(N=(N_0,N_1)\), \(E=(E_0,E_1)\), \(S=(S_0,S_1)\) and \(W=(W_0,W_1)\), ordered north, east, south, west, clockwise along each side. All coordinates are cyclic, so this definition also applies to a plaquette meeting either seam. The orientations are prescribed on this one block; no globally alternating tensor field on an odd torus is asserted.

Consider an actual translated four-site plaquette of a torus with both periods at least three, and prescribe its four tensor orientations as in the checkerboard blocking diagram. Read its original physical spins clockwise from the upper right. Let \(R_v\) be the resulting physical parity unitary, \(E_{P,v}\) the zero-parity physical inclusion, \(C_v\) the native boundary CNOT, and \(E_{V,v}\) the derived inclusion of the four retained boundary colors. Define \(J_v=E_{V,v}J\).

Let \(H_v,V_v\in \mathbb Z_2^2\) be the horizontal and vertical crossing pairs of a block \(v\). Its clockwise boundary is \(\alpha _v=(V_v,H_v,\operatorname {swap}V_{v-e_y}, \operatorname {swap}H_{v-e_x})\). The periodic blocked coefficient is \(\Psi _B(\tau )=\sum _{H,V}\prod _v B(\alpha _v;\tau _v)\). The fine coefficient \(\Psi _T\) is the sum over these crossing pairs and the four independent internal bonds of every block, with one factor \(T_0\) or \(T_1\) for each of the four corners.

Let \(w,h\) be positive integers. Divide the fine torus \(\mathbb Z_{2w}\times \mathbb Z_{2h}\) into the disjoint squares based at \((2x,2y)\), with \((x,y)\in \mathbb Z_w\times \mathbb Z_h\). Number their corners clockwise from the upper right, so that \(c_0=(1,1)\), \(c_1=(1,0)\), \(c_2=(0,0)\) and \(c_3=(0,1)\). The site correspondence is \(E(v,i)=2v+c_i\). Place \(T_1\) at corners \(0,2\) and \(T_0\) at corners \(1,3\). Contract every periodic bond with the identity and denote the resulting untwisted PEPS coefficient by \(\Psi _T(\sigma )\). This is the periodic version of the elementary tensor in [ 235 , Section 7, “Kitaev’s code state” ] .

For the binary tensor of [ 235 , Section 7 ] , let \(F(p)=(p_0+p_1,p_1+p_2,p_2+p_3)\) and \(K_3(s,p)=\delta _{s,F(p)}\). The physical parity permutation is \(R(\sigma )=(\sigma _0,\sigma _1,\sigma _2, \sigma _0+\sigma _1+\sigma _2+\sigma _3)\), and \(E_P\) includes three physical bits by appending zero. The color support coordinates are \(p\mapsto (F(p),p_0)\), with inverse \((s,t)\mapsto (t,t+s_0,t+s_0+s_1,t+s_0+s_1+s_2)\). Define \(J=2^{-1/2}K_3^\dagger \).

For a region \(R\) in a finite directed multigraph, assign an independent virtual index to every incidence at every vertex of \(R\); write \(\eta \) for such an assignment, and \(\eta _{t(e)}\), \(\eta _{h(e)}\) for its values at the tail and head incidences of a bond \(e\). Fix the indices on region-side incidences of crossing bonds to boundary data \(\theta \), with \(\theta _{t(e)}\), \(\theta _{h(e)}\) its values there. With matrices \(O_e\) and site coefficients \(A_v\), put

Pair this coefficient with any joint boundary tensor \(B\) by \(\sum _\theta B(\theta )c(\theta )\).

A matrix product tensor \(A\) is \(G\)-injective if the map \(\mathcal{P}(A)X=\sum _i\operatorname{tr}[A^iX]|i\rangle \) is \(G\)-injective for the conjugation action of \(U\) on the operators on the bond space.

A matrix product tensor \(A\) with bond space \(\mathbb {C}[G]\) is \(G\)-isometric if \(L_gA^iL_g^{-1}=A^i\) for all \(g\) and \(i\) and \(\mathcal{P}(A^\dagger )\mathcal{P}(A)=c\, \sigma \) for one constant \(c{\gt}0\), where \(\sigma \) is the twirl onto the commutant of the left-regular representation. For \(c=1\) this is the condition \(\mathcal{P}(A)^{-1}=\mathcal{P}(A^\dagger )\) of the source; the constant absorbs the normalization of the tensor [ 235 , Definition 6.1 ] , [ 96 ] .

Choose nonempty finite multiplicity sets \(M_i\), with \(m_i=|M_i|\). In paired endpoint coordinates the bond isometry is

Thus \(J\) carries the vector of entries of \(\sqrt{m_i}X_i\) to the vector of entries of \(X_i\otimes I_{M_i}\), block by block. This is the bond transformation of [ 235 , Section 7 ] .

Let \(I\) be a finite set of bond labels, and let \(M_i\) be a nonempty finite multiplicity set of cardinality \(m_i\). Define

Consequently, \(J(\sum _i\sqrt{m_i}x_i|i\rangle ) =\sum _{i,a}x_i|i,a,a\rangle \). The label \(i\) may include an irreducible representation and both matrix indices, as in the bond sums of [ 235 , Section 7 ] .

Let \(V_i\) and \(M_i\), indexed by a finite set, be finite-dimensional coordinate spaces, with \(m_i=\dim M_i{\gt}0\). On the direct sum of matrix spaces, define

On each matrix unit this reads

For an irreducible representation of dimension \(d_i\), the regular representation has multiplicity \(m_i=d_i\). Thus \(J\) is precisely the bond transformation used in the semi-regular bond-dimension reduction of [ 235 , Section 7 ] . In a fixed block \(i\), the four open indices give \(J(X_i)_{(k,a),(\ell ,b)}=m_i^{-1/2}(X_i)_{k,\ell }\delta _{a,b}\):

\(=m_i^{-1/2}\)

Given matrix representations \(D_i:G\to M_{d_i}(\mathbb C)\), define

This is a representation. If the \(D_i\) are precisely the inequivalent irreducible representations, these are the dimension multiplicities in the regular representation used in [ 235 , Section 7 ] . The definition itself does not assert that every irreducible occurs.

For a coefficient matrix \(M\) of a pure state on \(\mathcal H_R\otimes \mathcal H_S\), define

The inverse is taken to be zero when \(M=0\). For nonzero \(M\), \(\widehat M\) represents a normalized pure state.

For a region \(R\) in a finite PEPS graph, let \(T_R(\mu ;\sigma )\) be the contraction of its site tensors with physical configuration \(\sigma \), fixing the crossing-bond labels \(\mu \) and summing only the internal-bond labels. It defines the linear map from the crossing virtual indices to the physical system of \(R\). No edge wholly outside \(R\) enters this contraction. This is the finite-region tensor used when blocking and cutting the PEPS in [ 235 , Section 6, proof of Theorem 6.9 ] .

Let \(L,S\) be disjoint vertex regions of a finite simple graph. Their joining bonds are the bonds internal to \(L\cup S\) but internal to neither part. For joining labels \(\mu \) and prescribed outer boundary labels \(\theta \), let \(\beta _L(\mu ,\theta )\) and \(\beta _S(\mu ,\theta )\) read the original crossing labels of the two parts: they equal \(\mu \) on joining bonds and \(\theta \) on the exterior boundary. These are the actual labels in the two-column blocking of [ 235 ] , lines 2489–2507.

Two pairs are equivalent if \((g,h)=(xg'x^{-1},xh'x^{-1})\) for some \(x\in G\). Their equivalence classes are the pair-conjugacy classes of [ 235 , Definition 5.8 ] . Commutation is constant on each class, so the classes of commuting pairs are well-defined.

Let \(A\) be a tensor family on a finite graph and let \(F_v:\mathbb {C}^d\to \mathbb {C}^e\) be a linear map at every vertex. Its physical deformation \(A^F\) keeps the virtual bond dimensions and has components

For a region \(R\), write \(F_R=\bigotimes _{v\in R}F_v\). When every \(F_v\) is invertible, \(F_R\) is invertible with inverse \(\bigotimes _{v\in R}F_v^{-1}\).

Let \(F\) be a rectangular physical matrix. Its product over a finite site set \(\Lambda \) has coefficients

The transformed site tensor is \((FA)^q_{t,r,b,l}=\sum _s F_{q,s}A^s_{t,r,b,l}\). Input and output physical dimensions may differ.

For physical coefficient vectors \(\psi \in \mathbb C^P\) and \(\omega \in \mathbb C^Q\), their tensor product has complete pair-indexed coefficients \((\psi \otimes \omega )(p,q)=\psi (p)\omega (q)\).

For \(u\in G\), define the permutation operator \(V_u\) on \(\mathbb {C}[G^4]\otimes \mathbb {C}[G^4]\) by

The four changed spins carry \(R_u,L_u,R_u,L_u\), respectively, where \(L_u|z\rangle =|uz\rangle \) and \(R_u|z\rangle =|zu^{-1}\rangle \). These are the incoming/outgoing spin actions of [ 235 , Section 7, “The double models” ] .

The virtual legs carry colors \(h_1\) (left), \(h_2\) (top), \(h_3\) (right) and \(h_4\) (down) in \(G\), and the dual tensor is \(1\) at the physical configuration \((h_2h_1^{-1},\, h_3h_2^{-1},\, h_3h_4^{-1},\, h_4h_1^{-1})\) of differences of adjacent colors and \(0\) elsewhere [ 17 , Appendix A, “The Toric Code and quantum double models” ] . The group acts on the virtual system by shifting all four colors, \((U_gx)(h_1,h_2,h_3,h_4)=x(h_1g,h_2g,h_3g,h_4g)\), the right-regular representation on each leg, which the relabeling \(h\mapsto h^{-1}\) turns into the left-regular one [ 96 ] .

For \(\sigma =(\sigma _{NW},\sigma _{NE},\sigma _{SE},\sigma _{SW})\), set

Both operators act on the full sixteen-spin space.

Define the color-difference map \(F:G^4\to G^4\) and the tensor \(K\) by

Its linear map is \(\mathsf K|p,q,r,s\rangle =|F(p,q,r,s)\rangle \). No normalization is included in \(K\).

On the full four-spin space \(\mathbb {C}[G^4]\), set

The product is ordered as displayed; \(G\) need not be abelian.

A tensor carries the four edge spins \(g_1,g_2,g_3,g_4\) of a plaquette (top left, top right, bottom right, bottom left), and is \(1\) when its virtual labels are \(g_1g_2\) (top), \(g_2^{-1}g_3\) (right), \(g_4g_3\) (down) and \(g_1g_4^{-1}\) (left), and \(0\) otherwise [ 17 , Appendix A, “The Toric Code and quantum double models” ] .

A configuration is flat on \(R\) if there are \(k_v\in G\) such that \(k_v^{-1}\alpha _{v,e}=k_w^{-1}\alpha _{w,e}\) on every internal edge \(e=(v,w)\). Write \(D_R\) for the diagonal projector onto these configurations. This is the trivial internal holonomy condition in regular coordinates [ 235 , Section 6 ] .

For arbitrary finite coordinate sets \(X_v\), put \(X_R=\prod _{v\in R}X_v\) and \(\mathcal H_R=\mathbb {C}^{X_R}\). Write a full configuration as \((\alpha ,\tau )\in X_R\times X_{V\setminus R}\). Extend \(K\) on \(\mathcal H_R\) by \(\widehat K_R=K\otimes I_{V\setminus R}\), so that \(\widehat K_R((\alpha ,\tau ),(\beta ,\rho ))=K(\alpha ,\beta )\delta _{\tau ,\rho }\). For \(R\subseteq U\), its extension inside \(U\) is \(K\otimes I_{U\setminus R}\).

Let \(E_R\) and \(\partial E_R\) denote the internal and boundary edges, and let \(I_R=E_R\sqcup \partial E_R\). Extend \(u\in G^{E_R}\) to \(\bar u\in G^E\) by \(1\) outside \(E_R\). Given \(\eta \in G^{I_R}\), define \(b(u,\eta )\) on an oriented internal edge \(e=(v,w)\) by

At the unique regional endpoint of a boundary edge its label is \(\eta _e\). This parametrization is a bijection. Its inverse has \(u_e=\alpha _{w,e}\alpha _{v,e}^{-1}\) and reference \(\eta _e\) at the tail if the tail lies in \(R\), and at the head otherwise. For \(\theta \in G^{\partial E_R}\), let \(\eta ^\theta \) equal \(1\) internally and \(\theta \) on the boundary, and write \(b(u,\theta )=b(u,\eta ^\theta )\). The boundary restriction of an arbitrary reference is \(\theta =\eta |_{\partial E_R}\).

Write \(U_k|\alpha \rangle =|L_k\alpha \rangle \) and \(W_r|\alpha \rangle =|T_r\alpha \rangle \) for the permutation operators. The inverse of \(T_r\) is \(T_{r^{-1}}\). If \(r^{e,g}\) equals \(g\) at \(e\) and \(1\) elsewhere, define

Here \(e\) may be internal, boundary, or disjoint from \(R\); in the last case \(B_{R,e}=I\). The operator \(P_R\) is the tensor product of the local regular invariant projectors \(P_v\).

The edges crossing the boundary of \(R\) are the same edges crossing the boundary of \(V\setminus R\). Thus a boundary configuration \(\eta \) for \(R\) determines the boundary configuration \(\eta ^c\) for the complement by reading the same edge labels through this identification.

Let \(U\) be any representation of a finite group \(G\) on a virtual space with finite basis \(V\), and write \(R_q\) for its four-leg representation. Take the physical basis to be \(V^4\) and define \(A_0\) by

Then \(\Pi R_q=\Pi \) and \(\Pi \) is the identity on invariant vectors, so \(A_0\) is \(G\)-injective. Unitarity is not required for these statements [ 235 , Definition 5.1 ] .

Set \(b=n+1\) and \(r=|G|^n\), and let \(P\) be the averaging projector on \(\mathbb C[G^b]\). The unnormalized bipartite boundary vector has coefficients

Its normalized coefficient matrix is \(C=P/\sqrt r\), and its reduced density operator is \(\rho =P/r\). These are the virtual vectors used in the proof of [ 235 , Theorem 6.9 ] .

For \(b=n+1\) regular boundary legs, let \(r\in G^n\) denote relative coordinates. Define the boundary embedding

For an open map \(T\), write \(C=TU\) and \(Q=C^\dagger C\). Its supported boundary Hamiltonian is \(H=-\log Q\), with the logarithm defined by spectral calculus and the value at a zero eigenvalue set to zero. For \(G\)-injective \(T\), the operator \(Q\) is positive definite and this is the usual spectral logarithm. The logarithm gives the inverse relation below when \(T\) is \(G\)-injective.

Let \(G\) be a finite group. Let \(\mathcal C\) be the finite set of irreducible characters occurring in its left-regular representation on \(\mathbb C^G\). Every irreducible complex character occurs in this set, and each label has a unitary irreducible representative.

Let \(G\) be a finite group, \(\chi :G\to \mathbb C\), and \(p,x,y\in G\). Define the matrix \(A_\chi (p;x,y)\) by

This is the accessible-coordinate matrix in [ 235 , charge-detection proof ] , lines 2470–2486. Here \(p\) is the internal translation, and \(x,y\) are the unknown vertex translations. No original-spin measurement is asserted.

Write \(D_\chi \) for the matrix of the orthogonal projection onto \(\mathcal H_\chi \) in the canonical basis of \(\ell ^2(G\times G)\). Write \(T_{z,w}\) for the matrix of independent left translations. These are the regular-coordinate operators used in [ 235 , charge detection ] , lines 2470–2486.

Choose an internal charge edge \(e\) and a non-tree flux edge \(b\). Retain \(y,r,z\) and every reference register except \(a_e\mapsto z_ba_e\); call this permutation \(C_{e,b}\). Its original half-edge permutation is \(E^{-1}C_{e,b}E\). For actual operators \(u\), let \(K_{u,e,b}=k_{\mathrm{tail}(e)}\rho _bk_{\mathrm{tail}(e)}^{-1}\). The transported charge parameter is \(p'=pK_{u,e,b}^{-1}\).

Let \(G\) be a finite group and \(\chi :G\to \mathbb C\). The normalized vectorization of the diagonal charge-pair operator is

The initial vector in the accessible creation diagram is \(\Omega =\Phi _{1,1}\): the two uniform diagonal pairs \(a=c\) and \(b=d\). These are the actual four registers of [ 235 , charge-pair creation ] , lines 2534–2558. A comparison with the original six-spin contraction remains separate.

Let \(R\) be a connected finite block, with a spanning tree and root. For two internal bonds \(e,f\), exchange \(a_e\) and \(a_f\) in its actual accessible coordinates \((y,a,r,z)\), retaining all other registers. For actual group operators \(u\), let \(k\) be the derived rooted tree gauge. The corresponding transported parameter is \(p'=p k_{\mathrm{tail}(e)}k_{\mathrm{tail}(f)}^{-1}\). This is the register exchange used for charge motion in [ 235 ] , lines 2489–2507.

Let \(G\) be a finite group, \(\chi :G\to \mathbb C\), and \(p\in G\). On the two group registers, define the diagonal operator

Its coefficient vector is \(v_{\chi ,p}(g,h)=\chi (ph^{-1}g)\); put \(\widehat v_{\chi ,p}=|G|^{-1}v_{\chi ,p}\). This is the accessible operator of [ 235 , charge-pair creation ] , lines 2505–2535. The identities below concern the two registers; an original-spin creation operation requires a separate physical contraction comparison.

Let \(e\) be an internal bond of a finite region \(R\). For an operator \(Q\) on its two endpoint spins, define its regional action by summing over only those two physical inputs and retaining all other spins. Thus this action is \(Q\otimes I\) under the endpoint-and-complement ordering of the regional spins.

Let \(G\) be a finite group, let \(\chi \) be a unitary irreducible character of dimension \(d\), and let \(p,k,x\in G\). On two group registers put

Here \(x\) is the common translation arising in the accessible coordinates. Define \(D=|v\rangle \langle v|\), with measurement outcomes \(D\) and \(1-D\). These are the coefficient vectors and measurement in [ 235 , interference calculation ] , lines 2582–2615. The statements below concern these two registers. Their realization on the six original spins and the geometric braid require the corresponding contraction identities and physical operations; see [ 96 ] .

For distinct internal bonds \(e_0,e_1\), split the two references \((a_{e_0},a_{e_1})\) from all other tree coordinates. A matrix \(Q\) on these two group registers acts as \(Q\otimes I\) in this decomposition. Transport this operation through the actual physical coordinate permutation to obtain \(U_R(Q)\).

Regard the actual matrix \(A_\chi (p;x,y)\) as a vector in \(\ell ^2(G\times G)\), with its ordinary Hilbert–Schmidt inner product. For a coefficient function \(\chi :G\to \mathbb C\), define

Every internal parameter and both unknown vertex translations are included. This is an auxiliary accessible-coordinate subspace for [ 235 , charge detection ] , lines 2470–2486; no original-spin detector is asserted.

Define the unitary register permutation \(T_{z,w}f(r,s)=f(z^{-1}r,w^{-1}s)\). Let \(P_\chi \) be the orthogonal projection onto \(\mathcal H_\chi \).

For a permutation \(C\) of the accessible coordinate tuples, define its physical half-edge permutation to be \(E^{-1}CE\).

For \(R\subseteq T\), let \(\mathcal S_{R;T}^0\) consist of vectors on the physical half-edge factors of \(T\) whose every slice at fixed coordinates on \(T\setminus R\) lies in \(\mathcal S_R^0\). Define \(J_Tx\) on the whole graph by \((J_Tx)(\alpha )=x(\alpha |_T)\), so it is constant in exterior physical coordinates. The space \(\mathcal S_{R;T}^0\) itself has physical factors only at the vertices of \(T\).

Let \(R\) be a finite region of an ordered simple graph, let \(e_0,e_1\) be internal bonds, and let \(A_v\) be the original site coefficients. For a physical configuration \(s\) and crossing labels \(\theta \), put

Replacing \(A_v\) by the local regular averaging projectors gives the canonical column \(B_{\chi ,p}\). This is a single correlated weight on two bonds, as in [ 235 , charge-pair creation ] , lines 2505–2558.

Let \(e_0,e_1\) be ordered edges of a finite graph, and let \(u\) be a bond assignment. Resolve the remote label \(q\) by inserting \(\mathbf1_{\eta _{e_1}=q}\) on \(e_1\), and put \(\chi (pq^{-1}z\eta _{e_0})\) on \(e_0\). Sum the resulting closed tensor states over \(q\). Denote this correlated state by \(\Psi _z(u;\chi ,p)\). This definition retains the same original site coefficients in every summand. It is auxiliary to the literal charge-pair formula in [ 235 ] , lines 2505–2535.

Let \(G\) be finite, let \(R\) be a finite graph region, and choose a spanning tree \(T\) and root \(o\). Write \(J\) for the internal bonds outside \(T\) and \(E\) for the actual coordinate bijection from half-edge configurations to \((y,a,r,z)\), with \(z\in G^J\). For a permutation \(\phi \) of \(G^J\), define \(C_\phi (y,a,r,z)=(y,a,r,\phi z)\) and \(P_\phi =E^{-1}C_\phi E\). Let \(Q_\phi \) be the permutation matrix which sends \(|\alpha \rangle \) to \(|P_\phi \alpha \rangle \).

Label \(\chi ^\vee \) by its function \(g\mapsto \chi (g^{-1})\) in the actual finite family of regular irreducible characters.

Let \(e=(v_-,v_+)\) be an ordered edge of a finite simple graph, and put \(R_e=\{ v_-,v_+\} \). The distinguished incident leg at either endpoint is \(e\). All other incident edges are boundary edges of \(R_e\), so a native boundary configuration \(\theta \) supplies the remaining labels \(\theta _-,\theta _+\). The physical coordinate map is \(\sigma \mapsto (\sigma (v_-),\sigma (v_+))\). For a site family \(a\), a function \(\chi :G\to \mathbb C\), and \(p\in G\), insert \(\chi (pk)\) on the shared group label by replacing only the tail tensor:

This is the diagonal insertion in [ 235 , Definition of chargeons ] , local source lines 2432–2453.

Set \(U_g=I-(\delta _1-\widehat\alpha _g)(\delta _1-\widehat\alpha _g)^\dagger \). On the pair \((a,b)\), let \(V_g\) act as \(U_g\otimes I\) in the coordinates \((ab^{-1},b)\).

Let \(G\) be a finite group and \(C_G(g)\) the centralizer of \(g\). Write \(\kappa =|G|\) and \(c_g=|C_G(g)|\). Define

These amplitudes retain the multiplicities in the accessible-register sum of [ 235 , Theorem 6.17 ] .

Let \(G\) be a finite group, \(K\) a finite set, and \(\Gamma \) a finite simple graph. Write \(t(e)\) and \(h(e)\) for its ordered edge endpoints. Put \(X=G\times G^K\), and let \(M_g\) simultaneously translate all coordinates of \(X\) on the left. At each vertex use the averaging tensor

Its state \(\Psi _K\) is the sum over one virtual label \(\eta _e\in X\) per edge of the product of the vertex tensors. Let \(\Psi _0\) be the same construction with \(X=G\) and \(M_g=L_g\).

On each physical half-edge apply the basis permutation \(|(a,(b_k)_k)\rangle \mapsto |a,(a^{-1}b_k)_k\rangle \). Group the distinguished physical coordinates into \(\sigma \) and the residual coordinates into \(\tau \), and denote the resulting physical unitary by \(U\). It is a product of vertex-local operations followed by a rearrangement of tensor factors. For \(D=|G^K|\), define

These are the physical coordinates and discarded factors in the bond-separation construction of [ 235 , Observations 6.5–6.6 ] .

Let \(a_v\) be a site tensor on the bundled virtual alphabet \(G\times G^K\) with arbitrary finite physical alphabet \(P_v\). For positive numbers \(c_v\), define the local physical map and its global composition with the relative coordinate unitary by

Every map \(J_v\) acts only on the original physical system at \(v\).

Let \(G\) be a group, let \(K\) be a set of surplus indices, and let \(\Gamma \) be a finite simple graph. For a bundled incident configuration \(\eta _f=(u_f,z_f)\in G\times G^K\), define \(E_v\eta =(u,r)\) with \(r_f=u_f^{-1}z_f\). Its inverse is \((u,r)\mapsto (u_f,u_fr_f)_f\), and \(E_v(g\eta )=(gu,r)\). Given original physical coefficients \(a_v(u;s)\) with physical alphabet \(P_v\), define the augmented tensor by

The physical alphabet of \(C_v\) is \(P_v\times (G^K)^{\operatorname {Inc}(v)}\). Thus its first factor is the original physical system and its second factor records the relative surplus labels. For \(K\) a singleton, this is the identity-tensor factor in [ 235 , Observation 6.6 ] .

Let \(G\) be a finite group and let \(\Gamma =(V,E)\) be a finite simple graph, with each edge oriented from \(s(e)\) to \(t(e)\). A half-edge configuration assigns \(\alpha _{v,e}\in G\) to every incident vertex-edge pair. Vertex and bond translations are \((k\alpha )_{v,e}=k_v\alpha _{v,e}\) and \((\alpha r)_{v,e}=\alpha _{v,e}r_e\), respectively. Define the operator quotient \(u(\alpha )_e=\alpha _{t(e),e}\alpha _{s(e),e}^{-1}\). For \(u,\eta \in G^E\), let \(\alpha (u,\eta )\) have tail labels \(\eta _e\) and head labels \(u_e\eta _e\). The map \((u,\eta )\mapsto \alpha (u,\eta )\) is a bijection, with inverse \(\alpha \mapsto (u(\alpha ),(\alpha _{s(e),e})_e)\). Write \(\alpha (u)=\alpha (u,1)\). Then \(\alpha (u,\eta )=\alpha (u)\eta \), \(u(\alpha r)=u(\alpha )\), and \(u(k\alpha )_e=k_{t(e)}u(\alpha )_ek_{s(e)}^{-1}\).

With \(P_v\) the regular averaging projector, the canonical closed coefficient vector with inserted operators \(u\) is

These are the full-region contractions of Definition 26.1.4.123.

For a matrix product tensor \(A\) with bond space \(\mathbb {C}[G]\), write each \(A^{i}\) as a matrix in the basis of group elements. In this basis the adjoint \((A^{i})^{\dagger }\) is the conjugate transpose, and the parent Hamiltonian of \(A\) is that of these matrices.

Express every bond in a group basis, and enumerate the \(b=n+1\) crossing bonds. Let \(A\) and \(B\) be the open tensors of the region and its complement, with their columns indexed by \(G^b\) in the same crossing-bond order. Let \(K\) be the matrix of physical PEPS coefficients on these two systems. These are the actual cut maps used before the virtual reduction in [ 235 , Proof of Theorem 6.9 ] .

For a site tensor \(a(\eta ,s)\) whose virtual labels are group elements on an arbitrary incident-leg set, define \(a^F(\eta ,t)=\sum _sF_{ts}a(\eta ,s)\). Its physical coefficient map is \(T_{a^F}=FT_a\).

Let \(R\) be a finite region of the original graph, \(E_R\) its incident edges, and \(\partial R\) its crossing edges. Write \(E_v\) for the incident edges at \(v\) and put \(\kappa =|G|\). The canonical physical configuration is \(\alpha =(\alpha _v)_{v\in R}\), with \(\alpha _v\in G^{E_v}\). The local canonical site map is the regular averaging projector \(P_v\); it is \(G\)-isometric with factor one. Define the canonical open-region coefficient matrix by

Every internal bond is summed once, and no exterior bond label occurs. Encoding group labels in finite coordinates preserves the boundary condition. These are the exposed virtual coordinates of [ 235 , Observation after Lemma 6.3 ] .

For arbitrary inserted group elements \(u_e\), including those on crossing bonds, define

Here the half-edge modification \(\eta ^u\) is that of Definition 26.1.4.122. Identity insertions recover \(B_R\). This is the literal finite contraction of the local averaging-projector sites with the inserted regular operators.

Number the \(n+1\) actual crossing bonds. The spanning-tree physical coordinates identify the region rows with \((d,a)\), where \(d\in G^n\) is the relative boundary label and \(a\) consists of its reference label, internal references, normalized vertex labels, and cycle residuals. Pull back \(\rho _{\phi _R}\) to the native half-edge physical rows, with the fixed normalized ancillary vector \(\phi _R\) of Theorem 26.1.4.60. Denote this density by \(\rho _R^0\).

Let \(G\) be a finite group and \(R\) a finite region. Choose a spanning tree \(T\) of its induced graph and a root \(o\in R\). Write \(I\) for the internal edges and \(\partial R\) for the crossing edges, and orient each internal edge \(e=(u,v)\) by the ordering of its endpoints. The tree gradient \(r\mapsto (r_vr_u^{-1})_{e\in T}\) is a bijection from vertex labels satisfying \(r_o=1\) to arbitrary labels on \(T\). Such normalized vertex labels are themselves in bijection with arbitrary labels on \(R\setminus \{ o\} \).

Physical labels \(\alpha _{v,e}\in G\), one for each incident half-edge at every \(v\in R\), are in bijection with coordinates \(y\in G^{\partial R}\), \(a\in G^I\), normalized vertex labels \(r\), and \(z\in G^{I\setminus T}\). The reconstruction is

This is a reversible change of regular basis labels associated with the accessible virtual systems and blocking operations in [ 235 , §6.2–6.3 ] .

Let \(R\) be a nonempty finite region with a spanning tree \(T\), root \(o\), and an internal bond \(e\) outside \(T\). In the actual coordinates \((y,a,r,z)\), apply the class creation unitary \(U_g\) to \(z_e\) and the identity to every other coordinate. Transport this operator through the coordinate bijection and denote the resulting half-edge operator by \(Q_{g,e}\).

Let \(E_v\) be the set of virtual legs incident to a site \(v\). In the group basis, its coefficient map \(A_v\) has entries \(A_v(s,\eta )=a_v(\eta ;s)\), for \(\eta \in G^{E_v}\). Simultaneous left translation acts on functions by \((U_gf)(\eta )=f(g^{-1}\eta )\), and its invariant projector has entries

The local Gram kernel is \(\sum _s\overline{a_v(\eta ;s)}a_v(\theta ;s)\).

Let \(G\) be a group, let \(\Gamma \) have linearly ordered vertices, and let \(R\) be a finite set of vertices. For \(g\in G\), put \(k_v=g^{-1}\) on \(R\) and \(k_v=1\) outside \(R\). For ordered edges \(e=(s,t)\), define the swept assignment by

This is an auxiliary formulation of the vertex-invariance calculation in [ 235 , Section 6.6 ] , the virtual string-deformation figure, local source lines 2361–2386.

Take vertices \(0,\ldots ,10\) and the unordered edges

Set \(R=\{ 0,1\} \), \(S=\{ 1,2\} \) and \(T=\{ 0,1,2\} \). The core is the path \(0\)–\(1\)–\(2\). Its end vertices each have three exterior legs and its middle vertex has two. The eight remaining vertices are distinct exterior endpoints, so an arbitrary boundary tensor may correlate all eight open legs of \(T\).

Let \(J\) index group labels, and choose distinct \(e_0,e_1,e_2\in J\). Define a permutation \(C\) of \(G^J\) which retains the other coordinates and sends the selected directed labels \((t_0,t_1,t_2)\) to

Its inverse sends \((a,b,c)\) to \((c^{-1}b^{-1}a^{-1},c^{-1}b^{-1},c^{-1})\).

Define \(\Phi =C^{-1}H^2C\) on directed bond labels. For a common orientation flag \(\varepsilon \in \{ 0,1\} \), let \(I_0z=z\) and \(I_1z=z^{-1}\) coordinatewise, and define \(\Phi _\varepsilon =I_\varepsilon ^{-1}\Phi I_\varepsilon \).

Order the endpoints of each bond \(e\), writing them as \(s(e){\lt}t(e)\). For an assignment \(u_e\in G\), the inserted regular matrix makes the label at \(t(e)\) equal to \(u_e\) times the label at \(s(e)\). Equivalently, replace the coefficient at vertex \(v\) by

Contracting the internal bonds with a fixed crossing configuration gives the actual open map \(T_R^u\).

For distinct indices \(e_0,e_1\) in a family of group labels \(z\), let \(\Phi \) retain every label except \(z_{e_1}\) and replace that label by \(z_{e_0}z_{e_1}z_{e_0}^{-1}\). Its inverse replaces \(z_{e_1}\) by \(z_{e_0}^{-1}z_{e_1}z_{e_0}\).

For distinct selected indices, let \(H\) conjugate the second label by the first and then interchange the two coordinates. It sends \((a,b)\) to \((aba^{-1},a)\) and retains every other label.

Let \(I,J\) be finite incident-leg sets, with distinguished shared legs \(i_0,j_0\). Write \(I_0=I\setminus \{ i_0\} \) and \(J_0=J\setminus \{ j_0\} \). Splitting the shared labels gives the bijection

If \(\iota _I(k,\theta _I)\) inserts \(k\) at the shared leg, the actual canonical charged column is

Here \(P_I,P_J\) are the local regular averaging projectors.

Let \(D_\chi \) be the orthogonal projector onto the span of all translated charge matrices \(A_{\chi ,p;x,y}\). In the shared-leg coordinates define \(\widehat D_\chi =D_\chi \otimes I\) on the remaining labels, and transport it back by the preceding bijection.

For local tensors \(a:G^I\to \mathcal H_A\) and \(b:G^J\to \mathcal H_B\), define

This is the actual contraction of the two original physical tensors along the character-weighted shared bond.

For original site tensors \(A_v\), replace \(P_v[\alpha _v,(u\eta )_v]\) by \(A_v((u\eta )_v,\sigma _v)\) in the preceding incident-label sum. Denote the resulting actual original-spin coefficient vector by \(\Psi _R(u,F;\theta )\).

Let \(R\) be a finite region with a chosen spanning tree \(T\) and root \(o\). Let \(k\) be its actual tree gauge for \(u\), and write \((y,a,r,z)\) for its accessible coordinates. For an arbitrary scalar weight \(F\) on incident label configurations, define

where \(P_v\) is the actual regular invariant projector. The reconstructed label \(\eta _x\) is \(k_t x^{-1}a_e\) on an internal edge, \(k_t x^{-1}y_e\) on an outgoing boundary edge, and \(u_e^{-1}k_hx^{-1}y_e\) on an incoming boundary edge. These definitions retain the literal diagonal weights of [ 235 ] , lines 2569–2581, in the accessible coordinates of lines 1765–1920.

For a positive density \(\rho \) and \(\alpha {\gt}0\), \(\alpha \ne 1\), define

where the real power is defined spectrally. Set \(S_1(\rho )=S(\rho )\) and \(S_0(\rho )=\log \operatorname {rank}\rho \).

For a finite-dimensional complex representation of a finite group, let \(P_i\) be its isotypic projectors, \(d_i\) the irreducible dimensions, and \(m_i\) their multiplicities. Define \(\Theta =\sum _i(d_i/m_i)^{1/4}P_i\), with the positive real fourth root. For the multiplicity-one representation in [ 235 , Section 7 ] , this is the displayed operator \(\bigoplus _i d_i^{1/4}I_{d_i}\).

Let \(T\) map a finite-dimensional virtual space to the original physical space. Suppose \(T^*T=cP\) and \(TP=T\), where \(c{\gt}0\). For a Hermitian idempotent \(D\) commuting with \(P\), put

These are the finite-dimensional relations used in the local isometry argument of [ 235 , Section 6.6 ] , lines 2582–2615.

A representation of \(G\) is semi-regular if it contains every irreducible representation of \(G\), that is, if every irreducible representation admits a nonzero intertwining map into it [ 235 , Section “MPS: the \(G\)-injective case” ] . The left-regular representation is semi-regular.

For a finite index set \(I\), let \(G_I\) be the group of matrices sending each standard basis vector to another standard basis vector with sign \(+1\) or \(-1\). Its defining representation is unitary, and \(G_I\) is finite. These groups supply the finite-dimensional matrix representations used in the construction of [ 235 , Theorem 4.1 ] .

For orthogonal site projectors \(J_v\), define \(J_v^{[U]}=J_v\) on \(U\) and \(J_v^{[U]}=I\) outside \(U\). Put \(M_U=\bigotimes _{v\in V}J_v^{[U]}=\widehat{J_U}\). Every selected-site product is an orthogonal projector, \(M_RM_S=M_{R\cup S}\), and \((\bigotimes _{v\in R}J_v^{[U]})J_R=J_R\).

Let \(U\) be a representation of a finite group on a finite virtual space, and let \(\Theta \) be its fourth-root isotypic weight operator. Obtain the weighted site from the canonical averaging site by placing \(\Theta \) on each of its four virtual legs. For the multiplicity-one representation, this is the tensor displayed in [ 235 , Section 7 ] , divided by \(|G|\).

On a torus of horizontal period at least five and vertical period at least three, fix any position \(v=(x,y)\). The strip \(R_v\) consists of the four columns \(x,x+1,x+2,x+3\) in the two rows \(y,y+1\). Number its sites along the lower row from left to right, then along the upper row from right to left. The corresponding path \(T_v\) contains the six horizontal bonds and the rightmost vertical bond. Let \(e_0,e_1,e_2\) be the remaining ordered vertical bonds from left to right. These are actual non-tree internal bonds of \(R_v\). This eight-site geometry is auxiliary to the adjacent-flux construction in [ 235 , braiding passage ] .

Write \(G=\mathbb Z_2\) additively and put \(V=G^4\). The primal tensor of [ 17 , Appendix A, “The Toric Code and quantum double models” ] defines the map \(P:\mathbb {C}^V\to \mathbb {C}^V\) with entries

In the basis \(|0\rangle ,|1\rangle \), let \(Z=\operatorname {diag}(1,-1)\). The virtual representation of \(G\) has generator \(J=Z^{\otimes 4}\). Its invariant subspace is

The orthogonal projection onto this subspace is

Let every site \(v\) of the torus carry a tensor \(A_v\) with four virtual indices, ordered top, right, down, left, and let every bond carry a \(D\times D\) matrix \(O\): \(O^{\mathrm h}_v\) on the horizontal bond from \(v\) to \(v+(1,0)\) and \(O^{\mathrm v}_v\) on the vertical bond from \(v+(0,1)\) to \(v\). Split every bond into its two ends, and give a labelling of the ends the weight \(O_{ij}\) on each bond whose head reads \(i\) and whose tail reads \(j\). The network is the sum over all labellings of the ends of the product of these weights and of the entries of the site tensors, each reading the ends of its four legs. For a representation \(g\mapsto U_g\) on one bond, \(U_g\) acts on the virtual system of a site by \(U_g\) on the incoming legs (top and left) and \(U_g^{-1}\) on the outgoing legs (right and down), each along the orientation of its edge [ 235 , Section “Two dimensions: \(G\)-injective PEPS” ] .

For \(v=(x,y)\), take the translated four-column, two-row strip with lower-left vertex \((x-1,y+1)\). Its charge bond joins \((x+1,y+1)\) to \((x+2,y+1)\) on the lower row. Its leftmost plaquette contains the endpoint of the literal flux string. The tree consists of all horizontal bonds and the rightmost vertical bond. This is auxiliary geometry for the charge–flux figure of [ 235 ] , lines 2560–2581.

At \(v=(x,y)\), the remote reference is the native horizontal bond joining \((x+1,y)\) to \((x+2,y)\), below the eight-site crossing strip. Both periodic seam orientations are retained. This is auxiliary native geometry for [ 235 ] , lines 2505–2581.

On a finite torus with both periods at least three, the motion square based at \(v=(x,y)\) has the four sites \((x,y),(x+1,y),(x+1,y+1),(x,y+1)\). Its distinct upper and lower charge bonds are the native right edges based at \((x,y+1)\) and \((x,y)\), respectively. Both ordered endpoints lie in the actual four-site region, including at either periodic seam. This is the charge-motion diagram of [ 235 ] , lines 2489–2507.

At a translated reference vertex \(v=(x,y)\), let \(e_c\) be the native horizontal bond joining \((x+1,y+1)\) to \((x+2,y+1)\). This is the central charge diagonal in [ 235 ] , lines 2569–2581 and the virtual charge–flux deformation figure. Its orientation is the actual ordered edge orientation.

The coordinate exchange \(S(x,y)=(y,x)\) is a graph isomorphism from the \(w\times h\) torus to the \(h\times w\) torus. Its inverse is the coordinate exchange in the opposite direction.

On a finite torus with both periods at least three, let \(u\) assign an element of a group to each ordered edge. Denote its positive rightward and upward transports by \(D_0(u;p)\) and \(D_1(u;p)\). For \(a\in \{ 0,1\} \), define \(u[a,p,t]\) by replacing the ordered coefficient of the corresponding bond by \(t\) when its positive direction agrees with the ordering, and by \(t^{-1}\) otherwise. Every other coefficient is retained. These auxiliary coordinates include bonds crossing either periodic seam; they support the string operations in [ 235 , §6.6 ] .

On a finite simple torus, choose either the right edge or the up edge of a native vertex \(v\). Its ordered endpoints form the measured two-site region. The diagonal insertion is \(\chi (pk)\) on this actual shared bond, including bonds that cross either coordinate seam.

On a \(w\times \ell \) torus let \(H_v=U_h\) when \(v+(1,0)\) crosses the horizontal seam and \(H_v=\mathbb 1\) elsewhere; let \(V_v=U_g\) when the vertical bond crosses the vertical seam and \(V_v=\mathbb 1\) elsewhere. Define the vector \(|\mathcal M(A\mid g,h)\rangle \) by contracting the site tensors with these bond operators. Its coefficient at \(\sigma \) is

This is [ 235 , Definition 5.6 ] , which uses a square lattice; the same contraction defines the vector on a rectangle. The four-site diagram suppresses the physical legs at fixed \(\sigma \). Crossings without a vertex do not denote contractions.

For \(n,m{\gt}1\), the discrete \(n\times m\) torus has vertex set \(\mathbb {Z}/n\mathbb {Z}\times \mathbb {Z}/m\mathbb {Z}\). Its graph joins \((x,y)\) to \((x+1,y)\) and to \((x,y+1)\), with the coordinates read cyclically; every edge is of one of these two forms, so every edge is horizontal or vertical. Translation by \((a,b)\) is the graph automorphism \(\tau _{a,b}(x,y)=(x+a,y+b)\). The induced action on edges sends an edge with unordered endpoint pair \(\{ u,v\} \) to the edge with unordered endpoint pair \(\{ \tau _{a,b}(u),\tau _{a,b}(v)\} \); in particular it preserves the horizontal and vertical orientation classes. For a general graph isomorphism \(\phi :G\simeq G'\), the same endpoint action gives a bijection on edges and a bijection from the edges incident to \(v\) to the edges incident to \(\phi (v)\).

On a torus of horizontal period at least four and vertical period at least three, let \(R_v\) be the six-site block of two adjacent plaquettes based at \(v=(x,y)\). Its counterclockwise outer walk makes two rightward steps, one upward step, two leftward steps, and one downward step. Its visited sites are exactly \(R_v\).

For \(a,b\in G\), define the routed assignment \(u_{a,b}\) by giving upward transport \((ab)^{-1}\) to the left vertical bond and \(b^{-1}\) to the middle vertical bond. Every other transport is the identity. Each ordered bond receives the specified element when its ordering agrees with the upward direction, and its inverse otherwise. Thus this definition also applies at the vertical seam. This is the routed geometry of the joint flux measurement in [ 235 , braiding passage ] .

Let \(X\) be the vertex set of a torus with positive periods and let \(\mathcal B=X\times \{ \mathrm h,\mathrm v\} \) be its horizontal and vertical bond positions. Write \(\sigma _v=(t_v,r_v,b_v,l_v)\) for the four original physical coordinates at \(v\). Their bond regrouping is

Each pair is ordered head, tail: horizontal bonds point right and vertical bonds point down. The inverse reads

Thus regrouping is a bijection of the actual physical basis configurations. Its induced coefficient map is \((\mathcal R\psi )(\beta )=\psi (\sigma (\beta ))\), and is a linear isometric equivalence of the finite-dimensional physical Hilbert spaces. This is the grouping into the two physical bond endpoints used in [ 235 , Section 7 ] .

Enumerate the oriented sides of \(p_v\) by \(i=0,1,2,3\) in traversal order. For \(g\in G\), let \(u^{v,i,g}\) put directed transport \(g\) on side \(i\) and identity transport on all other bonds. The ordered bond receives \(g\) or \(g^{-1}\) according to its orientation. Then

This realizes the four attachment directions in the final paragraph of [ 235 , proof of Theorem 6.15 ] .

On a native torus whose two periods are at least three, let \(v=(x,y)\). Define the plaquette walk \(p_v\) and its physical region \(R_v\) by

Coordinates are periodic, so this definition includes plaquettes crossing either seam. It is the four-site region used in [ 235 , proof of Theorem 6.15 ] .

For \(g\in G\), put directed transport \(g\) on the first side of \(p_v\) and identity transport on every other bond. In the ordered-edge convention, the distinguished bond therefore carries \(g\) when its orientation agrees with the walk and \(g^{-1}\) otherwise. Denote this actual assignment by \(u^{v,g}\). Its plaquette holonomy is

This is the one-bond open regular string of [ 235 , Section 6.5 and proof of Theorem 6.15 ] .

On the torus vertices, retain only consecutive coordinate values without wraparound. The result is the ordinary rectangular grid, a subgraph of the periodic square lattice.

In the group basis, \(L_x|k\rangle =|xk\rangle \). The coordinate map \(\sum _k c_k[k]\mapsto (c_k)_k\) intertwines the group-algebra regular representation with these permutation matrices. They are therefore semi-regular by Theorem 26.1.10. The oriented four-leg action sends \((t,r,b,l)\) to \((xt,xr,xb,xl)\); its invariant projector has entries

Thus the four regular coordinates agree with the oriented virtual action in Definition 6.1 of [ 235 ] .

Choose positive dimensions \(d_i\) and put \(V=\bigoplus _i\mathbb C^{d_i}\) and \(V'=\bigoplus _i(\mathbb C^{d_i}\otimes \mathbb C^{d_i})\). Let \(F:V\otimes V\to V'\otimes V'\) be the full endpoint multiplicity-restoring map. For \(Q:V'\to \mathbb C^G\), put \(C_Q=Q\otimes \overline Q\). On the physical bond spaces of a torus with positive periods, define

For a plaquette based at \(v=(x,y)\), write \(k_v(g)=g\) if the shared upward edge from \((x+1,y)\) to \((x+1,y+1)\) agrees with the native endpoint order, and \(k_v(g)=g^{-1}\) otherwise. Thus its directed upward transport is always \(g\), including the vertical seam.

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 \(v+(0,1)\).

Start at \(B_1\), follow the actual partner connector to \(B_2\), traverse its clockwise endpoint loop, and return along the reversed connector. Then traverse the clockwise endpoint loop at \(B_1\). Thus, if \(P\) is the connector transport, the reverse-ordered joint holonomy is \(f_{B_1}(P^{-1}f_{B_2}P)\).

Let the horizontal and vertical torus periods be at least seven and six, respectively, and fix any translated position \(v\). Write \(p(i,j)=(v_x+i,v_y+j)\), with coordinates taken modulo the periods. Complete the displayed four-column, three-row patch \(\{ p(i,j):0\leq i\leq 3,\ 0\leq j\leq 2\} \) by the side sites

and the lower sites

Denote this finite set by \(R_v\). This is an auxiliary completion of the swept-string diagram in [ 235 , fluxon-braiding passage ] . The distant lower partner lies beyond the final string detour; it is not placed at the endpoint of the displayed lower bond.

Take the four plaquette bases, in order, to be

At each base take the clockwise four-bond plaquette loop. The \(A_2\)–\(A_1\) connector goes up, then right four times, then down. The \(B_1\)–\(B_2\) connector goes down three times. For the eventual bond assignments, a group element \(s\) written along a rightward or downward matrix arrow in the source figure corresponds to native directed transport \(s^{-1}\). The value on an ordered bond is this directed transport or its inverse, according to its endpoint order.

Let the horizontal and vertical torus periods be at least eight and seven, respectively. Fix any translated position \(v\) and write \(p(i,j)=(v_x+i,v_y+j)\). Consider the successive plaquette bases

Each successive pair is joined by an actual unit bond of the native torus graph. Their ordered traversal defines a twelve-step walk \(\rho _v\). This chosen route passes above, to the right of, and below the \(A_1\) plaquette before continuing towards \(B_2\). It is an auxiliary finite geometry for the physical movement discussed in [ 235 , fluxon-braiding passage ] , lines 2340–2423.

For twelve physical matrices \(M_0,\ldots ,M_{11}\), let \(P_0=I\) and \(P_{n+1}=M_nP_n\) for \(0\leq n{\lt}12\). Thus the latest operation acts on the left. This is the chronological product for the explicit endpoint route of [ 235 , §6.6, local lines 2340–2423 ] .

Let \(F_v\) be the union of the twenty-site four-endpoint witness \(R_v\) and the four physical corners of each of the thirteen route plaquettes. All its vertices have offsets \(-1\leq i\leq 5\) and \(-2\leq j\leq 3\) from \(v\). This seven-column, six-row envelope contains both plaquettes of every successive elementary movement step.

Assume that the torus periods are at least eight and seven. Begin with the four-endpoint string assignment and take twelve steps through the following thirteen plaquette bases, translated by an arbitrary vertex:

At a source plaquette write \(B,V,T,L\) for its bottom rightward, right upward, top rightward and left upward transports. A right, upward, left or downward step respectively replaces its shared transport by

All other ordered bonds, including the partner strings, are retained. Let \(u_n\) denote the resulting recursively defined assignment after \(n\) steps. These are auxiliary coefficient assignments for the flux motion of [ 235 , §6.6, local lines 2340–2423 ] ; no complete physical braid is asserted by their definition.

Relative to the translation vertex, the final endpoint bases in the order \(A_2,A_1,B_1,B_2\) are

Retain the actual original connector from \(A_2\) to \(A_1\). Compare the final neighbouring B partners along the rightward edge from \((0,-2)\) to \((1,-2)\). In each joint loop, traverse the partner loop along the connector first and then the loop at the chosen base. Consequently its holonomy is the first based flux times the transported partner flux.

On a torus with periods at least eight and seven, take the translated six-site two-plaquette block whose lower-left site is \((0,-2)\) relative to the translation vertex. It contains the actual final B reunion connector and both partner loops.

For any group \(G\) and \(g,h\in G\), prescribe the initial rightward and upward transports by

Assign each native right or up edge the prescribed transport if its forward direction agrees with the ordered orientation, and its inverse otherwise. Denote this ordered assignment by \(u^{(0)}\), and put \(u^{(j)}=(u^{(0)})^{S_j,g}\) for \(j=1,2\). The second string continues below the displayed patch to retain its distant partner. A displayed rightward or downward matrix labelled \(s\) has transport \(s^{-1}\) in the corresponding direction. This is a finite realization of the virtual figure in [ 235 , Section 6.6 ] , lines 2361–2386.

Let the torus periods satisfy \(w\geq 5\) and \(H\geq 4\). For \(v=(x,y)\), with all coordinates read periodically, define

These finite sets specify the two successive sweeps in the virtual string-deformation figure of [ 235 , Section 6.6 ] , local source lines 2361–2386.

For directed transports \(t=(t_0,t_1,t_2)\), define

If \(C(t)=(a,b,c)\) denotes the three plaquette holonomies, then \(t'=C^{-1}H^2C(t)\), with \(H(a,b)=(aba^{-1},a)\) acting on the first two plaquettes. The same ordered-bond orientation correction applies to both literal assignments.

In the actual translated three-plaquette strip, prescribe upward transports \(t_0,t_1,t_2\) on its first three vertical bonds. The rightmost vertical bond and all other bonds carry the identity. In the native ordered convention, the three selected coefficients are \(k_y(t_i)\), where

This literal assignment is exactly the tree-cycle assignment with residual \(k_y(t_i)\) on \(e_i\), and identity on every other non-tree bond. Thus a single orientation flag treats all three chords, including at the vertical seam. This construction is auxiliary to the adjacent fluxes in [ 235 , braiding passage ] .

On a finite torus of horizontal period at least four and vertical period at least three, fix any position \(v=(x,y)\). Let \(u_g^0\) give upward transport \(g\) on the vertical bond at \(v\), and let \(u_g^1\) give upward transport \(g\) on both vertical bonds at \(v\) and \(v+(1,0)\). Every other bond carries the identity. In the ordered-edge convention, both selected bonds receive the same element

The literal assignments are precisely the tree-cycle assignments of the translated six-site block. This specifies the insertion in [ 235 , Theorem 6.16 ] at either periodic seam as well.

At \(v=(x,y)\) the two columns are \(L=\{ (x,y),(x,y+1)\} \) and \(S=\{ (x+1,y),(x+1,y+1)\} \). The upper and lower horizontal native bonds join them. All coordinates are torus coordinates, including at either seam.

On a finite torus of horizontal period at least four and vertical period at least three, fix any position \(v\). Translate the six-site numbering of the two-plaquette block by \(v\), with both coordinates taken modulo their periods. Let \(R_v\) be the resulting set of sites. Its induced graph is identified with the fixed two-plaquette graph. Transport the fixed spanning path and its two remaining bonds through that identification. These constructions permit either periodic seam to pass through the block. They describe the local geometry in [ 235 , Theorem 6.16 ] .

For a simple torus graph with periods at least three, express every native bond in the ordered-edge orientation. A reversed native arrow replaces its group label \(u\) by \(u^{-1}\). The horizontal periodic seam carries \(h_0\) and the downward vertical seam carries \(g_0\).

Group the fine torus into disjoint two-by-two squares. Give each pair of horizontal crossing bonds the same permutation \(s_h\), and each pair of vertical crossing bonds the same permutation \(s_v\); leave all internal bonds unchanged. The paired regular action is \(L_g^{(2)}=L_g\otimes L_g\), acting on \((a,b)\) by \((ga,gb)\). Horizontal insertions act on the receiving left leg, and vertical insertions on the top leg in the native downward orientation.

Let \(C=\mathbb Z_w\times \mathbb Z_h\) with \(w,h{\gt}0\), and identify the fine torus \(\mathbb Z_{2w}\times \mathbb Z_{2h}\) with \(C\times \{ 0,1,2,3\} \) by its disjoint four-site tiling. The paired crossing bonds \(h_z,v_z\) and the four internal bonds \(x_{z,i}\) give the fine outgoing bonds by

This is a bijection. Reading the top, right, bottom, and left fine bonds gives exactly \(\lambda _i(\alpha _z,x_z)\), including periodic seams. There is one horizontal and one vertical bond per site even when a coarse period is one or two.

Identify \(G\times G\) with \(G\times G^{\{ *\} }\) by \((a,b)\mapsto (a,(*\mapsto b))\). Express the actual paired coarse tensor in these incident bundle coordinates, without changing its coefficients or physical alphabet \(P^4\). Reading the four incident edges and the two labels on each edge identifies this boundary with all eight separate fine boundary legs.

Let \(\Psi _a\) be the original fine bond-indexed torus state. Let \(W\) be its full physical regrouping unitary, and let \(T_B=\bigotimes _vB_v\) be the product map of the actual bundled coarse tensors. Define \(S_a=W^{-1}(\operatorname {ran}T_B)\). Restricting \(W\) gives an isometric equivalence \(S_a\cong \operatorname {ran}T_B\). All physical coordinates remain in their original ambient spaces.

Suppose \(w,h\geq 3\), so the four incident edges at every coarse vertex are distinct. A four-leg tensor is expressed on incident labels by reading its top, right, bottom, and left edges. Applying this change to \(B_v\) defines its coarse graph tensor, with physical alphabet \(P^4\).

Let \(\Gamma \) be the coarse torus graph. At each coarse vertex, retain the original physical label \(s_v\) and every relative surplus half-edge label \(r_{v,e}\). Their full physical space is identified unitarily with the tensor product of the original coarse physical space and the surplus space. For positive site factors \(c_{A,v},c_{B,v}\) put

The disjoint tiling identifies fine physical configurations \(\sigma \) with block configurations \(\tau _{v,i}=\sigma _{(v,i)}\). Let \(W\) be its permutation matrix on the complete physical spaces.

Let \(X\) and \(P\) be finite virtual and physical alphabets. Number the four corners clockwise from the upper right by \(0,1,2,3\). Write \(\alpha =(t,r,b,l)\in (X^2)^4\), with top and bottom pairs ordered left to right and right and left pairs ordered top to bottom. For \(x=(x_0,x_1,x_2,x_3)\in X^4\), define the four fine leg configurations by

The eight separate boundary labels are indexed by \(\{ 0,1,2,3\} \times \{ 0,1\} \). For tensors \(a_i:X^4\times P\to \mathbb C\), their actual blocked tensor is

On a periodic coarse lattice, with horizontal and vertical paired bond assignments \(h,v\), the boundary at \(z\) is \((v_z,h_z,v_{z-\hat y},h_{z-\hat x})\). This is the four-site grouping in [ 235 , Observation 6.6 ] .

Number the six sites in the order \((0,0),(1,0),(2,0),(2,1),(1,1),(0,1)\). Let \(T\) be the path joining consecutive numbered sites, and let \(\Gamma \) add to \(T\) the bonds \(e_0=\{ 0,5\} \) and \(e_1=\{ 1,4\} \). These are respectively the leftmost and middle vertical bonds. This is the six-site geometry used in the construction of [ 235 , proof of moving fluxons ] .

On a torus of horizontal period at least four and vertical period at least three, place the six sites at the displayed coordinates. Their numbering identifies \(\Gamma \) with the actual induced torus graph on the resulting region \(R\). Transport \(T,e_0,e_1\) by this identification, retaining the ordered bond endpoints.

Relative to the four-endpoint patch based at \(v\), the successive plaquette bases are \(v+(1,1)\), \(v+(2,1)\), and \(v+(2,2)\).

Let \(U:G\to \mathrm U(D)\) be a unitary representation. Regard its matrices as operators on the standard Hilbert space \(\mathbb C^D\), and let \(\mathcal A_U\) be their generated unital star algebra. Every \(\mathcal A_U\)-invariant subspace carries the restriction of \(U\). These are the algebra and invariant sectors used in [ 235 , Section 4.1 ] .

Let \(A\) be a tensor on a finite graph, with physical space \(\mathcal P_v=\mathbb C^d\) at each vertex and independent virtual half-edge space \(\mathcal V_v\) at that vertex. Write \(A_v:\mathcal V_v\to \mathcal P_v\) for its site map and \(T_A=\bigotimes _v A_v\) for their product. The common one-site image space \(\mathcal K_A\) consists of those physical vectors whose one-site slices lie in \(\operatorname {ran}A_v\), for every vertex \(v\) and every choice of physical indices at all other vertices. If every \(A_v\) is injective, choose left inverses \(L_v\) and set \(P_v=A_vL_v\). These retractions need not be orthogonal. This is the independent site-image condition preceding the virtual bond constraints in [ 17 , Section IV.C.1 ] .

For a physical configuration \(s\), write \(f(s_v)=(t_v,r_v,b_v,l_v)\) for the four virtual labels fixed by the primal tensor. Call \(s\) compatible when every site \(v=(x,y)\) satisfies

Its seam phase is

where coordinates are periodic and each product includes all sites on the indicated seam.

On one virtual bond set

For a torus with positive periods \(L,M\), let \(\Psi _{g,h}\) be the contraction of the primal tensor with \(U_h\) on each horizontal bond crossing the seam from the last column to the first, and \(U_g\) on each vertical bond crossing the seam from the last row to the first. Horizontal edges point right and vertical edges point down.

Let \(N\ge 2\). Suppose that, for every \(i\in \mathbb {Z}/N\mathbb {Z}\), the translated length-two local term commutes with its cyclic right neighbor:

Then all translated length-two local terms commute.

Let \(Q:\mathbb C^I\to \mathbb C^O\) be an isometry. On a physical bond, the matrix \(Q\otimes \overline Q\) is an isometry and carries the vector of entries of \(X\) to the vector of entries of \(QXQ^\dagger \). This is the coordinate transport needed to express the irreducible blocks of [ 235 , Section 7 ] in the original regular basis.

The boundary transformation acts on each pair by \((a,b)\mapsto (a,a+b)\) and satisfies \(C^2=1\).

The vector \(\omega \) is invariant under \(\tau ^*(g)\otimes \tau (g)\) for every \(g\). For an operator \(M\) on \(E\), the pairing \(f\otimes e\mapsto f(Me)\) sends \(\omega \) to \(\operatorname{tr}M\) and \((\tau ^*(g)\otimes \tau (h))\, \omega \) to \(\operatorname{tr}[U_{g^{-1}}MU_h]\). If \(\tau \) is semi-regular and \(M=\Delta \), the latter is \(\delta _{g,h}\) [ 235 , Section “Two dimensions: \(G\)-injective PEPS” ] .

Let \(V=\bigoplus _{i\in I}V_i\) be a finite orthogonal direct sum. Any injective relabelling of orthonormal endpoint coordinates defines an isometric inclusion. In particular, the natural inclusion

is an isometry. If a bond vector is obtained by arranging the entries of matrices \(X_i\) in their respective summands, its image under \(E_V\) is the vector of entries of \(\bigoplus _i X_i\).

If \(A\) is \(G\)-injective for a representation \(\rho \) of a finite group on a finite-dimensional virtual space, then \(\rho (g)A^i\rho (g)^{-1}=A^i\) and there is a linear map \(L_A\) from physical vectors to virtual operators such that

This is [ 235 , Definition 4.2 and Lemma 4.7 ] with the left inverse extended by group averaging.

Let \(L_A\) be any linear map and let \(L_B\mathcal{P}(B)=\sigma \). The map \(L_C|j,k\rangle =|G|\, (L_A|k\rangle )\, \Delta \, (L_B|j\rangle )\) of the concatenation \(C^{jk}=B^jA^k\) sends the vector \((\operatorname{tr}[B^jY_k])_{j,k}\) to \(|G|\sum _k(L_A|k\rangle )\, \Delta \, \sigma (Y_k)\), for every family of operators \(Y_k\).

Let \(K\ge 1\). A vector on \(K+2\) sites whose restrictions to the first and to the last \(K+1\) sites lie in \(\{ \sum _{\sigma }\operatorname{tr}[A^{\sigma _1}\cdots A^{\sigma _{K+1}}X]|\sigma \rangle \mid X\} \) lies in the corresponding space on \(K+2\) sites. Consequently, if \(2\le L\le N\), every vector of \(\mathcal G_{N,L}(A)\) has the form \(\sum _{\sigma }\operatorname{tr}[A^{\sigma _1}\cdots A^{\sigma _N}X]|\sigma \rangle \) [ 235 , Section “Parent Hamiltonians” ] .

If \(U_gC^iU_g^{-1}=C^i\) for all \(i\) and \(g\), then \(\operatorname{tr}[C^i\sigma (X)]=\operatorname{tr}[C^iX]\) for every operator \(X\): an operator closing an invariant tensor may be replaced by its twirl without changing the state. Moreover \(\sigma (YZ)=\sigma (Y)Z\) whenever \(Z\) commutes with every \(U_g\), and the concatenation of two invariant tensors is invariant [ 235 , Section “Parent Hamiltonians” ] .

Let \(I\) be a finite set, \(R\) a semiring, \(A\) an \(I\times I\) matrix over \(R\), and \(\pi \) a permutation of \(I\). If \(A_{\pi i,\pi j}=A_{i,j}\) for every \(i,j\), then \(AP_\pi =P_\pi A\), where \(P_\pi \) is the permutation matrix. This elementary identity supplies the permutation commutation in the coordinate symmetry arguments of [ 235 , charge-pair creation, lines 2505–2558 ] .

Let \(E\) and \(Q\) be finite sets, let \(F\) be a rectangular physical matrix, and suppose \(Fx_{e,q}=y_{e,q}\) for every \(e\in E\) and \(q\in Q\). For arbitrary complex weights \(w_q\),

If \(T\iota =F\iota \) for a bond inclusion \(\iota \), then \(T^{\otimes E}\) and \(F^{\otimes E}\) agree on the range of \(\iota ^{\otimes E}\).

For any finite family of positive integers \(d_i\), the single-copy index space has cardinality \(\sum _i d_i\), and

Prescribe internal bond operators in a finite region \(R\) and retain one common assignment \(u\) on exterior and crossing bonds. Altering the internal operators leaves every exterior site tensor unchanged. If the original sites are invariant under simultaneous regular translation, every tree-cycle assignment \(\omega \) satisfies

where \(\mathcal B_u\) is the native boundary transport for the identity vertex gauge. It depends on \(u\) but not on \(\omega \). Here a graph on \(R\) specifies which internal bonds carry the cycle assignment; a spanning-tree hypothesis is unnecessary for these identities. These are the shared contraction facts in the operations of [ 235 , Theorems 6.16–6.17 ] .

For the left-regular representation \(|G|\, \Delta =\mathbb {1}\). Hence the left inverse of a concatenation \(C^{ij}=A^iB^j\) of matrix product tensors built from left inverses \(L_A\) and \(L_B\) is \(L_C|i,j\rangle =(L_B|j\rangle )(L_A|i\rangle )\), and if the contracted link of two tensors in two dimensions carries the left-regular representation, the left inverse of the contracted tensor applies \(L_A\otimes L_B\) and contracts the two ends of the link directly [ 235 , Lemma 6.2 ] .

An operator on the regional half-edge space which commutes with every independent vertex translation commutes with the product of the local regular invariant projectors.

Consider the actual canonical regular contraction on any finite graph region \(R\). Choose group elements \(r_e\) on its bonds, with \(r_e=1\) on every crossing bond. Simultaneously multiply both exposed half-edge labels of each internal bond \(e\) on the right by \(r_e\). Every canonical regional column remains unchanged. In particular this holds for one internal bond, with both its endpoints in \(R\). The same statement holds in the presence of arbitrary inserted left group multiplications.

Let the physical coefficients be \(a_v\), and suppose that local inverse coefficients \(F_v\) satisfy

where \(\Pi _v\) is the regular averaging projector. Then every original regional ground vector exposed by the product of the \(F_v\) has this internal right invariance. Consequently, every complementary slice of an arbitrary global vector satisfying the original regional parent condition has this invariance after exposure. Such inverse coefficients exist for regular \(G\)-injective tensors.

No \(G\)-isometry, flat-connection reconstruction, or global closure-span representation is assumed. This is a local identity of the contractions appearing in the accessible coordinate argument of [ 235 ] . This auxiliary regular bond identity does not establish global closure spanning; see [ 96 ] .

If every bond carries a diagonal matrix, the network is the sum over one index per bond of the product of the diagonal entries and of the site tensors. In particular, with the identity on every bond it is the contraction of the site tensors, and for \(w,h\ge 3\) and \(A_v=a^{\sigma (v)}\) it is the coefficient of the torus PEPS of \(a\) at \(\sigma \).

Let \(P_v\) and \(Q_v\) be matrices at every site. Replacing the matrix \(O\) of every bond by \(P_{\text{head}}\, O\, Q_{\text{tail}}\) gives the same network as keeping \(O\) and replacing every site tensor \(A_v\) by the tensor that has absorbed \(P_v\) into its incoming legs and \(Q_v\) into its outgoing legs.

Let \(G\) be a finite group and assign to every site \(v\) of the torus two elements \(a_v,b_v\in G\). Call \(f\) from the sites to \(G\) a parallel section if \(f_{v+(1,0)}=a_vf_v\) and \(f_{v+(0,1)}=b_vf_v\) for every \(v\). Parallel sections exist if and only if \(b_{v+(1,0)}a_v=a_{v+(0,1)}b_v\) for every \(v\) and the ordered products \(a_{(w-1,0)}\cdots a_{(0,0)}\) and \(b_{(0,h-1)}\cdots b_{(0,0)}\) are the identity. If they exist, there are exactly \(|G|\) of them, and any two differ by right multiplication with one group element.

Every pair of physical coefficient vectors satisfies

For any complex bond factors \(F_e\), the literal product identity is

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.

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 \((x,y+1)\).

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.

The isotypic projectors \(P_i\) commute with every \(U_g\), are linearly independent, and

If \(U\) is semi-regular, then the operators \(U_g\) are linearly independent, and so are the twirls \(\sigma (U_g)\) of one representative \(g\) of each conjugacy class [ 235 , Section “Parent Hamiltonians” ] .

Let \(\rho \) be a unitary representation of a finite group on a finite-dimensional complex inner product space. If \(S,T\) are invariant irreducible subspaces with distinct characters, then \(S\perp T\). Consequently, a family of invariant irreducible subspaces with pairwise distinct characters is an orthogonal family. This is the orthogonality in the unitary block decomposition of [ 235 , Lemma 4.4 and Section 7 ] .

Let \(x,y\) be orthogonal unit vectors in a finite-dimensional complex Hilbert space. The matrix

is unitary and satisfies \(Sx=y\) and \(Sy=x\). It fixes every vector orthogonal to both \(x\) and \(y\). If a unitary \(U\) fixes \(x\) and \(y\), then \(SU=US\). This is an auxiliary Hilbert-space construction for the coherent flux-pair creation in [ 235 , Theorem 6.17 ] .

The oriented four-leg action of \(U\) is the primal tensor’s parity representation. The one-bond representation is semi-regular.

Put \(q(A)=1-P_{G_2(A)}\) for the canonical two-site parent interaction. On the three-site space \(\mathcal{H}_A\otimes \mathcal{H}_X\otimes \mathcal{H}_B\), with sites \(0,1,2\) labelled \(A,X,B\), the two adjacent translated length-two parent interactions are the two local actions

This identifies the two translated parent interactions with the local actions in [ 13 , Definition D.2 ] .

If \(G\) is abelian, every pair-conjugacy class is a singleton and all pairs commute. A finite abelian group therefore has \(|G|^2\) commuting pair-conjugacy classes.

In a chosen finite region with an actual spanning tree and root, a cycle permutation commuting with simultaneous conjugation has one original-spin unitary for the regular virtual action and \(G\)-isometric site maps. It acts on the coordinates derived from every actual input. Its adjoint implements the inverse permutation, with the same derived tree background and the literal exterior retained. Both statements hold on all outgoing boundary columns and on the actual closed contraction. A controlled two-chord permutation, or its inverse, retains every ordered bond other than its selected second chord. This auxiliary statement uses SCP10, lines 1765–1920 and Theorem 6.16, lines 2271–2305; the chosen tree and cycle permutation remain explicit.

Fix a finite region, an actual spanning tree of its induced graph, and a root. The bond assignment \(u\) determines its normalized tree gauge \(k_u\) and its cycle residual tuple \(\omega _u\). Reconstructing those coordinates on internal edges and retaining the literal exterior recovers \(u\) itself.

Let \(\phi \) be a permutation of the cycle tuples commuting with simultaneous conjugation, and suppose the original site maps are \(G\)-isometric for the regular virtual action. There is one unitary on the original region spins, chosen before every \(u\) and every boundary configuration, sending the actual contraction with assignment \(u\) to that with internal coefficients reconstructed from \(k_u\) and \(\phi (\omega _u)\). Every exterior and crossing operator is unchanged. The identity extension outside the region acts on the actual closed contraction in the same way.

This is an auxiliary consequence of the accessible-coordinate and physical movement constructions of SCP10, local lines 1765–1920 and 2271–2305. The tree and the conjugation-equivariant permutation remain explicit. An occupied-to-vacant native plaquette interpretation requires its actual holonomy calculation; the full four-endpoint braid is not asserted here. See [ 96 ] .

Assume the regular virtual action and original \(G\)-isometric site tensors. For each actual translated horizontal block there is one six-spin unitary, chosen before every actual bond assignment and every outgoing boundary configuration. It multiplies the derived middle cycle residual on the left by the derived left residual and reconstructs with the original tree gauge. It acts on both the actual open contraction and the actual closed state, retaining every ordered bond other than the middle chord, including all crossing and exterior bonds. If the right plaquette is vacant, its new middle ordered coefficient is \(k_{\mathrm{head}}\omega _{\mathrm{left}} k_{\mathrm{tail}}^{-1}\). Equivalently, its directed upward transport becomes \(T L B^{-1}\), where \(T\), \(L\), and \(B\) are the original top-left rightward, left upward, and bottom-left rightward transports, respectively. The finite-torus scope is the same as above. This auxiliary operation uses actual derived coordinates; it does not assert the complete prescribed braid [ 96 ] .

On a finite torus of width at least four and height at least three, take the actual translated three-by-two block with its derived spanning path. For arbitrary ordered bond operators, the right plaquette is vacant exactly when the middle upward bond has trivial residual in the root-normalized tree gauge. Both periodic seams are included. This is an auxiliary finite-torus criterion for SCP10, Theorem 6.16, lines 2271–2305; the unrestricted geometry and the complete prescribed braid remain separate [ 96 ] .

On a finite torus with both periods at least four, choose any position and axis. The actual translated six-site tile supplies its spanning path, root and two chords. One original-spin unitary is chosen before every actual bond assignment, direction, boundary configuration and target-vacancy proof. Its forward action and adjoint action implement the actual cycle permutation and its inverse. If the target plaquette is vacant, these operations are exactly the literal shared-bond updates

Here \(B,V,T,L\) are the original bottom rightward, right upward, top rightward and left upward transports of the source plaquette. Every other ordered coefficient, including each exterior and crossing bond, is retained. Both periodic seams are included, and the ordered coefficient is inverted precisely when the native positive step reverses the sorted edge. The action holds on the actual open contraction and closed state. This is a finite-torus specialization of the elementary physical movement in SCP10, Theorem 6.16, lines 2271–2305, not the complete four-endpoint braid [ 96 ] .

Let a five-edge spanning path have successive vertices \(v_0,\ldots ,v_5\), with remaining chords \(v_0v_5\) and \(v_1v_4\) having the same ordered orientation. Write \(D_{ij}\) for the original transport from \(v_i\) to \(v_j\). If the actual loop \(v_1v_2v_3v_4v_1\) is vacant, the forward operation replaces \(D_{14}\) by \(D_{45}^{-1}D_{05}D_{01}^{-1}\). If the actual loop \(v_0v_1v_4v_5v_0\) is vacant, the inverse operation replaces \(D_{14}\) by \(D_{34}D_{23}D_{12}\). No gauge, residual, coefficient or state equality is assumed. These auxiliary calculations underlie SCP10, Theorem 6.16, lines 2271–2305.

Let \(n{\gt}0\) be an integer, let \(z\in \mathbb Z/n\mathbb Z\), and let \(\bar z\in \{ 0,\ldots ,n-1\} \) be its standard representative. If \(x\in \mathbb R\) has the same image as \(\bar z\) in \(\mathbb R/n\mathbb Z\), there is an integer \(j\) with \(j=x\) and \(j\equiv z\pmod n\).

For any real period \(p\), let \(f:[0,1]\to \mathbb R\) be continuous. If its image in the additive circle satisfies \([f(t)]=[f(0)]+[td]\) for every \(t\in [0,1]\), then \(f(t)=f(0)+td\) throughout the interval. These are auxiliary arithmetic and covering-space identities used for the cell and edge paths of the block domains in [ 235 , §6.3 ] .

Every ambient \(D_R\) commutes with all shared right translations and all left vertex translations, hence with every \(B_e\) and \(A_v\). The matrices \(D_R,D_S\) commute for arbitrary \(R,S\), as do \(A_v,A_w\) and \(A_v,B_e\). Thus \(C_iC_j=C_jC_i\) for all constraint indices. Any two finite ordered products of these constraints commute. In particular, \(Q_RQ_S=Q_SQ_R\) for arbitrary regions \(R,S\). This is the regular virtual-coordinate calculation in the proof of [ 235 , Theorem 6.12 ] .

The regional matrices \(B_{R,e}\) and \(D_R\) are self-adjoint idempotents. On the ambient space, every \(C_i\) is an orthogonal projector, as is every finite ordered product \(C_{i_1}\cdots C_{i_n}\), allowing repetitions and the empty product. Its fixed space is \(\bigcap _{j=1}^n\ker (C_{i_j}-I)\). In particular, \(Q_R\) is an orthogonal projector. It fixes exactly the vectors fixed by every \(A_v\) with \(v\in R\), every \(B_e\) with \(e\subseteq R\), and \(D_R\).

Let \(U\) be a matrix representation of a finite group and let \(P\) be the normalized average of its four-leg action. The corresponding site map is \(G\)-isometric, with factor one on the invariant virtual subspace: \(\langle Px,Py\rangle =\langle x,y\rangle \) for invariant \(x,y\). This is an auxiliary projector consequence of the isometry condition in [ 235 , Definition 6.1 ] .

Let \(D\) be a Hermitian idempotent on a finite-dimensional complex space. Then \(D\) and \(I-D\) are positive orthogonal projectors, their product in either order is zero, and their sum is \(I\). This auxiliary identity supplies the two outcomes of the interference measurement in [ 235 , Section 6.6 ] , source lines 2605–2615.

A matrix commutes with every \(U(g)\) if and only if it is of the form \(\bigoplus _k I_{m_k}\otimes B_k\), with \(B_k\in M_{d_k}(\mathbb C)\).

For any graph PEPS tensor, its canonical regional interaction is an orthogonal projector with \(h_R=I-\Pi _{\mathcal S_R}\). Its extension to the physical configuration space agrees with the region/complement extension under the natural coordinate identification; in particular \(\widehat h_R=I-\widehat{\Pi _{\mathcal S_R}}\).

Let the site tensors of a finite simple graph be regular \(G\)-isometric. Let \(R\) have a specified spanning tree and root, and let \(e_0,e_1\) be two distinct internal bonds. For every irreducible character \(\chi \) occurring in the regular representation and every \(p\in G\), there is a unitary \(W\) on the original spins in \(R\) such that

Here \(\Psi _0\) is the actual closed contraction without insertions. Consequently \(\Psi _{\chi ,p}\ne 0\) and \(\langle \Psi _{\chi ,p},\Psi _{\chi ,p}\rangle =\langle \Psi _0,\Psi _0\rangle \). The local operation is uniform in every crossing configuration.

This auxiliary finite-region consequence of SCP10, lines 2505–2558, retains the supplied region, spanning tree and two distinct internal bonds. It does not identify the prescribed charge–flux braid, its flux-string background, or the reunion experiment of lines 2560–2615; see [ 96 ] .

Let \(R\) be a region of a finite simple graph, with finite group \(G\) on its bonds. Let \(e_0,e_1\) be two internal bonds, \(u\) an arbitrary bond assignment, and \(\theta \) a fixed crossing configuration. Resolving the label on \(e_1\) by the diagonal indicators \(\mathbf{1}_{\eta _{e_1}=q}\) gives

Both sides are the actual open contractions of the original site tensors. The character weight remains correlated between the two labels. This is the finite-contraction identity underlying the creation of charge pairs in [ 235 , Section 6.6 ] , source lines 2505–2535.

Let \(R\) have connected induced graph and let \(p\) be a specified closed walk based at a vertex of \(R\). Suppose the original local tensors are regular \(G\)-isometric for a finite group \(G\). There is a complete projective measurement on the original physical spins in \(R\), indexed by the conjugacy classes of \(G\) and one unused outcome, such that every actual twisted open-block column \(\psi _{u,\theta }\) has outcome \([\operatorname {Hol}_u(p)]\) with certainty. The operators are Hermitian and positive semidefinite, satisfy \(Q_iQ_j=\delta _{ij}Q_i\), and sum to the identity on the whole physical space. This is an auxiliary closed-walk form of the conjugacy calculation in [ 235 , proof of Theorem 6.15 ] .

Let \(d_i{\gt}0\) and let \(M(a)=\bigoplus _i\sqrt{d_i}\, D_i(a)\) be a family of matrices. Every finite coherent sum of products of these matrices belongs to the product of matching-sector bond spaces. The multiplicity-restoring map sends it to the same coherent sum with bond matrices \(\bigoplus _i D_i(a)\otimes I_{d_i}\). If an isometric coordinate change identifies these repeated blocks with another matrix family, one fixed full-domain product isometry gives the corresponding identity for every choice of coefficients and bond labels, and preserves all inner products. These are auxiliary matrix statements for the construction in Section 7 of [ 235 ] ; an actual contraction formula is still required when applying them to a tensor-network state.

For every group \(G\) and \(g,x\in G\), the conjugacy-class quotient satisfies \(C[xgx^{-1}]=C[g]\). This is the class invariance used in [ 235 , Theorem 6.15 and Section 6.6 ] .

Let \(R\) induce a nonempty connected graph, with \(V_R\) vertices and \(I_R\) internal edges. Suppose the local Gram kernels are \(A_v^\dagger A_v=c_vP_v\). For unnormalized regular bonds and no inserted bond operators, the open-region tensor satisfies

The boundary may be empty. In particular, positive local Gram factors give an isometry on the invariant boundary space after dividing by \(\sqrt{\kappa _R}\), also when the region contains cycles. This is the untwisted connected-region contraction underlying [ 235 , Lemma 6.2 and proof of Theorem 6.9 ] .

Let \(R\) be a nonempty connected finite region whose site tensors are \(G\)-injective for the simultaneous regular action on their incident legs. Then the actual open tensor of \(R\) is \(G\)-injective for the simultaneous regular action on its crossing legs. Internal cycles and an empty boundary are allowed. This is the regular-representation connected-region consequence of [ 235 , Lemma 5.2 and the following observation ] ; the argument does not assume local isometry.

Let every site be regular \(G\)-isometric, and let \(R\) induce a nonempty connected graph. The open-region map \(T_R\) is \(G\)-isometric for simultaneous regular translation on the crossing bonds. Its derived positive factor is

where \(I_R\) is the number of internal edges. This includes blocks with internal cycles and blocks with no crossing bonds. It is the actual finite-region form of the stability and blocking argument in [ 235 , Lemma 6.2 and §6.2 ] , with the positive-factor convention recorded in [ 96 ] .

Consider a finite graph PEPS with regular bonds and a \(G\)-isometric tensor at every site. Assume that both \(R\) and its complement induce nonempty connected graphs, and that \(b{\gt}0\) bonds cross the cut. For the untwisted contraction there is a positive normalization of the actual physical PEPS vector to a unit vector. Its reduced state on \(R\) has rank \(r=|G|^{b-1}\) and a flat spectrum: \(\rho _R^2=r^{-1}\rho _R\). Its von Neumann entropy is

This finite untwisted statement is a restricted consequence of the calculation in [ 235 , Theorem 6.9 ] ; it does not assert the theorem for arbitrary torus closures. The remaining scope is recorded in [ 96 ] .

Assume the actual complement walks satisfy the relative-word identities of (120). Number the \(b=n+1\) crossing bonds and use reference boundary coordinates \(y=(t,d)\) and \(\theta =(a,r)\), where \(d,r\in G^n\). For the actual controlled complementary matrix at the reindexed input \(F_R^{-1}\theta \), put \(x=t a^{-1}m_{f_0}^{-1}\). Its coefficient is

The normalized boundary factor is \(|G|^{-n/2}\mathbf1_{\{ d=r\} }\). Its remaining factor is \(|G|^{n/2-|S|}\mathbf1_{\{ z=x\omega _Sx^{-1}\} }\). The trees, numbering, and control walks may be fixed before an arbitrary finite family of bond assignments. The same factor then separates from every coherent linear combination of their actual complementary matrices; the residual coefficients are combined with the same weights.

This statement retains the virtual reference label \(a\) in the residual coefficient. It is an auxiliary coefficient statement, rather than a conclusion about the full physical bipartite state; for commuting torus closures, the complementary loops of Theorem 26.1.4.75 realize its geometric walk condition.

Retain the same actual complement walk identities, and suppose every actual cycle residual in \(R\) is the identity. Contract the two canonical region matrices, applying the fixed controlled physical permutation only in \(S\). Reindexing their common boundary sum by \(F_R\) preserves the original contraction.

Let \(d_R,d_S\in G^n\) be the relative boundary labels. In the spanning-tree physical coordinates the actual contracted coefficient is

In particular neither reference boundary label nor the common multiplier \(m_{f_0}\) occurs in the result.

Let \(N_R\) be the number of choices of the reference boundary label, internal reference labels, and normalized vertex labels in \(R\). The fixed region ancillary vector is \(\phi _R=N_R^{-1/2}\mathbf1_{\{ z_R=\mathbf1\} }\); it has norm one. For any finite sector family obeying the stated actual residual and walk conditions, and any coefficients \(\lambda _i\), the coherent cut sum has the exact factorization

Both normalized factors are fixed before the sector family and its coefficients; all sector dependence is on the complementary cycle coordinates. These are auxiliary canonical-cut statements associated with [ 235 , Theorem 6.9 ] . The actual geometric conditions and transport to the original physical tensors remain separate steps, as recorded in [ 96 ] .

Suppose that a family of complex matrices \(A^i\) spans a unital C*-subalgebra \(S\subset M_I(\mathbb C)\), with \(I\) finite. There exist a finite group \(G\) and a unitary representation \(U\) whose commutant is \(S\), such that the tensor \(A\) is \(G\)-injective. This gives the passage from the unital version of [ 235 , Theorem 4.1 ] to its Definition 4.2. The unital restriction is recorded in [ 55 ] .

Two edges of an ordered simple graph are equal precisely when their unordered endpoint pairs agree. A coordinate that is constant on the endpoints of one edge distinguishes it from an edge whose endpoint has a different coordinate. Let \(R\) have a chosen tree \(T\). Any bond assignment supported on internal non-tree edges is recovered by restricting it to those edges and extending it by the identity. These are auxiliary facts for the accessible systems and local movement of [ 235 , Theorem 6.16 ] .

In the setting of Definition 26.7.8, suppose every original local tensor is regular \(G\)-isometric. There is a single projective measurement \((Q_C)_C\), together with an additional outcome \(Q_\perp \), on the original physical spins in \(R\). Every operator is Hermitian and positive semidefinite, and the full family satisfies

For arbitrary inserted regular-bond operators \(u_f\), let \(\omega _e(u)\) be the cycle residual obtained from the rooted spanning-tree gauge. Denote the actual twisted open-block column with crossing labels \(\theta \) by \(\psi _{u,\theta }\). Then

Thus every nonzero such column yields the conjugacy class of its cycle residual with certainty. This establishes the conjugacy calculation \(ab^{-1}=xgx^{-1}\) in [ 235 , proof of Theorem 6.15 ] for a connected block with a chosen spanning-tree cycle. Its geometric scope and the native plaquette specialization are recorded in [ 96 ] .

Fix spanning-tree physical coordinates \((y,a,r,z)\) of a finite region and any function \(W:G^C\to G^{\partial R}\) of its cycle labels. The permutation \(\sigma _W(y,a,r,z)=(W(z)^{-1}y,a,r,z)\) is reversible, with inverse \((y,a,r,z)\mapsto (W(z)y,a,r,z)\), and its matrix \(U_W\) on the original half-edge physical basis is unitary. In particular, fixed actual complementary walks define such a matrix by evaluating their cycle words; this construction is independent of all bond assignments.

For the actual canonical block \(M_R^u\), let \(b_u\) be its boundary transport and \(\omega \) its cycle residual tuple. Its transformed coefficients satisfy

The cycle constraint is retained without any assumption on the relation between \(W\) and the crossing transports. This is an algebraic coordinate operation associated with [ 235 , §6.3 ] .

Fix a rooted spanning tree in a finite region. Independent vertex translations \(\lambda _v\) of its actual physical half-edge labels transform the spanning-tree coordinates by

Reconstructing these coordinates gives precisely the original half-edge labels multiplied on the left by \(\lambda _v\) at vertex \(v\). For a fixed family of actual region walks, let \(U\) be the native boundary permutation controlled by their constructed cycle words. This matrix commutes with every independent vertex translation. Their finite average is the actual product local invariant projector

In particular \(U\) maps the local invariant support \(\operatorname {range}P\) into itself.

If the original local physical maps \(A_v\) are regular \(G\)-isometric, there exists a unitary \(W\) on their physical region such that

The fixed words and their physical implementation are chosen without closure labels. No equivariance premise on an arbitrary control function or surjectivity of the canonical projector is required. These are auxiliary physical support identities for the accessible complement calculation in [ 235 , §6.2–6.3 ] , local source lines 1765–1920 and 1935–1990.

Let \(S\) be the complement of \(R\), and identify their crossing bonds by the same native edges. Choose a region gauge \(k_R\) and a rooted spanning tree in \(S\), with normalized gauge \(k_S\) and cycle residuals \(\omega _S\). Let \(F_R,F_S\) be the actual boundary transports and set \(M=F_S\circ F_R^{-1}\) under this crossing identification. Write \(m_f=M(\mathbf1)_f\), so that \(M(\theta )_f=m_f\theta _f\). Fix a reference crossing bond \(f_0\) and an actual walk \(p_f\) entirely in the induced graph on \(S\) for each crossing bond \(f\). Assume

Define \(W(z)_f=\operatorname {ev}_zW_T(p_f)\). With the complementary boundary input reread from \(F_R^{-1}\theta \), the actual coefficients satisfy

Here \(\theta _f\) denotes the label on the corresponding crossing bond of \(R\). This is an auxiliary conditional statement in the complement argument of [ 235 , §6.3 ] . Constructing the required complement walks with (120) remains the geometric step described in [ 96 ] .

On every irreducible subrepresentation of \(U\) with character \(\chi _i\), \(\Delta \) acts as the scalar \(\frac{d_i}{m_i|G|}\). Thus, in a basis in which \(U_g\cong \bigoplus _iD^i(g)\otimes \mathbb {1}_{m_i}\),

[ 235 , Section “MPS: the \(G\)-injective case” ] .

For every \(g\in G\),

For a unitary representation \(U_{h^{-1}}=U_h^\dagger \) [ 235 , Section “MPS: the \(G\)-injective case” ] .

The operator in (304) is the two-sided inverse of \(\Delta \). On a summand with character \(\chi \), it acts by \(|G|m_\chi /d_\chi \). Both \(\Delta \) and its inverse commute with every group operator. These assertions hold without semi-regularity or unitarity, for every finite-dimensional complex representation of a finite group.

Let \(\rho \) be a unitary finite-dimensional representation of a finite group. Every invariant subrepresentation is unitary. Then \(\Delta \) is positive, and its matrix in every orthonormal basis is positive definite. For unitary representations, the character projector of any irreducible summand is self-adjoint and positive. More generally, every nonnegative weighted sum of the occurring character projectors is positive. This supplies the positive bond weights appearing in [ 235 , Lemma 4.4 ] .

Grouping the bond-end labels by their anchoring vertex identifies bond-end assignments with maps \(X\to V^4\), and the labelled contraction equals the native four-leg contraction. Hence, for an arbitrary region, writing \(c^{\rm nat}(\theta )\) and \(\Delta ^{\rm nat}\) for the open coefficient and the delta-completed tensors read through the top, right, down, left ordering, the native contraction of \(\Delta ^{\rm nat}\) with the bond matrices \(O^{\rm int}\) equals \(c^{\rm nat}(\theta )\). The collar is nonempty and embedded. Every swept site and its four neighbors lie in it; hence each noninternal bond has both endpoints outside the swept support. For an empty virtual alphabet the open coefficient of a nonempty region is zero, since the region contains an incidence.

For a dual path \(p\), let \(O^p_e\) be \(U_{u_h^p(v)}\) on the horizontal bond at \(v\) and \(U_{u_v^p(v)}\) on the vertical bond at \(v\), as in Definition 26.9.211. Let \(p,q\) be two labelled unit-step paths with the same lifted endpoints in the embedded patch. There is \(k:X\to G\) with \(k=1\) off \(R\) such that the matrices \(O^{p,\rm int}\), \(O^{q,\rm int}\) of Theorem 26.9.228 for the collar \(Y\) satisfy

on every bond. If the site tensors are virtually invariant on \(R\), then \(p\) and \(q\) have equal open coefficients on \(Y\) for every fixed physical configuration on \(Y\). The group and its matrix representation are arbitrary. Equality persists after contraction against any joint boundary tensor, without a factorization hypothesis or a nonempty virtual-alphabet assumption.

Suppose \(p\sim _R q\) and every \(A_v\) with \(v\in R\) is virtually \(G\)-invariant. Then \(\Psi _p^E(\sigma )=\Psi _q^E(\sigma )\) for every physical configuration and every unchanged exterior assignment \(E\). The site tensors and their physical spaces may vary with \(v\); exterior tensors are arbitrary. This holds for all positive periods, including one and two, and for nonabelian \(G\). It is the finite generated-deformation form of [ 235 , Lemma 6.14 ] .

Let \(A\) be regular \(G\)-isometric. On \(S_z\) there is one complete projective measurement \((Q_C^{\rm ccw})_C\), with an additional outcome \(Q_\perp ^{\rm ccw}\), such that for every native assignment \(u\)

All projectors are Hermitian and positive, distinct outcomes are orthogonal, and their sum is the identity. Inversion permutes conjugacy classes by \(\iota ([s])=[s^{-1}]\). Relabelling \(Q_C^{\rm cw}=Q_{\iota (C)}^{\rm ccw}\) and leaving the extra outcome unchanged gives, simultaneously for every \(g\) and every finite dual path \(p:a\to b\),

Thus the source convention detects \([g]\) and \([g^{-1}]\) at the distinct endpoints of every such state. The two classes need not coincide. These are the native regular torus instances of [ 235 , Theorem 6.15 ] .

If \(p\sim _R q\), then \(d(p)=d(q)\). Thus equal torus endpoints alone do not imply that two paths are related by these moves.

Under the hypotheses of Theorem 26.9.220, let \(p:a\to b\) be any finite dual path. Every positive parent interaction at a plaquette \(z\notin \{ a,b\} \) annihilates the actual inserted state \(\Psi _p\), including the canonical parent projector. If \(a=b\), every canonical plaquette parent term annihilates \(\Psi _p\).

Every finite dual path \(p:a\to b\) satisfies

If \(a\ne b\), the clockwise labels are \(g\) at \(a\), \(g^{-1}\) at \(b\), and identity elsewhere, as in [ 235 , Definition 6.13 ] . A closed dual path has identity plaquette holonomy everywhere, including when it winds around the torus.

For arbitrary native bond matrices \((O_h,O_v)\) and \(k:X\to G\), make the substitutions

The closed contraction is unchanged if each site tensor absorbs the induced four-leg action of its \(k(v)\). It suffices to assume virtual \(G\)-invariance only at sites where \(k(v)\ne 1\). If \(k=1\) outside \(R\), all bonds with both endpoints outside \(R\) and all exterior site tensors remain unchanged.

Let \(w,h\geq 3\), and let \(A\) be a homogeneous virtually \(G\)-invariant tensor for an arbitrary finite-dimensional representation \(U\). If \(H_u(z)=1\), every complementary physical slice of the inserted state lies in the original plaquette PEPS range \(\mathcal G_{S_z}\). Consequently, for every positive local interaction \(P_z\) with kernel \(\mathcal G_{S_z}\),

This includes the canonical orthogonal projector onto \(\mathcal G_{S_z}^{\perp }\). In particular, zero signed curl of native powers implies (496).

If \(A\) is regular \(G\)-isometric, every actual native dual-string state \(\Psi _p\) is nonzero. Its coefficient matrix \(M_{u^p,S}\) across every physical cut \(S\) is also nonzero.

Suppose \(w,h\geq 3\). Passing from native bonds to the ordered edges of the torus graph gives rightward transport \(u_h(v)\) and upward transport \(u_v(v)^{-1}\), including on both seams. Hence the counterclockwise plaquette walk has holonomy \(H_u(z)^{-1}\) and its reverse has holonomy \(H_u(z)\). For a dual path, the counterclockwise exponent is \(\delta _b(z)-\delta _a(z)\).

If \(p\sim _R q\), there is an integer function \(f\) vanishing outside \(R\) such that \(n(q)=n(p)+\nabla f\). A square move has \(f=\delta _{(x+1,y+1)}\). Retracing any finite path contracts by backtrack removal, and the four-sided dual square loop contracts by one square move followed by backtrack removal.

Fix a dual plaquette \((x_0,y_0)\) and integers \(0\leq W{\lt}w\), \(0\leq H{\lt}h\). Embed the closed rectangle \(D=\{ 0,\ldots ,W\} \times \{ 0,\ldots ,H\} \) by \(\kappa (i,j)=(x_0+i,y_0+j)\) modulo \((w,h)\), and put \(R=\{ \kappa (i+1,j+1):0\leq i{\lt}W,\ 0\leq j{\lt}H\} \). Every two labelled unit-step paths in \(D\) with the same lifted endpoints have projections related by \(\sim _R\). This is the rectangular bulk specialization of [ 235 , Lemma 6.14 ] .

For every permitted edge \(e:a\to b\), \(C_a e\sim _R C_b\). More generally, \(ce\sim _R d\) implies \(de^{-1}\sim _R c\), and \(cp\sim _R cq\) implies \(p\sim _R q\).

Let \(p,q\) be two paths as above, let \(U\) be a group representation, and fix \(g\) in the group. Assume virtual \(G\)-invariance at every primal site of \(R\). Insert \(U_g\) and \(U_{g^{-1}}\) according to the signed crossings of the paths on bonds incident to \(R\), and keep arbitrary fixed matrices on all remaining bonds. The two actual native contractions are equal for each fixed physical configuration. Site tensors outside \(R\) are unrestricted.

Let \(x\in \mathbb Z/n\) and \(W{\lt}n\). If \(0\leq i,j\leq W\) and \(x+i=x+j\) in \(\mathbb Z/n\), then \(i=j\). Consequently, if \(W{\lt}w\) and \(H{\lt}h\), then \(\kappa \) is injective on the closed lifted rectangle \(\{ 0,\ldots ,W\} \times \{ 0,\ldots ,H\} \).

Let \(\varphi :\Gamma \simeq \Gamma '\) be a graph isomorphism between linearly ordered vertex sets, and let \(T\) be a graph on the vertices of \(\Gamma \). For an ordered edge \(e=(v,w)\) of \(\Gamma \), sort the mapped endpoints to obtain \(\varphi e\). Its endpoints are adjacent in the transported graph \(T\circ \varphi ^{-1}\) if and only if \(v,w\) are adjacent in \(T\). In particular, a non-tree bond remains outside the transported tree. This elementary assertion is used in the finite blocking and flux geometries of [ 235 ] .

Let \(K\) be compact and locally path connected, let \(Y\) be Hausdorff, and let \(f_i:K\to Y\) be a finite family of continuous maps. Then \(\bigcup _i f_i(K)\) is locally path connected.

Let \(S\subset M_I(\mathbb C)\) be a unital C*-subalgebra, with \(I\) finite. There exist a finite group \(G\) and a unitary representation \(U\) such that \(S=\{ A\in M_I(\mathbb C): AU(g)=U(g)A\text{ for every }g\in G\} \). This is the unital version of [ 235 , Theorem 4.1 ] . The source’s definition of C*-algebra does not require the identity; the printed nonunital assertion fails because every commutant contains the identity. See [ 55 ] .

Let \(I\) be a finite set, let \(J_i\) be finite sets indexed by \(i\in I\), and let \(P_i\) be a condition on \(J_i\). In every commutative semiring,

Thus multiplying the separate numbers of admissible choices counts the admissible simultaneous choices.

Consider the native four-endpoint string assignment on a torus with \(w\ge 7\) and \(H\ge 6\). The upward movement tile based at \(v+(1,1)\) has a common reconstruction gauge equal to \(1\) on its lower row and \(g\) on its other two rows. Its lower canonical insertion is \(h^{-1}\). Extending this insertion to the middle chord changes only the bond based at \(v+(1,2)\), whose rightward transport becomes \(gh^{-1}g^{-1}\). The lower bond retains the continuing partner string, and all other ordered operators are retained. For an original regular \(G\)-isometric site tensor there exists one unitary on these six original spins, chosen before \(g,h\) and every boundary label, which maps all actual input columns to the output columns. Its extension by the identity maps the actual closed input state to the output state. This is a direct elementary crossing auxiliary to [ 235 , Theorem 6.16 and Section 6.6 ] ; it is not identified with the entire prescribed braid.

If \(\| \phi \| =1\), then \(J_\phi \) is an isometry and \(\rho _\phi \) is a positive density of trace one and rank \(|D|\), satisfying \(\rho _\phi ^2=|D|^{-1}\rho _\phi \). Moreover,

These identities persist under bijective changes of physical coordinates. A unitary acting on the complementary indices leaves the reduced matrix unchanged. If a nonzero coefficient matrix has reduced matrix \(a\rho _\phi \), its trace is positive and trace normalization gives exactly \(\rho _\phi \).

Let \(\rho \) be a finite-dimensional positive semidefinite matrix of trace one. If \(\rho ^2=r^{-1}\rho \) for a positive real number \(r\), then \(S(\rho )=\log r\). This is the spectral step used in the proof of [ 235 , Theorem 6.9 ] .

Let \(\rho \geq 0\) have trace one and satisfy \(\rho ^2=r^{-1}\rho \), with \(r{\gt}0\). For every \(\alpha {\gt}0\), \(\operatorname {tr}(\rho ^\alpha )=r^{1-\alpha }\) and \(S_\alpha (\rho )=\log r\). If \(\operatorname {rank}\rho =r\), the same entropy formula also holds at order zero.

Let \(G\) be finite and let the torus periods be at least three. Suppose the site coefficients are invariant under simultaneous left translation of their incident group labels. Every flat inserted-bond network has the same physical state as a network with two commuting closure seams. For a single four-leg tensor, this is the actual closure vector \(\psi _{g,h}\) of [ 235 , Section 5 ] . Neither injectivity nor isometry is needed for this equality.

Suppose the regular virtual space has an orthonormal basis indexed by \(\{ (i,a,u):1\le a,u\le d_i\} \). Then \(|G|=\sum _i d_i^2\). If every \(d_i\) is positive, the dimension of the single-copy virtual space is

Moreover, \(D_{\mathrm{single}}{\lt}|G|\) if and only if some \(d_i{\gt}1\). These are the dimension formulas in [ 235 , Section 7 ] .

In an actual regular Fourier decomposition \(L_g\cong \bigoplus _iD_i(g)\otimes I_{d_i}\), with every \(D_i\) irreducible and \(d_i{\gt}0\), some \(d_i{\gt}1\) if and only if \(G\) is noncommutative. Equivalently, \(\sum _i d_i{\lt}|G|\) if and only if \(G\) is noncommutative. This is a consequence of the representation construction in [ 235 , Section 7 ] .

Let \(\nu _i\) and \(\mu _i\) be finite label spaces, with every \(\mu _i\) nonempty. Write \(E\) for matching-sector inclusion and \(F\) for the full endpoint multiplicity-restoring map. There exists an isometry \(T\) from the entire physical input endpoint-pair space to the entire output endpoint-pair space such that

Hence \(T\psi =F\psi \) for every \(\psi \in \operatorname {range}E\), and in particular

for every family of block matrices \(X_i\). Thus the supported transformation in [ 235 , Section 7 ] may be realized by an isometry on the full physical bond space.

Put \(V'=\bigoplus _i(V_i\otimes \mathbb C^{M_i})\) and define

Then \(F^\dagger F=E_VE_V^\dagger \), so \(F\) preserves inner products on the matching-sector subspace. More precisely, whenever \(E_VE_V^\dagger \phi =\phi \),

It sends the full endpoint vector of \(\bigoplus _i\sqrt{m_i}X_i\) to that of \(\bigoplus _i(X_i\otimes I_{M_i})\). For \(m_i=\dim V_i\), this is the local physical bond transformation used in [ 235 , Section 7 ] .

Let a finite group \(G\) act on a virtual space \(\mathcal V\). Suppose \(T,S:\mathcal V\to \mathcal P\) are \(G\)-injective for that same representation and their physical ranges are finite-dimensional. Then an invertible physical map \(F:\mathcal P\to \mathcal P\) satisfies \(FT=S\). This is a local consequence of [ 235 , Definition 5.1 ] .

Let \(U\) be a semi-regular representation of a finite group \(G\), and let \(A\) be \(G\)-injective for its four-leg virtual representation. On a nonempty rectangular torus, let \(\mathcal V_A\) be the span of the vectors \(|\mathcal M(A\mid C)\rangle \) indexed by commuting simultaneous-conjugacy classes \(C\). Then

This is the dimension of the commuting closure span appearing in [ 235 , Theorem 5.9 ] .

Let \(A\) be a tensor with virtual system \(W_A\otimes E^*\) and group action \(\sigma _A\otimes \tau ^*\), where \(E^*\) is its right leg and \(W_A\) collects its left, up and down legs, and let \(B\) be a tensor with virtual system \(E\otimes W_B\) and group action \(\tau \otimes \sigma _B\), where \(E\) is its left leg and \(W_B\) collects its up, right and down legs. Let \(\tau \) be semi-regular, and let \(C\) be the tensor obtained by contracting the right leg of \(A\) with the left leg of \(B\), with the up, down and physical indices blocked:

If \(A\) and \(B\) are \(G\)-injective, then \(C\) is \(G\)-injective for \(\sigma _A\otimes \sigma _B\). Precisely, if \(L_A\mathcal{P}(A)=\Pi _{\sigma _A\otimes \tau ^*}\) and \(L_B\mathcal{P}(B)=\Pi _{\tau \otimes \sigma _B}\), then the map \(L_C\) that applies \(L_A\otimes L_B\) and contracts the two inner legs through \(\Delta \), \((x\otimes f)\otimes (e\otimes y)\mapsto |G|\, f(\Delta e)\, x\otimes y\), satisfies \(L_C\mathcal{P}(C)=\Pi _{\sigma _A\otimes \sigma _B}\). If moreover \(\sigma _A\) and \(\sigma _B\) are semi-regular, then so is \(\sigma _A\otimes \sigma _B\) [ 235 , Lemma 5.2 ] .

Let \(T:\mathcal V\to \mathcal P\) and \(S:\mathcal V\to \mathcal Q\) be \(G\)-injective for the same representation of a finite group. If \(\mathcal P\) and \(\mathcal Q\) are finite-dimensional and have equal dimensions, there is a linear isomorphism \(F:\mathcal P\to \mathcal Q\) satisfying \(FT=S\).

Let \(G\) be finite, let \(\rho \) be a complex representation of \(G\), and let \(T\) be \(G\)-injective. Write \(\Pi =|G|^{-1}\sum _{g\in G}\rho (g)\) for the invariant projector. For virtual vectors \(x,y\), one has \(Tx=Ty\) if and only if \(\Pi x=\Pi y\). This is a consequence of [ 235 , Definition 5.1 ] .

Let \(G\) be a finite group acting linearly on a complex virtual space \(\mathcal V\), and let \(P_G=|G|^{-1}\sum _{g\in G}U_g\) be the projector onto \(\mathcal V^G\). If \(T:\mathcal V\to \mathcal P\) is \(G\)-injective, then

In finite dimension, \(\dim \operatorname {im}T=\dim \mathcal V^G\). These are local consequences of [ 235 , Definition 5.1 ] . The virtual projector can be inserted before the physical map:

\(=\)

A tensor is \(G\)-injective if and only if \(\mathcal{P}(A)U_g=\mathcal{P}(A)\) for every \(g\) and \(\mathcal{P}(A)\) has a left inverse \(L\) with \(L\, \mathcal{P}(A)=\Pi _U\) [ 235 , Section “Two dimensions: \(G\)-injective PEPS” ] .

Let a tensor have virtual system \(W\otimes E^*\otimes E\) with group action \(\sigma \otimes \tau ^*\otimes \tau \), as for the up leg of \(A\) and the down leg of \(B\) in the block \(C\) above. Contracting the legs \(E^*\) and \(E\) gives the map \(x\mapsto \sum _i\mathcal{P}(x\otimes e_i^*\otimes e_i)\). If \(L\mathcal{P}=\Pi _{\sigma \otimes \tau ^*\otimes \tau }\) and \(\operatorname{tr}M\ne 0\), then contracting the corresponding legs of \(L\) through \(M\) and dividing by \(\operatorname{tr}M\) gives a left inverse of the contracted map up to \(\Pi _\sigma \). Consequently, if the tensor is \(G\)-injective and \(\tau \) is semi-regular, the contracted map is \(G\)-injective for \(\sigma \), with \(M=\Delta \), whose trace is \(1\) [ 235 , Observation 5.3 ] .

Let a finite group \(G\) act linearly on a finite-dimensional complex Hilbert space \(\mathcal V\), and let \(T:\mathcal V\to \mathcal P\) be \(G\)-injective, where \(\mathcal P\) is a finite-dimensional complex Hilbert space. There exists an invertible physical map \(F:\mathcal P\to \mathcal P\) such that \(FT\) is invariant and

In particular, for the regular virtual representation this is a normalized \(G\)-isometric tensor, with normalization factor one. This local consequence of [ 235 , Definition 5.1 and the definition of \(G\)-isometric PEPS ] concerns an invertible physical transformation; it asserts no equality of closed states.

Let \(T:\mathcal V\to \mathcal P\) be \(G\)-injective and let \(F:\mathcal P\to \mathcal Q\) be linear. The composite \(FT\) is invariant under the original virtual representation. If \(F|_{\operatorname {im}T}\) is injective, then \(FT\) is \(G\)-injective. For a finite group \(G\), the converse holds: \(FT\) is \(G\)-injective if and only if \(\operatorname {im}T\cap \ker F=\{ 0\} \). In particular an invertible physical change of coordinates preserves \(G\)-injectivity in both directions. These are local consequences of [ 235 , Definition 5.1 ] .

Let \(T:\mathcal V\to \mathcal P\) and \(S:\mathcal V\to \mathcal Q\) be \(G\)-injective for the same virtual representation. There is a unique linear map \(F:\operatorname {im}T\to \operatorname {im}S\) such that \(F(Tx)=Sx\) for every \(x\in \mathcal V\), and this map is an isomorphism.

An injective physical map preserves \(G\)-injectivity of a site tensor, and an isometry preserves its \(G\)-isometry. Both statements hold for four virtual legs and for arbitrary incident-leg sets. Consequently invertible physical maps at every vertex preserve regular \(G\)-injectivity of a complete finite-graph site family; physical isometries preserve regular \(G\)-isometry. In finite bond coordinates the resulting tensor family is exactly the physical deformation of the original group-labelled family.

Let \(T:\mathcal V\to \mathcal P\) and \(S:\mathcal W\to \mathcal P\) be injective on the invariant spaces of possibly different finite groups, and suppose their physical ranges agree. Applying the same injective physical map \(F:\mathcal P\to \mathcal Q\) to both tensors leaves their virtual comparison unchanged.

Suppose a finite group acts on \(\mathcal V\) through \(\rho \), the middle and right blocks \(B,C\) are \(G\)-injective, and the left block satisfies \(\rho (g)A^i\rho (g)^{-1}=A^i\). Then

In particular this holds when all three blocks are \(G\)-injective. The endpoints may have different tensors and physical alphabets. This is the matrix-valued strip argument used in [ 235 , Theorems 4.8 and 5.4 ] ; identifying an arbitrary graph’s regional open contractions with these strip maps is a separate geometric step.

Let a finite group act on \(\mathcal V\) through a common representation \(\rho \). Suppose every block is \(G\)-injective and \(n\ge 2\). Then

No equality of local intersections is a hypothesis. The blocks and their physical alphabets may vary along the strip. This is the open-boundary intersection iteration in [ 235 , Theorems 4.11 and 5.4 ] . A PEPS graph application additionally requires identifying its regional contractions with these operator-valued strip tensors.

For the trivial representation, \(G\)-injectivity is injectivity of \(\mathcal{P}(A)\), that is, linear independence of the physical vectors \(A^{\bullet }_{\alpha \beta \gamma \delta }\) indexed by the virtual configurations.

Under the hypotheses of Definition 26.3.1, \(\Phi \) is the unique linear map \(F:\mathcal V^G\to \mathcal W^H\) satisfying \(S(Fx)=Tx\) for every \(x\in \mathcal V^G\). Reversing \(T\) and \(S\) gives \(\Phi ^{-1}\). In finite dimension, \(\dim \mathcal V^G=\dim \mathcal W^H\). This is a local consequence of the invariant-space injectivity in [ 235 , Definition 5.1 ] .

Let \(T\) be \(G\)-isometric for a unitary representation \(\rho \) of a finite group, and put \(P_\rho =|G|^{-1}\sum _g\rho (g)\). For some \(c{\gt}0\), the map \(J=c^{-1/2}T^*\) is an isometric equivalence \(\operatorname {ran}T\cong \operatorname {ran}P_\rho \), with \(JT=\sqrt c\, P_\rho \). No surjectivity onto either ambient coordinate space is required. This is the support identification used in [ 235 , Observations 6.4 and 6.6 ] .

Let \(\sigma _{A}\) and \(\sigma _{B}\) be unitary representations on the remaining legs of \(A\) and \(B\), and let the contracted link carry the left-regular representation. If \(A\) is \(G\)-isometric for \(\sigma _{A}\otimes L\) with constant \(c_{A}\) and \(B\) is \(G\)-isometric for \(L\otimes \sigma _{B}\) with constant \(c_{B}\), then the contracted tensor \(C\) is \(G\)-isometric for \(\sigma _{A}\otimes \sigma _{B}\) with constant \(c_{A}c_{B}\), and its left inverse is \((c_{A}c_{B})^{-1}\mathcal{P}(C)^{\dagger }\), the contraction of the adjoints of the two factors. In the source every leg carries the left-regular representation or its contragredient, so the unitarity of \(\sigma _{A}\) and \(\sigma _{B}\) follows from its hypotheses [ 235 , Lemma 6.2 ] , [ 96 ] . In particular, contracting the right leg of one four-leg regular PEPS tensor with the left leg of another preserves \(G\)-isometry with constant \(c_Ac_B\), leaving the other six virtual legs open.

Let \(T\) be \(G\)-isometric with normalization factor \(c{\gt}0\). A physical linear isometry \(F\) makes \(FT\) \(G\)-isometric with the same factor \(c\). This includes rectangular isometries. An invertible physical isometry preserves \(G\)-isometry in both directions.

Let \(A,B\) be G-isometric physical maps for the same finite-dimensional unitary representation of a finite group. There are \(c_A,c_B{\gt}0\) and a physical map \(F\) such that \(FA=\sqrt{c_A/c_B}\, B\), and \(F\) preserves inner products on \(\operatorname {ran}A\). For finite graph tensors with this property at every vertex, the product of the local maps restricts to an isometry \(I\) on the actual product physical range, with

The graph states are their actual bond contractions. All local physical alphabets may vary with the vertex.

Let \(G\) be finite, let \(\rho \) be a unitary representation on a finite coordinate Hilbert space, and let \(T\) be \(G\)-isometric for \(\rho \). Write \(P_\rho =|G|^{-1}\sum _g\rho (g)\). There is a positive number \(c\) such that the explicit map \(J=c^{-1/2}T^*\) satisfies \(JT=\sqrt c\, P_\rho \) and, for every \(x,y\in \operatorname {ran}T\),

Thus \(J\) restricts to a Hilbert-space isometry on the actual physical range. No surjectivity onto the ambient physical space is required. This is the support-restricted operation in [ 235 , Observation 6.4 ] , with the positive local normalization factor retained.

Let \(A\) be \(G\)-isometric with \(\mathcal{P}(A)\) onto the physical system. If \(V\) is a unitary on the virtual level that commutes with \(U_g\) for every \(g\), there is a unitary \(W\) on the physical system with \(W\mathcal{P}(A)=\mathcal{P}(A)V\). Conversely, for every unitary \(W\) on the physical system, \(V=\mathcal{P}(A)^{-1}W\mathcal{P}(A)\) satisfies \(VU_g=U_gV=V\) for every \(g\), preserves the inner products of invariant vectors, and \(\mathcal{P}(A)V=W\mathcal{P}(A)\) [ 235 , Lemma 6.3 ] .

Let \(T:\mathbb C^I\to \mathbb C^J\) be \(G\)-isometric for a representation \(\rho \). Let \(e:I'\to I\) and \(f:J'\to J\) be bijections, and let \(E:\mathbb C^{I'}\to \mathbb C^I\) and \(F:\mathbb C^{J'}\to \mathbb C^J\) be their coordinate permutation maps. Suppose \(E\rho '(g)=\rho (g)E\) for every \(g\) and \(FT'=TE\). Then \(T'\) is \(G\)-isometric for \(\rho '\), with the same positive isometry factor as \(T\).

On a regular torus with horizontal period at least four and vertical period at least three, let \(R\) be the translated six-site horizontal two-plaquette block. Reconstruct both internal assignments from the same vertex labels \(k\) and the single-bond or two-bond insertion of \(g\). There is one unitary \(W\) on the original spins in \(R\), chosen before \(k\), \(g\), all boundary configurations and all common exterior and crossing operators, which sends the single insertion to the two-bond insertion. Its adjoint effects the reverse movement. The identity extension outside \(R\) gives the corresponding identity of actual closed contractions. Both periodic seams are included.

This is a finite-torus common-background specialization of the local movement construction in SCP10, Theorem 6.16, and its accessible-coordinate argument (local source lines 1765–1920 and 2271–2305). It does not identify different crossing presentations or establish the prescribed four-endpoint braid; see [ 96 ] .

Let both periods of a finite regular torus be at least four, and suppose the original site tensor is \(G\)-isometric for the regular virtual action. There is one family of six-spin unitaries, indexed by the actual horizontal or vertical movement tile and its starting vertex. For every tile, common vertex gauge, insertion label, boundary configuration and literal exterior and crossing assignment, the forward unitary extends the single-chord insertion to the second chord. Its adjoint reverses the movement. Thus the family supplies rightward, leftward, upward and downward elementary operations on the actual open contractions and the corresponding globally contracted states. Both periodic seams are included.

This is a finite-torus common-background consequence of SCP10, Theorem 6.16 and the accessible-coordinate construction (local lines 1765–1920 and 2271–2305). Applying the family to a complete endpoint route still requires the literal assignment at every intermediate step; no four-endpoint braid or reunion protocol is asserted here. See [ 96 ] .

When all physical alphabets are \(\{ 0,\ldots ,d-1\} \), numbering the virtual alphabet identifies this contraction with the original tensor coefficient.

Let \(U\) and \(L\) be matrix representations of a finite group and suppose \(L(g)=QU(g)Q^\dagger \) for every \(g\). On any finite simple graph, the product of the physical bond operations \(Q\otimes \overline Q\) carries the actual averaging contraction for \(U\) to the actual averaging contraction for \(L\), after regrouping physical coordinates by edges.

Let \(G\) be finite and \(U\) a matrix representation on \(\mathbb C^X\). Dress the averaging site by the same virtual matrix \(W\) on each leg. Regroup the physical labels into head-tail bond pairs \(\beta _e\). Then

This identity holds for arbitrary \(W\) and \(K_e\). If \(W\) commutes with every \(U(g)\) and \(K_e=U(u_e)\), it becomes

Taking \(W=I\) gives the ordinary averaging-site contraction. No connectedness or compatibility among the group insertions is required. These finite-simple-graph coefficient identities extend the calculation in [ 235 , Section 7 ] ; the normalized site convention is recorded in [ 96 ] . No physical isometry for arbitrary \(K_e\) is asserted.

When every \(K_e=I\), the contraction is the ordinary closed graph contraction. For \(X=G\) and \(K_e=L(u_e)\), where \(L\) is the left-regular representation, it is exactly the ordinary contraction with the head label changed from \(\eta _e\) to \(u_e\eta _e\) and the tail label retained. This equality holds for vertex-dependent physical alphabets. For uniform numbered alphabets it is exactly the original twisted tensor-network coefficient, with the existing oriented regular insertion convention of [ 235 , Section 6.4 ] .

Let \(\Gamma \) be a finite simple graph, let \(D_i\) be matrix representations with positive dimensions \(d_i\), and suppose an isometric coordinate change identifies \(\bigoplus _i D_i\otimes I_{d_i}\) with the regular representation. Form the weighted averaging sites from \(U=\bigoplus _i D_i\) and \(W=\bigoplus _i d_i^{1/4}I_{d_i}\). There is one full-domain physical bond isometry, independent of all group insertions, which sends their actual closed contracted state to the actual regular averaging-site state with the same insertions. Its product preserves all inner products. No connectedness or commutativity of the edge labels is required. This is an auxiliary closed-state extension of Section 7 of [ 235 ] , with the Fourier coordinate change supplied; it does not assert surjectivity for arbitrary open boundary tensors.

Let \(G\) be finite and let \(\Gamma \) be a finite simple graph. There are matrix representations \(D_i\) of dimensions \(d_i\), a unitary semi-regular representation \(U=\bigoplus _i D_i\), and one full-domain physical bond isometry whose product \(\mathcal T\) satisfies

Here \(\Theta =\bigoplus _i d_i^{1/4}I_{d_i}\), and the states are the actual closed averaging-site contractions with the representation insertion \(U(u_e)\), respectively \(L(u_e)\), on each oriented bond. Every displayed character has multiplicity one in \(U\). Moreover,

for every finite-dimensional complex semi-regular representation \(\sigma \), without requiring \(\sigma \) to be unitary. The representation and the physical isometry are chosen before all insertion assignments and all physical vectors. No Fourier, coefficient or Gram identity is supplied as a hypothesis. This is an auxiliary inserted-state consequence of [ 235 , Section 7 ] on closed finite simple graphs; it does not assert equivalence of arbitrary open-boundary images or an isometry for arbitrary matrices mixing sectors.

Let \(G\) be a finite group and \(\Gamma \) a finite simple graph. There are positive dimensions \(d_i\), unitary irreducible representations \(D_i\) with distinct characters, Fourier coordinates \(Q\), and a full-domain physical bond isometry \(T\), all chosen from \(G\), with the following properties. The single-copy representation \(U=\bigoplus _iD_i\) is semi-regular with character multiplicity one. Its fourth-root-weighted averaging contraction \(\Psi \) is nonzero. The product of \((Q\otimes \overline Q)T\) over the disjoint physical bond pairs sends \(\Psi \) to the actual regular averaging contraction and preserves every physical inner product. Moreover,

with strict inequality exactly when some \(d_i{\gt}1\), equivalently when \(G\) is noncommutative. For every finite-dimensional complex semi-regular representation \(\sigma \), \(\sum _i d_i\leq \dim \sigma \); \(\sigma \) need not be unitary.

Every virtual leg is contracted along an edge, and the site has one virtual and one physical coordinate for each incident edge. In degree four this is the construction of [ 235 , Section 7 ] . Disconnected graphs, isolated vertices, and the empty graph are allowed. Open virtual boundary legs, parallel bonds, self-loops, and infinite contractions are outside this statement.

Let \(D_i\) be matrix representations of positive dimensions \(d_i\), and set \(U=\bigoplus _iD_i\) and \(W=\bigoplus _i d_i^{1/4}I\). The actual dressed graph state has matching-sector support on every physical bond. The product of the multiplicity-restoring maps

sends this state to the actual averaging-site graph state for \(\bigoplus _i(D_i\otimes I_{d_i})\). Support and the state equality are conclusions, not hypotheses. This is the finite-simple-graph bond calculation of [ 235 , Section 7 ] ; group-derived Fourier choices and full-domain physical isometries are separate consequences.

The two factors of the incident-label bijection read exactly the original internal and crossing labels. Prescribing \(\theta \) therefore gives

with each internal assignment counted once. When the physical alphabet is uniformly numbered, enumerating \(X\) into finite indices gives exactly the original numbered open-region coefficient of the same site tensors. No boundary-state equality or Gram identity is a hypothesis.

Write \(\Psi _R(\theta )\) for the original open tensor contraction on \(R\). With vertex-dependent physical alphabets and regular group bond labels,

Every exterior label remains fixed, and each joining label is summed exactly once. No factorization identity is assumed.

Recovering the site coordinates reads the second endpoint coordinate at each tail and the first at each head. Regrouping evaluates the original coefficient at these recovered coordinates and preserves every inner product.

Let \(\Gamma \) be a finite simple graph and let \(T_v\) be arbitrary site matrices with a common finite virtual edge alphabet. Denote the actual contracted state by \(\Psi _T\) and put \(\mathcal T=\bigotimes _vT_v\). Then \(\Psi _T\in \operatorname {ran}\mathcal T\) and \(\mathcal F\Psi _T=\Psi _{FT}\). Products of physical matrices compose vertexwise and their adjoints are the products of the local adjoints. If \((F_vT_v)^*(F_vT_v)=T_v^*T_v\) at each vertex, then \(\mathcal F\) preserves every inner product on \(\operatorname {ran}\mathcal T\).

Let \(G\) be finite and let \(\Gamma \) be any finite simple graph with vertex set \(\mathcal V\). The actual regular averaging-site contraction is nonzero. More precisely, at the all-identity physical bond configuration its regrouped coefficient equals

This coefficient has strictly positive real part. The statement includes empty graphs, disconnected graphs, and isolated vertices; no connectedness or nonempty-region premise is imposed. It is an auxiliary consequence of the actual regular bond expansion in [ 235 , Section 7 ] , without a parent-Hamiltonian assertion.

Identity virtual weights leave every site unchanged. If \(W\) commutes with every \(U(g)\), the actual dressed graph state, regrouped by physical bonds, is

This includes \(W=I\) and the actual fourth-root operator \(W=\Theta \). No connectedness, valence, or topological assumption is required.

The matrix of \(P_{n}\) is Hermitian. If \(\Gamma _{n}\Gamma _{n}^{\dagger }\Gamma _{n}=c\, \Gamma _{n}\) for some \(c{\gt}0\), then \(c^{-1}P_{n}\) is idempotent with range \(\mathcal{S}_{n}\), so it is the orthogonal projector onto \(\mathcal{S}_{n}\) and the parent interaction is \(h=\mathbb {1}-c^{-1}P_{n}\).