31 Physics Examples of the Multi-Block Compression Theorem

Matrix product states and operators describe periodic spin chains by contracting local tensors around a ring  [ 17 ] . At every positive chain length \(N\), an MPS tensor \(A\) and an MPO tensor \(M\) give

The physical indices \(i,j\) range over \(\{ 0,\ldots ,d-1\} \). Each local matrix acts on its tensor’s virtual bond space, with rows and columns indexed by \(\{ 0,\ldots ,D-1\} \) for bond dimension \(D\). The periodic boundary contraction is the ordinary matrix trace, with no boundary insertion. The first MPO physical index is the output and the second is the input; operator products act from right to left. Thus the local tensor \(\sum _jM^{ij}\otimes P^{jk}\) represents \(O_N(M)O_N(P)\). MPS vectors need not be normalized. A nonzero positive periodic MPDO operator gives a density operator after division by its trace.

Physical examples of matrix product operators, topological symmetries, and fixed points illustrate the multi-block asymmetric compression theorem (Theorem 29.5.1). The cases treated here, in chapter order, are the GHZ and product states  [ 202 , 201 ] in the conventions of Ref.  [ 17 , Appendix A ] ; the CZX symmetry of Chen, Liu, and Wen  [ 11 ] , whose matrix product unitary is displayed in Ref.  [ 17 , Appendix A, “The CZX MPU” ] , together with its decorated form  [ 200 ] , whose anomaly class is shown here to be nontrivial; Kramers–Wannier duality  [ 212 , 1 , 239 , 214 ] ; the Fibonacci  [ 8 , Appendix D.1 ] and Ising  [ 8 , Appendix D.2 ] fusion categories and their anyon chains  [ 195 ] , together with the review’s Fibonacci operator  [ 17 , Appendix A, “The MPO for the Fibonacci model” ] ; the periodic group matrix product operators built from a normalized three-cocycle  [ 199 ] , evaluated for the group of order two, for the cyclic groups with the cocycles of  [ 200 ] , and for \(\mathbb {Z}_2\times \mathbb {Z}_2\); symmetry tensors for \(\mathbb {Z}_2\times \mathbb {Z}_2\) and \(\mathbb {Z}_3\) with their condensation defects, constructed here after the anomalous group symmetries discussed in Refs.  [ 199 , 200 , 231 ] , whose anomaly class this chapter determines for \(\mathbb {Z}_3\), where it is nontrivial, and shows to be nontrivial for the full dressed \(\mathbb {Z}_2\times \mathbb {Z}_2\) representation, whose detector values are those of the type-II class; equality with that class is not claimed  [ 115 ] ; and mixed Bell-bond fixed points addressing the length-dependence question of Ref.  [ 13 , Theorem 4.14 ] . The general notions used by the symmetry examples, namely fusion rules, units, invertible and non-invertible labels, Perron–Frobenius dimensions, symmetric matrix product states with their nonnegative integer representations, and the zipper condition, are defined in Chapter 30; this chapter supplies the tensors, fusion tables, and computations.

31.1 The GHZ state as a compression target

The Greenberger–Horne–Zeilinger (GHZ) state  [ 202 , 201 ] is the equal superposition of the two constant product configurations; for \(N{\gt}0\) sites, its unnormalized form on \((\mathbb {C}^2)^{\otimes N}\) is

Here \(|0\rangle ,|1\rangle \) form the orthonormal local basis, following the product-state and GHZ conventions of Ref.  [ 17 , Appendix A ] . The standard tensor generating this state has bond dimension two and is the direct sum \(C_0\oplus C_1\) of two scalar sectors, as in Ref.  [ 17 , Appendix A ] and Definition 15.2.7. The upper-triangular source tensor \(B\) of bond dimension four (Definition 31.1.1) satisfies \(|V^{(N)}(B)\rangle = |\mathrm{GHZ}_N\rangle \) for all \(N{\gt}0\), providing an explicit instance of the multi-block compression theorem (Theorem 29.5.1).

Throughout this section and the next, for a tensor \(T=\{ T^i\} _{i=0}^1\) of bond dimension \(D\), the periodic trace boundary defines

where \(\{ 0,\ldots ,N{-}1\} =\{ 0,\ldots ,N-1\} \) for \(N{\gt}0\), \(\{ 0,\ldots ,0{-}1\} =\varnothing \), and \(|\sigma \rangle =\bigotimes _{n=0}^{N-1}|\sigma (n)\rangle \). At \(N=0\), the matrix product is \(\mathbb {1}_D\), so the unique coefficient is \(D\) and the physical space is \(\mathbb {C}\). For a word \(w=(i_1,\ldots ,i_{|w|})\), write \(T^w=T^{i_1}\cdots T^{i_{|w|}}\), with \(T^\varnothing =\mathbb {1}_D\).

Use the ordinary trace boundary of (1). The state and norm agree with the standard GHZ tensor at positive lengths (Theorems 31.1.2 and 31.1.3). The compression is given in Theorem 31.1.4; the intertwiner obstruction is stated in Theorem 31.1.6.

The source generates the GHZ state at every positive length, with the same amplitudes and the same norm as the standard tensor; the two tensors differ only in their empty-length coefficients, which count their bond dimensions.

The amplitude identity rests on the compression of the source onto its two scalar sectors. The source has bond dimension four while both sectors are scalar, so two of its four diagonal positions are zero slots; the compression theorem accounts for them in the dimension count and delivers the periodic trace identity behind Theorem 31.1.2.

The compression is genuinely a word-level statement. A sitewise intertwiner for a sector would be a vector fixed by one letter of the source and annihilated by the other; the following theorem shows that the sector \(C_0\) admits none in either direction.

31.2 A product state from a repeated block

A product state \(|\phi \rangle ^{\otimes N}\) is the simplest matrix product state: it is generated by a tensor of bond dimension one whose entries are the local amplitudes of \(|\phi \rangle \), as in the product-state construction of Ref.  [ 17 , Appendix A ] . Here \(|\phi \rangle =|0\rangle +2|1\rangle \), so the bond-one tensor \(A\) has local amplitudes \((1,2)\). The direct sum \(A\oplus A\) doubles every amplitude, and the upper-triangular source \(B\) of Definition 31.2.1 generates the same vector:

The tensors \(A\) and \(B\) in Definition 31.2.1 generate these states via the trace boundary (1); at \(N=0\) the tensor power is the scalar \(1\). The example shows a target with a repeated block, where the compression theorem records the multiplicity of the block, and a source that is not gauge equivalent to the target even though the two generate equal periodic vectors at every length.

Again the boundary is the ordinary trace. The multiplicity factor in the product state and its norm are given in Theorems 31.2.2 and 31.2.3. Theorems 31.2.1.1 and 31.2.1.2 establish the compression and show that it does not arise from a simultaneous similarity to the split tensor.

31.2.1 Compression and the absence of a gauge

The source \(B\) and target \(A\oplus A\) satisfy \(|V^{(N)}(B)\rangle =|V^{(N)}(A\oplus A)\rangle \) at all lengths \(N\geq 0\) by (5). However, they are not gauge equivalent: \(B^i\neq X(A^i\oplus A^i)X^{-1}\) for any invertible \(X\) (Theorem 31.2.1.2), even though \(\operatorname{tr}(B^w)=2\operatorname{tr}(A^w)\) for all nonempty words \(w\) (Theorem 31.2.1.1). The compression theorem acts only on the two diagonal copies of \(A\), yielding the trace identity \(\operatorname{tr}(B^w)=2\operatorname{tr}(A^w)\) that proves the factor two in (6).

31.2.2 A Jordan extension detected by a boundary

A Jordan extension of a repeated block of arbitrary bond dimension shares every periodic trace with the direct sum of two copies of the block, while a single boundary matrix \(Q\) detects it:

The ordinary trace and the boundary insertion \(Q\) are compared in Theorem 31.2.2.2. If some nonempty word of \(A\) has nonzero trace, the insertion distinguishes the two tensors despite their identical periodic traces (Theorem 31.2.2.3).

31.3 The CZX symmetry

The CZX model of Chen, Liu and Wen  [ 11 ] is a two-dimensional symmetry-protected topological phase whose anomalous boundary \(\mathbb {Z}_2\) symmetry on a ring of \(N\) qubits is the CZX operator: controlled-\(Z\) gates on the nearest-neighbor bonds of the ring followed by a Pauli \(X\) on every site. The review presents this operator as a bond-two matrix product unitary and its canonical-form reduction  [ 17 , Appendix A, “The CZX MPU” ] ; the symmetry-protected context is [ 17 , Section III.B.1 ] . The decorated representative with additional Pauli \(Z\) factors in  [ 200 , Section III.D ] is a different tensor.

For \(N{\gt}0\), close the virtual indices by the ordinary trace: \(O_N(A)_{st}=\operatorname{tr}(A^{s_0t_0}\cdots A^{s_{N-1}t_{N-1}})\). Theorem 31.3.2.1 gives the periodic kernel and its relation to the input-phase convention used below. Unitarity, the square and normality are established in Theorems 31.3.2.2–31.3.2.4. The decorated representative has its own anomaly-class statement in Theorem 31.3.3.8.

31.3.1 The bond-two tensor and the periodic operator

On a periodic chain of \(N{\gt}0\) qubits, the undecorated CZX operator acts on computational basis vectors by

Here \(t=(t_0,\ldots ,t_{N-1})\in \{ 0,1\} ^N\), \(\bar t_j=1-t_j\), and all site subscripts are taken modulo \(N\). The cyclic phase exponent is

Define the diagonal operator \(D_N\) by \(D_N|t\rangle =(-1)^{q(t)}|t\rangle \), and let \(X|a\rangle =|1-a\rangle \). With operator products acting from right to left,

For \(N\geq 3\), \(D_N\) is the product of controlled-\(Z\) gates on the cyclic nearest-neighbor bonds. Formula (18) also specifies the short rings: it includes a self-loop for \(N=1\) and counts the two-site bond twice for \(N=2\).

The tensor of the review differs from the one defined below by a transposition of the bond and by the gate order \(D_NX^{\otimes N}\), which differs from (19) by the sign \((-1)^N\): the review’s printed matrices place the phase on the output bits, the tensor of Definition 31.3.1.1 on the input bits.

The periodic boundary is the ordinary, unnormalized trace. Thus the operator generated by \(M\) has matrix coefficients

There are no open boundary vectors. For bond indices \(\alpha ,\beta \in \{ 0,1\} \), the local entries are

Unitarity and (25) give the inverse and physical adjoint:

Consequently \(U_N^4=\mathbb {1}\). On even positive rings, \(U_N\) is a self-adjoint involution; on odd rings, \(U_N^2=-\mathbb {1}\). The phase identity (26) also gives \(D_NX^{\otimes N}=(-1)^NU_N\), explaining the gate-order difference.

At length one, \(q(t)=t_0^2=t_0\), hence \(D_1=Z\) and \(U_1=XZ\), where \(Z|a\rangle =(-1)^a|a\rangle \). At length two, \(q(t)=2t_0t_1\), hence \(D_2=\mathbb {1}\) and \(U_2=X\otimes X\). The exponent \(q(t)\) includes a self-loop at \(N=1\) and counts the two-site edge twice at \(N=2\).

At length zero the trace convention gives the scalar \(\operatorname{tr}(\mathbb {1}_2)=2\), not a unitary operator. The stacked tensor similarly gives \(\operatorname{tr}(\mathbb {1}_4)=4\), whereas the bond-one target gives \(\operatorname{tr}(\mathbb {1}_1)=1\). All periodic operator and trace comparisons below therefore retain their positive-length or nonempty-word hypotheses.

31.3.2 The tensor printed in the review

The review  [ 17 , Appendix A, “The CZX MPU” ] prints the CZX matrix product unitary through its two nonzero matrices and reduces its square to canonical form in two steps, following the construction of the canonical form in  [ 17 , Section IV ] . The presentation below uses the review’s coordinates. Throughout, the first physical index is the output, and for a configuration \(s\in \{ 0,1\} ^N\) the exponent \(q(s)\) is (18).

The review describes the operator in words as controlled-\(Z\) gates followed by a Pauli \(X\) on every site, which is \(X^{\otimes N}D_N=U_N\); by (29) the two readings differ by the global sign \((-1)^N\)  [ 180 ] .

The review says that the projector \(Q\) gives an invariant subspace. Read on columns this fails, since \(b^{00}(1,-1)^{\mathsf T}=(-2,0)^{\mathsf T}\), and it is the range of \(\mathbb {1}-Q\) that is invariant. The review’s discussion of the canonical form notes that a nontrivial invariant subspace exists if and only if one exists for the action on rows, and the identity \(Qb^{ij}=Qb^{ij}Q\) says exactly that the range of \(Q\), read on rows, is invariant. In either reading \(Q\) cuts out the block \((-1)\delta _{ij}\), which is the canonical form the review states.

This is the review’s conclusion that the canonical form \((-1)\delta _{ij}\) globally means \(O(A)^2=(-1)^N\mathbb {1}\): the stacked product generates the square of the generated operators, so the statement agrees with Theorem 31.3.2.3, reached here through the two reduction steps instead of the explicit kernel. The main text of the review instead describes the one-dimensional block of the square as the identity operator. The block \((-1)\delta _{ij}\) generates \((-1)^N\mathbb {1}\), which is the identity for even \(N\); rescaling the tensor by a scalar \(c\) with \(c^2=-1\) turns the block into \(\delta _{ij}\) at every length  [ 180 ] .

31.3.3 The decorated representative

Garre-Rubio and Schuch  [ 200 , Section III.D ] write the CZX symmetry with an additional Pauli \(Z\) on every site, \(U=\prod _iZ_i\prod _i\mathrm{CZ}_{i,i+1}\prod _iX_i\), as a bond-two tensor built from delta tensors and the unnormalized Hadamard matrix. This representative has order two. Its fusion tensors reduce two stacked layers to one, and the two ways of fusing three layers differ by the sign \(\omega =-1\)  [ 200 , Section III.A ] .

The source states (34) with at least one intermediate site; the case \(k=0\) is the identity \(S^aS^b=S^aV\hat VS^b\).

31.3.4 Asymmetric compression of the square

By (25), the periodic operator satisfies \(O_N(B)=U_N^2=(-1)^N\mathbb {1}\), and \(\operatorname{tr}(B^w)=(-1)^{|w|}\operatorname{tr}(\delta ^w)\) for all nonempty words \(w\). This periodic comparison with the bond-one identity tensor with weight \(-1\) lies beyond the unit-weight setting. The sitewise theory of  [ 199 , Appendix A ] imposes a closedness requirement for arbitrary boundary conditions. The following example distinguishes a word-level compression from a sitewise intertwiner.

For a word \(w=((i_1,j_1),\ldots ,(i_{|w|},j_{|w|}))\), write \(B^w=B^{i_1j_1}\cdots B^{i_{|w|}j_{|w|}}\), and use the same convention for the other tensors. The empty word evaluates to the identity matrix on the corresponding bond space.

31.3.5 A non-simple family with an oscillating coefficient

Let \(M\) be the bond-two tensor of Definition 31.3.1.1, with the first physical index denoting the output. For a bit configuration \(t=(t_0,\ldots ,t_{N-1})\), write \(\bar t_n=1-t_n\) and \(E(t)=\sum _{n=0}^{N-1}t_nt_{n+1}\) (with cyclic subscripts). By Theorem 31.3.1.2, the periodic operator acts as

In terms of the tensor entries,

Here \(\delta _{s,\bar t}\) is one exactly when the two configurations agree, and \(I_N=\mathbb {1}^{\otimes N}\) denotes the identity on the \(N\)-qubit space. Theorems 31.3.1.3 and 31.3.1.4 give unitarity together with the square and adjoint

Define the operator family \(O_N=I_N+U_N\). Its basis action is

The identity and the CZX operator are generated by normal, inequivalent blocks of bond dimensions one and two. Their sum therefore has a non-simple matrix product representation of bond dimension three. Before considering its virtual compression, its physical square follows from (39):

The identity coefficient is two on even rings and zero on odd rings.

At length zero, the unnormalized trace boundary assigns the scalar value \(U_0=[2]\) to the bond-two tensor and \(O_0=[3]\) to its direct sum with the bond-one identity, so the square law (39) holds only for positive lengths.

Stacking represents physical multiplication: expanding the intermediate physical configuration \(r\) and using multiplicativity of the Kronecker product and \(\operatorname{tr}(A\otimes D)=\operatorname{tr}(A)\operatorname{tr}(D)\) gives

All matrix products are ordered by increasing site index. Thus the periodic multiplication law is established independently of the asymmetric fundamental theorem.

Taking \(|w|=N{\gt}0\), (43) identifies (46) with the physical square law (41). The compression resolves the oscillating integer coefficient into the signed weights \(+1\) and \(-1\): the coefficient \(1+(-1)^N\) is their length-\(N\) power sum, as in Theorem 29.7.2. The two copies of \(M\) each have unit weight and contribute the constant coefficient two. This is the general power-sum mechanism, not an instance of the positive diagonal matrices in the characterization of matrix product density operator renormalization fixed points of  [ 13 , Theorem 4.14(ii) ] . In particular, the even-ring projector property does not establish an MPDO renormalization-fixed-point condition.

31.3.6 Scalar three-cocycles on the group of order two

The scalar three-cocycles on the group of order two, whose two gauge classes are distinguished below, are the candidates for the anomaly class of a representation of that group; for the decorated CZX representation the class is the nontrivial one, by Theorem 31.3.3.8. For scalar associators, the three-cocycle and fusion-gauge conventions follow [ 200 ] .

31.4 The anomalous \(\mathbb {Z}_2\) symmetry of the Levin–Gu edge

The edge Hamiltonian of the Levin–Gu model, unitarily equivalent to the XX chain, commutes with the \(\mathbb {Z}_2\) symmetry \(U_N=\prod _nCZ_{n,n+1}\prod _nZ_nX_n\) on a ring of \(N\) qubits  [ 213 , Section II ] . The source writes it as a bond-two matrix product operator, reduces the product of two copies to a bond-three tensor \(A_0\) padded by a zero block, and states that \(A_0\) is not injective, cannot be further decomposed into a direct sum of injective matrix product operators, and generates the identity on periodic chains. This section reads these statements through the multi-block asymmetric compression theorem (Theorem 29.5.1). The associator of the fusion tensors, \(\omega (1,1,1)=-1\) in  [ 213 , Section II ] , is proved below. The unrestricted scalar relation on the full virtual space is corrected to a relation on the physical images, as explained in  [ 181 ] .

The first physical index is the output and the second the input, as in the source. For a word \(w\) in the index pairs, \(B^w\) denotes the product of the matrices of \(B\) along \(w\), as in Section 31.3.

The periodic operator \(U_N\) of \(A_1\) has the matrix coefficients \(U_N(s,t)=\operatorname{tr}(A_1^{s_0t_0}\cdots A_1^{s_{N-1}t_{N-1}})\), the ordinary unnormalized trace:

The periodic kernel and the identity \(U_N^2=I\) are established in Theorems 31.4.2 and 31.4.4. Theorems 31.4.3 and 31.4.7 give the compression and the printed bond-three reduction; Theorem 31.4.9 excludes compressions of the reduced tensor onto the identity with zero remainder. Arbitrary direct-sum decompositions and the associator mentioned above are not settled by these results.

In a picture, the compression pair closes the bond of the stacked square onto the bond-one identity tensor, letter by letter:

The next theorem states that the two stacked copies of \(A_1\) contract, on every ring of \(N\geq 1\) sites, to the identity:

The identity of Theorem 31.4.11 moves \(Y_{0,0}\) from the west of two stacked copies of \(A_0\) to the east of one copy:

31.5 Kramers–Wannier duality

Kramers–Wannier duality originates in the two-dimensional classical Ising model, where it relates the high- and low-temperature regimes  [ 212 ] ; in the quantum transverse-field Ising chain it exchanges the bond and field terms of the Hamiltonian. On a periodic chain the duality is a noninvertible operator, rather than a unitary change of variables on the full Hilbert space. Aasen, Mong and Fendley realize it as a lattice defect  [ 1 ] ; Seiberg and Shao place it in a noninvertible symmetry algebra together with translation and the spin flip  [ 239 ] ; Lootens, Delcamp, Ortiz and Verstraete write it as a matrix product operator intertwiner between the two Hamiltonians  [ 214 ] . The review discusses this operator among its matrix product operator symmetries  [ 17 , Section III.A.6 ] ; its relation to fusion algebras is Remark 30.14.4. Here the duality is built from an explicit two-by-two tensor whose periodic operator is unnormalized; it reproduces the primal-to-dual kernel of  [ 1 ] up to the factor \(2^{N/2}\). Its square and its action on ground states are characterized both as operator identities and as asymmetric compressions of stacked tensors.

Close the virtual bonds by the ordinary trace on a ring of \(N{\gt}0\) sites, without dividing by the bond dimension. The resulting operator is \(K_N(a,b)=\operatorname{tr}(W^{a_0b_0}\cdots W^{a_{N-1}b_{N-1}})\). Its normalization relative to the source defect follows from Theorem 31.5.1.1 and Corollary 31.5.1.7. Theorem 31.5.3.2 proves that \(W\) is normal, and Theorem 31.5.1.12 that \(T\) and \(\eta T\) are; Corollary 31.5.1.4 gives the exchange of Ising couplings, and Lemma 31.5.2.1 describes its action on product and GHZ states. The compression of its square is Theorem 31.5.1.13.

Consider \(N\geq 1\) qubits on a periodic chain, with sites \(I_N=\{ 0,\ldots ,N-1\} \) and all site indices understood modulo \(N\). The Hilbert space \(\mathcal H_N=(\mathbb {C}^2)^{\otimes N}\) has computational basis \(\{ |b\rangle :b\in \{ 0,1\} ^{I_N}\} \). Write \(\bar b\) for the componentwise complement of \(b\), and \(b\oplus e_j\) for the configuration obtained by flipping its bit at site \(j\). The Pauli operators act by

They are self-adjoint involutions, satisfy \(X_jZ_j=-Z_jX_j\), and commute at distinct sites. Define global spin flip \(\eta \) and one-site left translation \(T\) by

Thus \(\eta =\prod _jX_j\), \(\eta ^2=\mathbb {1}\), \(T^N=\mathbb {1}\), \(T^\dagger =T^{-1}\), and \(T\eta =\eta T\). In this convention, \(TX_jT^{-1}=X_{j-1}\) and \(TZ_jT^{-1}=Z_{j-1}\).

For real couplings \(J,h\), the periodic transverse-field Ising Hamiltonian is

The sum includes one bond per site. In particular, the bond at \(N=1\) is \(Z_0^2=\mathbb {1}\), and at \(N=2\) the two bond terms both equal \(Z_0Z_1\). The duality exchanges the bond terms with the field terms.

31.5.1 The duality kernel and its square

The translation tensor is the shift matrix product unitary of  [ 17 , Appendix A, “The shift MPU” ] , taken with the opposite direction of translation, and the flipped translation tensor composes it with the global spin flip.

The site labels in the diagrams run from \(1\) to \(N\), a cyclic relabeling of \(I_N\).

Operator identities

Define the raw periodic operator \(K_N\) by the ordinary virtual trace, without dividing by the bond dimension:

We also write \(D=K_N\) for this raw operator, retaining \(D\) in the square identity below. The normalized operators will be denoted \(\mathcal D_N\) and \(U_N\), not \(D\).

For subsequent calculations, put \(q(a,b)=\sum _j a_j(b_j+b_{j+1})\), so \((K_N)_{ab}=(-1)^{q(a,b)}\). Exponents may be calculated modulo two. Unless indicated otherwise, site sums and products run over \(I_N\).

Under the identification of the dual lattice described after Corollary 31.5.1.7, these are the two local relations of the duality defect in  [ 1 ] .

The local identities also exchange strings and endpoint operators. For consecutive sites \(j_r=j_0+r\) and \(1\leq m\leq N\),

Indeed, iterating (57) and (58) cancels every interior factor \(Z_j^2\). At \(m=N\) the endpoints coincide, and these relations give \(K_N\eta =\eta K_N=K_N\). They are intertwining identities; no inverse of \(K_N\) is involved.

Define the spin-flip sectors \(\mathcal H_N^\pm =\{ v:\eta v=\pm v\} \) and their orthogonal projections \(P_\pm =(\mathbb {1}\pm \eta )/2\). Two useful normalizations of the duality operator are

Each \(H_N(J,h)\) preserves both spin-flip sectors: the field terms commute with \(\eta \), and the two signs acquired by a bond term cancel. Thus \(U_N\) implements a unitary equivalence between \(H_N(J,h)\) and \(H_N(h,J)\) on \(\mathcal H_N^+\), but not on the full Hilbert space.

The corresponding normalized identities are

The last identity uses \(P_+T=TP_+\) and \(T^N=\mathbb {1}\). The two summands \(T\) and \(\eta T\) act on the same Hilbert space. On \(\mathcal H_N^+\) they add, giving \(K_N^2=2^{N+1}T\); on \(\mathcal H_N^-\) they cancel. The negative sector is annihilated, not mapped by flipped translation.

The translation factor depends on identifying the dual lattice with the original one. In Ref.  [ 1 ] , identify the input spin \(h_j\) with \(b_j\) and the dual spin \(\hat h_{j+1/2}\) with \(a_j\). Its primal-to-dual kernel is then \(\mathcal D_N\), since

The source returns to the original lattice with the same matrix elements read with the arguments exchanged, that is, by the transpose \(\mathcal D_N^{\mathsf T}\). Since the kernel is real, \(\mathcal D_N^{\mathsf T}=\mathcal D_N^\dagger \).

With the spin-flip defect equal to \(\eta \) and the square of the source read as \(\mathcal D_N^{\mathsf T}\mathcal D_N\), the first three lines of (71) are the fusion algebra of  [ 1 ] , and the last line is its relation \(\mathcal D_\sigma (\mathcal D_\sigma ^2-2)=0\). The source computes the composite on the original lattice only; the identity \(\mathcal D_N\mathcal D_N^{\mathsf T}=\mathbb {1}+\eta \) on the dual lattice is not printed there. Repeating the identified endomorphism instead gives \(\mathcal D_N^2=(\mathbb {1}+\eta )T\).

The partial-isometry convention of Ref.  [ 239 ] has the scale of \(U_N\). Since the kernel is real, \(U_N^\dagger =2^{-(N+1)/2}K_N^{\mathsf T}\).

After exchanging the Pauli operators \(X\) and \(Z\) at every site, the relations (72) and (73) are those that Ref.  [ 239 ] prints for its noninvertible translation, with its translation read as \(T^{-1}\) and its projection \((1+\eta )/2\) read as \(P_+\). The proposition shows that \(U_N^\dagger \) satisfies these relations. That reference defines its operator by an explicit product of local factors; Proposition 31.5.1.11 below shows that, after the exchange of \(X\) and \(Z\), this product is exactly \(U_N^\dagger \), with no scalar factor. The noninvertibility here concerns the periodic bosonic spin chain, not fermionic Majorana translation. The translation in (69) must be retained when comparing this lattice relation with a size-independent fusion algebra.

Positive length is essential in these calculations. At \(N=1\), \(K_1=\mathbb {1}+X_0\) and \(U_1=P_+\), consistently with the self-bond convention. The empty virtual word instead has trace \(\operatorname{tr}\mathbb {1}_2=2\), so extending the raw trace definition to \(N=0\) would give \(K_0=2\). Its square would be \(4\), whereas the right-hand side of (69) would be \(2\).

Asymmetric compression

The square identity (69) has a virtual decomposition, obtained by applying the multi-block compression theorem to the stacked tensor \(B\) of Definition 31.5.1. For the matrix-unit tensors of that definition, expanding the cyclic trace gives

The stacked tensor \(B\) generates \(K_N^2\): multiplication of periodic operators sums the intermediate physical index at every site.

Normality of a tensor below means that words of some fixed positive length span its virtual matrix algebra. This differs from operator normality in (62) and does not impose a normalization on the spectral radius of a transfer map.

31.5.2 Action on ground states

The product state and the Greenberger–Horne–Zeilinger (GHZ) state  [ 202 , 201 ] give elementary finite-chain examples of the action. Use the unnormalized vectors

For \(h{\gt}0\), \(p_N\) is proportional to the unique ground state of \(H_N(0,h)\). For \(J{\gt}0\) and \(N\geq 2\), the ground space of \(H_N(J,0)\) is \(\operatorname{span}_{\mathbb {C}}\{ |\mathbf0\rangle ,|\mathbf1\rangle \} \), and \(g_N\) spans its spin-flip-even subspace. These facts follow respectively by minimizing each field term \(-hX_j\) and each diagonal bond term \(-JZ_jZ_{j+1}\). They do not require a thermodynamic limit.

In particular, the antisymmetric combination \(|\mathbf0\rangle -|\mathbf1\rangle \) is annihilated. These finite-chain identities alone assert neither spontaneous symmetry breaking nor a thermodynamic spectral-gap property.

The finite-chain actions in (79) have different virtual decompositions. The product-state action splits into the two GHZ sectors with a local weight \(2\). The GHZ action has two unweighted copies of the same product-state target, together with a nonzero remainder. The product-state and GHZ tensors are the elementary matrix product state examples of  [ 17 , Appendix A ] ; here their explicit matrices allow the asymmetric compression and its sitewise intertwiners to be compared directly.

For the product-state action, the matrices are \(2Q_+\) and \(2Q_-\), where

They satisfy \(Q_\pm ^2=Q_\pm \), \(Q_+Q_-=Q_-Q_+=0\) and \(\operatorname{tr}Q_\pm =1\). Hence a nonempty word of length \(N\) has trace \(2^N\) when all letters agree and trace zero otherwise. This recovers \(K_Np_N=2^Ng_N\) and identifies the factor \(2\) as a local tensor weight. By contrast, the factor \(2\) in \(K_Ng_N=2p_N\) is the multiplicity of two unweighted targets. Replacing \(p_N\) or \(g_N\) by a normalized vector changes the global scalar as in Lemma 31.5.2.1; it does not change the weights of these fixed tensors.

In the following two results, \(B\) denotes the relevant action tensor, rather than the stacked square tensor of Definition 31.5.1.

A word reduction need not provide a pair of sitewise intertwiners. Here the periodic coefficients have merged into two copies of the same product state, but the virtual action still has a one-sided obstruction.

This obstruction rules out a sitewise split copy of the product-state target: such a splitting would require a nonzero left intertwiner. It does not obstruct the word reductions or the periodic trace formula in Theorem 31.5.2.4. Those statements retain the finite-chain action \(K_Ng_N=2p_N\) while distinguishing it from a split virtual direct sum.

31.5.3 The duality as a bimodule over the spin-flip algebra

By (69) (Corollary 31.5.1.7), \(D^2=2^N(\mathbb {1}+\eta )T\), where the non-trivial translation factor \(T\) prevents the duality \(D\) from closing in a fusion algebra with \(\mathbb {1}\) and \(\eta \). In the framework of  [ 1 ] , it is instead fused with the symmetry operators on either side and with its own return map. These relations are recorded below for the raw operator and derived at the tensor level from the compression theorems, without computing a gauge.

Explicitly, the stacked tensors have the letters \(\sum _mW^{s'm}\otimes O_1^{ms}=W^{s',1-s}\) and \(\sum _mO_1^{s'm}\otimes W^{ms}=W^{1-s',s}\), and a direct check on the four matrices of Definition 31.5.1 gives

with \(X^{-1}=X\) and \(Y^{-1}=Y\). The proof above does not use these matrices.

The biorthogonal compression pairs of this theorem are fusion tensors for the channels \(\mathbb {1}\) and \(\eta \) in the product of the duality with its adjoint. The stacked square of Theorem 31.5.1.13 has no zero slot, whereas the product with the adjoint has two.

31.6 Fibonacci fusion operators

The Fibonacci fusion category has two simple objects, the vacuum \(1\) and \(\tau \), with the fusion rule \(\tau \otimes \tau =1\oplus \tau \). Its string-net construction produces matrix product operators whose fusion rule, fusion tensors, and \(F\)-symbols are given in  [ 8 , Appendix D.1 ] ; the review displays the string-net operator built from the same Fibonacci data in  [ 17 , Appendix A, “The MPO for the Fibonacci model” ] , and the general string-net construction of such operator symmetries and their intertwiners is in  [ 215 ] . The physical space used here is the two-label space of the golden chain  [ 195 ] , one label \(1\) or \(\tau \) per site, rather than the full triple-index alphabet of the categorical string-net construction. The entries of the blocks defined below are obtained from the \(F\)-symbols of  [ 8 , Appendix D.1 ] . From these data one obtains two operators on the chain whose fusion relation and action on periodic states reduce to finite matrix computations.

The printed \(G\)-symbol tensor, including its quantum-dimension factors, is given in Definition 31.6.1. Its restriction to edge label \(\tau \) is related to the two-label blocks by the diagonal bond similarity of Theorem 31.6.9. The review uses a different physical normalization; its relation to these blocks is the congruence of Theorem 31.6.11.

For every \(N{\gt}0\), close the bond indices by the ordinary, unnormalized trace of the ordered product of local matrices. The first physical index is the output and the second is the input.

The word-span certificates are in Theorem 31.6.12. The admissibility projector and its action as the fusion unit are identified in Lemma 31.6.1.1 and Theorem 31.6.1.2. The fusion relation is proved for the two-label blocks in Theorem 31.6.4.2 and for the source’s blocks on both edge labels in Theorem 31.6.10. Commutation with the golden-chain Hamiltonian is established for \(N\geq 2\) in Theorem 31.6.2.4; the regular action on the two explicit state tensors is in Theorem 31.6.3.4.

The physical space at each site is \(\mathbb {C}^2\), with computational labels \(0\) for the vacuum and \(1\) for \(\tau \). At system size \(N\geq 1\), write \(\mathcal{H}_N=(\mathbb {C}^2)^{\otimes N}\) and use the orthonormal basis \(|x\rangle =|x_0,\ldots ,x_{N-1}\rangle \). Site indices are cyclic: \(x_N=x_0\). Throughout, the golden-number convention is \(\varphi =(1+\sqrt5)/2\) and \(\sigma =\varphi ^{-1/2}=\sqrt{(\sqrt5-1)/2}\). In particular, \(\sigma ^2=\varphi ^{-1}\) and \(\sigma ^4+\sigma ^2=1\).

For an MPO tensor \(T^{yx}\), the first physical index is the output index. Its periodic operator is defined by

For the tensors \(B_1\) and \(B_\tau \) defined below, write \(O_1=O_N(B_1)\) and \(O_\tau =O_N(B_\tau )\) when \(N\) is fixed. Their non-invertible fusion relation is

as proved below for every positive system size. The label \(\tau \) is not invertible in the sense of Definition 30.13.2. The vacuum operator \(O_1\) is a projection onto admissible configurations, not the identity on \(\mathcal{H}_N\), as for the unit of any fusion algebra (Remark 30.14.1).

Every entry of the tensors below, and of the changes of bond coordinates in the proofs, lies in a quartic ring generated by \(\sigma \). The identities of this section are first verified in that ring and then embedded into \(\mathbb {C}\). The exceptions are the string-net tensors of Definition 31.6.1 and the bond similarity relating them to the blocks, whose entries involve \(\varphi ^{\pm 1/4}\); those identities are proved over \(\mathbb {C}\).

The review writes the operator with the prefactor \(1/\sqrt{d_Ad_D}\) arising from the normalization of the fusion basis, where \(d_A\) and \(d_D\) are the quantum dimensions of the labels on the adjacent bonds  [ 17 , Appendix A, “The MPO for the Fibonacci model” ] ; that prefactor is omitted in the blocks defined here, and its effect on the periodic operators is Theorem 31.6.11.

The blocks above are built from the categorical data of the Fibonacci theory, which  [ 8 , Appendix D.1 ] and the review print in the following form. The former gives the operator tensor only as a diagram and prints no entries of \(B_1\) or \(B_\tau \); the placement of the \(F\)-symbols used here, without the closed-loop factors of the source and without the prefactor of the review, is recorded in  [ 52 ] . Theorem 31.6.9 relates these blocks to the source’s tensor built from the \(G\)-symbols, and Theorem 31.6.10 carries their fusion rules to that tensor on both edge labels.

While the review states \(O_aO_b=\sum _cN_{ab}^cO_c\) for the prefactored review tensor \(R_f\)  [ 17 , Section 3 ] , that operator already contains the prefactor \(1/\sqrt{d_{x'}d_{q''}}\) and fails the fusion rules in the orthonormal plaquette basis. Theorem 31.6.11 shows that the fusion rules hold after composing it with \(W\); see  [ 52 ] .

31.6.1 Admissible configurations and the vacuum operator

Define the adjacency matrix \(K=\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}\) and, for a cyclic word \(x\in \{ 0,1\} ^N\), set

A cyclic word \(x\) is admissible if \(p_N(x)=1\), equivalently if it contains no cyclic neighboring pair \(00\).

By (108), the range of \(O_N(B_\tau )\) lies in the admissible subspace. By (109), it vanishes on the orthogonal complement of that subspace. Thus the vacuum acts as the identity on the admissible subspace and is the unit of the algebra generated there by \(O_N(B_\tau )\).

This vacuum block is distinct from the weighted string-net projector of  [ 8 , Appendix D.1 ] , which combines the vacuum and \(\tau \) blocks with weights \(w_1=1/(1+\varphi ^2)\) and \(w_\tau =\varphi /(1+\varphi ^2)\).

At \(N=0\) the empty traces give \(O_0(B_1)=2\) and \(O_0(B_\tau )=3\), so \(\Pi _0=(2+3\varphi )/(1+\varphi ^2)\), a scalar different from \(0\) and \(1\), hence \(\Pi _0^2\neq \Pi _0\); [ 8 , Section 3.2 ] asks for idempotence at every length, and the empty chain is excluded as a degenerate reading; see  [ 53 ] .

31.6.2 The golden chain

The golden chain of Feiguin et al.  [ 195 ] acts on the labels \(x_0,\ldots ,x_{N-1}\in \{ 1,\tau \} \) of a periodic chain. Its three-site term projects the two anyons at links \(i\) and \(i+1\) onto their trivial fusion channel, with the amplitudes

where \(p_i\) changes only the label \(x_i\). The Hamiltonian is \(H_N=-\sum _ip_i\). The review states that such anyonic chains commute with the whole algebra of periodic operators  [ 17 ] .

31.6.3 Two periodic states and the Fibonacci action

For an MPS tensor \(D\), write \(|V^{(N)}(D)\rangle =\sum _x\operatorname{tr}(D^{x_0}\cdots D^{x_{N-1}})|x\rangle \). These vectors are not normalized unless this is stated explicitly.

The normalized chain vector is obtained by dividing by the positive square root of (120). For example, \(|V^{(1)}(C)\rangle =-\sigma ^2|1\rangle \) and its squared norm is \(\sigma ^4\). In particular, the two periodic vectors are not independent at every length.

The first nontrivial action also has a direct contraction proof. The all-one input selects the letters \(B_\tau ^{01}\) and \(B_\tau ^{11}\). Both have zero row \(1\), so a cyclic bond configuration using that coordinate contributes zero. Restricting to the ordered coordinates \((0,2)\) gives exactly \(C^0\) and \(C^1\). Hence

There is no additional normalization factor.

Relative to the ordered pair of these unnormalized state families, the action coefficients are the Fibonacci matrix \(N_\tau =\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}\), realizing the nonnegative-integer action discussed in  [ 199 ] . The positive-length hypothesis is essential: at \(N=0\) the periodic trace is the bond dimension, yielding \(O_0(B_1)=2\), \(O_0(B_\tau )=3\), and empty amplitudes \(1\) and \(2\) for \(A\) and \(C\).

31.6.4 Asymmetric compression of the stacked square

The stacked square in (93) contracts the physical index between two copies of \(B_\tau \) while retaining both virtual indices. Its periodic operator is therefore the square of \(O_N(B_\tau )\). The compression maps onto its two target blocks also have entries in \(\mathbb {Z}[\sigma ]\), so the fusion rule below is again an exact computation in the ring of golden integers.

At length zero, the traces of the stacked tensor and the two targets are \(9\), \(2\), and \(3\), respectively. The omitted four-dimensional zero block has empty trace four, although all its positive-length traces vanish. This explains the positive-length restriction in the fusion rule without changing the trace convention. The source asks for the fusion rules at every length. At the empty chain the periodic operators are the scalars \(O_0(B_1)=2\) and \(O_0(B_\tau )=3\), so \(O_0(B_1)^2=4\neq 2\) and \(O_0(B_1)O_0(B_\tau )=O_0(B_\tau )O_0(B_1)=6\neq 3\): the unit laws of Theorem 31.6.1.2 fail there as well, and the empty chain is excluded as a degenerate reading; see  [ 53 ] .

31.6.5 The categorical anomaly

The Fibonacci fusion ring \(\mathbb {Z}[\tau ]/(\tau ^2-\tau -1)\) has no unital ring homomorphism to \(\mathbb {Z}\): the two possible images of \(\tau \) over \(\mathbb {C}\), \(\varphi =(1+\sqrt5)/2\) and \(-1/\varphi \), are both irrational. This arithmetic fact is the form in which the anomaly of the Fibonacci fusion rules  [ 17 , Section III.A.6 ] enters the argument below. The general notions used here, fusion algebras with a unit, symmetric families of matrix product states, and their nonnegative integer representations, are those of Chapter 30; the multiplicities with which the normal blocks of an invariant state reappear under the symmetry operators form such a representation (Theorem 30.15.8). In the Fibonacci case the same source exhibits an invariant pair of states carrying the regular action: two blocks \(x_1,x_\tau \) with \(\tau \cdot x_1=x_\tau \) and \(\tau \cdot x_\tau =x_1+x_\tau \).

Here the two blocks are concrete state tensors on the two-label physical space of this section: the all-\(\tau \) product tensor and a bond-two tensor generating its image under \(O_\tau \). The explicit tensors are defined on the two-label physical alphabet rather than the three-index alphabet of the source. Their raw periodic vectors carry the regular action. The operator \(O_1\) is the admissibility projector introduced earlier in this section, not the identity on the ambient physical space. Its action as the identity on these states is therefore part of the symmetry statement.

The no-go proved below concerns a single word-normal tensor \(A\) of positive bond dimension whose periodic vectors satisfy \(O_1|V^{(N)}(A)\rangle =|V^{(N)}(A)\rangle \) and \(O_\tau |V^{(N)}(A)\rangle =m|V^{(N)}(A)\rangle \) for a length-independent scalar \(m\) at every \(N{\gt}0\). Then \(m\in \mathbb {N}\) must satisfy \(m^2=m+1\), which has no integer solutions. This is the Fibonacci case of Corollary 30.15.10, a specialization of the statement of  [ 199 ] to one normal tensor with a length-independent scalar action; the source does not give the proof for this tensor. The specialization does not classify families with more than one block or exclude direct sums of the regular representation. It is a non-invertible analogue of the obstruction to an invariant single-block state from a non-trivial three-cocycle discussed in the same source.

The explicit regular action is recorded in Theorem 31.6.5.4, and the single-block obstruction with the hypotheses just stated is Theorem 31.6.5.6. Neither result classifies all invariant families with several blocks.

The pair of states is the all-\(\tau \) product tensor and the chain tensor of Definition 31.6.3.1, and the identities \(O_1\Psi _1=\Psi _1\), \(O_\tau \Psi _1=\Psi _\tau \), and \(O_\tau \Psi _\tau =\Psi _1+\Psi _\tau \) are those of Theorem 31.6.3.4. The content beyond that theorem is the remaining identity \(O_1\Psi _\tau =\Psi _\tau \) and the packaging of all four as the regular representation of the fusion matrices.

The admissibility projector and the chain amplitudes are derived earlier in this section. The action here uses that same two-label physical space and the same unnormalized contractions. Categorical fusion data alone would not establish these operator and state identities or identify them with a construction on a different physical alphabet. As noted after Theorem 31.6.3.4, the identities do not extend to length zero.

The no-go rests on the integrality of a length-independent scalar action on a normal tensor, Theorem 30.15.4 of Chapter 30.

The scalar hypothesis concerns the raw periodic vectors of one fixed tensor at every positive length. The result does not rule out eigenvectors at an individual length, length-dependent eigenvalues, or state tensors that are not word-normal.

31.7 The Ising anyon chain

The Ising fusion category has three simple objects, the vacuum \(1\), the fermion \(\psi \), and the non-Abelian anyon \(\sigma \); its F-symbols define the string-net operators of Bultinck et al.  [ 8 , Appendix D.2 ] . Chains of interacting anyons were introduced for Fibonacci anyons by Feiguin et al.  [ 195 ] , and the algebra of topological defects of the Ising chain is described by Aasen, Mong, and Fendley  [ 1 ] . Matrix product operators realize the corresponding topological symmetries of anyon chains, as reviewed in Ref.  [ 17 , Section III.A.6 ] . The matrix product operator tensors for each simple object are taken in the F-symbol convention of Ref.  [ 8 , Appendix D.2 ] ; the sigma tensor is scaled by \(\sqrt2\) so that every entry lies in \(\mathbb {Z}[\sqrt2]\) and all local computations are exact. Weighted direct sums of these tensors and the weighted Verlinde action on a fusion-tree product state address the question of whether the structure coefficients of a renormalization fixed point can depend on the chain length  [ 13 , Theorem 4.14 ] .

Definition 31.7.1 gives the printed \(G\)-symbol tensor with all quantum-dimension factors. The diagonal bond similarity of Theorem 31.7.1.9 relates it to the \(F\)-symbol blocks used here. The additional factor \(\sqrt2\) on the \(\sigma \) block multiplies its periodic operator by \(2^{N/2}\).

For every \(N{\gt}0\), close the bond indices by the ordinary, unnormalized trace of the ordered product of local matrices. The first physical index is the output and the second is the input.

The vacuum and fermion kernels are identified in Theorem 31.7.1.2; the fermion is a partial isometry whose initial space is the admissible-path subspace (Corollary 31.7.1.3). The fusion identities with the \(\sigma \) block are in Theorems 31.7.1.6 and 31.7.1.8, with the scaling factors retained. The word-span certificates are in Theorem 31.7.2.1. For weighted direct sums, the compression and periodic trace formula are given in Theorems 31.7.5.1 and 31.7.6.1; the weighted action on the state family is in Theorem 31.7.8.6.

The quantum dimensions of \(1\), \(\psi \), and \(\sigma \) are \(1,1,\sqrt2\), respectively; they are the Perron–Frobenius dimensions of the fusion rules below (Example 30.14.3). For every simple object \(a\), the vacuum satisfies \(1\otimes a=a\otimes 1=a\). The remaining fusion rules are

With rows and columns ordered as \((1,\psi )\), the nontrivial F-symbols are

All other allowed components are one, and forbidden components vanish.

The physical site is \(\mathbb {C}^{10}\), with orthonormal basis \(|h\rangle \), \(h=0,\ldots ,9\). A label \(h\) is a locally admissible fusion vertex \((\ell (h),\rho (h),r(h))\), meaning that \(r(h)\) occurs in \(\ell (h)\otimes \rho (h)\). On \(N\) sites the ambient Hilbert space is \(\mathcal H_N=(\mathbb {C}^{10})^{\otimes N}\). For an MPO tensor \(T\), the first physical index is the output index, and its periodic contraction is

Here \(h\) and \(k\) are words of length \(N\), and \(|h\rangle \) denotes the corresponding tensor-product basis vector. Unless stated otherwise, \(N\geq 1\), and site indices are cyclic.

The entries of the three tensors lie in the ring \(\mathbb {Z}[\sqrt2]\), so their local products and changes of auxiliary coordinates can be calculated in this ring before embedding into \(\mathbb {C}\). We fix the ring and its embedding before writing down the tensors.

31.7.1 The periodic operators of the three tensors

The vacuum and fermion tensors are evaluated on closed paths first, where their periodic operators become projections, and the rescaled sigma tensor is evaluated on a product state second.

Closed paths and the vacuum and fermion operators

Local admissibility does not impose compatibility between neighboring vertices. A word \(h\) is a closed path when \(r(h_v)=\ell (h_{v+1})\) at every site, including the last-to-first bond. Define its closure indicator and sign by

The map \(f\) acts on words site by site. The closed-path subspace is \(\mathcal K_N=\operatorname{span}_{\mathbb {C}}\{ |h\rangle :q_N(h)=1\} \).

The kernels show \(\operatorname{range}S_N=\operatorname{range}Q_N=\mathcal K_N\) and \(\ker S_N=\ker Q_N=\mathcal K_N^\perp \). On \(\mathcal K_N\), the operator \(S_N\) is a self-adjoint signed permutation involution.

At one site, the only closed labels are \(0,3,6,7\). Using physical matrix units in \(M_{10}(\mathbb {C})\),

Thus the vacuum label acts as the identity on closed paths, not on the entire physical space. These identities follow from the concrete tensor letters rather than from the abstract fusion rules alone.

The sigma normalization and a product-state action

The rescaling \(\widehat A_\sigma =\sqrt2A_\sigma \) occurs at every site. By multilinearity of the periodic contraction,

All three constant paths in this identity lie in \(\mathcal K_N\). The input is a normalized product state, while the unscaled output has squared norm two. No sigma-square operator identity is needed for this calculation.

Positive length matters here. At length zero, the empty matrix product has trace equal to its auxiliary dimension. Thus \(O_0(A_1)=O_0(A_\psi )=3\) and \(O_0(\widehat A_\sigma )=4\), as operators on a one-dimensional physical space. The projection and product-state identities above do not extend to that case.

The fusion algebra of the three periodic operators

The operator blocks of the Fibonacci string-net satisfy the Fibonacci fusion rules  [ 8 , Appendix D.1 ] , and the Ising operators are built in the same way from the Ising F-symbols  [ 8 , Appendix D.2 ] . For the Ising category the source prints only the fusion rules themselves. The theorems below prove the corresponding operator identities for the tensors of Definition 31.7.4. These tensors use the F-symbols directly, without the square roots of quantum dimensions that the source divides them by; by Theorem 31.7.1.9 this choice does not change the periodic operators  [ 69 ] . The same abstract algebra is realized by the topological defects of the Ising lattice model  [ 1 ] .

Each product is computed by one change of auxiliary coordinates of the product tensor.

Bond similarity for the source tensors

31.7.2 One-site injectivity by exact arithmetic

The embedding of \(\mathbb {Z}[\sqrt2]\) into \(\mathbb {C}\) from Theorem 31.7.3 preserves the matrix-unit spanning criterion of Theorem I.2.4. Here that criterion establishes one-site injectivity, not the spectral-radius-one normalization required of a normal tensor (NT) in the CPSV convention.

31.7.3 The weighted direct sum and its local product

We now apply asymmetric compression to products of these operators. Consider \(\Theta =\lambda _1A_1\oplus \lambda _\psi A_\psi \oplus \lambda _\sigma A_\sigma \). First retain the two summands with labels \(1\) and \(\psi \) and rescale \((\lambda _1,\lambda _\psi )=(1/2,3/10)\) to the integer weights \(5,3\). Write \(\Theta _2=5A_1\oplus 3A_\psi \).

The local product \(\Theta _2\star (\sqrt2A_\sigma )\) represents a product of operators, as in Section 29.6. Here \(\star \) denotes the local MPO product of Definition 31.7.2, distinguished from the action of an MPO on an MPS. Theorem 29.6.2 applies to this product. Section 31.7.8 instead applies the full weighted direct sum to a product state using Corollary 29.6.4. Neither calculation alone establishes a positive vertical decomposition of an MPDO renormalization fixed point.

In the periodic diagrams below, \(L\) denotes the number of sites.

31.7.4 The gauge and the block decomposition

The change of auxiliary coordinates is a signed permutation of the twenty-four coordinates of \(B\). Its first four rows are the F-move isometry for \(1\otimes \sigma =\sigma \), the next four are the F-move isometry for \(\psi \otimes \sigma =\sigma \), and the remaining sixteen form a coordinate complement. The single negative sign is the F-symbol \([F^{\psi \sigma \psi }_\sigma ]=-1\).

31.7.5 Compression onto the \(\sigma \) tensor

In the following diagram, \(C_s=\mu _s\sqrt2A_\sigma \), and \(h\) denotes a single letter of the pair alphabet.

31.7.6 The length-dependent product coefficient

For a tensor \(T\) and a word \(w=(i_1,\ldots ,i_{|w|})\), write \(T^w=T^{i_1}\cdots T^{i_{|w|}}\).

The two-copy space can be written as \(\mathbb {C}^2\otimes P_\sigma \). Let \(\Pi _1\) and \(\Pi _\psi \) be the coordinate projections on \(\mathbb {C}^2\). In the following diagram, the ring denotes the multiplicity operator \(\Theta =5\Pi _1+3\Pi _\psi \), not the direct-sum MPO tensor previously denoted by \(\Theta \). Each \(\sqrt2A_\sigma \) node acts on \(P_\sigma \). On the combined bond space the ring acts as \(\Theta \otimes \mathbb {1}_{P_\sigma }\), and a tensor letter acts as \(\mathbb {1}_2\otimes (\sqrt2A_\sigma )^h\). One weight insertion per site therefore gives

31.7.7 Coefficients from the Ising fusion rules

The power sum \(5^N+3^N\) is one value of a family of structure coefficients built from the Ising fusion multiplicities and one inserted weighted label, which we now define in general. The definition and the theorems of this subsection write \(L\) for the chain length that the surrounding prose calls \(N\).

31.7.8 Action on a fusion-tree product state

We next apply the full weighted direct sum \(\Theta _3=5A_1\oplus 3A_\psi \oplus 2A_\sigma \) to the constant closed path with label \(6=(\sigma ,1,\sigma )\). This is an operator acting on a state, rather than the operator product of Section 31.7.3. The elementary basis action determines the resulting vector; Corollary 29.6.4 describes its local compression.

Both \(Q_N\) and \(S_N\) fix \(|6\rangle ^{\otimes N}\): label \(6\) is closed, fixed by \(f\), and has positive sign. The sigma contribution follows from (177):

The factor \(2^N\) comes from two factors of \(\sqrt2\) at each site, not from a length-independent normalization.

At length zero the action tensor has trace \(10\), whereas its four one-dimensional target blocks have total empty trace \(4\). The difference is the empty trace of the six-dimensional zero block. At positive length that block contributes zero, which explains the nonempty-word and positive-size hypotheses.

31.7.9 Contraction of a matrix-unit factor

The following identities contract a finite matrix-unit factor before any fusion decomposition of the remaining matrices. They apply to every physical letter; no selection of matrix entries is required.

Taking \(|V|=|K|=16\) and \(|I|=4\) gives stacked bond dimension \(4096\), while the matrix remaining after this contraction has dimension \(1024\) and is tensored with a \(4\)-dimensional matrix unit. A fusion decomposition of that remaining matrix is a separate assertion.

31.7.10 The three-object weighted direct sum

Retaining all three hidden bond objects gives four fusion channels, since \(\sigma \otimes \sigma =1\oplus \psi \). The following result concerns the local product of the weighted direct sum with the \(\sigma \) strand. The literal round-45B boundary tensors require the additional register contraction described in  [ 152 ] .

31.8 Group matrix product operators from a three-cocycle

Garre-Rubio, Lootens and Molnár  [ 199 ] build, from a finite group \(G\) and a three-cocycle \(\omega \), a matrix product operator representation of \(G\) on a periodic chain whose sites carry the regular representation \(\mathbb {C}[G]\). Each periodic operator acts as \(U_g=\bigl(\bigotimes _iL_g\bigr)\prod _iW_g^{i,i+1}\) (Theorem 31.8.2), combining on-site left shifts \(L_g\) with diagonal two-site phase gates \(W_g\) determined by \(\omega \); its place among matrix product operator symmetries, and the anomaly carried by the class of \(\omega \), are described in Remark 30.12.3. For cyclic groups, Garre-Rubio and Schuch  [ 200 ] give one explicit cocycle in each cohomology class.

The periodic display of  [ 199 ] places two left shifts on each pair of neighboring sites, which read literally acts on every site by the shift by \(g^2\). The local tensor of that construction, and its example for the group of order two, carry one left shift per site; the operator below follows the tensor, as recorded in  [ 79 ] .

Following  [ 199 ] , write \(\omega _g(l,b)=\omega (g,b,b^{-1}l)\) for the diagonal entries of \(W_g\). By (217), the tensor \(\hat T_g\) copies its input \(b\) to the outgoing bond, weights the pair formed by the incoming bond and the input by \(\omega _g\), and applies \(L_g\) to the input to give the output.

The boundary is the ordinary trace, with no additional boundary matrix. The monomial kernel is given in Theorem 31.8.2. The cocycle equation implies the multiplication law (Theorem 31.8.3), while unit-modulus values imply unitarity (Theorem 31.8.5). For the chosen fusion tensors, the associator is \(\omega ^{-1}\) (Theorem 31.8.1.7). This does not assert that all the tensors are normal: for a nontrivial group and a normalized cocycle, the identity tensor is not normal (Theorem 31.8.1.8).

The product law is the following identity of periodic operators, with \(U_g\) applied after \(U_h\).

The single-site tensors are not injective: their \(|G|^2\) physical matrices span a space of dimension at most \(|G|\). The operator laws above are therefore stated directly rather than through an injective representation.

31.8.1 Fusion tensors and the inverse cocycle

The source equips the tensors \(\hat T_g\) with explicit fusion tensors and states that they realize the inverse cocycle  [ 199 ] . The bond of \(\hat T_g\) carries the input label of the preceding site, so along the stacked product \(\hat T_g\hat T_h\) the bond pair over a label \(l\) of \(\hat T_{gh}\) is \((hl,l)\).

The projector at a periodic cut

For the periodic tensors, the source replaces the right zipper and exact factorization by an identity with a projector on the incoming product bond  [ 199 ] . Write \(B^{ij}=(\hat T_g\hat T_h)^{ij}\), \(T^{ij}=\hat T_{gh}^{ij}\), \(V=F^{\lt}_{g,h}\) and \(W=F^{\gt}_{g,h}\), with the fusion tensors of Definition 31.8.1.1. The support identities hold for every three-cochain \(\omega :G^3\to \mathbb {C}^\times \); neither normalization nor unit modulus is needed.

For example, take \(G=\mathbb {Z}_2\), \(g=h=e\), and \(\omega =1\). At physical indices \((0,0)\), incoming pair \((1,0)\), and fused outgoing label \(0\), the entries of \(B^{00}W\) and \(WT^{00}\) are respectively \(1\) and \(0\). Here \(P_{e,e}\) has diagonal \((1,0,0,1)\), so even this trivial-cocycle example requires the projector. For the empty word, \(\operatorname{tr}(P_{e,e})=2\ne 4=\operatorname{tr}(1)\); the periodic support assertion concerns nonempty chains.

31.8.2 The group of order two and the decorated CZX operator

For \(G=\mathbb {Z}_2=\{ e,s\} \) with the nontrivial cocycle \(\omega _1\) of Definition 31.3.6.1, the source identifies the gate as \(CZ(1\otimes Z)\), the shift as \(X\), and the operator as \(\prod _iCZ_{i,i+1}Z_i\prod _iX_i\)  [ 199 ] . Identify \(e,s\) with the qubit labels \(0,1\) and write \(\bar t\) for the bitwise complement of \(t\in \{ 0,1\} ^N\) and \(q(s)=\sum _is_is_{i+1}\).

Hamiltonians symmetric under the CZX operators

Write \(X_j\) and \(Z_j\) for the Pauli operators on qubit \(j\) of a periodic chain of \(N\geq 2\) qubits, with sites taken modulo \(N\), and \(K_j=Z_{j-1}X_jZ_{j+1}\). The source records the Hamiltonian \(H_{\mathrm R}(\mu )=\sum _iCZ_{i,i+2}X_{i+1}-\mu \sum _iZ_iZ_{i+1}\) of Roose et al. as invariant under the operator \(U=\prod _iCZ_{i,i+1}Z_i\prod _iX_i\) of Theorem 31.8.2.1  [ 199 ] . The review states that the cluster Hamiltonian in a critical transverse field commutes with the operator of the nontrivial three-cocycle, which it writes, as the virtual symmetry of the CZX state, as \(O_Z=\prod _iCZ_{i,i+1}\prod _iX_i\), without the factors \(Z_i\)  [ 17 ] . Let \(H(\lambda )=-\sum _jK_j-\lambda \sum _jX_j\) be the cluster Ising Hamiltonian.

For \(\lambda \neq 0\) the conjugate in item 1 is \(-\sum _jX_j-\lambda \sum _jK_j=\lambda H(\lambda ^{-1})\), so \(O_Z\) exchanges \(H(\lambda )\) with a multiple of \(H(\lambda ^{-1})\).

The review prints no formula for the cluster Hamiltonian in a critical field. The factors \(Z_i\) of \(U\) reverse the sign of the transverse field at the symmetric point: \(O_Z\) fixes \(H(1)\) and \(U\) fixes \(H(-1)\).

31.8.3 The cyclic groups

For \(n\geq 1\) and \(j\in \mathbb {N}\), Garre-Rubio and Schuch  [ 200 , Section IV.D ] write the three-cocycles of \(\mathbb {Z}_n\) as \(\omega _j\) below, with \(j=0,\ldots ,n-1\) labelling the cohomology classes.

Garre-Rubio and Schuch  [ 200 , Section IV.C ] show that interchanging the domain walls created by a symmetry-breaking element \(g\) of order \(o(g)\) multiplies the state by the phase \(\prod _{k=1}^{o(g)}\omega ^{-1}(g,g^k,g)\). Since \(g^{o(g)}=e\) this phase is the inverse of the cyclic invariant \(I_{g,o(g)}(\omega )\) of Definition 19.5.10.42, so it depends only on the class of \(\omega \).

31.8.4 The group \(\mathbb {Z}_2\times \mathbb {Z}_2\)

For \(\mathbb {Z}_2\times \mathbb {Z}_2\) the construction is applied to the mixed three-cocycle of type II whose exponent couples the first factor to the second, the cocycle targeted in Section 31.9. It is the type-II component in the decomposition of the three-cocycles of \(\mathbb {Z}_2\times \mathbb {Z}_2\) printed in  [ 199 , Section 6.1 ] , where it takes the value \(-1\) exactly on the triples \((ab^i,a^jb,a^kb)\) with \(a=(1,0)\) and \(b=(0,1)\). That source does not apply the periodic construction to it.

31.9 The anomalous \(\mathbb {Z}_2\times \mathbb {Z}_2\) symmetry and its condensation defect

The operators \(U_{g,L}\) on a periodic chain of \(L\) two-qubit sites are defined by (224). The construction targets the mixed three-cocycle of type II whose exponent \((-1)^{a_1b_2c_2}\) couples the first factor of the group to the second, motivated by anomalous \(\mathbb {Z}_2\times \mathbb {Z}_2\) symmetry-protected topological phases  [ 10 ] . The four operators do not form an exact representation. After an on-site dressing they do (Theorem 31.9.1.4), and the anomaly class of the dressed representation has the detector values of the type-II class (Theorem 31.9.3.1); in particular it is nontrivial. Matrix product operator symmetries of this kind are reviewed in  [ 17 , Section III.A.6 ] . The local tensors are assembled from the letters of the CZX matrix product unitary  [ 17 , Appendix A, “The CZX MPU” ] , whose symmetry-protected context is  [ 17 , Section III.B.1 ] . The group matrix product operators of  [ 199 ] and the anomalous representations of  [ 200 ] are exact representations, whereas the tensors defined below compose only projectively on odd rings, built from the CZX letters of Section 31.3. The sum \(\mathcal C_L = \sum _{g\in \mathbb {Z}_2\times \mathbb {Z}_2} U_{g,L}\) over the group is a condensation defect in the sense of  [ 231 ] . Throughout this section the chain length is written \(L\) rather than \(N\), because \(N\) denotes the direct-sum tensor that generates the condensation defect.

On a periodic chain of \(L{\gt}0\) two-qubit sites, define

Here \(g=(g_1,g_2)\in G=\mathbb {Z}_2\times \mathbb {Z}_2\), \(t=(t_0,\ldots ,t_{L-1})\) with \(t_n=(t_{n,1},t_{n,2})\in \{ 0,1\} ^2\), and \((g\oplus t)_n=g\oplus t_n\) is componentwise addition modulo two. Site indices are cyclic, so \(t_L=t_0\). The vectors \(|t\rangle \) form the computational basis of \((\mathbb {C}^2\otimes \mathbb {C}^2)^{\otimes L}\). Write \(e=(0,0)\), \(y=(0,1)\), \(x=(1,0)\), and \(xy=(1,1)\). The product \(gh\) denotes \(g\oplus h\). The phase exponent in (224) is evaluated on the input bits.

Let \(X_n^{(r)}\) flip qubit \(r\) at site \(n\), and define the diagonal operator \(D_{2,L}\) by

Then

For \(L\geq 3\), \(D_{2,L}\) is the product of controlled-\(Z\) gates between neighboring second qubits. Thus \(U_{x,L}\) flips the first bits and applies a controlled phase to the second bits; it does not flip the second bits and is not the CZX operator on those qubits. The operator \(U_{xy,L}\) flips both bits, with the diagonal phase acting first. Operator products act from right to left.

The cyclic convention in (224) also specifies the short rings. For \(L=1\), the phase exponent is \(t_{0,2}^2=t_{0,2}\), so \(D_{2,1}=Z^{(2)}\) and \(U_{xy,1}=X^{(1)}X^{(2)}Z^{(2)}\), where \(Z|b\rangle =(-1)^b|b\rangle \). For \(L=2\), the exponent is \(2t_{0,2}t_{1,2}\), so \(D_{2,2}=\mathbb {1}\). There is one self-loop at length one and two cyclic bonds at length two; these are not replaced by the edges of a simple graph.

The local tensors are matrix product operator tensors whose first physical index is the output, or matrix row, and whose second is the input, or matrix column.

Every entry lies in \(\{ 0,\pm 1\} \). The matrices \(T_0,T_1\) are those of the bond-two symmetry tensor in Section 31.3, but the physical output constraints are different. The tensor \(M_{xy}\) is obtained from \(M_x\) by relabeling the outgoing physical index as \(o\mapsto o\oplus y\). The boundary convention is the ordinary, unnormalized trace:

There are no open boundary vectors.

Theorem 31.9.3 establishes the word spans, with one-site injectivity for the bond-one tensors and two-site injectivity for the bond-two tensors. The multiplication phases are retained in Theorem 31.9.1.2. The dressed exact representation is given in Theorem 31.9.1.4, and its nontrivial anomaly detectors in Theorem 31.9.3.1. These detector values concern the dressed family; they do not identify the full cohomology class of the original family.

The first property is that the periodic contraction of each tensor reproduces the operator of (224).

Equation (232) also fixes the gate order: \(D_{2,L}\prod _nX_n^{(1)}X_n^{(2)} =(-1)^LU_{xy,L}\). At length zero, the trace of the empty product is \(D_g\). The bond-one tensors therefore give the scalar \(1\), while the bond-two tensors give \(2\), not a unitary operator. All unitarity and group-multiplication assertions here concern positive lengths.

The basis action also determines multiplication. For \(g,h\in G\),

The last equality uses (232) and the fact that the parity of \(g_1+h_1\) is the first bit of \(gh\). Thus \(\lambda (g,h)=(-1)^{g_1h_2}\) supplies a length-dependent scalar factor, reflecting the non-trivial projective 2-cocycle class of \(\mathbb {Z}_2\times \mathbb {Z}_2\)  [ 10 , 199 ] . The tensor proof of this law, through word traces, is given in Theorem 31.9.1.2.

The second property concerns the word spans of the four tensors, which the compression results of Section 31.9.2 require.

31.9.1 The condensation defect and its periodic fusion table

The condensation defect is the operator \(A_L=U_{e,L}+U_{x,L}+U_{y,L}+U_{xy,L}\). It is generated by the direct-sum tensor \(N=M_e\oplus M_x\oplus M_y\oplus M_{xy}\) of Definition 16.1.8, with bond dimension six.

The multiplication law (233) of the four operators is now derived from the tensors themselves, through word traces of their stacked products.

The four operators are related to the group matrix product operators of Section 31.8 for the mixed cocycle of Definition 31.8.4.1 by an on-site dressing. Identify the two-qubit label \(2i_1+i_2\) with the group element \((i_1,i_2)\), so that each site carries \(\mathbb {C}[\mathbb {Z}_2\times \mathbb {Z}_2]\), and write \(U^{\omega }_{g,L}\) for the periodic operator of that construction on a ring of \(L\) sites.

31.9.2 The sixteen pair blocks of the stacked defect

The operator identities of the fusion table have word-trace counterparts, to which the multi-block compression theorem applies. The stacked product \(B=N\cdot N\) has local matrices

Its bond dimension is thirty-six, and its periodic operator is \(A_L^2\) for \(L{\gt}0\). After grouping bond indices by \((g,h)\in G^2\), it is block diagonal with sixteen blocks \(B_{g,h}=M_g\cdot M_h\), each generating \(U_{g,L}U_{h,L}\).

For twelve of the sixteen pairs, the stacked tensor is conjugate to its target by an invertible integer matrix, with no additional zero diagonal blocks. These are the split pairs. Ten of them, with at most one factor from \(\{ x,xy\} \) and with \(\lambda (g,h)=+1\), are equal to their targets. The two pairs \((x,y)\) and \((xy,y)\) have weight \(-1\) and are conjugate to \(-M_{xy}\) and \(-M_x\), respectively, by \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\). In the following diagram, \(B\) denotes one such pair block and \(s=(g,h)\).

The remaining four pairs are the ones with both factors of bond dimension two. Their stacked blocks have bond dimension four but targets of bond dimension one, so the compression must discard three bond directions.

The sixteen pair-block compressions assemble into one compression of the full stacked tensor.

31.9.3 The anomaly class of the full group

By Theorem 31.9.1.2 the four operators \(U_{g,L}\) compose only projectively on odd rings, and no rescaling of the four tensors removes the signs, since \(\lambda (x,y)\neq \lambda (y,x)\). The dressed tensors \(M'_g\) of Theorem 31.9.1.4 form an exact representation by normal tensors, so the anomaly three-cochain of Definition 19.5.1.3 is defined for them. According to  [ 199 ] , the values \(\omega (g,g,g)\) in a normalized gauge at the three elements of order two distinguish the eight classes of \(H^3(\mathbb {Z}_2\times \mathbb {Z}_2,\mathbb {C}^\times )\cong \mathbb {Z}_2^3\), and the type-II class has the values \((+1,+1,-1)\) at \(x\), \(y\) and \(xy\). The theorem below states these detector values for the dressed representation. Theorem 31.9.4.16 proves that the three products distinguish all cohomology classes. The distinction between the dressed representation and the original odd-ring projective family remains as described in  [ 115 ] .

31.9.4 The eight scalar representatives

31.10 The anomalous \(\mathbb {Z}_3\) symmetry and its condensation defect

A finite group acting on a spin chain by matrix product operators may be anomalous (Remark 30.12.3): the obstruction to gauging it is a three-cocycle, and for a cyclic group Garre-Rubio and Schuch  [ 200 ] give the cocycle in closed form together with the interchange statistics of the domain walls it governs. The phase-decorated shift defined below realizes their cocycle formula for \(\mathbb {Z}_3\) by an explicit local tensor; its anomaly class is computed in Theorem 31.10.2.4. The general construction of a matrix product operator representation from a finite group is given in Garre-Rubio, Lootens and Molnár  [ 199 ] , and matrix product operator symmetries are reviewed in  [ 17 , Section III.A.6 ] . The sum \(\mathcal C_N = \mathbb {1}+ U_N + U_N^\dagger \) of the three group operators is a condensation defect in the terminology of Roumpedakis, Seifnashri and Shao  [ 231 ] .

Let \(\omega =(-1+\mathrm{i}\sqrt{3})/2\) be the primitive cube root of unity. Thus \(\omega ^2+\omega +1=0\), \(\omega ^3=1\), and \(\overline{\omega }=\omega ^{-1}=\omega ^2\). On a periodic chain of \(N{\gt}0\) qutrits, use basis vectors \(|t\rangle \) indexed by \(t=(t_0,\ldots ,t_{N-1})\in \{ 0,1,2\} ^N\). Set \(\chi (a)=[a\neq 0]\) and

where \(t_N=t_0\) and the exponent uses the integer representatives \(0,1,2\). Write \(X|a\rangle =|a+1\rangle \), with addition modulo three, and \((Xt)_j=t_j+1\). The phase-decorated shift acts by

Thus \(U_N = X^{\otimes N}\operatorname {diag}(\omega ^{F(t)})\). Every cyclic bond is included: at \(N=1\) there is a self-loop, and at \(N=2\) both oriented bonds contribute.

The group-cocycle context for anomalous symmetries is discussed in Garre-Rubio and Schuch  [ 200 ] ; see also Sun  [ 241 ] .

Theorem 31.10.2 identifies the periodic kernels and unitarity. The word-span statements, including injectivity after two-site blocking, are in Theorem 31.10.5; the exact group law is in Theorem 31.10.1.4. The nontrivial anomaly is established by the cyclic invariant in Theorem 31.10.2.4. The condensation operator satisfies \(\mathcal C_N^2=3\mathcal C_N\) for every \(N{\gt}0\) (Theorem 31.10.3.5).

31.10.1 Fusion of the anomalous representation

Write \(M_0=\delta \), \(M_1=M\) and \(M_2=M^\dagger \), with group indices added modulo three. The physical calculations give \(O_N(M_g)O_N(M_h)=O_N(M_{g+h})\) for every \(N{\gt}0\). The stacked product tensor has entries

Its bond dimension is the product of the two bond dimensions, and its periodic operator is the physical product (Theorem 29.6.2). Consequently, the group law gives equality of nonempty-word traces before any compression pair is chosen. The asymmetric reductions realizing these equalities on the virtual spaces are described next. In the statements and diagrams below, \(L\) denotes the ring length, as \(N\) did above.

The four nontrivial products have targets \(M^\dagger \) for \(M\cdot M\), \(\delta \) for \(M\cdot M^\dagger \) and \(M^\dagger \cdot M\), and \(M\) for \(M^\dagger \cdot M^\dagger \). The compressions come first, and the periodic fusion rules follow from them.

Stacking \(M\) over \(M\) and compressing gives the first fusion rule, \(U_L^2=O_L(M^\dagger )\):

31.10.2 The group family

Garre-Rubio and Schuch  [ 200 , Section IV.A ] define a matrix product unitary representation of a finite group by injective tensors whose periodic operators satisfy \(U_gU_h=U_{gh}\) and \(U_e=\mathbb {1}\). The three tensors above, indexed by \(\mathbb {Z}_3\), satisfy these operator laws; injectivity holds after blocking two sites.

The periodic construction of Section 31.8, applied to the cocycle \(\omega _1\) of \(\mathbb {Z}_3\) from Definition 31.8.3.1, gives a second exact representation of \(\mathbb {Z}_3\), of bond dimension three. The two families are conjugate by one diagonal unitary built from on-site and two-site gates (Theorem 31.10.2.3).

Reading off the two terms of (273), \(V_N=D^{\otimes N}\prod _kC_{k,k+1}\), where \(D=\operatorname {diag}(1,1,\omega )\) and the commuting two-site gate \(C\) multiplies \(|1,2\rangle \) by \(\omega \) and fixes the other basis vectors, so \(V_N\) is a circuit of depth two. No purely on-site diagonal gauge relates the two families. A diagonal unitary \(\prod _k\omega ^{w(t_k,t_{k+1})}\) changes the exponent of the shift by \(\sum _k\bigl(w(t_k+1,t_{k+1}+1)-w(t_k,t_{k+1})\bigr)\). For an on-site gate \(w(a,b)=\alpha (a)\) this is a sum of single-site terms, whereas the difference \([a\neq 0]\, b-[a{\lt}b]\) of the two bond exponents is not, modulo three, the sum of a function of \(a\) and a function of \(b\), which it would have to be for the exponents to agree on every chain. What relates the two families is the identity \(O_N(T_g)=V_N\, U^{\omega _1}_g\, V_N^\dagger \) of Theorem 31.10.2.3, with \(V_N=D^{\otimes N}\prod _kC_{k,k+1}\) a product of commuting diagonal gates. The class of Theorem 31.10.2.4 is computed from fusion tensors of the family \(\{ T_g\} \). The construction for \(\omega _1\) has fusion tensors of its own, and they give the same cyclic product (Theorem 31.10.2.6), as invariance of the class under the conjugation by \(V_N\) would predict; that invariance is not proved here.

Both  [ 200 ] and  [ 199 ] define the anomaly by the same move between the two fusion trees of \((g,h,k)\) as the three-cochain used here. The construction of  [ 199 ] from a cocycle \(\omega \) carries \(\omega ^{-1}\), as that source states for its fusion tensors (Theorem 31.8.1.7). Built from \(\omega _1\) it therefore realizes \(\omega _2=\omega _1^{-1}\), the class of the family, and not \(\omega _1\): the orientation is a property of the construction, not a discrepancy between the two definitions of the anomaly.

31.10.3 The condensation defect

The condensation defect of the anomalous symmetry is the sum of its three group operators, \(\mathbb {1}+U_N+U_N^\dagger \), in the terminology of  [ 231 ] . It is realized by the direct sum of the three tensors. On the physical side the defect squares to three times itself; on the virtual side the stacked square splits into nine pair blocks, each compressing onto one group tensor.

The physical identity has a corresponding virtual decomposition. The stacked square has nine pair blocks, each reducing to one of the three group tensors. The asymmetric compression records these nine targets separately.

31.11 The \(\mathbb {Z}_3\) clock symmetry

The anomalous symmetry of Section 31.10 is an exact representation of \(\mathbb {Z}_3\), and its condensation defect satisfies \(A^2=3A\) at every length (Theorem 31.10.3.5). This section treats a second family of \(\mathbb {Z}_3\) symmetry operators whose representatives multiply only up to phases depending on the length, and shows that the structure constants of its condensation defect then genuinely depend on the length. The length dependence comes from the per-site phases of the chosen representatives. The expected triviality of the Else–Nayak class remains to be established for these dressed operators; the trivial cocycle in Theorem 31.11.2 alone does not identify that class.

As in Section 31.10, let \(\omega =\exp (2\pi \mathrm{i}/3)\), let \(X|s\rangle =|s+1\rangle \) on a qutrit, and let \(CZ|a,b\rangle =\omega ^{ab}|a,b\rangle \) and \(Z|a\rangle =\omega ^a|a\rangle \). On a periodic chain of \(N{\gt}0\) qutrits the clock operators are

The clock tensor has the shape of the group tensor of Definition 31.8.1: it copies the input \(i\) to the outgoing bond, weights the pair formed by the incoming bond and the input by \(\omega ^{f(l,i)}\), and applies \(X^g\) to the input to give the output.

Both tensors use the ordinary trace on rings of \(N{\gt}0\) sites. Theorem 31.11.2 identifies their kernels and proves unitarity. The multiplication table retains its length-dependent phases (Theorem 31.11.3); the resulting square of the condensation defect and the obstruction to length-independent fusion constants are given in Theorems 31.11.6 and 31.11.7. The anomaly-class question remains separate from these operator identities.

The first relation of the next theorem, \(U_N^2=\omega ^NV_N\), reads as follows on a ring of \(N\) sites.

31.12 Mixed Bell bonds and renormalization fixed points

A renormalization fixed point of a matrix product density operator is a tensor whose one-site and two-site closures with an arbitrary boundary matrix are carried into each other by trace-preserving completely positive maps, in both directions  [ 13 , Definition 4.1, Example 4.12, Proposition 4.13, Theorem 4.14 ] . The characterization there expresses the products of the vertical components of such a tensor through structure coefficients, and the question left open after its Theorem 4.14 is whether these coefficients can depend on the length of the chain. The mixed Bell-bond tensor, the flagged tensor, the one-label tensor, and the dimer twist below address that question. They are not taken from the literature; only the flag factor of the flagged tensor is, namely the normalized tensor of [ 13 , Example 4.12 ] . The first two are mixed states built from Bell bonds: in the first, incoming bonds carry the mixed state \(\beta _R=(25/32)P_++(7/32)P_-\); in the second, each site also carries a flag qubit \(|f\rangle \) (\(f\in \{ 0,1\} \)), with a controlled flip coupling the Bell sectors to an even-parity flag state. The vertical components of these two tensors yield the one-label tensor and the dimer twist, whose structure coefficients are computed explicitly by the multi-block compression theorem. These results establish tensor decompositions and periodic trace identities for the examples, not a classification of length-dependent fixed points.

The density formulas are (292) and Theorem 31.12.2.1. The channel equations, including arbitrary boundary matrices, are proved in Theorem 31.12.3.1; their necessary physical-trace identity is Theorem 31.12.3.2. The one-label and dimer word-span certificates are Theorems 31.12.6.2 and 31.12.7.3, with compression and trace identities in Theorems 31.12.6.3, 31.12.6.4, 31.12.7.5, and 31.12.7.6.

Write \(E_{ij}=|i\rangle \! \langle j|\) and \(|\Phi _\pm \rangle =(|00\rangle \pm |11\rangle )/\sqrt2\), with \(P_\pm =|\Phi _\pm \rangle \! \langle \Phi _\pm |\). For an MPO tensor \(M^{ij}\in M_{D}(\mathbb {C})\), its periodic physical operator is

An MPDO requires positivity for every \(N{\gt}0\); trace-one normalization is a separate condition. Throughout the physical state formulas below, \(N{\gt}0\) and site indices are cyclic.

31.12.1 Independent mixed Bell bonds

The local physical space is \(\mathcal H_R=\mathbb {C}^2_L\otimes \mathbb {C}^2_R\), with basis index \(2l+r\). On the incoming bond \((R_{n-1},L_n)\), put \(\beta _R=(25/32)P_++(7/32)P_-\). Let \(Q_N\) regroup sites into incoming bonds:

Thus \(Q_N^\dagger \beta _R^{\otimes N}Q_N\) is a normalized density operator of rank \(2^N\).

An MPO representation uses four real matrices:

The auxiliary indices \(a,b\in \{ 0,1,2,3\} \) label these matrices; their ordering is not the physical pair ordering \(2l+r\). Define

To contract this tensor, introduce \(w=\left(\begin{smallmatrix} 16/25 & 9/25 \\ 9/25 & 16/25 \end{smallmatrix}\right)\). The matrix-unit entries give

For physical strings \(p_n=(l_n,r_n)\) and \(q_n=(l'_n,r'_n)\), the auxiliary trace pairs the right half of site \(n-1\) with the left half of site \(n\). Consequently,

This also gives a support-isometry description. Define \(B_N|a_0,\ldots ,a_{N-1}\rangle =\bigotimes _n|a_n,a_{n+1}\rangle \). Distinct bit strings have distinct images, so \(B_N^\dagger B_N=\mathbb {1}\). Set \(h=(25/32)w= \left(\begin{smallmatrix} 1/2 & 9/32 \\ 9/32 & 1/2 \end{smallmatrix}\right)\). Formula (290) becomes

The eigenvectors of \(h\) are \((1,\pm 1)/\sqrt2\), with eigenvalues \(25/32\) and \(7/32\). For \(J|a\rangle =|a,a\rangle \), one has \(JhJ^\dagger =\beta _R\) and \(Q_NB_N=J^{\otimes N}\). Hence

This proves positivity, trace one, and rank \(2^N\) directly. At \(N=1\) the bond closes within the site, giving \(\rho ^{(1)}(R)=\beta _R\). At \(N\ge 2\) the two qubits at a site belong to distinct independent bonds. Since both one-qubit marginals of \(\beta _R\) are \(\mathbb {1}_2/2\), the one-site marginal is \(\mathbb {1}_4/4\).

31.12.2 Mixed Bell bonds with an even-parity flag

The second model has local space \(\mathcal H_T=\mathbb {C}^2_L\otimes \mathbb {C}^2_R\otimes \mathbb {C}^2_F\), with basis index \(4l+2r+f\). Put \(x=7/8\), \(y=1/8\), and define the bond state \(\sigma =xP_++yP_-\). The flag state is the classical even-parity mixture

where \(Z=E_{00}-E_{11}\). There are \(2^{N-1}\) even strings, so \(\eta _N\) is positive and has trace one. For \(N\ge 2\) its rank is \(2^{N-1}\); it is not a pure GHZ state. At \(N=1\) it is \(|0\rangle \! \langle 0|\).

A cell consists of the incoming bond and its receiving flag. With \(X_F=E_{01}+E_{10}\) on the flag, define

This gate fixes the flag outside the minus Bell sector and flips it inside that sector. Orthogonality of \(P_-\) and \(\mathbb {1}_4-P_-\) gives \(V^\dagger =V\) and \(V^2=\mathbb {1}_8\), including on the complement of the two Bell vectors. Let \(\widetilde Q_N\) regroup sites into incoming cells by \(\bigotimes _n|l_n,r_n,f_n\rangle \mapsto \bigotimes _n|(r_{n-1},l_n),f_n\rangle \). The physical state is obtained by applying \(V\) on each cell to \(\sigma ^{\otimes N}\otimes \eta _N\). Here and below, the separated bond and flag factors are identified with cell coordinates by the canonical permutation of tensor factors.

For its MPO representation, set \(\tau _k(f)=(-1)^{kf}\) and \(C_k(u,v)=\tfrac 12(x+(-1)^{k+u+v}y)\). Explicitly,

An auxiliary index is \((p,p',k)\), encoded as \(4p+2p'+k\). Define

Both the physical and auxiliary dimensions are eight.

The factorization (298) is an identity of closed operators, not an on-site tensor equivalence or an auxiliary gauge equivalence. The positive-length restriction matters for both models: at length zero the auxiliary trace gives \(\rho ^{(0)}(R)=4\) and \(\rho ^{(0)}(T)=8\), whereas the corresponding empty factor expressions give \(1\) and \(2\).

31.12.3 Trace-preserving fixed-point maps

For a tensor \(M\) and an auxiliary boundary matrix \(X\), define the one-site and two-site physical closures by

The channel fixed-point condition of [ 13 , Definition 4.1 ] requires trace-preserving completely positive maps, abbreviated tpCPMs, \(\mathcal S\) and \(\mathcal T\) such that, for every \(X\),

The symbol \(T\) denotes the physical dimer tensor, whereas \(\mathcal T\) denotes a refinement channel. Closed density factorizations alone do not establish these equations for arbitrary boundary matrices.

This is a statement about the physical-trace transfer matrix \(F_M\), not the doubled-index completely positive transfer map. For \(R\), the identity also follows directly from \(F_R=(25/32)tt^{\mathsf T}\), where \(t=(4/5,4/5,0,0)^{\mathsf T}\) and \(t^{\mathsf T}t=32/25\).

31.12.4 Vertical components and their multiplication

Rotation exchanges the physical and auxiliary pairs: \(\mathcal V(M)^{ab}_{ij}=M^{ij}_{ab}\). For a vertical component \(K\), its periodic operator at length \(L{\gt}0\) is

It acts on strings of the original auxiliary indices, rather than on the original physical chain.

For \(R\), the normalized vertical component is \(K^{ab}=A^a\otimes A^b\), so \(\mathcal V(R)=(25/32)K\). This rotation and scalar factor are essential: \(K\) is not the physical tensor \(R\). If its indices \(a,b\) are instead treated as physical indices, then

Its \((00,00)\) entry satisfies \((F_K^2-F_K)_{00,00}=(337-400)/625=-63/625\). Theorem 31.12.3.2 therefore excludes the channel fixed-point property for \(K\) in this physical reading.

Writing \(|\psi _L\rangle =\sum _{\boldsymbol a} \operatorname{tr}(A^{a_0}\cdots A^{a_{L-1}})|\boldsymbol a\rangle \) gives \(O_L(K)=|\psi _L\rangle \! \langle \psi _L|\). A nonzero cyclic product of the matrix-unit letters specifies a cyclic bit path, and its squared amplitude is the product of the corresponding entries of \(w\). Thus \(\langle \psi _L | \psi _L \rangle =\operatorname{tr}(w^L)\), and

The eigenvalue \(7/25\) controls vertical multiplication; it is not the physical Bell probability \(7/32\).

For \(T\), restrict the physical ket and bra flags to the same value \(f\). The unnormalized vertical component \(D_f\) is defined by

It satisfies \(D_f=\nu \widehat K_f\), where \(\widehat K_f\) is the normalized flag component, \(\nu =5/8\), and \(\nu ^2=(x^2+y^2)/2=25/64\). For \(a=(p,p',k)\) and \(b=(q,q',k')\), its normalized letter is

The nonzero coefficient magnitudes of \(D_f\) are \(1/4,3/16\), whereas those of \(\widehat K_f\) are \(2/5,3/10\). Neither component is the physical tensor \(T\).

For an MPDO in CPSV canonical form, the fixed-point characterization requires a vertical canonical decomposition with a basis of normal tensors \(K_\alpha \), positive diagonal multiplicity matrices \(\mu _\alpha \), and positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\). Set \(m_\alpha =\operatorname{tr}\mu _\alpha \) and \(c_{\alpha ,\beta ,\gamma }^{(L)}=\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^L)\). The required multiplication identities, for every \(L{\gt}0\), are

The vector \(m\) is an idempotent for the multiplication induced by \(c^{(1)}\):

For \(R\), \(\mu _0=(25/32)\) and \(\chi _{000}=\operatorname{diag}(1,7/25)\); indeed \((32/25)(25/32)^2=25/32\). For the normalized dimer components, \(m_0=m_1=5/8\). The fusion calculation below gives unnormalized weights \(7/16,1/16\), which become \(7/10,1/10\) after division by \(\nu \). Each output sector receives two contributions of each kind in (310), giving \(2(7/10+1/10)(5/8)^2=5/8\).

31.12.5 Representative rescaling

Scalar rescalings of the normal tensors in a basis change the structure coefficients by geometric factors, so the length-dependence question must be posed modulo this freedom; the entries below record the rescaling rule for the structure coefficients of [ 13 , Theorem 4.14 ] .

Consider the structure coefficients \(c_{\alpha ,\beta ,\gamma }^{(L)}=\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) and trace scalars \(m_\alpha \) of [ 13 , Theorem 4.14(ii) ] .

Scaling a normal-tensor representative by a positive factor \(c \ne 1\) alters the transfer map’s spectral radius to \(c^2\), though the rescaled idempotent coefficient equations remain preserved.

31.12.6 The one-label fixed point

The two remaining subsections apply the multi-block compression theorem to the vertical components of \(R\) and \(T\), in the unnormalized representatives fixed above. The adapted basis of the inner bond and the reordering behind the factorization \(B^a=\sigma \otimes M^a\) used below are set out in Section I.2, together with the two supporting results the proofs invoke, Theorems I.2.3 and I.2.4.

The vertical component \(K\) of \(R\) has letters \(K^{ij}=A^i\otimes A^j\), with \(A\) given in (287). Its stacked product has letters \(A^i\otimes F_K\otimes A^k\). The inner matrix \(F_K=\sum _jA^j\otimes A^j\) has eigenvalues \(1,7/25,0,0\), with eigenvectors \(\Phi _+,\Phi _-,01,10\). Thus the stacked tensor contains two weighted copies of the same target, not two inequivalent targets. In this subsection write \(M=K\); it remains a vertical component, not the physical tensor \(R\).

The following schematic records the inner spectral factors. Its \(\sigma \) denotes the inner matrix \(\Pi _0+(7/25)\Pi _1\), not a trace-one physical density. After reordering, \(B^a=\sigma \otimes M^a\): the spectral factor acts on the inner bond and \(M^a\) on the outer bond. In the alternating drawing, each \(M\) node means \(\mathbb {1}_4\otimes M^a\) and each spectral ring means \(\sigma \otimes \mathbb {1}_4\), both on the full bond space. Their product is \(B^a\). Only the virtual legs are closed; the physical pair legs remain open.

31.12.7 The graded quantum-dimer twist

For the dimer, write \(M_f=D_f\) for the unnormalized vertical components of (308). Their matrix space acts on \(\mathbb {C}^2\otimes \mathbb {C}^2\); their physical letter indices are the horizontal auxiliary pairs of \(T\). The matrices \(C_k\) are the two-by-two matrices in (296), equivalently the restrictions of \(xP_++(-1)^kyP_-\) to the matched bond subspace.

Stacking sectors \(f,f'\) gives targets \(M_{f+f'}\) and \(M_{f+f'+1}\), with addition modulo two, and weights \(x/2=7/16\) and \(y/2=1/16\). Unlike the one-label example, these contributions multiply different targets. For sector zero multiplied by itself, \(c_{000}^{(L)}=(7/16)^L\) and \(c_{001}^{(L)}=(1/16)^L\) in this unnormalized component convention. These are the trace powers of \(\chi _{000}=(7/16)\) and \(\chi _{001}=(1/16)\). The normalized BNT coefficients instead use \(7/10,1/10\), as explained in Subsection 31.12.4.

The inner idempotents select different outer target sectors. The exact reordered letter identity is

For fixed \(f,f'\), define matrices on the full inner–outer bond space by

Orthogonality of the idempotents gives \(B_{f,f'}^a=\mathcal M^a\Sigma =\Sigma \mathcal M^a\). The alternating ring therefore represents the periodic contraction of these stacked letters. Its spectral factor \(\Sigma \) is not the physical Bell density \(\sigma \) in (298).