32 Matrix Product Operator Algebras

A matrix product operator algebra is a finite family of matrix product operators satisfying the fusion rule (96) with structure constants $N_{ab}^c\in \mathbb {N}$ [ 8 , 199 , 222 ] . Chapter 30 introduces these algebras and their non-invertible examples, defines symmetric matrix product states, and proves the integrality of their action coefficients. Transfer of the symmetry to the virtual level rests on the multi-block asymmetric compression theorem (Theorem 29.5.1): the stacked tensor of a product of two matrix product operators, or of an operator acting on a state, need not be in canonical form and in general admits no sitewise intertwiner with the tensors of the fusion products (Remark 29.1.2), so the Fundamental Theorem for tensors in canonical form does not apply to it in general; the asymmetric theorem, which requires equality of periodic traces only, does.

Throughout, $T_a$ denotes a matrix product operator tensor of bond dimension $\chi _a$, $O_a^{(N)}$ its periodic operator on $N$ sites, and $|V^{(N)}(A)\rangle $ the periodic vector of a matrix product state tensor $A$. For two operator tensors, $T_aT_b$ denotes the stacked tensor $(T_aT_b)^{ij}=\sum _mT_a^{im}\otimes T_b^{mj}$ of bond dimension $\chi _a\chi _b$, and $T_a\cdot A$ denotes the tensor $(T_a\cdot A)^i=\sum _jT_a^{ij}\otimes A^j$. A reduction of a tensor $B$ onto a tensor $C$ is a pair of maps $(V,W)$ with $VW=\mathbb {1}$ and $VB^wW=C^w$ for every word $w$ (Definition 2.4.5), with compression $V$ and embedding $W$.

32.1 Fusion tensors and action tensors

Equality of periodic traces in the fusion rule (96) decomposes the stacked tensor $T_aT_b$ into copies of the blocks $T_c$.

Definition 32.1.1 Fusion tensors

Let $(T_a)_{a\in I}$ be a fusion algebra of normal tensors of positive bond dimension. A family of fusion tensors for the pair $(a,b)$ consists of maps

\begin{align} V_{ab}^{c,\mu } & \colon \mathbb {C}^{\chi _a}\otimes \mathbb {C}^{\chi _b}\to \mathbb {C}^{\chi _c}, & W_{ab}^{c,\mu } & \colon \mathbb {C}^{\chi _c}\to \mathbb {C}^{\chi _a}\otimes \mathbb {C}^{\chi _b}, \label{eq:mpoalg_fusion_maps} \end{align}

for $c\in I$ and $1\leq \mu \leq N_{ab}^c$, such that the pairs are mutually biorthogonal, $V_{ab}^{c,\mu }W_{ab}^{d,\nu }=\delta _{cd}\delta _{\mu \nu } \mathbb {1}_{\chi _c}$, each pair is a reduction of $T_aT_b$ onto $T_c$,

\begin{align} V_{ab}^{c,\mu }(T_aT_b)^{\mathbf w}W_{ab}^{c,\mu } & =T_c^{\mathbf w} \label{eq:mpoalg_fusion_reduction} \end{align}

for every word $\mathbf w$ in the pair alphabet, and the remainder $(T_aT_b)^{ij}-\sum _{c,\mu }W_{ab}^{c,\mu }T_c^{ij}V_{ab}^{c,\mu }$ generates a nilpotent algebra. This is the decomposition of [ 199 , Section 2 ] and [ 8 ] , with the nilpotent remainder of [ 199 , Section 4.4 ] in place of an exact identity.

$\begin{tenkz}[rows={op,op}, cols=5] \tnfuse[at=(1,1), name=V, skin=pill]{V_{ab}^{c,\mu}} \tn[at=(1,2), skin=mpo, ports={90:physical:$i_1$, 270:physical}]{T_a} \tn[at=(2,2), skin=mpo, ports={90:physical, 270:physical:$j_1$}]{T_b} \tn[at=(1,3), skin=dots, ports={180:virtual, 0:virtual}]{} \tn[at=(2,3), skin=dots, ports={180:virtual, 0:virtual}]{} \tn[at=(1,4), skin=mpo, ports={90:physical:$i_n$, 270:physical}]{T_a} \tn[at=(2,4), skin=mpo, ports={90:physical, 270:physical:$j_n$}]{T_b} \tnfuse[at=(1,5), name=W, skin=pill]{W_{ab}^{c,\mu}} \tnwire{open w}{V.180} \tnwire{W.0}{open e} \end{tenkz}$ $=$ $\begin{tenkz}[rows={op}, cols=3, west=open, east=open] \tn[skin=mpo, ports={90:physical:$i_1$, 270:physical:$j_1$}]{T_c} & \tn[skin=dots, ports={180:virtual, 0:virtual}]{} & \tn[skin=mpo, ports={90:physical:$i_n$, 270:physical:$j_n$}]{T_c} \end{tenkz}$

Theorem 32.1.2 Existence of fusion tensors

Every pair $(a,b)$ of labels of a fusion algebra of normal tensors of positive bond dimension admits a family of fusion tensors (Corollary 29.6.1.2). This is the periodic-trace reduction under the fusion rule (96). It allows a nilpotent remainder and does not assert the exact local reconstruction of [ 199 , Appendix A ] . The exact fusion and action decompositions from length-independent arbitrary-boundary hypotheses are proved in Theorem 30.2.8.

Proof ▶

Apply Theorem 29.6.2 with the target family consisting of $N_{ab}^c$ copies of $T_c$ for each $c$, all with weight $1$. Its compression pairs, indexed by $(c,\mu )$, are mutually biorthogonal reductions with a nilpotent remainder; this is Theorem 29.6.1.1 for the stacked product.

For a group-like algebra, with labels in a group $G$ and $N_{gh}^k=\delta _{k,gh}$, the fusion tensors reduce to one pair $(V_{g,h},W_{g,h})$ for each pair of group elements; for representations by normal tensors this is Corollary 29.6.3.

Definition 32.1.3 Action tensors

Let $(A_x)_{x\in K}$ be a symmetric family of normal tensors of positive bond dimension whose coefficients $M_{a,x}^y$ are nonnegative integers. A family of action tensors for $(a,x)$ consists of maps $V_{ax}^{y,i}\colon \mathbb {C}^{\chi _a}\otimes \mathbb {C}^{D_x}\to \mathbb {C}^{D_y}$ and $W_{ax}^{y,i}\colon \mathbb {C}^{D_y}\to \mathbb {C}^{\chi _a}\otimes \mathbb {C}^{D_x}$, for $y\in K$ and $1\leq i\leq M_{a,x}^y$, that are mutually biorthogonal reductions of $T_a\cdot A_x$ onto the blocks $A_y$ with a nilpotent remainder.

$\begin{tenkz}[rows={op,ket}, cols=5] \tnfuse[at=(1,1), name=V, skin=pill]{V_{ax}^{y,i}} \tn[at=(1,2), skin=mpo, ports={90:physical:$s_1$}]{T_a} \tn[at=(2,2), label pos=s, ports={180:virtual, 0:virtual}]{A_x} \tn[at=(1,3), skin=dots, ports={180:virtual, 0:virtual}]{} \tn[at=(2,3), skin=dots, ports={180:virtual, 0:virtual}]{} \tn[at=(1,4), skin=mpo, ports={90:physical:$s_n$}]{T_a} \tn[at=(2,4), label pos=s, ports={180:virtual, 0:virtual}]{A_x} \tnfuse[at=(1,5), name=W, skin=pill]{W_{ax}^{y,i}} \tnwire{open w}{V.180} \tnwire{W.0}{open e} \end{tenkz}$ $=$ $\begin{tenkz}[rows={ket}, cols=3, west=open, east=open] \tn[label pos=nw, ports={90:physical:$s_1$}]{A_y} & \tn[skin=dots, ports={180:virtual, 0:virtual}]{} & \tn[label pos=nw, ports={90:physical:$s_n$}]{A_y} \end{tenkz}$

Theorem 32.1.4 Existence of action tensors

Under the hypotheses of Theorem 30.15.8, every pair $(a,x)$ admits a family of action tensors; for integer coefficients this is the action clause of Corollary 29.6.1.2. This is the existence of action tensors in [ 199 , Section 3 ] .

Proof ▶

By Theorem 30.15.8 the coefficients are nonnegative integers, so (99) writes $O_a^{(N)}|V^{(N)}(A_x)\rangle $ as a sum of periodic vectors of blocks repeated with multiplicities $M_{a,x}^y$. Corollary 29.6.4 applied to this family with unit weights gives the reductions.

32.2 Gauge freedom of fusion and action tensors

Fusion tensors are unique up to a dressed scalar when the stacked product reduces onto a single normal block, and up to the multiplicity gauge under the zipper condition (Remark 32.2.4).

Corollary 32.2.1 Uniqueness of a single-target reduction

Let $B$ be a tensor, $C$ a normal tensor, and $(V,W)$, $(\widetilde V,\widetilde W)$ two reductions of $B$ onto $C$ whose residual tensors $B-WCV$ and $B-\widetilde WC\widetilde V$ are nilpotent with a common bound $N_0$. Then there is a unique $\lambda \in \mathbb {C}^\times $ with

\begin{align} \widetilde VB^{\mathbf w} & =\lambda \, VB^{\mathbf w}, & B^{\mathbf w}\widetilde W & =\lambda ^{-1}B^{\mathbf w}W \label{eq:mpoalg_single_target} \end{align}

for every word with $|\mathbf w|{\gt}2N_0$. This applies to the fusion tensors of a group-like algebra, where $T_gT_h$ reduces onto the single block $T_{gh}$, to the composite reductions of Theorem 32.3.1, and to the action tensors of an invertible label on a permuted block. It does not apply to a single slot of a reduction onto multiple blocks, whose residual is not nilpotent.

Proof ▶

Existence of $\lambda $ is Theorem 29.11.28 written for compressions on the left and embeddings on the right, and for fusion tensors Corollary 29.6.1.3. Uniqueness is Lemma 19.5.1.2: $VB^{\mathbf w}W= C^{\mathbf w}$ is nonzero for arbitrarily long words.

Theorem 32.2.2 Uniqueness of zipper fusion tensors up to multiplicity gauge

Let the blocks $T_c$ of a fusion algebra be normal and pairwise not related by a gauge transformation, and let $(V_{ab}^{c,\mu },W_{ab}^{c,\mu })$ and $(\widetilde V_{ab}^{c,\mu },\widetilde W_{ab}^{c,\mu })$ be two families of fusion tensors for $(a,b)$ with vanishing remainder,

\begin{align} (T_aT_b)^{\mathbf w} & =\sum _{c,\mu }W_{ab}^{c,\mu }T_c^{\mathbf w}V_{ab}^{c,\mu } =\sum _{c,\mu }\widetilde W_{ab}^{c,\mu }T_c^{\mathbf w} \widetilde V_{ab}^{c,\mu } \label{eq:mpoalg_zipper_decomposition} \end{align}

for every nonempty word $\mathbf w$. Then for each $c$ there is a unique $Y_{ab}^c\in \mathrm{GL}_{N_{ab}^c}(\mathbb {C})$ with $\widetilde V_{ab}^{c,\mu }=\sum _\nu (Y_{ab}^c)_{\mu \nu }V_{ab}^{c,\nu }$ and $\widetilde W_{ab}^{c,\mu }=\sum _\nu W_{ab}^{c,\nu } \bigl((Y_{ab}^c)^{-1}\bigr)_{\nu \mu }$ on the support of the decomposition, so the ambiguity is $\bigoplus _c\mathrm{GL}_{N_{ab}^c}(\mathbb {C})$ with no word dressing. This is the uniqueness of fusion tensors under the zipper condition in [ 8 ] , and the basis change of [ 222 , Section 4 ] ; it is Corollary 29.8.1.4.

$\begin{tenkz}[rows={op,op}, cols=3, west=open, east=open] \tn[skin=mpo, ports={90:physical:$i_1$, 270:physical}]{T_a} & \tn[skin=dots, ports={180:virtual, 0:virtual}]{} & \tn[skin=mpo, ports={90:physical:$i_n$, 270:physical}]{T_a} \\ \tn[skin=mpo, ports={90:physical, 270:physical:$j_1$}]{T_b} & \tn[skin=dots, ports={180:virtual, 0:virtual}]{} & \tn[skin=mpo, ports={90:physical, 270:physical:$j_n$}]{T_b} \end{tenkz}$ $=\; \displaystyle \sum _{c,\mu }$ $\begin{tenkz}[rows={op,op}, cols=5, bonds=none] \tnfuse[at=(1,1), name=W, skin=pill]{W_{ab}^{c,\mu}} \tn[at=(1,2), name=t1, skin=mpo, ports={90:physical:$i_1$, 270:physical:$j_1$, 180:virtual, 0:virtual}]{T_c} \tn[at=(1,3), name=td, skin=dots, ports={180:virtual, 0:virtual}]{} \tn[at=(1,4), name=tn, skin=mpo, ports={90:physical:$i_n$, 270:physical:$j_n$, 180:virtual, 0:virtual}]{T_c} \tnfuse[at=(1,5), name=V, skin=pill]{V_{ab}^{c,\mu}} \tn[at=(2,2), skin=none]{} \tn[at=(2,3), skin=none]{} \tn[at=(2,4), skin=none]{} \tnwire{open w}{W.180@1} \tnwire{open w}{W.180@2} \tnwire{W.0}{t1.180} \tnwire{t1.0}{td.180} \tnwire{td.0}{tn.180} \tnwire{tn.0}{V.180} \tnwire{V.0@1}{open e} \tnwire{V.0@2}{open e} \end{tenkz}$

Proof ▶

This is the argument of Theorem 29.8.1.3. Multiply (4) by $\widetilde V_{ab}^{c,\mu }$ on the left and $W_{ab}^{e,\nu }$ on the right and use biorthogonality of both families: $T_c^{\mathbf w}\, \widetilde V_{ab}^{c,\mu }W_{ab}^{e,\nu } =\widetilde V_{ab}^{c,\mu }W_{ab}^{e,\nu }\, T_e^{\mathbf w}$ for every nonempty word. For normal blocks, the words of a fixed injective length span the full matrix algebra, so an intertwiner between $T_c$ and $T_e$ is zero unless $T_c$ and $T_e$ are gauge equivalent, and is a scalar when $e=c$. This gives $\widetilde V_{ab}^{c,\mu }W_{ab}^{e,\nu }= \delta _{ce}(Y_{ab}^c)_{\mu \nu }\mathbb {1}$, and the same argument with the families exchanged gives the inverse matrix.

Theorem 32.2.3 Uniqueness of zipper action tensors up to multiplicity gauge

Let the blocks $A_y$ be normal and pairwise not related by a gauge transformation. Two families of action tensors for $(a,x)$ with vanishing remainder, $(T_a\cdot A_x)^w=\sum _{y,i}W_{ax}^{y,i}A_y^w V_{ax}^{y,i}$ for every nonempty word, are related by a unique $X_{ax}^y\in \mathrm{GL}_{M_{a,x}^y}(\mathbb {C})$ for each $y$, in the sense of Theorem 32.2.2. This is the gauge freedom of the action tensors in [ 199 , Section 3.1 ] , and it is Corollary 29.8.1.5.

Proof ▶

The proof of Theorem 32.2.2 applies to the tensor $T_a\cdot A_x$ and the blocks $A_y$.

Remark 32.2.4 Uniqueness fails with a nilpotent remainder

With only a nilpotent remainder, reductions are not unique up to scalars. Let $C$ be normal of bond dimension $D$, let $R^i\in \mathbb {C}^{D\times 1}$ be arbitrary, and let

\begin{align} B^i & =\begin{pmatrix} C^i & R^i \\ 0 & 0 \end{pmatrix}, & W & =\begin{pmatrix} \mathbb {1}_D \\ 0 \end{pmatrix}, & V_y & =\begin{pmatrix} \mathbb {1}_D & y \end{pmatrix} \label{eq:mpoalg_nonunique} \end{align}

for $y\in \mathbb {C}^D$. Then $V_yW=\mathbb {1}_D$, $V_yB^wW=C^w$ for every word $w$, and the remainder $B^i-WC^iV_y$ has vanishing products of length two, so every $(V_y,W)$ is a reduction of $B$ onto $C$ with nilpotent remainder. For $y\neq 0$ the compressions $V_y$ and $V_0$ are not proportional, although $V_yB^w=V_0B^w$ for every nonempty word; with a single target, Corollary 32.2.1 recovers uniqueness after dressing.

With several targets even the dressed identities fail. Let $(V_\mu ,W_\mu )$ and $(V_\nu ,W_\nu )$ be two slots of a compression of $B$ onto blocks $C_\mu $ and $C_\nu $ of the same bond dimension $D$, with $\mu $ before $\nu $ in the block upper triangular order, and conjugate the slot coordinates by $S=\begin{pmatrix} \mathbb {1} & Z \\ 0 & \mathbb {1} \end{pmatrix}$ for an arbitrary $Z\in M_{D}(\mathbb {C})$. The new family is still block upper triangular, hence again a compression with nilpotent remainder, and $\widetilde V_\mu =V_\mu -ZV_\nu $. Then $\widetilde V_\mu B^{\mathbf w}=V_\mu B^{\mathbf w}-ZV_\nu B^{\mathbf w}$ on every long word, which is not a combination of the $V_\kappa B^{\mathbf w}$ with scalar coefficients unless $Z$ is scalar. This holds whether $\nu $ is a duplicate copy of the same block or a distinct block. The zipper condition of [ 8 ] , with vanishing remainder, rules out both ambiguities and restores uniqueness up to the multiplicity gauge (Theorem 32.2.2).

32.3 The associator of a group-like algebra

Let $G$ be a group and $(T_g)_{g\in G}$ a fusion algebra of normal tensors of positive bond dimension with $N_{gh}^k=\delta _{k,gh}$, so that $O_g^{(N)}O_h^{(N)}=O_{gh}^{(N)}$ for every $N\geq 1$. Choose fusion tensors $(V_{g,h},W_{g,h})$ for every pair. The triple product $T_gT_hT_k$ has two composite reductions onto $T_{ghk}$,

\begin{align} V^{(gh)k} & =V_{gh,k}\, (V_{g,h}\otimes \mathbb {1}_{\chi _k}), & V^{g(hk)} & =V_{g,hk}\, (\mathbb {1}_{\chi _g}\otimes V_{h,k}), \label{eq:mpoalg_composite_compressions} \end{align}

with the corresponding composite embeddings.

Theorem 32.3.1 Composite reductions

✓

Let $(V_1,W_1)$ be a reduction of a tensor $C$ onto $B$ whose residual letters $N^i=C^i-W_1B^iV_1$ have vanishing products of length $N_1$, and let $(V_2,W_2)$ be a pair of matrices, not necessarily a reduction of $B$ onto $A$, whose residual letters $M^i=B^i-W_2A^iV_2$ have vanishing products of length $N_2$. Then the residual letters $R^i=C^i-W_1W_2A^iV_2V_1$ of the composite pair $(V_2V_1,W_1W_2)$ have vanishing products of length $2N_1+N_2$. The residual bound is unchanged when a reduction is tensored with an identity strand, $(V\otimes \mathbb {1},W\otimes \mathbb {1})$ on $T_aT_d$ or $(\mathbb {1}\otimes V,\mathbb {1}\otimes W)$ on $T_dT_a$, and likewise for an operator tensor acting on a state tensor, and it is unchanged under the bond associator. In particular, if $(V,W)$ reduces $T_aT_b$ onto $T_c$ and $(V',W')$ reduces $T_cT_d$ onto $T_e$, both with residual bound $K$, then $(V'(V\otimes \mathbb {1}),(W\otimes \mathbb {1})W')$ reduces $T_aT_bT_d$ onto $T_e$ with residual bound $3K$. The reduction property, without the residual bound, is Lemma 19.5.1.1, and for the two fusion trees of a triple product it is Lemma 19.5.1.4.

Proof ▶

The letters satisfy $R^i=N^i+W_1M^iV_1$, and $V_1N^{\mathbf q}W_1=0$ for every nonempty word $\mathbf q$ because $(V_1,W_1)$ is a reduction. Three inductions on words give the following.

$V_1R^{\mathbf w}W_1=M^{\mathbf w}$, and $V_1R^{\mathbf w}N^{\mathbf q}W_1=0$ for nonempty $\mathbf q$. Append a letter $x$ to $\mathbf w$: since $R^{\mathbf wx}=R^{\mathbf w}N^x+R^{\mathbf w}W_1M^xV_1$, both identities for $\mathbf wx$ follow from those for $\mathbf w$ and from $V_1W_1=\mathbb {1}$.
$N^{\mathbf p}R^{\mathbf w}W_1=0$ when $|\mathbf p|+|\mathbf w|\geq N_1+N_2$. Prepend a letter $x$: $N^{\mathbf p}R^{x\mathbf w}W_1=N^{\mathbf px}R^{\mathbf w}W_1 +N^{\mathbf p}W_1M^{x\mathbf w}$ by the first step, and the last term vanishes because $|\mathbf p|\geq N_1$ or $|x\mathbf w|\geq N_2$.
$R^{\mathbf w}N^{\mathbf q}=0$ when $|\mathbf w|+|\mathbf q|\geq 2N_1+N_2$. Append a letter $x$: $R^{\mathbf wx}N^{\mathbf q}=R^{\mathbf w}N^{x\mathbf q} +R^{\mathbf w}W_1M^xV_1N^{\mathbf q}$, and the last term vanishes because $|\mathbf q|\geq N_1$, or else $|\mathbf w|\geq N_1+N_2$ and the second step applies.

The third step with $\mathbf q$ empty is the claim. The residual of $(V\otimes \mathbb {1},W\otimes \mathbb {1})$ on $T_aT_d$ has words $\sum _{\mathbf m}(T_a-WT_cV)^{(\mathbf i,\mathbf m)}\otimes T_d^{(\mathbf m,\mathbf j)}$, which vanish at the same length, and an invertible intertwiner of letters conjugates the residual letters.

Remark 32.3.2 Sharpness of the hypotheses and of the bound

In Theorem 32.3.1 the factors must have nilpotent residuals: a single slot of a reduction onto several blocks has a residual that is not nilpotent, and composing it with the identity reduction of $T_cT_d$ for a trivial $T_d$ returns it unchanged. The bound $N_1+N_2$ fails: for one letter acting as the shift $f_0\to f_1\to e_1\to e_2\to f_3\to f_4$ on $\mathbb {C}^6$, with $(V_1,W_1)$ the compression onto $\operatorname {span}\{ e_1,e_2\} $ and $(V_2,W_2)$ the compression of the shift $e_1\to e_2$ onto $\operatorname {span}\{ e_2\} $ with $A=0$, one has $N_1=3$, $N_2=2$, and $R^5\neq 0$. Counting the lengths of the nonvanishing terms in the expansion of the proof more carefully gives the bound $\max (N_1,2N_1+N_2-2)$, which this example attains.

Definition 32.3.3 Associator

The associator of the chosen fusion tensors is the function $\omega \colon G^3\to \mathbb {C}^\times $ defined by

\begin{align} V^{(gh)k}(T_gT_hT_k)^{\mathbf w} & =\omega (g,h,k)\, V^{g(hk)}(T_gT_hT_k)^{\mathbf w} \label{eq:mpoalg_associator} \end{align}

for all sufficiently long words $\mathbf w$; equivalently, $(T_gT_hT_k)^{\mathbf w}W^{g(hk)}=\omega (g,h,k)\, (T_gT_hT_k)^{\mathbf w}W^{(gh)k}$ for the composite embeddings, which is the orientation of the defining display of [ 197 ] . It exists and is nonzero by Corollary 32.2.1 applied to the two composite reductions of Theorem 32.3.1, and it is unique because $T_{ghk}^{\mathbf w}\neq 0$ for some long word. This is the three-cocycle of [ 199 , Section 4.4 ] , and it is the anomaly three-cochain of Definition 19.5.1.3, characterized by Theorem 19.5.1.5.

$\begin{tenkz}[rows={op,op,op}, cols=5, bonds=none] \tn[at=(1,1), name=O, skin=pill, wires=3, ports={180@1:virtual, 0@1:virtual, 0@3:virtual}]{V_{gh,k}} \tn[at=(1,2), name=I, skin=pill, wires=2, ports={180@1:virtual, 0@1:virtual, 0@2:virtual}]{V_{g,h}} \tn[at=(1,3), name=g1, skin=mpo, ports={90:physical:$i_1$, 270:physical, 180:virtual, 0:virtual}]{T_g} \tn[at=(2,3), name=h1, skin=mpo, ports={90:physical, 270:physical, 180:virtual, 0:virtual}]{T_h} \tn[at=(3,3), name=k1, skin=mpo, ports={90:physical, 270:physical:$j_1$, 180:virtual, 0:virtual}]{T_k} \tn[at=(1,4), name=gd, skin=dots, ports={180:virtual, 0:virtual}]{} \tn[at=(2,4), name=hd, skin=dots, ports={180:virtual, 0:virtual}]{} \tn[at=(3,4), name=kd, skin=dots, ports={180:virtual, 0:virtual}]{} \tn[at=(1,5), name=gn, skin=mpo, ports={90:physical:$i_n$, 270:physical, 180:virtual, 0:virtual}]{T_g} \tn[at=(2,5), name=hn, skin=mpo, ports={90:physical, 270:physical, 180:virtual, 0:virtual}]{T_h} \tn[at=(3,5), name=kn, skin=mpo, ports={90:physical, 270:physical:$j_n$, 180:virtual, 0:virtual}]{T_k} \tnwire{open w}{O.180@1} \tnwire{O.0@1}{I.180@1} \tn[at=(3,2), name=J, skin=none, ports={180:virtual, 0:virtual}]{} \tnwire{O.0@3}{J.180} \tnwire{J.0}{k1.180} \tnwire{I.0@1}{g1.180} \tnwire{I.0@2}{h1.180} \tnwire{g1.270}{h1.90} \tnwire{h1.270}{k1.90} \tnwire{gn.270}{hn.90} \tnwire{hn.270}{kn.90} \tnwire{g1.0}{gd.180} \tnwire{gd.0}{gn.180} \tnwire{h1.0}{hd.180} \tnwire{hd.0}{hn.180} \tnwire{k1.0}{kd.180} \tnwire{kd.0}{kn.180} \tnwire{gn.0}{open e} \tnwire{hn.0}{open e} \tnwire{kn.0}{open e} \end{tenkz}$ $=\omega (g,h,k)$ $\begin{tenkz}[rows={op,op,op}, cols=5, bonds=none] \tn[at=(1,1), name=O, skin=pill, wires=2, ports={180@1:virtual, 0@1:virtual, 0@2:virtual}]{V_{g,hk}} \tn[at=(3,1), skin=none]{} \tn[at=(1,2), name=J, skin=none, ports={180:virtual, 0:virtual}]{} \tn[at=(2,2), name=I, skin=pill, wires=2, ports={180@1:virtual, 0@1:virtual, 0@2:virtual}]{V_{h,k}} \tn[at=(1,3), name=g1, skin=mpo, ports={90:physical:$i_1$, 270:physical, 180:virtual, 0:virtual}]{T_g} \tn[at=(2,3), name=h1, skin=mpo, ports={90:physical, 270:physical, 180:virtual, 0:virtual}]{T_h} \tn[at=(3,3), name=k1, skin=mpo, ports={90:physical, 270:physical:$j_1$, 180:virtual, 0:virtual}]{T_k} \tn[at=(1,4), name=gd, skin=dots, ports={180:virtual, 0:virtual}]{} \tn[at=(2,4), name=hd, skin=dots, ports={180:virtual, 0:virtual}]{} \tn[at=(3,4), name=kd, skin=dots, ports={180:virtual, 0:virtual}]{} \tn[at=(1,5), name=gn, skin=mpo, ports={90:physical:$i_n$, 270:physical, 180:virtual, 0:virtual}]{T_g} \tn[at=(2,5), name=hn, skin=mpo, ports={90:physical, 270:physical, 180:virtual, 0:virtual}]{T_h} \tn[at=(3,5), name=kn, skin=mpo, ports={90:physical, 270:physical:$j_n$, 180:virtual, 0:virtual}]{T_k} \tnwire{open w}{O.180@1} \tnwire{O.0@1}{J.180} \tnwire{J.0}{g1.180} \tnwire{O.0@2}{I.180@1} \tnwire{I.0@1}{h1.180} \tnwire{I.0@2}{k1.180} \tnwire{g1.270}{h1.90} \tnwire{h1.270}{k1.90} \tnwire{gn.270}{hn.90} \tnwire{hn.270}{kn.90} \tnwire{g1.0}{gd.180} \tnwire{gd.0}{gn.180} \tnwire{h1.0}{hd.180} \tnwire{hd.0}{hn.180} \tnwire{k1.0}{kd.180} \tnwire{kd.0}{kn.180} \tnwire{gn.0}{open e} \tnwire{hn.0}{open e} \tnwire{kn.0}{open e} \end{tenkz}$

Theorem 32.3.4 The associator is a three-cocycle

The associator satisfies the three-cocycle relation (394),

\begin{align} \omega (gh,k,l)\, \omega (g,h,kl) & =\omega (g,h,k)\, \omega (g,hk,l)\, \omega (h,k,l), \label{eq:mpoalg_pentagon_group} \end{align}

which is the pentagon equation for a group-like algebra [ 199 , Section 4 ] . This is Theorem 19.5.1.6.

Proof ▶

Write $X=(T_gT_hT_kT_l)^{\mathbf w}$ for a long word $\mathbf w$ and $V^P$ for the composite compression of the fourfold product onto $T_{ghkl}$ along a bracketing $P$. By Theorem 32.3.1, applying (7) to one triple inside a bracketing, tensored with identities and followed by the remaining compressions, replaces $(xy)z$ by $x(yz)$ at the cost of $\omega (x,y,z)$. Along the two paths of the pentagon this gives

\begin{align} V^{((gh)k)l}X & =\omega (gh,k,l)\, V^{(gh)(kl)}X =\omega (gh,k,l)\, \omega (g,h,kl)\, V^{g(h(kl))}X, \notag \\ V^{((gh)k)l}X & =\omega (g,h,k)\, V^{(g(hk))l}X =\omega (g,h,k)\, \omega (g,hk,l)\, V^{g((hk)l)}X \notag \\ & =\omega (g,h,k)\, \omega (g,hk,l)\, \omega (h,k,l)\, V^{g(h(kl))}X. \label{eq:mpoalg_pentagon_paths} \end{align}

Since $V^{g(h(kl))}XW^{g(h(kl))}=T_{ghkl}^{\mathbf w}$ is nonzero for some long word, comparing the two lines of (9) gives (8).

Theorem 32.3.5 Gauge dependence of the associator

Replacing the fusion tensors by $(\beta _{g,h}^{-1}V_{g,h}, \beta _{g,h}W_{g,h})$, so that $V_{g,h}B^{\mathbf w}=\beta _{g,h}\, V'_{g,h}B^{\mathbf w}$ for the new compression $V'_{g,h}$, with $\beta \colon G^2\to \mathbb {C}^\times $ replaces $\omega $ by $\beta \mathbin {\triangleright }\omega $ of Definition 19.5.10.32. Consequently the class of $\omega $ under the relation of Definition 19.5.10.37 does not depend on the choice of fusion tensors, and depends only on the family $(T_g)_{g\in G}$. This is the coboundary freedom of [ 199 , Section 4 ] , and it is Theorem 19.5.1.7.

Proof ▶

The rescaled pairs are again fusion tensors (Theorem 19.2.4.24). Substituting them into (6) multiplies $V^{(gh)k}$ by $(\beta _{g,h}\beta _{gh,k})^{-1}$ and $V^{g(hk)}$ by $(\beta _{g,hk}\beta _{h,k})^{-1}$, so (7) holds for the new tensors with $\omega $ multiplied by $\beta _{g,hk}\beta _{h,k}/(\beta _{g,h}\beta _{gh,k})$, which is (395). By Corollary 32.2.1 every other choice of fusion tensors is boundary-dressed equal to such a rescaling.

Remark 32.3.6 A computed anomaly class

For a cyclic group generated by $g$ of order $n$, the product $\prod _{k{\lt}n}\omega (g,g^k,g)$ is unchanged by the gauge action of Theorem 32.3.5, so it depends only on the family $(T_g)_{g\in G}$. For the phase-decorated shift representation of $\mathbb {Z}_3$ it is a primitive cube root of unity (Theorem 31.10.2.4), so its associator does not have trivial gauge class.

Remark 32.3.7 F-symbols with multiplicity under the zipper condition

For a general fusion algebra the analogue of the associator is a matrix $F^{abc}_d$ relating the two families of composite fusion tensors of $T_aT_bT_c$ onto copies of $T_d$. Under the zipper condition of [ 8 ] , which asks that the fusion tensors intertwine the stacked tensor letter by letter, $F^{abc}_d$ is an invertible matrix between multiplicity spaces satisfying the $F$-move (120) of Theorem 21.3.91, and its inverse satisfies the pentagon equation (124) of Theorem 21.3.94; both are stated for the complete zipper fusion families of Definition 21.3.85. For a group-like algebra, the scalar playing the role of $F^{ghk}_{ghk}$ in the orientation of (120) is $\omega (g,h,k)^{-1}$, and the pentagon equation is (8).

Without the zipper condition, the fusion tensors of Theorem 32.1.2 carry a nilpotent remainder and, with several targets, are not determined up to the multiplicity gauge, even after dressing with long words (Remark 32.2.4). No $F$-move with multiplicity is stated in that setting; for group-like algebras, single-target uniqueness yields (7).

32.4 Cohomological obstruction to symmetric states

Definition 32.4.1 L-symbols of an action

In the setting of Section 32.3, let $(A_x)_{x\in K}$ be a symmetric family as in Theorem 30.15.11, so that $O_g^{(N)}|V^{(N)}(A_x)\rangle =|V^{(N)}(A_{g\cdot x})\rangle $, with action tensors $(V_{g,x},W_{g,x})$ reducing $T_g\cdot A_x$ onto $A_{g\cdot x}$. The tensor $T_gT_h\cdot A_x$ has two composite reductions onto $A_{gh\cdot x}$, and the $L$-symbol $L^x_{g,h}\in \mathbb {C}^\times $ is defined by

\begin{align} V_{g,h\cdot x}(\mathbb {1}\otimes V_{h,x})(T_gT_h\cdot A_x)^w & =L^x_{g,h}\, V_{gh,x}(V_{g,h}\otimes \mathbb {1}) (T_gT_h\cdot A_x)^w \label{eq:mpoalg_l_symbol} \end{align}

for all sufficiently long words $w$. This is the $L$-symbol of [ 199 , Section 4.4 ] .

Theorem 32.4.2 Compatibility of L-symbols with the associator

The $L$-symbols of Definition 32.4.1 are compatible with the associator in the sense of (417),

\begin{align} L^x_{g,hk}\, L^x_{h,k} & =\omega (g,h,k)\, L^{k\cdot x}_{g,h}\, L^x_{gh,k}, \label{eq:mpoalg_l_compatible} \end{align}

and replacing the fusion and action tensors by $(\beta _{g,h}^{-1}V_{g,h},\beta _{g,h}W_{g,h})$ and $((\gamma ^x_g)^{-1}V_{g,x},\gamma ^x_gW_{g,x})$ changes $(\omega ,L)$ to $(\beta \mathbin {\triangleright }\omega ,(\beta ,\gamma )\mathbin {\triangleright }L)$ of Definitions 19.5.10.32 and 19.5.10.50. This is the coupled pentagon equation for a group-like algebra [ 199 , Section 4 ] .

Proof ▶

Write $Y=(T_gT_hT_k\cdot A_x)^w$ for a long word $w$ and $V^P$ for the composite compression onto $A_{ghk\cdot x}$ along a bracketing $P$. By Theorem 32.3.1, (10) and (7) applied inside a bracketing give

\begin{align} V^{g(h(kx))}Y & =L^x_{h,k}\, V^{g((hk)x)}Y =L^x_{h,k}\, L^x_{g,hk}\, V^{(g(hk))x}Y, \notag \\ V^{g(h(kx))}Y & =L^{k\cdot x}_{g,h}\, V^{(gh)(kx)}Y =L^{k\cdot x}_{g,h}\, L^x_{gh,k}\, V^{((gh)k)x}Y \notag \\ & =L^{k\cdot x}_{g,h}\, L^x_{gh,k}\, \omega (g,h,k)\, V^{(g(hk))x}Y. \label{eq:mpoalg_l_paths} \end{align}

Since $V^{(g(hk))x}YW^{(g(hk))x}=A_{ghk\cdot x}^w$ is nonzero for some long word, comparing the two lines of (12) gives (11). The gauge statement follows by substituting the rescaled tensors into (7) and (10).

Theorem 32.4.3 No symmetric normal tensor under a nontrivial associator

Let $(T_g)_{g\in G}$ be a group-like fusion algebra of normal tensors whose associator does not have trivial gauge class. Then there is no normal tensor $A$ of positive bond dimension with $O_g^{(N)}|V^{(N)}(A)\rangle =|V^{(N)}(A)\rangle $ for all $g\in G$ and $N\geq 1$. This is the statement of [ 199 , Section 3.2 ] that an injective state without multiplicity cannot be invariant under a nontrivial algebra, specialized to group-like algebras; for group-like algebras it goes back to [ 11 ] .

Proof ▶

Action tensors exist by Theorem 32.1.4 with $K$ a single point fixed by every $g$. Then (11) reads $L_{g,hk}L_{h,k}=\omega (g,h,k)L_{g,h}L_{gh,k}$, that is, $\omega =dL$ in the notation of (395). Hence $\omega =L\mathbin {\triangleright }1$ has trivial gauge class, a contradiction.

32.5 Weak Hopf algebras, dualities, and density operators

Remark 32.5.1 Fusion tensors from a coproduct

For a matrix product operator representation of a cosemisimple pre-bialgebra $\mathcal A$, the operators are indexed by irreducible representations $\psi _a$ of the dual algebra, and the fusion tensors decompose the coproduct of $\mathcal A^*$, $(\psi _a\otimes \psi _b)\circ \Delta (f)=\sum _{c,\mu }W_{ab}^{c\mu }\psi _c(f) V_{ab}^{c\mu }$ with $V_{ab}^{c\mu }W_{ab}^{d\nu }= \delta _{cd}\delta _{\mu \nu }\mathbb {1}$ [ 222 , Section 4 ] . The maps are named as in Definition 32.1.1; the source writes the same identity as $\sum _{c,\mu }V_{ab}^{c\mu }\psi _c(f) W_{ab}^{c\mu }$ with $W_{ab}^{c\mu }V_{ab}^{d\nu }=\delta _{cd}\delta _{\mu \nu }\mathbb {1}$, so the letters $V$ and $W$ are swapped. Theorem 32.5.2 reads this identity backwards: the fusion tensors of a complete zipper fusion family define the coproduct. No part of the correspondence concerning units, counits, or antipodes is stated here.

Theorem 32.5.2 Coproduct of a complete zipper fusion family

✓

Let $(B_a)_{a\in \Lambda }$ be a complete zipper fusion family (Definition 21.3.85) with fusion tensors $X^c_{ab,\mu }$ and left inverses $X^{c+}_{ab,\mu }$. For $X=(X_c)_c\in \bigoplus _cM_{D_c}(\mathbb {C})$ put

\begin{align} \Delta (X)_{ab} & =\sum _{c,\mu }X^c_{ab,\mu }\, X_c\, X^{c+}_{ab,\mu } \in M_{D_a}(\mathbb {C})\otimes M_{D_b}(\mathbb {C}). \label{eq:mpoalg_fusion_coproduct} \end{align}

Then the elements $(B_c^{ik})_c$ span $\bigoplus _cM_{D_c}(\mathbb {C})$, and

\begin{align} \Delta \big((B_c^{ik})_c\big)_{ab} & =\sum _jB_a^{ij}\otimes B_b^{jk}. \label{eq:mpoalg_fusion_coproduct_letter} \end{align}

The map $\Delta $ is the only linear map with (14), so it depends only on the blocks. It is multiplicative, $\Delta (XY)=\Delta (X)\Delta (Y)$, and $\Delta (\mathbb {1})_{ab}=\sum _{c,\mu }X^c_{ab,\mu }X^{c+}_{ab,\mu }$ is the support projector of the fusion tensors. In the notation of [ 222 , Section 4 ] , $X^c_{ab,\mu }$ and $X^{c+}_{ab,\mu }$ are the source’s $V_{ab}^{c\mu }$ and $W_{ab}^{c\mu }$, (13) is the decomposition of the coproduct of the dual algebra, (14) is the coproduct of the matrix coefficients of the physical representation, and multiplicativity is the pre-bialgebra axiom.

Proof ▶

The simultaneous left inverse of the block letters writes every matrix unit of every block as a combination of the elements $(B_c^{ik})_c$, so they span. Equation (14) is the reconstruction of the product letters through the fusion support, and uniqueness follows from spanning. Multiplicativity follows from $X^{d+}_{ab,\nu }X^c_{ab,\mu }=\delta _{cd}\delta _{\mu \nu }\mathbb {1}$, which gives $\Delta (X)\Delta (Y)=\sum _{c,\mu }X^c_{ab,\mu }X_cY_cX^{c+}_{ab,\mu }$.

Theorem 32.5.3 Coassociativity of the fusion coproduct

✓

The coproduct (13) is coassociative, $(\Delta \otimes \mathbb {1})\circ \Delta =(\mathbb {1}\otimes \Delta )\circ \Delta $, where on the triple bond space

\begin{align} (\Delta \otimes \mathbb {1})(Y)_{abc} & =\sum _{e,\mu }(X^e_{ab,\mu }\otimes \mathbb {1})\, Y_{ec}\, (X^{e+}_{ab,\mu }\otimes \mathbb {1}), \notag \\ (\mathbb {1}\otimes \Delta )(Y)_{abc} & =\sum _{f,\lambda }(\mathbb {1}\otimes X^f_{bc,\lambda })\, Y_{af}\, (\mathbb {1}\otimes X^{f+}_{bc,\lambda }). \notag \end{align}

This is the coassociativity axiom of a coalgebra [ 222 , Section 3 ] . The source assumes it and derives the associativity of fusion trees and the $F$-symbols; here it is proved from the fusion tensors.

Proof ▶

Both sides are linear in the argument, and by Theorem 32.5.2 the elements $(B_c^{il})_c$ span the space of tuples $(Z_c)_c$ on which $\Delta $ acts. By (14) applied twice, both sides send $(B_c^{il})_c$ to $\sum _{j,k}B_a^{ij}\otimes B_b^{jk}\otimes B_c^{kl}$.

Remark 32.5.4 Kramers–Wannier duality is an intertwiner

The Kramers–Wannier operator of Section 31.5 does not belong to a fusion algebra on one lattice: by (69) its square is $(\mathbb {1}+\eta )T$ up to normalization, which contains the translation $T$ and is not a length-independent combination of the operators of a finite family. It is a matrix product operator intertwiner between two symmetric Hamiltonians, a bimodule between the symmetry algebras of the primal and dual chains, rather than a symmetry of one of them [ 214 , 215 ] . Fusion and action tensors for such intertwiners come from the same asymmetric compression theorem, applied to products of operators between different chains; the Kramers–Wannier case is worked out in Section 31.5.3.

Remark 32.5.5 Density operators

Renormalization fixed points of matrix product density operators are described by matrix product operator algebras of $C^*$-weak Hopf algebras through the pulling-through identity of the canonical regular element [ 232 ] ; the density-operator setting is that of Chapter 16. The fusion tensors of Section 32.1 apply there without change, since they use only equality of periodic traces.