Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2507.05185

An operator algebraic approach to fusion category symmetry on the lattice

David Evans, Corey Jones

Published Jul 7, 2025• 0 citations
arXivHTMLPDF

Abstract

We propose a framework for fusion category symmetry on the (1+1)D lattice in the infinite-volume limit by giving a formal interpretation of SymTFT decompositions. Our approach is based on axiomatizing physical boundary subalgebra of quasi-local observables, and applying ideas from algebraic quantum field theory to derive the expected categorical structures. We show that given a physical boundary subalgebra $B$ of a quasi-local algebra $A$, there is a canonical fusion category $\mathcal{C}$ that acts on $A$ by bimodules and whose fusion ring acts by locality preserving quantum channels on the quasi-local algebra such that $B$ is recovered as the fixed point operators. We show that a fusion category can be realized as symmetries on a tensor product quasi-local algebra if and only if all of its objects have integer dimensions, and that it admits an ``on-site" action on a tensor product spin chain if and only if it admits a fiber functor. We give a formal definition of a topological symmetric state, and prove two anomaly enforced gaplessness theorems, one for internal categorical symmetries and one for anomalous duality channels. Using the first, we show that for any fusion category $\mathcal{C}$ with no fiber functor there always exist gapless pure symmetric states on an anyon chain.

DOI
10.48550/arxiv.2507.05185

Related papers

Bulk-boundary correspondence of (1+1)D symmetric gapped phases

Yizhou Ma, Gen Yue, Tian Lan

We develop an operator-algebraic framework for boundary conditions and bulk-boundary correspondence in one-dimensional gapped phases with categorical symmetry. Working directly in the thermodynamic limit, we construct half-infinite fusion spin chains and commuting-projector boundary Hamiltonians from a unitary fusion category $\mathcal{C}$, an indecomposable semisimple right $\mathcal{C}$-module category $\mathcal{M}$, a Q-system $Q\in\mathcal{C}$ specifying the bulk phase, and a right $Q$-module $K\in\mathcal{M}_{Q}$, regarded as an object of $\mathcal{M}_{Q}^{\mathrm{op}}$, specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor $\mathcal{M}_{Q}^{\mathrm{op}}\to\mathrm{BCond}$ is an equivalence, so simple boundary conditions are classified by simple objects of $\mathcal{M}_{Q}$ and general boundary conditions by their finite direct sums. We also give a microscopic formulation of the boundary symmetry topological field theory using DHR bimodules of the boundary quasi-local algebra. For a half-infinite fusion spin chain, the boundary DHR category is monoidally equivalent to $(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}$, and the canonical action of the bulk DHR category on it agrees with the categorical action of $Z_1(\mathcal{C}^{\mathrm{rev}})$. Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of $(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}$ on $\mathcal{M}_{Q}^{\mathrm{op}}$. This yields a one-dimensional bulk-boundary correspondence: the enriched monoidal category describing the bulk is the enriched center of the enriched category describing the boundary.

arXiv:2606.19137•Jun 17, 2026•0 citations

On the structure of categorical duality operators

Corey Jones, Xinping Yang

We systematically study categorical duality operators on spin (and anyon) chains with respect to an internal fusion category symmetry C. We parameterize duality operators on the quasi-local algebra in terms of data dependent on the associated quantum cellular automata (QCA) on the symmetric subalgebra $B$. In particular, a QCA $α$ on $B$ defines an invertible C-C bimodule category $M_α$, and the duality operators extending $α$ form a simplex, with extreme points in bijective correspondence with the simple object of $M_α$. Then we consider the structure of external symmetries generated by a family of duality operators, and show that if the UV models are all defined on tensor product Hilbert spaces, these categories necessarily flow to weakly integral fusion categories in the IR.

arXiv:2603.09949•Mar 10, 2026•0 citations

Non-invertible SPTs: an on-site realization of (1+1)d anomaly-free fusion category symmetry

Meng, Chenqi, Yang, Xinping, Tian Lan +1 more

We investigate (1+1)d symmetry-protected topological (SPT) phases with fusion category symmetries. We emphasize that the UV description of an anomaly-free fusion category symmetry must include the fiber functor, giving rise to a local symmetry action, a charge category and a trivial phase. We construct an ``onsite'' matrix-product-operator (MPO) version of the Hopf algebra symmetry operators in a lattice model with tensor-product Hilbert space. In particular, we propose a systematic framework for classifying and constructing SPTs with non-invertible symmetries. An SPT phase corresponds to a Q-system in the charge category, such that the Q-system becomes a matrix algebra when the symmetry is forgotten. As an example, we provide an explicit microscopic realization of all three $\mathsf{Rep}^\dagger(D_8)$ SPT phases, including a trivial phase, and further demonstrate the $S_3$-duality among these three SPT phases.

arXiv:2412.20546•Dec 29, 2024•5 citations

Universal fusion category symmetries on tensor products of infinite-dimensional Hilbert spaces

Ian Bunner, Corey Jones

We show that anyon chains, after stabilizing with infinite-dimensional ancilla spaces, factorize locally as tensor products of infinite-dimensional Hilbert spaces. This implies that any unitary fusion category can be realized as symmetries on a tensor product of infinite-dimensional Hilbert spaces. We then show that any two anyon chains with the same symmetry category are related by a symmetry-compatible locality-preserving unitary after stabilizing with infinite-dimensional ancilla, showing that for a fixed fusion category, there is a single stable equivalence class of symmetry realizations on the lattice via anyon chains. As a corollary of our proof, we show that the physical boundary algebras of Levin-Wen type models are bounded spread isomorphic after stabilization if and only if they have the same bulk topological order.

arXiv:2605.21327•May 20, 2026•0 citations

Gapped phases with non-invertible symmetries: (1+1)d

Lakshya Bhardwaj, Lea E. Bottini, Daniel Pajer +1 more

We propose a general framework to characterize gapped infra-red (IR) phases of theories with non-invertible (or categorical) symmetries. In this paper we focus on (1+1)d gapped phases with fusion category symmetries. The approach that we propose uses the Symmetry Topological Field Theory (SymTFT) as a key input: associated to a field theory in d d spacetime dimensions, the SymTFT lives in one dimension higher and admits a gapped boundary, which realizes the categorical symmetries. It also admits a second, physical, boundary, which is generically not gapped. Upon interval compactification of the SymTFT by colliding the gapped and physical boundaries, we regain the original theory. In this paper, we realize gapped symmetric phases by choosing the physical boundary to be a gapped boundary condition as well. This set-up provides computational power to determine the number of vacua, the symmetry breaking pattern, and the action of the symmetry on the vacua. The SymTFT also manifestly encodes the order parameters for these gapped phases, thus providing a generalized, categorical Landau paradigm for (1+1)d gapped phases. We find that for non-invertible symmetries the order parameters involve multiplets containing both untwisted and twisted sector local operators, and hence can be interpreted as mixtures of conventional and string order parameters. We also observe that spontaneous breaking of non-invertible symmetries can lead to vacua that are physically distinguishable: unlike the standard symmetries described by groups, non-invertible symmetries can have different actions on different vacua of an irreducible gapped phase. This leads to the presence of relative Euler terms between physically distinct vacua. Along with the physical description of symmetric gapped phases, we also provide a mathematical one as pivotal 2-functors whose source 2-category is the delooping of the fusion category characterizing the symmetry and the target 2-category is the Euler completion of 2-vector spaces.

arXiv:2310.03784•SciPost Physics•Jan 24, 2025•68 citations

Compactification of quasi-local algebras on the lattice

Ikeda, Jun

We introduce a compactification construction for abstract quasi-local C*-algebras over countable metric spaces equipped with an isometric group action which is functorial with respect to bounded spread isomorphisms. In $1$D, the construction recovers Ocneanu's Tube algebra for fusion spin chains, and provides a canonical bridge between infinite-volume observables and observables with periodic boundary conditions. We exploit this connection to derive an obstruction for the implementability of such topological symmetries as Kramers-Wannier type dualities on symmetric subalgebras.

arXiv:2510.18021•Oct 20, 2025•0 citations
We propose a framework for fusion category symmetry on the (1+1)D lattice in the infinite-volume limit by giving a formal interpretation of SymTFT decompositions. Our approach is based on axiomatizing physical boundary subalgebra of quasi-local observables, and applying ideas from algebraic quantum field theory to derive the expected categorical structures. We show that given a physical boundary subalgebra BBB of a quasi-local algebra AAA, there is a canonical fusion category C\mathcal{C}C that acts on AAA by bimodules and whose fusion ring acts by locality preserving quantum channels on the quasi-local algebra such that BBB is recovered as the fixed point operators. We show that a fusion category can be realized as symmetries on a tensor product quasi-local algebra if and only if all of its objects have integer dimensions, and that it admits an ``on-site" action on a tensor product spin chain if and only if it admits a fiber functor. We give a formal definition of a topological symmetric state, and prove two anomaly enforced gaplessness theorems, one for internal categorical symmetries and one for anomalous duality channels. Using the first, we show that for any fusion category C\mathcal{C}C with no fiber functor there always exist gapless pure symmetric states on an anyon chain.
We develop an operator-algebraic framework for boundary conditions and bulk-boundary correspondence in one-dimensional gapped phases with categorical symmetry. Working directly in the thermodynamic limit, we construct half-infinite fusion spin chains and commuting-projector boundary Hamiltonians from a unitary fusion category C\mathcal{C}C, an indecomposable semisimple right C\mathcal{C}C-module category M\mathcal{M}M, a Q-system Q∈CQ\in\mathcal{C}Q∈C specifying the bulk phase, and a right QQQ-module K∈MQK\in\mathcal{M}_{Q}K∈MQ​, regarded as an object of MQop\mathcal{M}_{Q}^{\mathrm{op}}MQop​, specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor MQop→BCond\mathcal{M}_{Q}^{\mathrm{op}}\to\mathrm{BCond}MQop​→BCond is an equivalence, so simple boundary conditions are classified by simple objects of MQ\mathcal{M}_{Q}MQ​ and general boundary conditions by their finite direct sums. We also give a microscopic formulation of the boundary symmetry topological field theory using DHR bimodules of the boundary quasi-local algebra. For a half-infinite fusion spin chain, the boundary DHR category is monoidally equivalent to (CM∨)rev(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}(CM∨​)rev, and the canonical action of the bulk DHR category on it agrees with the categorical action of Z1(Crev)Z_1(\mathcal{C}^{\mathrm{rev}})Z1​(Crev). Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of (CM∨)rev(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}(CM∨​)rev on MQop\mathcal{M}_{Q}^{\mathrm{op}}MQop​. This yields a one-dimensional bulk-boundary correspondence: the enriched monoidal category describing the bulk is the enriched center of the enriched category describing the boundary.
We systematically study categorical duality operators on spin (and anyon) chains with respect to an internal fusion category symmetry C. We parameterize duality operators on the quasi-local algebra in terms of data dependent on the associated quantum cellular automata (QCA) on the symmetric subalgebra BBB. In particular, a QCA ααα on BBB defines an invertible C-C bimodule category MαM_αMα​, and the duality operators extending ααα form a simplex, with extreme points in bijective correspondence with the simple object of MαM_αMα​. Then we consider the structure of external symmetries generated by a family of duality operators, and show that if the UV models are all defined on tensor product Hilbert spaces, these categories necessarily flow to weakly integral fusion categories in the IR.
We investigate (1+1)d symmetry-protected topological (SPT) phases with fusion category symmetries. We emphasize that the UV description of an anomaly-free fusion category symmetry must include the fiber functor, giving rise to a local symmetry action, a charge category and a trivial phase. We construct an ``onsite'' matrix-product-operator (MPO) version of the Hopf algebra symmetry operators in a lattice model with tensor-product Hilbert space. In particular, we propose a systematic framework for classifying and constructing SPTs with non-invertible symmetries. An SPT phase corresponds to a Q-system in the charge category, such that the Q-system becomes a matrix algebra when the symmetry is forgotten. As an example, we provide an explicit microscopic realization of all three Rep†(D8)\mathsf{Rep}^\dagger(D_8)Rep†(D8​) SPT phases, including a trivial phase, and further demonstrate the S3S_3S3​-duality among these three SPT phases.
We show that anyon chains, after stabilizing with infinite-dimensional ancilla spaces, factorize locally as tensor products of infinite-dimensional Hilbert spaces. This implies that any unitary fusion category can be realized as symmetries on a tensor product of infinite-dimensional Hilbert spaces. We then show that any two anyon chains with the same symmetry category are related by a symmetry-compatible locality-preserving unitary after stabilizing with infinite-dimensional ancilla, showing that for a fixed fusion category, there is a single stable equivalence class of symmetry realizations on the lattice via anyon chains. As a corollary of our proof, we show that the physical boundary algebras of Levin-Wen type models are bounded spread isomorphic after stabilization if and only if they have the same bulk topological order.
We propose a general framework to characterize gapped infra-red (IR) phases of theories with non-invertible (or categorical) symmetries. In this paper we focus on (1+1)d gapped phases with fusion category symmetries. The approach that we propose uses the Symmetry Topological Field Theory (SymTFT) as a key input: associated to a field theory in d d spacetime dimensions, the SymTFT lives in one dimension higher and admits a gapped boundary, which realizes the categorical symmetries. It also admits a second, physical, boundary, which is generically not gapped. Upon interval compactification of the SymTFT by colliding the gapped and physical boundaries, we regain the original theory. In this paper, we realize gapped symmetric phases by choosing the physical boundary to be a gapped boundary condition as well. This set-up provides computational power to determine the number of vacua, the symmetry breaking pattern, and the action of the symmetry on the vacua. The SymTFT also manifestly encodes the order parameters for these gapped phases, thus providing a generalized, categorical Landau paradigm for (1+1)d gapped phases. We find that for non-invertible symmetries the order parameters involve multiplets containing both untwisted and twisted sector local operators, and hence can be interpreted as mixtures of conventional and string order parameters. We also observe that spontaneous breaking of non-invertible symmetries can lead to vacua that are physically distinguishable: unlike the standard symmetries described by groups, non-invertible symmetries can have different actions on different vacua of an irreducible gapped phase. This leads to the presence of relative Euler terms between physically distinct vacua. Along with the physical description of symmetric gapped phases, we also provide a mathematical one as pivotal 2-functors whose source 2-category is the delooping of the fusion category characterizing the symmetry and the target 2-category is the Euler completion of 2-vector spaces.
We introduce a compactification construction for abstract quasi-local C*-algebras over countable metric spaces equipped with an isometric group action which is functorial with respect to bounded spread isomorphisms. In 111D, the construction recovers Ocneanu's Tube algebra for fusion spin chains, and provides a canonical bridge between infinite-volume observables and observables with periodic boundary conditions. We exploit this connection to derive an obstruction for the implementability of such topological symmetries as Kramers-Wannier type dualities on symmetric subalgebras.