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:2504.06094

On the structure of DHR bimodules of abstract spin chains

Lucas Hataishi, David Jaklitsch, Corey Jones, Yamashita, Makoto

Published Apr 8, 2025• 0 citations
arXivHTMLPDF

Abstract

Abstract spin chains axiomatize the structure of local observables on the 1D lattice which are invariant under a global symmetry, and arise at the physical boundary of 2+1D topologically ordered spin systems. In this paper, we study tensor categorical properties of DHR bimodules over abstract spin chains. Assuming that the charge transporters generate the algebra of observables, we prove that the associated category has a structure of modular tensor category with respect to the natural braiding. Under an additional assumption of algebraic Haag duality, this category becomes the Drinfeld center of the half-line fusion category.

DOI
10.48550/arxiv.2504.06094

Related papers

DHR bimodules of quasi-local algebras and symmetric quantum cellular automata

Corey Jones

For a net of C*-algebras on a discrete metric space, we introduce a bimodule version of the DHR tensor category and show that it is an invariant of quasi-local algebras under isomorphisms with bounded spread. For abstract spin systems on a lattice L\subseteq \mathbb{R}^{n} satisfying a weak version of Haag duality, we construct a braiding on these categories. Applying the general theory to quasi-local algebras A of operators on a lattice invariant under a (categorical) symmetry, we obtain a homomorphism from the group of symmetric QCA to \mathbf{Aut}_{\mathrm{br}}(\mathbf{DHR}(A)) , containing symmetric finite-depth circuits in the kernel. For a spin chain with fusion categorical symmetry \mathcal{D} , we show that the DHR category of the quasi-local algebra of symmetric operators is equivalent to the Drinfeld center \mathcal{Z}(\mathcal{D}) . We use this to show that, for the double spin-flip action \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\curvearrowright \mathbb{C}^{2}\otimes \mathbb{C}^{2} , the group of symmetric QCA modulo symmetric finite-depth circuits in 1D contains a copy of S_{3} ; hence, it is non-abelian, in contrast to the case with no symmetry.

arXiv:2304.00068•Quantum Topology•May 21, 2024•11 citations

Quantum Cellular Automata and Categorical Dualities of Spin Chains

Corey Jones, Kylan Schatz, Dominic J. Williamson

Abstract Dualities play a central role in the study of quantum spin chains, providing insight into the structure of quantum phase diagrams and phase transitions. In this work, we study categorical dualities, which are defined as bounded-spread isomorphisms between algebras of symmetry-respecting local operators on a spin chain. We consider generalized global symmetries that correspond to unitary fusion categories, which are represented by matrix-product operator algebras. A fundamental question about dualities is whether they can be extended to quantum cellular automata on the larger algebra generated by all local operators in the the unit matrix-product operator sector. For on-site representations of Hopf algebra symmetries, this larger algebra is the usual tensor product quasi-local algebra. We present a solution to the extension problem using the machinery of Doplicher–Haag–Roberts bimodules. Our solution provides a crisp categorical criterion for when an extension of a duality exists. We show that the set of possible extensions form a torsor over the invertible objects in the relevant symmetry category. As a corollary, we obtain a classification result concerning dualities in the group case.

arXiv:2410.08884•Communications in Mathematical Physics•Mar 9, 2026•4 citations

Holography for bulk-boundary local topological order

Corey Jones, Pieter Naaijkens, David Penneys

In our previous article [arXiv:2307.12552], we introduced local topological order (LTO) axioms for quantum spin systems which allowed us to define a physical boundary (associated to a cut of the lattice) manifested by a net of boundary algebras in one dimension lower. This gives a formal setting for topological holography, where the braided tensor category of DHR bimodules of the physical boundary algebra captures the bulk topological order. In this article, we extend the LTO axioms to quantum spin systems equipped with a topological boundary (domain wall with the trivial phase), again producing a physical boundary algebra for the bulk-boundary system, whose category of (topological) boundary DHR bimodules recovers the topological boundary order. We perform this analysis in explicit detail for Levin-Wen and Walker-Wang bulk-boundary systems. Along the way, we introduce a 2D braided categorical net of algebras built from a unitary braided fusion category (UBFC). Such nets arise as boundary algebras of Walker-Wang models. We consider the canonical state on this braided categorical net corresponding to the standard topological boundary for the Walker-Wang model. Interestingly, in this state, the cone von Neumann algebras are type I with finite dimensional centers, in contrast with the type II and III cone von Neumann algebras from the Levin-Wen models studied in [arXiv:2307.12552]. The superselection sectors recover the underlying unitary category of our UBFC, and it was recently proven in [arXiv:2609.20725] that the superselection category also captures the fusion and braiding.

arXiv:2506.19969•Jun 24, 2025•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

Boundary states of a bulk gapped ground state in 222-d quantum spin systems

Yoshiko Ogata

We introduce a natural mathematical definition of boundary states of a bulk gapped ground state, in the operator algebraic framework of $2$-d quantum spin systems. With approximate Haag duality at the boundary, we derive a $C^*$-tensor category $\tilde{\mathcal{M}}$ out of such boundary state. Under a non-triviality condition of the braiding in the bulk, we show that the Drinfeld center (with an asymptotic constraint) of $\tilde{\mathcal{M}}$ is equivalent to the bulk braided $C^*$-tensor category derived in [14].

arXiv:2308.08087•Aug 16, 2023•1 citation

Boundary symmetries of (2+1)D topological orders

Kylan Schatz

Abstract For a G-crossed braided extension of a unitary modular tensor category $$\mathcal {C}$$ C —as in one representing a (2+1)D symmetry enriched topological order (SETO)—preservation of global on-site group symmetry after condensation by a commutative Q-system object $$A \in \mathcal {C}$$ A ∈ C necessitates the existence of a G-equivariant structure on A. When interpreted spatially, the condensation boundary has its own internal topological symmetries. We elaborate an algebraic framework for describing the internal topological symmetries of compatible (1+1)D gapped boundaries for (2+1)D topologically ordered systems in terms of hypergroup actions. Then, we investigate the coherence of global on-site bulk symmetries and boundary symmetries. We present a categorical obstruction to the preservation of symmetry in a way which is coherent in terms of lifts of categorical actions to a certain 2-group of bulk symmetries. We give a characterization of this obstruction in the case of condensation by a Lagrangian algebra and boundary symmetries given by subalgebras of the convolution algebra associated with a Lagrangian algebra object.

arXiv:2408.10832•Letters in Mathematical Physics•Jun 12, 2025•2 citations
Abstract spin chains axiomatize the structure of local observables on the 1D lattice which are invariant under a global symmetry, and arise at the physical boundary of 2+1D topologically ordered spin systems. In this paper, we study tensor categorical properties of DHR bimodules over abstract spin chains. Assuming that the charge transporters generate the algebra of observables, we prove that the associated category has a structure of modular tensor category with respect to the natural braiding. Under an additional assumption of algebraic Haag duality, this category becomes the Drinfeld center of the half-line fusion category.
For a net of C*-algebras on a discrete metric space, we introduce a bimodule version of the DHR tensor category and show that it is an invariant of quasi-local algebras under isomorphisms with bounded spread. For abstract spin systems on a lattice L\subseteq \mathbb{R}^{n} satisfying a weak version of Haag duality, we construct a braiding on these categories. Applying the general theory to quasi-local algebras A of operators on a lattice invariant under a (categorical) symmetry, we obtain a homomorphism from the group of symmetric QCA to \mathbf{Aut}_{\mathrm{br}}(\mathbf{DHR}(A)) , containing symmetric finite-depth circuits in the kernel. For a spin chain with fusion categorical symmetry \mathcal{D} , we show that the DHR category of the quasi-local algebra of symmetric operators is equivalent to the Drinfeld center \mathcal{Z}(\mathcal{D}) . We use this to show that, for the double spin-flip action \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\curvearrowright \mathbb{C}^{2}\otimes \mathbb{C}^{2} , the group of symmetric QCA modulo symmetric finite-depth circuits in 1D contains a copy of S_{3} ; hence, it is non-abelian, in contrast to the case with no symmetry.
Abstract Dualities play a central role in the study of quantum spin chains, providing insight into the structure of quantum phase diagrams and phase transitions. In this work, we study categorical dualities, which are defined as bounded-spread isomorphisms between algebras of symmetry-respecting local operators on a spin chain. We consider generalized global symmetries that correspond to unitary fusion categories, which are represented by matrix-product operator algebras. A fundamental question about dualities is whether they can be extended to quantum cellular automata on the larger algebra generated by all local operators in the the unit matrix-product operator sector. For on-site representations of Hopf algebra symmetries, this larger algebra is the usual tensor product quasi-local algebra. We present a solution to the extension problem using the machinery of Doplicher–Haag–Roberts bimodules. Our solution provides a crisp categorical criterion for when an extension of a duality exists. We show that the set of possible extensions form a torsor over the invertible objects in the relevant symmetry category. As a corollary, we obtain a classification result concerning dualities in the group case.
In our previous article [arXiv:2307.12552], we introduced local topological order (LTO) axioms for quantum spin systems which allowed us to define a physical boundary (associated to a cut of the lattice) manifested by a net of boundary algebras in one dimension lower. This gives a formal setting for topological holography, where the braided tensor category of DHR bimodules of the physical boundary algebra captures the bulk topological order. In this article, we extend the LTO axioms to quantum spin systems equipped with a topological boundary (domain wall with the trivial phase), again producing a physical boundary algebra for the bulk-boundary system, whose category of (topological) boundary DHR bimodules recovers the topological boundary order. We perform this analysis in explicit detail for Levin-Wen and Walker-Wang bulk-boundary systems. Along the way, we introduce a 2D braided categorical net of algebras built from a unitary braided fusion category (UBFC). Such nets arise as boundary algebras of Walker-Wang models. We consider the canonical state on this braided categorical net corresponding to the standard topological boundary for the Walker-Wang model. Interestingly, in this state, the cone von Neumann algebras are type I with finite dimensional centers, in contrast with the type II and III cone von Neumann algebras from the Levin-Wen models studied in [arXiv:2307.12552]. The superselection sectors recover the underlying unitary category of our UBFC, and it was recently proven in [arXiv:2609.20725] that the superselection category also captures the fusion and braiding.
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 introduce a natural mathematical definition of boundary states of a bulk gapped ground state, in the operator algebraic framework of 222-d quantum spin systems. With approximate Haag duality at the boundary, we derive a C∗C^*C∗-tensor category M~\tilde{\mathcal{M}}M~ out of such boundary state. Under a non-triviality condition of the braiding in the bulk, we show that the Drinfeld center (with an asymptotic constraint) of M~\tilde{\mathcal{M}}M~ is equivalent to the bulk braided C∗C^*C∗-tensor category derived in [14].
Abstract For a G-crossed braided extension of a unitary modular tensor category C\mathcal {C}C C —as in one representing a (2+1)D symmetry enriched topological order (SETO)—preservation of global on-site group symmetry after condensation by a commutative Q-system object A∈CA \in \mathcal {C}A∈C A ∈ C necessitates the existence of a G-equivariant structure on A. When interpreted spatially, the condensation boundary has its own internal topological symmetries. We elaborate an algebraic framework for describing the internal topological symmetries of compatible (1+1)D gapped boundaries for (2+1)D topologically ordered systems in terms of hypergroup actions. Then, we investigate the coherence of global on-site bulk symmetries and boundary symmetries. We present a categorical obstruction to the preservation of symmetry in a way which is coherent in terms of lifts of categorical actions to a certain 2-group of bulk symmetries. We give a characterization of this obstruction in the case of condensation by a Lagrangian algebra and boundary symmetries given by subalgebras of the convolution algebra associated with a Lagrangian algebra object.