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arXiv:2510.18021

Compactification of quasi-local algebras on the lattice

Ikeda, Jun

Published Oct 20, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2510.18021

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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.
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.
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.
We introduce and study (fermionic and bosonic) invertible quasi-local algebras over uniformly locally finite metric spaces XXX with infinite-dimensional local von Neumann algebras. We show that the group of Brauer equivalence classes of such algebras is isomorphic to both the group of phases of invertible states and the group of stable equivalence classes of quantum cellular automata over X×ZX\times \mathbb{Z}X×Z. Using K-theory of the symmetric monoidal category of invertible quasi-local algebras and bounded spread isomorphisms, we propose a definition of an ΩΩΩ-spectrum of invertible phases as conjectured by Kitaev. We then show that the c=12c=\frac{1}{2}c=21​ chiral Majorana fermion net and the (E8)1(E_{8})_{1}(E8​)1​ conformal net provide Brauer non-trivial invertible quasi-local algebras, thus providing explicit constructions of non-trivial invertible states and quantum cellular automata on Z2\mathbb{Z}^{2}Z2. In addition, we show that the time-slice nets of rational diagonal conformal field theories admit lattice degrees of freedom, which implies the discretization of any holomorphic conformal net is invertible.
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.
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 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.