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arXiv:2608.12157

Connecting the tensor-categorical formulation of anyon condensation with operator algebras and entropic order parameters

Hua-Chen Zhang

cond-mat.str-elhep-thquant-phSubmitted Aug 12, 2026• Announced Aug 13, 2026• 0 citations
v1 · Aug 12, 2026v2 · Sep 3, 2026v3 · Oct 6, 2026
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Abstract

Anyon condensation that describes the transition between topological quantum field theories can be formulated in the language of tensor categories or that of operator algebras. We deploy the formalism of Doplicher-Haag-Roberts bimodules over quasi-local $\mathrm{C}^{*}$-algebras recently developed in [1] to investigate anyon condensation, which is associated with an extension of a certain operator algebra. The connection between notions in the two formulations is thereby made manifest in an intuitive, diagrammatic manner. An entropic order parameter, as the quantum information-theoretic measure characterising a condensation, is naturally defined, and we give a very simple proof of a bound on it.

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Anyon condensation that describes the transition between topological quantum field theories can be formulated in the language of tensor categories or that of operator algebras. We deploy the formalism of Doplicher-Haag-Roberts bimodules over quasi-local C∗\mathrm{C}^{*}C∗-algebras recently developed in [1] to investigate anyon condensation, which is associated with an extension of a certain operator algebra. The connection between notions in the two formulations is thereby made manifest in an intuitive, diagrammatic manner. An entropic order parameter, as the quantum information-theoretic measure characterising a condensation, is naturally defined, and we give a very simple proof of a bound on it.
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We present a unified category-theoretic framework for quantum Hall hierarchy constructions and anyon superconductivity based on modular tensor categories over Rep(U(1))\mathrm{Rep}(\mathrm{U}(1))Rep(U(1)) and sRep(U(1)f)\mathrm{sRep}(\mathrm{U}(1)^f)sRep(U(1)f). Our approach explicitly incorporates conserved U(1)\mathrm{U}(1)U(1) charge and formulates doping via a generalized stack-and-condense procedure, in which an auxiliary topological order is stacked onto the parent phase, and the quasiparticles created by doping subsequently condense. Depending on whether this condensation preserves or breaks the U(1)\mathrm{U}(1)U(1) symmetry, the system undergoes a transition to a quantum Hall hierarchy state or to an anyon superconductor. For anyon superconductors, the condensate charge is determined unambiguously by the charged local bosons contained in the condensable algebra. Our framework reproduces all known anyon superconductors obtained from field-theoretic analyses and further predicts novel phases, including a charge-2e2e2e anyon superconductor derived from the Laughlin state and charge-kekeke anyon superconductors arising from bosonic Zk\mathbb{Z}_kZk​ Read-Rezayi states. By placing hierarchy transitions and anyon superconductivity within a single mathematical formalism, our work provides a unified understanding of competing and proximate phases near experimentally realizable fractional quantum Hall states.
Abstract We propose entropic order parameters that capture the physics of generalized symmetries and phases in QFT’s. We do it through an analysis of simple properties (additivity and Haag duality) of the net of operator algebras attached to space-time regions. We observe that different types of symmetries are associated with the breaking of these properties in regions of different non-trivial topologies. When such topologies are connected, we show that the non locally generated operators generate an Abelian symmetry group, and their commutation relations are fixed. The existence of order parameters with area law, like the Wilson loop for the confinement phase, or the ’t Hooft loop for the dual Higgs phase, is shown to imply the existence of more than one possible choice of algebras for the same underlying theory. A natural entropic order parameter arises by this non-uniqueness. We display aspects of the phases of theories with generalized symmetries in terms of these entropic order parameters. In particular, the connection between constant and area laws for dual order and disorder parameters is transparent in this approach, new constraints arising from conformal symmetry are revealed, and the algebraic origin of the Dirac quantization condition (and generalizations thereof) is described. A novel tool in this approach is the entropic certainty relation satisfied by dual relative entropies associated with complementary regions, which quantitatively relates the statistics of order and disorder parameters.
We investigate the algebraic theory of symmetry-enriched topological (SET) order in (2+1)D bosonic topological phases of matter and its applications to topological quantum computing. Our goal is twofold: first, to demonstrate how an abstract categorical approach can be applied to understand phenomena in (2+1)D topological phases of matter, and second, to show how ideas from physics can be useful for categorification.After reviewing modular tensor categories (MTCs) and their role as algebraic theories of anyons in topological phases of matter, we recall their associated quantum representations and their interpretation as quantum gates for a topological quantum computer. Next we recall the characterization of SET order in terms of GGG-crossed braided extensions of MTCs and the mathematical formalism of topological quantum computing (TQC) with anyons and symmetry defects.We then apply modular tensor category theory to construct algebraic models of symmetry defects in multi-layer (2+1)D topological order with layer-exchange permutation symmetry. Our main result frames a correspondence between bilayer symmetry enriched topological order and monolayer topological order on surfaces with genus, illuminating a connection between quantum symmetry and topological order that first appeared in the work of condensed matter theorists Barkeshi, Ji, and Qian.