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arXiv:2610.04704

Peeling off Abelian Anyons and Proliferation Transitions

Meng Cheng, Nathan Seiberg B

Published Oct 3, 2026• 0 citations
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Abstract

We study the invertible one-form global symmetry of a (2+1)-dimensional topological quantum field theory (TQFT). A universal description of this symmetry ``peels off'' all the Abelian anyons from the rest of the TQFT. This allows us to study this symmetry and some of its consequences, independently of the details of the rest of the system. We then describe these Abelian anyons using $\mathrm{U}(1)$ gauge fields with Chern-Simons terms and couple them to a scalar field, thus constructing a particular non-topological theory. This new theory, our transition field theory, exhibits a phase transition from the original TQFT to another phase. The latter is either gapless or another TQFT. This discussion generalizes and extends our earlier work (arXiv:2603.00245). In particular, we give a more detailed description of the global symmetries and their fates in the various phases.

DOI
10.48550/arxiv.2610.04704

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We study the invertible one-form global symmetry of a (2+1)-dimensional topological quantum field theory (TQFT). A universal description of this symmetry ``peels off'' all the Abelian anyons from the rest of the TQFT. This allows us to study this symmetry and some of its consequences, independently of the details of the rest of the system. We then describe these Abelian anyons using U(1)\mathrm{U}(1)U(1) gauge fields with Chern-Simons terms and couple them to a scalar field, thus constructing a particular non-topological theory. This new theory, our transition field theory, exhibits a phase transition from the original TQFT to another phase. The latter is either gapless or another TQFT. This discussion generalizes and extends our earlier work (arXiv:2603.00245). In particular, we give a more detailed description of the global symmetries and their fates in the various phases.
We study phase transitions out of a (2+1)d topological phase driven by the proliferation of anyons in a braided fusion subcategory. We describe a general theoretical framework for such transitions, in which the topological quantum field theory (TQFT) is coupled to dynamical matter associated with anyons in the subcategory. We interpret this construction using symmetry topological field theory (SymTFT), where the dynamics is localized on the symmetry boundary of the (3+1)d slab. We analyze several examples, and present explicit Chern-Simons-Higgs (CSH) field theories realizing the proliferation transitions. We also examine proliferation transitions that implement gauging of non-invertible one-form symmetry in the parent TQFT. This includes a general construction for Rep(G){\rm Rep}(G)Rep(G) one-form symmetry for a finite group GGG, and a CSH realization of the non-invertible one-form gauging SU(2)10→Spin(5)1{\rm SU}(2)_{10}\rightarrow {\rm Spin}(5)_1SU(2)10​→Spin(5)1​. We discuss which anyons proliferate at these transitions.
We construct phase transitions that proliferate condensable anyons in general 2+1d topological orders, including non-abelian ones. The central tool that provides a systematic approach to this question is the Symmetry Topological Field Theory (SymTFT). For a given topological order, we identify the relevant symmetry from the transparent lines generated by the condensable anyons, and thereby realize the topological order in terms of a 3+1d SymTFT sandwich. The proliferation phase transition is realized by coupling scalar fields to the anyons purely on the symmetry boundary of the SymTFT. We illustrate the construction for abelian theories, as well as non-abelian ones, D(S3)D(S_3)D(S3​) and SU(2)kSU(2)_kSU(2)k​ Chern-Simons theories, and extend it to anomalous anyons.
We consider phase transitions out of a general topological phase in 2+12+12+1 dimensions. We assume that the transition is triggered by a single Abelian anyon, which becomes light near the transition and whose worldlines proliferate after the transition. (This proliferation is often referred to as ``condensation.'') We describe the transition using a continuum field theory obtained by coupling the corresponding topological quantum field theory (TQFT) to a single complex scalar field associated with this anyon. With these assumptions, we find the most general relativistic field theory for such a transition. Even though for a given TQFT and a choice of anyon, there are infinitely many such field theories, the transition theory depends on only a single additional integer parameter. We analyze all these theories, their global symmetries, and their phases. In generic cases, the theory after the transition can be related to the original one via an Abelian hierarchy construction. In special cases, the theory after the transition is gapless, and with a particular deformation, it is related to the original TQFT by gauging an anomaly-free one-form global symmetry. We also explore the enrichment of this setup by a global U(1) symmetry. In some cases, enriching the original TQFT is incompatible with the full transition theory. Finally, we demonstrate our construction with many specific examples.
Symmetry-protected topological phases (SPT) are short-range entangled gapped states protected by global symmetry. Nontrivial SPT phases cannot be adiabatically connected to the trivial disordered state (or atomic insulator) as long as certain global symmetry GGG is unbroken. At low energies, most of the two-dimensional SPTs with Abelian symmetry can be described by topological quantum field theory (TQFT) of the multicomponent Chern-Simons type. However, in contrast to the fractional quantum Hall effect where TQFT can give rise to interesting bulk anyons, TQFT for SPTs only supports trivial bulk excitations. The essential question in TQFT descriptions for SPTs is to understand how the global symmetry is implemented in the partition function. In this paper, we systematically study TQFT of three-dimensional SPTs with unitary Abelian symmetry (e.g., ${\mathbb{Z}}_{{N}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{N}_{2}}\ifmmode\times\else\texttimes\fi{}...$). In addition to the usual multicomponent BFBFBF topological term at level-1, we find that there are new topological terms with quantized coefficients (e.g., ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}d{a}^{2}$ and ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}{a}^{3}\ensuremath{\wedge}{a}^{4}$) in TQFT actions, where a1,a2,...{a}^{1},{a}^{2},...a1,a2,... are 1-form U(1) gauge fields. These additional topological terms cannot be adiabatically turned off as long as GGG is unbroken. By investigating symmetry transformations for the TQFT partition function, we end up with the classification of SPTs that is consistent with the well-known group cohomology approach. We also discuss how to gauge the global symmetry and possible TQFT descriptions of Dijkgraaf-Witten gauge theory.
We find a series of non-Abelian topological phases that are separated from the deconfined phase of ZN{Z}_{N}ZN​ gauge theory by a continuous quantum phase transition. These non-Abelian states, which we refer to as the ``twisted'' ZN{Z}_{N}ZN​ states, are described by a recently studied $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons (CS) field theory. The $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ CS theory provides a way of gauging the global Z2{Z}_{2}Z2​ electric-magnetic symmetry of the Abelian ZN{Z}_{N}ZN​ phases, yielding the twisted ZN{Z}_{N}ZN​ states. We introduce a parton construction to describe the Abelian ZN{Z}_{N}ZN​ phases in terms of integer quantum Hall states, which then allows us to obtain the non-Abelian states from a theory of Z2{Z}_{2}Z2​ fractionalization. The non-Abelian twisted ZN{Z}_{N}ZN​ states do not have topologically protected gapless edge modes and, for N>2N>2N>2, break time-reversal symmetry.
Topological quantum computation relies on braiding non-Abelian anyons, but requires the underlying topological order to survive imperfect state preparation and environmental noise. We show that the instability of topological order to wavefunction deformations and to decoherence, with the latter probed by syndrome distributions, is generically captured by stat-mech models whose G×Rep(G)G\times \textrm{Rep}(G)G×Rep(G) symmetry naturally exposes the corrupting anyonic excitations. As an example, we combine this framework with Monte-Carlo simulations to resolve the stability of D4D_4D4​ topological order under deformations and quantum channels that proliferate multiple non-Abelian anyon species that are individually unable to condense. We show that beyond a finite threshold, proliferation of two non-Abelian anyon species parasitically condenses a shared Abelian-anyon fusion outcome---destroying the topological order. Our symmetry-based approach suggests the resulting trivial phase is distinct from that obtained by condensing all Abelian anyons; in other words, the trivial phase "remembers" which anyons condensed. This framework provides a first step into identifying the relevant symmetry for optimal decoders, conditioned on syndrome measurements, of non-Abelian topological order.