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:2608.12303

Proliferation Transitions for Non-Abelian Anyons

Sakura Schafer-Nameki, Yunqin Zheng, Andrea Antinucci

cond-mat.str-elhep-thmath.QAquant-phSubmitted Aug 12, 2026• Announced Aug 13, 2026• 0 citations• Revised Aug 21, 2026 (v2)
v1 · Aug 12, 2026v2 · Aug 21, 2026
arXivHTMLPDF

Abstract

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, $D(S_3)$ and $SU(2)_k$ Chern-Simons theories, and extend it to anomalous anyons.

Comments
25 pages, v2: added SU(2)_k exceptional transitions and references

Related papers

Note on proliferation transitions of non-Abelian anyons

Meng Cheng

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 ${\rm Rep}(G)$ one-form symmetry for a finite group $G$, and a CSH realization of the non-invertible one-form gauging ${\rm SU}(2)_{10}\rightarrow {\rm Spin}(5)_1$. We discuss which anyons proliferate at these transitions.

arXiv:2608.20604•Aug 20, 2026•0 citations

Proliferation Transitions for Quantum Doubles of Groups

Yuhan Gai, Sakura Schäfer‐Nameki

Proliferation transitions are phase transitions between a 2+1d topological order $\mathcal{T}$ and another $\mathcal{T}/\mathcal{A}$, obtained by condensing anyons in an algebra $\mathcal{A}\subset \mathcal{T}$. We consider such transitions in quantum doubles of groups, $\mathcal{T}=D(G, ω)$, for finite groups $G$ possibly with three-cocycle $ω$, for which condensable algebras are fully classified. Given a condensable algebra $\mathcal{A}$, we represent $\mathcal{T}$ in terms of a Symmetry Topological Field Theory (SymTFT), that is tailored to the $\mathcal{A}$ proliferation transition. The SymTFT bulk is constructed from the Müger center of the subcategory generated by the algebra, i.e. the category of transparent lines, which can be bosons or fermions. Properties of the transition can then be inferred directly from the algebra classification data. For gauged Landau-Ginzburg (LG*) type transitions, we show that they do not only depend on the anyons in $\mathcal{A}$ through their field content, but the cubic interactions are fixed in terms of the multiplication structure of $\mathcal{A}$. Applied to twin algebras, which have the same anyon content but different multiplication structures, we show that twins result in distinct proliferation transitions.

arXiv:2610.28253•Oct 20, 2026•0 citations

Non-Abelian Anyon Condensation: a Path-Integral Monte Carlo Approach

Rafael Flores-Calderón, Frank Pollmann, Michael Knap

Transitions out of non-Abelian topological order are difficult to describe in microscopic quantum models with numerical methods that remain tractable at large scales. We develop a sign-free path-integral framework for Kitaev quantum doubles $\mathcal D(G)$ by organizing single-link perturbations in terms of non-invertible electric and magnetic 1-form symmetries that proliferate distinct anyon species. For any finite group $G$, an exact $\textit{matterization}$ isometry introduces vertex degrees of freedom and maps the link-only model onto a $G$ gauge--Higgs theory. Furthermore, the 1-form symmetries of the fixed point allow us to define generalized Fredenhagen--Marcu order parameters that become finite when the corresponding anyons condense. For $G = S_3$, quantum Monte Carlo simulations show that proliferating a non-Abelian electric anyon drives a first-order transition in which all nontrivial electric anyons condense. In the purely magnetic limit, the model reduces to a $(2+1)$D pure $G$ gauge theory; for $G=S_3$, it exhibits a first-order confinement transition, diagnosed by the onset of a Wilson-loop area law and the restoration of an emergent magnetic 1-form symmetry. These results provide a unified numerical framework for non-Abelian anyon condensation, confinement, and generalized symmetry breaking.

cond-mat.str-elCross-listedarXiv:2609.19282•Sep 18, 2026•0 citations

Statistical Mechanics and Symmetries of Non-Abelian Anyon Proliferation: From Deformation to Decoherence

Avi Vadali, Robijn Vanhove, Ruben Verresen +2 more

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 $G\times \textrm{Rep}(G)$ symmetry naturally exposes the corrupting anyonic excitations. As an example, we combine this framework with Monte-Carlo simulations to resolve the stability of $D_4$ 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.

quant-pharXiv:2606.12527•Jun 12, 2026•0 citations

Peeling off Abelian Anyons and Proliferation Transitions

Meng Cheng, Nathan Seiberg B

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.

arXiv:2610.04704•Oct 3, 2026•0 citations

Proliferation transitions from a topological phase in 2 + 1 dimensions

Meng Cheng, Nathan Seiberg B

We consider phase transitions out of a general topological phase in $2+1$ 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.

arXiv:2603.00245•Physical review. B./Physical review. B•Jun 2, 2026•1 citation
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 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.
Proliferation transitions are phase transitions between a 2+1d topological order T\mathcal{T}T and another T/A\mathcal{T}/\mathcal{A}T/A, obtained by condensing anyons in an algebra A⊂T\mathcal{A}\subset \mathcal{T}A⊂T. We consider such transitions in quantum doubles of groups, T=D(G,ω)\mathcal{T}=D(G, ω)T=D(G,ω), for finite groups GGG possibly with three-cocycle ωωω, for which condensable algebras are fully classified. Given a condensable algebra A\mathcal{A}A, we represent T\mathcal{T}T in terms of a Symmetry Topological Field Theory (SymTFT), that is tailored to the A\mathcal{A}A proliferation transition. The SymTFT bulk is constructed from the Müger center of the subcategory generated by the algebra, i.e. the category of transparent lines, which can be bosons or fermions. Properties of the transition can then be inferred directly from the algebra classification data. For gauged Landau-Ginzburg (LG*) type transitions, we show that they do not only depend on the anyons in A\mathcal{A}A through their field content, but the cubic interactions are fixed in terms of the multiplication structure of A\mathcal{A}A. Applied to twin algebras, which have the same anyon content but different multiplication structures, we show that twins result in distinct proliferation transitions.
Transitions out of non-Abelian topological order are difficult to describe in microscopic quantum models with numerical methods that remain tractable at large scales. We develop a sign-free path-integral framework for Kitaev quantum doubles D(G)\mathcal D(G)D(G) by organizing single-link perturbations in terms of non-invertible electric and magnetic 1-form symmetries that proliferate distinct anyon species. For any finite group GGG, an exact matterization\textit{matterization}matterization isometry introduces vertex degrees of freedom and maps the link-only model onto a GGG gauge--Higgs theory. Furthermore, the 1-form symmetries of the fixed point allow us to define generalized Fredenhagen--Marcu order parameters that become finite when the corresponding anyons condense. For G=S3G = S_3G=S3​, quantum Monte Carlo simulations show that proliferating a non-Abelian electric anyon drives a first-order transition in which all nontrivial electric anyons condense. In the purely magnetic limit, the model reduces to a (2+1)(2+1)(2+1)D pure GGG gauge theory; for G=S3G=S_3G=S3​, it exhibits a first-order confinement transition, diagnosed by the onset of a Wilson-loop area law and the restoration of an emergent magnetic 1-form symmetry. These results provide a unified numerical framework for non-Abelian anyon condensation, confinement, and generalized symmetry breaking.
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.
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 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.