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arXiv:2610.28253

Proliferation Transitions for Quantum Doubles of Groups

Yuhan Gai, Sakura Schäfer‐Nameki

Published Oct 20, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2610.28253

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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.
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 present AlgebrasTQD, an open-source code written in GAP (Groups, Algorithms, and Programming), for computing condensable algebras and the corresponding topological boundaries and interfaces of twisted quantum doubles Dω(G)D^ω(G)Dω(G). These algebras can be used to classify 1+1d gapped and gapless phases whose symmetry is related by gauging to a finite group GGG with anomaly ω∈H3(G,U(1))ω\in H^3(G,U(1))ω∈H3(G,U(1)), to study 2+1d anyon proliferation and to construct logical gates in non-Abelian topological quantum codes. The functions in AlgebrasTQD compute all condensable algebras A(H,F,γ,ε)A(H,F,γ,ε)A(H,F,γ,ε) with their algebra structure and anyons, sort them into isomorphism classes and identify the twin algebras that are inequivalent but share the same anyon decomposition. AlgebrasTQD also determines all the subalgebra relations, the corresponding Hasse diagram and the types of 1+1d phases for any chosen, possibly non-invertible, group-theoretical symmetry. For non-maximal algebras, the code computes the reduced topological order with its 333-cocycle and the map of anyons across the topological interface. Finally, AlgebrasTQD includes functions that construct certain non-Clifford logical gates via non-Abelian surface codes. We summarize the formulas that are implemented in the AlgebrasTQD code, and we illustrate its use with Abelian and non-Abelian groups including Z23\mathbb{Z}_2^3Z23​ with ωIIIω_\text{III}ωIII​, D8D_8D8​, A4A_4A4​ with and without anomaly, twin algebras and twin phases, and the Clifford-hierarchy gates in quantum doubles of dihedral groups. All arithmetic is exact and AlgebrasTQD can be used with any finite group GGG and 3-cocycle ωωω, limited only by the time and size of the calculation. The code is available at https://github.com/AlisonWarman/AlgebrasTQD
Condensable algebras in 2+1d non-chiral topological orders characterize gapped boundary conditions and interfaces. Applied to the Symmetry Topological Field Theory, they allow classification of symmetric gapped phases and impose sharp constraints on possible phase transitions. A condensable algebra is specified not only by its underlying set of anyons, which end on the boundary or interface, but also by its algebra structure. We introduce the concept of twin condensable algebras, which have the same anyon decomposition, but inequivalent algebra structure. We revisit the classification of condensable algebras in Z(VecGω)\mathcal{Z}(\text{Vec}_G^ω)Z(VecGω​), i.e. in group-theoretical topological orders for finite groups GGG with anomaly ωωω. In this context we are able to identify twin algebras that arise from different mechanisms, such as subgroup data, SPT cocycles, and symmetry actions. In particular, we construct infinite families of examples of twins from so-called Gassmann triples, and exhibit cases in which the reduced topological orders are inequivalent despite having identical anyon content. Physically, twin algebras describe distinct symmetric phases that have isomorphic spaces of ground states, but inequivalent order parameters. Such twin phases never exhibit relative spontaneous symmetry breaking, and can be used to construct phase transitions without hidden symmetry breaking, which are intrinsically beyond Landau transitions.
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.
Abstract We classify condensable E 2-algebras in a modular tensor category C up to 2-Morita equivalence. Physically, this classification provides an explicit criterion to determine when distinct condensable E 2-algebras yield the same condensed topological phase under a two-dimensional anyon condensation process. The relations between different condensable algebras can be translated into their module categories, interpreted physically as gapped domain walls in topological orders. As concrete examples, we interpret the categories of quantum doubles of finite groups and examples beyond group symmetries. Our framework fully elucidates the interplay among condensable E 1-algebras in C , condensable E 2-algebras in C up to 2-Morita equivalence, and Lagrangian algebras in C ⊠ C ¯ .
Self-dual S3S_3S3​ gauge theory in 2+1d: lattice model and topological phase transitions
Electric-magnetic self-duality of the Z2\mathbb{Z}_2Z2​ gauge theory, realized microscopically as a half-lattice-translation exchanging electric charge and magnetic flux, has been an influential example of a duality symmetry with an exact lattice realization. We construct the first non-Abelian generalization of this construction: a lattice model of the S3S_3S3​ quantum double D(S3)\mathcal{D}(S_3)D(S3​) on a tensor product Hilbert space in which the Z2em\mathbb{Z}^{\mathrm{em}}_2Z2em​ anyon-permutation symmetry, exchanging the non-Abelian chargeon CCC and fluxon FFF, is realized via lattice translation. Consequently we find that the zigzag boundary termination of the model realizes, without fine-tuning, a gapless critical edge state described by the tetracritical Ising CFT. The bulk admits three independent Z2em\mathbb{Z}_2^{\mathrm{em}}Z2em​-preserving bosonic perturbations, driving D(S3)\mathcal{D}(S_3)D(S3​) into distinct gapped phases. We analyze these transitions by three independent methods: category-theoretic anyon condensation, microscopic lattice Hamiltonians, and Chern-Simons-Higgs theory, which all agree, yielding a unified picture. These examples motivate a minimal-condensation principle: proliferating a bosonic anyon generically drives condensation of a minimal condensable algebra containing it, with symmetry-related condensates appearing as degenerate vacua that spontaneously break the anyon-permutation symmetry. Our model construction extends to an infinite family of self-dual dihedral quantum doubles D(D2n)\mathcal{D}(D_{2n})D(D2n​). Notably, each model is sign-problem-free, opening the door to large-scale numerical exploration of the phases of non-Abelian Chern-Simons-Higgs theories.