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arXiv:2610.28148

Field theories and critical excitations at anyon condensation transitions

Luisa Eck, Xie Chen

Published Oct 20, 2026• 0 citations
arXivHTMLPDF

Abstract

Anyon condensation transitions remain challenging to characterize especially when the condensing anyons are non-abelian. We develop two complementary approaches for extracting information about the transition from the categorical data of the parent topological order and the condensate. The first adapts a Ginzburg-Landau procedure developed for $1+1$D transitions in the SymTFT setting to $2+1$D anyon condensation transitions. The construction applies when the anyons participating in the condensate have integer quantum dimensions. The second gives a criterion for identifying which point and loop excitations become critical at a continuous transition. The key step is to determine which anyon loops remain condensed throughout. All other loops become critical, while a point excitation becomes critical if it braids trivially with every condensed loop. We illustrate both proposals in several abelian and non-abelian examples.

DOI
10.48550/arxiv.2610.28148

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Anyon condensation transitions remain challenging to characterize especially when the condensing anyons are non-abelian. We develop two complementary approaches for extracting information about the transition from the categorical data of the parent topological order and the condensate. The first adapts a Ginzburg-Landau procedure developed for 1+11+11+1D transitions in the SymTFT setting to 2+12+12+1D anyon condensation transitions. The construction applies when the anyons participating in the condensate have integer quantum dimensions. The second gives a criterion for identifying which point and loop excitations become critical at a continuous transition. The key step is to determine which anyon loops remain condensed throughout. All other loops become critical, while a point excitation becomes critical if it braids trivially with every condensed loop. We illustrate both proposals in several abelian and non-abelian examples.
Topological phases - systems with a global ordering in the entanglement - exhibit exotic excitations known as anyons which can display highly nontrivial (non-Abelian) statistics. Through transitions between such phases, these anyons can undergo rather surprising changes. Perhaps most surprisingly, one anyon can split into two or more anyons, some of which might condense or become confined. The authors propose here a complete set of order parameters which allow one to extract the full information about how the properties of anyons change through phase transitions, and to apply them to a range of topological phase 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.
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.
Anyon superconductivity arises from the condensation of mobile anyons rather than from a conventional Cooper instability, yet a systematic wavefunction description remains lacking. We develop a hierarchy-wavefunction construction for superconducting states derived from parent topological orders and identify their off-diagonal long-range order, condensate charge, chiral central charge, and residual topological order through the plasma analogy and topological field theories. We construct Abelian and non-Abelian examples descending from the semion state, a ν=2/3ν=2/3ν=2/3 hierarchy state, the ν=1/3ν=1/3ν=1/3 Laughlin state, ν=1ν=1ν=1 integer quantum Hall state, and Pfaffian state. Remarkably, for the semion case, the superconducting many-semion wavefunction is equivalent to a state of fermionized anyons filling two effective Landau levels, recovering Laughlin's original construction of semion superconductor. Finally, we show that hierarchy wavefunctions emerge naturally in the dilute, long-distance limit of the anyon-Hilbert-space formulation, which applies to ideal Chern bands and moiré bands of twisted bilayer MoTe2_22​. Our results establish a unified wavefunction-level framework linking anyon condensation, superconducting order, and topological field theory.
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.
A bstract We develop a comprehensive framework for realizing anyon condensation of topological orders within the string-net model by constructing a Hamiltonian that bridges the parent string-net model before and the child string-net model after anyon condensation. Our approach classifies all possible types of bosonic anyon condensation in any parent string-net model and identifies the basic degrees of freedom in the corresponding child models. The Kogut-Susskind lattice gauge theory model is a special case of our model if the full degrees of freedom of the model are truncated from SU(2) representations to quantum group SU(2) k . Compared with the traditional UMTC perspective of topological orders, our method offers a finer categorical description of anyon condensation at the microscopic level. We also explicitly represent relevant UMTC categorical entities characterizing anyon condensation through our model-based physical quantities, providing practical algorithms for calculating these categorical data.