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arXiv:2406.14320 · SciPost Physics Core

Anyon condensation in mixed-state topological order

Ken KIKUCHI, Kah-Sen Kam, Fu-Hsiang Huang

Published Oct 20, 2025• 3 citations
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Abstract

We discuss anyon condensation in mixed-state topological order. The phases were recently conjectured to be classified by pre-modular fusion categories. Just like anyon condensation in pure-state topological order, a bootstrap analysis shows condensable anyons are given by connected étale algebras. We explain how to perform generic anyon condensation including non-invertible anyons and successive condensations. Interestingly, some condensations lead to pure-state topological orders. We clarify when this happens. We also compute topological invariants of equivalence classes.

DOI
10.21468/scipostphyscore.8.4.072

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We discuss anyon condensation in mixed-state topological order. The phases were recently conjectured to be classified by pre-modular fusion categories. Just like anyon condensation in pure-state topological order, a bootstrap analysis shows condensable anyons are given by connected étale algebras. We explain how to perform generic anyon condensation including non-invertible anyons and successive condensations. Interestingly, some condensations lead to pure-state topological orders. We clarify when this happens. We also compute topological invariants of equivalence classes.
Instead of constructing anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relations between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A-modules in C. More general situation is also discussed in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin-Wen types of lattice models and some chiral topological phases.
Instead of studying anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relation between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A-modules in C. A more general situation is also studied in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin–Wen types of lattice models and some chiral topological phases.
The classification and characterization of topological phases of matter is well understood for ground states of gapped Hamiltonians that are well isolated from the environment. However, decoherence due to interactions with the environment is inevitable—thus motivating the investigation of topological orders in the context of mixed states. Here, we take a step toward classifying mixed-state topological orders in two spatial dimensions by considering their (emergent) generalized symmetries. We argue that their 1-form symmetries and the associated anyon theories lead to a partial classification under two-way connectivity by quasilocal quantum channels. This allows us to establish mixed-state topological orders that are intrinsically mixed, i.e., that have no ground-state counterpart. We provide a wide range of examples based on topological subsystem codes, decohering G -graded string-net models, and “classically gauging” symmetry-enriched topological orders. One of our main examples is an Ising string-net model under the influence of dephasing noise. We study the resulting space of locally indistinguishable states and compute the modular transformations within a particular coherent space. Based on our examples, we identify two possible effects of quasilocal quantum channels on anyon theories: (1) anyons can be incoherently proliferated—thus reducing to a commutant of the proliferated anyons, or (2) the system can be “classically gauged,” resulting in the symmetrization of anyons and an extension by transparent bosons. Given these two mechanisms, we conjecture that mixed-state topological orders are classified by premodular anyon theories, i.e., those for which the braiding relations may be degenerate.
[NOTE, June 2026: Computations in this paper used mtctools <= 0.2.x, whose condensable-algebra finder has since been found to have false negatives on conformal-embedding (E-type) algebras (e.g. 1+(j=3) in SU(2)_10, within the level range covered here), and whose condensed-phase identification did not resolve the sixteen-fold way. The boundary counts below are therefore lower bounds where E-type algebras were missed; re-check against mtctools 0.3.0 and the corrected release at DOI 10.5281/zenodo.20618695. The text below is the original and is retained for the record.] We develop a systematic computational framework connecting anyon condensation in modular tensor categories to the classification of non-invertible symmetry boundaries in the Symmetry Topological Field Theory (SymTFT) paradigm. For each condensable algebra in a doubled MTC, we identify the corresponding symmetry boundary condition and classify the resulting non-invertible symmetry structure. We present explicit results for SU(2) Chern-Simons theories at levels k=1-8, SU(3) at levels k=1-3, and G₂ at level 1, covering 40+ modular tensor categories and identifying 100+ distinct symmetry boundary types.
We generalize the hierarchy construction to generic 2+1D topological orders (which can be non-Abelian) by condensing Abelian anyons in one topological order to construct a new one. We show that such construction is reversible and leads to a new equivalence relation between topological orders. We refer to the corresponding equivalence class (the orbit of the hierarchy construction) as "the non-Abelian family." Each non-Abelian family has one or a few root topological orders with the smallest number of anyon types. All the Abelian topological orders belong to the trivial non-Abelian family whose root is the trivial topological order. We show that Abelian anyons in root topological orders must be bosons or fermions with trivial mutual statistics between them. The classification of topological orders is then greatly simplified, by focusing on the roots of each family: those roots are given by non-Abelian modular extensions of representation categories of Abelian groups.
In this paper we would like to demonstrate how the known, physically-motivat-ed rules of anyon condensation proposed by Bais et al. can be recovered by the mathematics of twist-free commutative separable Frobenius algebra (CSFA). In some simple cases, those physical rules are also sufficient conditions defining a twist-free CSFA. This allows us to make use of the generalized ADE classification of CSFA’s and modular invariants to classify anyon condensation, characterize the topological boundaries between topological field theories and thus describe all gapped domain walls and gapped boundaries of a large class of topological orders. In fact, this classification is equivalent to the classification we proposed in ref. [1].