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.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.

arXiv:2406.14320SciPost Physics CoreOct 20, 20253 citations

Proliferation Transitions for Non-Abelian Anyons

Sakura Schafer-Nameki, Yunqin Zheng, Andrea Antinucci

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.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.

cond-mat.str-elCross-listedarXiv:2608.12303Aug 13, 20260 citations

Anyons and quantum groups

Alberto Lerda A, S. Sciuto

arXiv:hep-th/9301100Nuclear Physics BJul 1, 199389 citations

Continuous topological phase transition between Z 2 topologically ordered phases

Qi Zhang, Wen-Tao Xu

Topological phase transitions beyond anyon condensation remain poorly understood. A notable example is the transition between the toric code (TC) and double semion (DS) phases, which has two distinct Z 2 topological orders in (2+1) dimensions [ ( 2 + 1 ) D ]. Previous studies reveal that the transition between them can be either first order or via an intermediate phase, thus the existence of a direct continuous transition between them remains a long-standing problem. Motivated by the fact that both phases can arise from condensing distinct anyons in the Z 4 topological order, we introduce a perturbed Z 4 quantum double (QD) model to study the TC-DS transition. We confirm the existence of a continuous ( 2 + 1 ) D XY* transition between the TC and DS phases by mapping it to a two-coupled quantum Ising model. Importantly, using the condensation order parameters and the area law coefficients of the Wilson loops, we further reveal that Z 4 anyons, fractionalized from the Z 2 topological orders, become deconfined at the transition between Z 2 topologically ordered phases. Our results open a path toward developing a theoretical framework for topological phase transitions beyond anyon condensation.Topological phase transitions beyond anyon condensation remain poorly understood. A notable example is the transition between the toric code (TC) and double semion (DS) phases, which has two distinct Z 2 topological orders in (2+1) dimensions [ ( 2 + 1 ) D ]. Previous studies reveal that the transition between them can be either first order or via an intermediate phase, thus the existence of a direct continuous transition between them remains a long-standing problem. Motivated by the fact that both phases can arise from condensing distinct anyons in the Z 4 topological order, we introduce a perturbed Z 4 quantum double (QD) model to study the TC-DS transition. We confirm the existence of a continuous ( 2 + 1 ) D XY* transition between the TC and DS phases by mapping it to a two-coupled quantum Ising model. Importantly, using the condensation order parameters and the area law coefficients of the Wilson loops, we further reveal that Z 4 anyons, fractionalized from the Z 2 topological orders, become deconfined at the transition between Z 2 topologically ordered phases. Our results open a path toward developing a theoretical framework for topological phase transitions beyond anyon condensation.

arXiv:2508.08376Physical review. B./Physical review. BNov 12, 20253 citations

Some Properties of Anyons

H. Mori

Springer proceedings in physicsJan 1, 19920 citations

D(S_3)

and

SU(2)_k

Chern-Simons theories, and extend it to anomalous anyons.