Anyon condensation in mixed-state topological order

TL;DR

The paper develops a framework for anyon condensation in mixed-state TOs described by pre-modular fusion categories, showing that condensable anyons correspond to connected étale algebras $A\in\mathcal{B}_1$ and that condensation yields a new phase $\mathcal{B}_2\simeq(\mathcal{B}_1)_A^0$ separated by the wall $\mathcal{B}_A$. It proves that the condensed theory exists as a category of dyslectic right $A$-modules and identifies when condensations produce a pure-state TO via a theorem tying this to the presence and properties of transparent anyons. The work provides a systematic protocol for condensation, discusses the role of unitarity, and demonstrates the approach with explicit examples, including Toric Code-like and symmetric group representations, highlighting how the minimal anyon theory $\mathcal{A}^ ext{min}$ acts as a topological invariant of equivalence classes of mixed-state TOs. It also shows that successive condensations are path-independent and can yield well-known pure-state TOs, linking condensed phases to modularizations and invariants that classify mixed-state TOs. Overall, the results bridge condensed-m matter physics, category theory, and topological invariants, with potential implications for engineering robust quantum phases and understanding environment-induced transitions in topological systems.

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.

Figures (14)

• Figure 1: Two electrons condense to a Cooper pair in a new vacuum.
• Figure 2: Bosonic statistics of a Cooper pair.
• Figure 3: Unit morphism of an algebra.
• Figure 4: The particle $A$ is trivial.
• Figure 5: Separability of $A$.