Anyon condensation and its applications

TL;DR

This work surveys anyon condensation in 2D topologically ordered phases, focusing on bosonic condensates and the resulting topological symmetry breaking (TSB). It develops a unified picture where identifications, splittings, and confinement determine the condensed phase and connects these changes to gapped boundaries and edge physics, including the emergence of anyon-permuting symmetries and defect lines. The review also discusses critical points in TSB, highlighting dualities to Ising and Potts models for abelian condensates and the more challenging non-abelian cases, supported by lattice and tensor-network approaches. Overall, the paper establishes TSB as a foundational framework for relating bulk topological orders, their boundaries, and symmetry-related phenomena, while outlining open questions in non-abelian condensation and criticality.

Abstract

Bose condensation is central to our understanding of quantum phases of matter. Here we review Bose condensation in topologically ordered phases (also called topological symmetry breaking), where the condensing bosons have non-trivial mutual statistics with other quasiparticles in the system. We give a non-technical overview of the relationship between the phases before and after condensation, drawing parallels with more familiar symmetry-breaking transitions. We then review two important applications of this phenomenon. First, we describe the equivalence between such condensation transitions and pairs of phases with gappable boundaries, as well as examples where multiple types of gapped boundary between the same two phases exist. Second, we discuss how such transitions can lead to global symmetries which exchange or permute anyon types. Finally we discuss the nature of the critical point, which can be mapped to a conventional phase transition in some -- but not all -- cases.

Figures (3)

• Figure 3: A pair of $\sigma_1 \sigma_2$ bound states, initially with both $\sigma_1$ particles, and both $\sigma_2$ particles, fusing to the vacuum, is braided with a $\sigma_2$ particle. (a) Before braiding, the net fermion parity of the two pairs is even. (b) After braiding a $\sigma_2$ anyon around one of the $\overline{\sigma}_1\sigma_2$ anyons, the fermion parity of the remaining two $\sigma$'s in layer 2 (represented by a black dashed line) has changed. The net fermion parity of the two $\sigma_1\sigma_2$ pairs is now odd, meaning that one of these pairs has changed its fermion parity. A similar argument holds in other initial fusion channels.
• Figure 4: (a) Creating a boundary between topological orders $\mathcal{C}$ and $\mathcal{D}$ by condensing the boson $\gamma$ in the region $x>0$. $\gamma$ anyons can "dissappear" (indicated by an x) at this boundary; a confined anyon $a$ brought to $x>0$ incurs an energy cost proportional to its separation from the boundary (indicated by the thick dashed red line). (b) By folding the plane along the line $x=0$, we see that this is equivalent to a boundary $\mathcal{C} \times \overline{\mathcal{D}}$ with the vacuum.
• Figure 5: Mixed boundary conditions between the Toric code and the vacuum: along the top and bottom edges of the square (solid red lines) $e$ anyons have condensed; on the left and right edges (dashed green lines) $m$ anyons have. Dashed yellow lines show two distinct paths by which a pair of $e$ particles created in the bulk can be brought to the $e$ boundaries, where they can disappear.