Proliferation transitions from a topological phase in $2+1$ dimensions

TL;DR

This paper develops a comprehensive continuum field theory framework for topological phase transitions in 2+1 dimensions triggered by anyon condensation, revealing a single-parameter family of theories and their symmetry properties.

Contribution

It introduces the most general relativistic field theory for anyon-driven topological phase transitions, depending on only one integer parameter, and analyzes their global symmetries and phases.

Findings

Transition theories depend on a single integer parameter.

Post-transition phases relate to original TQFT via hierarchy or gauging.

Enrichment by U(1) symmetry can be incompatible with the transition.

Abstract

We consider phase transitions out of a general topological phase in $2+1$ dimensions. We assume that the transition is triggered by a single Abelian anyon, which becomes light near the transition and whose worldlines proliferate after the transition. (This proliferation is often referred to as ``condensation.'') We describe the transition using a continuum field theory obtained by coupling the corresponding topological quantum field theory (TQFT) to a single complex scalar field associated with this anyon. With these assumptions, we find the most general relativistic field theory for such a transition. Even though for a given TQFT and a choice of anyon, there are infinitely many such field theories, the transition theory depends on only a single additional integer parameter. We analyze all these theories, their global symmetries, and their phases. In generic cases, the theory after the transition can be related to the original one via an Abelian hierarchy construction. In special cases, the theory after the transition is gapless, and with a particular deformation, it is related to the original TQFT by gauging an anomaly-free one-form global symmetry. We also explore the enrichment of this setup by a global U(1) symmetry. In some cases, enriching the original TQFT is incompatible with the full transition theory. Finally, we demonstrate our construction with many specific examples.