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

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.

Figures (1)

• Figure 1: Theories at three different scales. At very short distances, denoted in violet, we have a UV theory. It can be either a lattice or a continuum theory. In Section \ref{['enrichmentsec']}, we will examine the consequences of a global $\mathrm{U}(1)_A$ symmetry acting at that scale. We follow the renormalization group flow to the IR. In the extreme IR, denoted in red, we consider two phases. One of them is described by a TQFT $\cal T$. The other phase is either a TQFT $\cal T'$ or a gapless phase. The main focus of this note is the transition continuum field theory that operates at intermediate energies and is denoted in pink. It captures the two extreme IR phases and the phase transition between them. We assume that in addition to various topological modes, this theory includes only a single complex scalar field $\Phi$. This scalar field creates an anyon $a$ in the phase with $\cal T$. The transition happens when we dial the mass square $\mu^2$ of $\Phi$ from positive to negative. As we do that, the anyon $a$ becomes light, and its worldlines proliferate in the other phase. In terms of $\Phi$, this is a Higgs transition.