Effective theory of vortices in two-dimensional spinless chiral $p$-wave superfluids

TL;DR

This paper develops an effective gauge theory for vortices in two-dimensional $p_x+ip_y$ superfluids, capturing vortex dynamics and predicting a universal exchange phase of $ ext{exp}(irac{\pi}{8})$, with implications for topological quantum computation.

Contribution

It introduces a combined $ ext{U}(1) imes ext{Z}_2$ gauge theory that preserves particle-hole symmetry and predicts a universal vortex exchange phase in 2D chiral superfluids.

Findings

Reproduces vortex dynamics including Magnus force.

Predicts a universal exchange phase of $ ext{exp}(irac{\pi}{8})$.

Shows screening of non-universal corrections in charged superfluids.

Abstract

We propose a $\mathbb{U}(1) \times \mathbb{Z}_2$ effective gauge theory for vortices in a $p_x+ip_y$ superfluid in two dimensions. The combined gauge transformation binds $\mathbb{U}(1)$ and $\mathbb{Z}_2$ defects so that the total transformation remains single-valued and manifestly preserves the the particle-hole symmetry of the action. The $\mathbb{Z}_2$ gauge field introduces a complete Chern-Simons term in addition to a partial one associated with the $\mathbb{U}(1)$ gauge field. The theory reproduces the known physics of vortex dynamics such as a Magnus force proportional to the superfluid density. More importantly, it predicts a universal Abelian phase, $\exp(i\pi/8)$, upon the exchange of two vortices. This phase is modified by non-universal corrections due to the partial Chern-Simon term, which are nevertheless screened in a charged superfluid at distances that are larger than the penetration depth.