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

We propose a $\mathbb{U}(1)\times {\mathbb{Z}}_2$ effective gauge theory for vortices in a $p_x+i{p}_y$ 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, $\exp (i\pi \mathrm{/}8)$, 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.