Edge modes, edge currents, and gauge invariance in $p_x+ip_y$ superfluids and superconductors

TL;DR

This paper explores the boundary phenomena in 2D $p_x+ip_y$ superfluids, linking edge modes and currents to gauge invariance and bulk topological actions, with implications for understanding Majorana fermions.

Contribution

It demonstrates that the gapless edge mode is essential for gauge invariance in the effective action, clarifying the role of boundary states in topological superfluids.

Findings

Edge modes are necessary for gauge invariance.

Boundary currents are not required for gauge invariance.

Bulk Chern-Simons actions relate to boundary effects.

Abstract

The excitation spectrum of a two-dimensional $p_x+ip_y$ fermionic superfluid, such as a thin film of $^3$He-A, includes a gapless Majorana-Weyl fermion which is confined to the boundary by Andreev reflection. There is also a persistent ground-state boundary current which provides a droplet containing $N$ particles with angular momentum $\hbar N/2$. Both of these boundary effects are associated with bulk Chern-Simons effective actions. We show that the gapless edge-mode is required for the gauge invariance of the total effective action, but the same is not true of the boundary current.