Edge Current due to Majorana Fermions in Superfluid $^3$He A- and B-Phases

TL;DR

This paper proposes a method to detect Majorana fermions in superfluid helium-3 by measuring edge currents, with temperature-dependent signatures indicating their presence, and suggests surface roughness as an alternative detection approach.

Contribution

It introduces a self-consistent analytic approach to connect edge currents with Majorana fermions in superfluid helium-3 phases, providing observable signatures.

Findings

Edge current depletion follows a T^2 power law in A-phase.

Edge spin current depletion follows a T^3 power law in B-phase.

Surface roughness can be used to observe Majorana fermions.

Abstract

We propose a method utilizing edge current to observe Majorana fermions in the surface Andreev bound state for the superfluid $^3$He A- and B-phases. The proposal is based on self-consistent analytic solutions of quasi-classical Green's function with an edge. The local density of states and edge mass current in the A-phase or edge spin current in the B-phase can be obtained from these solutions. The edge current carried by the Majorana fermions is partially cancelled by quasiparticles (QPs) in the continuum state outside the superfluid gap. QPs contributing to the edge current in the continuum state are distributed in energy even away from the superfluid gap. The effect of Majorana fermions emerges in the depletion of the edge current by temperature within a low-temperature range. The observations that the reduction in the mass current is changed by $T^2$-power in the A-phase and the reduction in the spin current is changed by $T^3$-power in the B-phase establish the existence of Majorana fermions. We also point out another possibility for observing Majorana fermions by controlling surface roughness.