Majorana Excitations, Spin- and Mass Currents on the Surface of Topological Superfluid $^3$He-B

TL;DR

This paper investigates the surface Majorana fermions in topological superfluid $^3$He-B, analyzing their spectral properties, spin and mass currents, and how superflow and temperature influence these surface states and observable superfluid characteristics.

Contribution

It provides detailed calculations of surface spectra, spin and mass currents, and their dependence on flow, temperature, and confinement in topological superfluid $^3$He-B, highlighting the role of Majorana surface states.

Findings

Surface states are Majorana fermions with transverse spin polarization.

Ground-state spin current is confined on the surface due to negative energy states.

Superfluid mass current shows power-law suppression at finite temperatures due to Majorana modes.

Abstract

The B-phase of superfluid $^3$He is a 3D time-reversal invariant (TRI) topological superfluid with an isotropic energy gap separating the ground-state and bulk continuum states. We report calculations of surface spectrum, spin- and mass current densities originating from the Andreev surface states for confined $^3$He-B. The surface states are Majorana Fermions with their spins polarized transverse to their direction of propagation along the surface. The negative energy states give rise to a ground-state helical spin current confined on the surface. The spectral functions reveal the subtle role of the spin-polarized surface states in relation to the ground-state spin current. By contrast, these states do not contribute to the $T=0$ mass current. Superfluid flow through a channel of confined $^3$He-B is characterized by the flow field, $p_s$. The flow field breaks 2D rotational symmetry and time reversal (T). However, the Bogoliubov-Nambu Hamiltonian remains invariant under $T$ combined with a 180 degree rotation about the normal to the film. As a result the B-phase in the presence of a superflow remains a topological phase with a gapless spectrum of Majorana modes on the surface. Thermal excitation of the Doppler shifted Majorana branches leads to a power law suppression of the superfluid mass current, providing a direct signature of the Majorana branches of surface excitations in the fully gapped 3D topological superfluid, $^3$He-B. Results are reported for the superfluid fraction (mass current) and helical spin current for confined $^3$He-B, including the temperature dependences, as well as dependences on confinement, interactions between quasiparticles and pressure.