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

The B-phase of superfluid ${}^{3}$He is a 3D time-reversal-invariant topological superfluid with an isotropic energy gap, $\ensuremath{\Delta}$, separating the ground state and bulk continuum states. We report calculations of surface spectrum and 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, ${\mathbf{p}}_{\ensuremath{\parallel}}$. 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, ${\mathbf{p}}_{s}=\frac{\ensuremath{\hbar}}{2}\ensuremath{\nabla}\ensuremath{\varphi}$. The flow field breaks $\mathrm{SO}{(2)}_{{\mathrm{L}}_{\mathrm{z}}+{\mathrm{S}}_{\mathrm{z}}}$ rotational symmetry and time reversal ($T$). However, the Bogoliubov-Nambu Hamiltonian remains invariant under the combined symmetry, ${U}_{\mathrm{z}}(\ensuremath{\pi})\ifmmode\times\else\texttimes\fi{}T$, where ${U}_{\mathrm{z}}(\ensuremath{\pi})$ is a $\ensuremath{\pi}$ rotation about the surface normal. 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 for $0lT\ensuremath{\lesssim}0.5{T}_{c}$, 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 dependencies, as well as dependencies on confinement, pressure and interactions between quasiparticles.