Majorana excitations, spin and mass currents on the surface of topological superfluid3He-B

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, p . 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 =h 2 ∇ϕ. The flow field breaks SO(2) L z +S z rotational symmetry and time reversal (T). However, the Bogoliubov-Nambu Hamiltonian remains invariant under the combined symmetry, U z (π) × T, where U z (π) is a π 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 0 < T 0.5T 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 dependences, as well as dependences on confinement, pressure and interactions between quasiparticles.

Symmetry protected topological superfluid3He-B

“…The numerical calculation based on the quasiclassical theory confirmed the appearance of a mini-gap generated by the hybridization of two surface states in 3 He-B confined in a slab geometry, where 3 He is confined in two directions [144]. For 3 He confined in one direction, i.e., in parallel plates, however, no finite excitation gap is generated by the hybridization even for strong confinement [170].…”

“…Note that the hidden Z 2 symmetry in the B-phase of confined 3 He remains even in the presence of superfluid flow [170], where the phase bias that generates the flow field is parallel to the surface. The flow field explicitly breaks the SO(2) Lz+Sz symmetry and the time-reversal symmetry, simultaneously.…”

“…In addition, the chiral component is induced through g in the case of 3 He-B. Wu and Sauls [170] demonstrated that the wavefunction of surface Majorana fermions in Eq. (141) can be extracted from this quasiclassical projectors.…”

“…Detection of the spin current will be more difficult than that of the mass current because we have to separate spin states for the detection of the spin current. Wu and Sauls [170] have pointed out that the surface Mojorana bound state in 3 He-B can be observed by the thermal depletion of mass current. Superfluid flow applied to 3 He-B in a narrow channel induces mass current on surfaces of the channel owing to the time-reversal symmetry breaking.…”

“…Expanding Eq. (298), one can derive the low-temperature behavior of the total spin current as [121,170] …”