Surface Majorana fermions and bulk collective modes in superfluidHe3−B

Abstract

The theoretical study of topological superfluids and superconductors has so far been carried out largely as a translation of the theory of noninteracting topological insulators into the superfluid language, whereby one replaces electrons by Bogoliubov quasiparticles and single-particle band Hamiltonians by Bogoliubov--de Gennes Hamiltonians. Band insulators and superfluids are, however, fundamentally different: While the former exist in the absence of interparticle interactions, the latter are broken symmetry states that owe their very existence to such interactions. In particular, unlike the static energy gap of a band insulator, the gap in a superfluid is due to a dynamical order parameter that is subject to both thermal and quantum fluctuations. In this work, we explore the consequences of bulk quantum fluctuations of the order parameter in the $B$ phase of superfluid $^{3}\mathrm{He}$ on the topologically protected Majorana surface states. Neglecting the high-energy amplitude modes, we find that one of the three spin-orbit Goldstone modes in $^{3}\mathrm{He}\ensuremath{-}B$ couples to the surface Majorana fermions. This coupling in turn induces an effective short-range two-body interaction between the Majorana fermions, with coupling constant inversely proportional to the strength of the nuclear dipole-dipole interaction in bulk $^{3}\mathrm{He}$. A mean-field theory suggests that the surface Majorana fermions in $^{3}\mathrm{He}\ensuremath{-}B$ may be in the vicinity of a metastable gapped time-reversal-symmetry-breaking phase.