Topological superfluid 3He-B in magnetic field and Ising variable

TL;DR

This paper explores how magnetic fields affect the topological and surface properties of superfluid 3He-B, revealing the emergence of an Ising variable and protected gapless fermions at domain boundaries.

Contribution

It introduces the concept of an Ising variable characterizing surface states and analyzes the impact of magnetic fields on topological invariants and surface fermions in 3He-B.

Findings

Magnetic field breaks time reversal symmetry, gapping surface fermions.

Surface domain boundaries host protected gapless fermions.

Surface fermion mass is proportional to symmetry-breaking perturbation.

Abstract

The topological superfluid 3He-B provides many examples of the interplay of symmetry and topology. Here we consider the effect of magnetic field on topological properties of 3He-B. Magnetic field violates the time reversal symmetry. As a result, the topological invariant supported by this symmetry ceases to exist; and thus the gapless fermions on the surface of 3He-B are not protected any more by topology: they become fully gapped. Nevertheless, if perturbation of symmetry is small, the surface fermions remain relativistic with mass proportional to symmetry violating perturbation -- magnetic field. The 3He-B symmetry gives rise to the Ising variable I=+/- 1, which emerges in magnetic field and which characterizes the states of the surface of 3He-B. This variable also determines the sign of the mass term of surface fermions and the topological invariant describing their effective Hamiltonian. The line on the surface, which separates the surface domains with different I, contains 1+1 gapless fermions, which are protected by combined action of symmetry and topology.