Topological superfluid $^3$He-B: fermion zero modes on interfaces and in the vortex core

TL;DR

This paper explores how topology determines robust fermionic zero modes in superfluid $^3$He-B and similar systems, revealing their boundary and vortex core properties through index theorems.

Contribution

It provides a topological framework for understanding fermion zero modes in superfluid $^3$He-B and related systems, connecting invariants with physical boundary and vortex phenomena.

Findings

Fermion zero modes are protected by topological invariants.

Gapless fermions appear on boundaries and vortex cores.

Index theorem links zero modes with topological invariants.

Abstract

Many quantum condensed matter systems are strongly correlated and strongly interacting fermionic systems, which cannot be treated perturbatively. However, topology allows us to determine generic features of their fermionic spectrum, which are robust to perturbation and interaction. We discuss the nodeless 3D system, such as superfluid $^3$He-B, vacuum of Dirac fermions, and relativistic singlet and triplet supercondutors which may arise in quark matter. The systems, which have nonzero value of topological invariant, have gapless fermions on the boundary and in the core of quantized vortices. We discuss the index theorem which relates fermion zero modes on vortices with the topological invariants in combined momentum and coordinate space.