$Z_2$ index for gapless fermionic modes in the vortex core of three dimensional paired Dirac fermions

TL;DR

This paper investigates the topological properties of gapless fermionic modes along vortex lines in three-dimensional Dirac superconductors, introducing a $Z_2$ index that characterizes the existence of zero modes and their robustness under perturbations.

Contribution

The paper introduces a $Z_2$ index based on spectral symmetry for classifying zero modes in vortex cores of 3D Dirac fermion superconductors, extending the Jackiw-Rossi model.

Findings

Odd vorticity vortices host a single zero mode described by a $Z_2$ index.

Perturbations like chemical potential and Zeeman couplings lead to two Majorana fermions bound to an odd vortex.

Topologically trivial s-wave superconductors do not support zero modes under symmetry-breaking perturbations.

Abstract

We consider the gapless modes along the vortex line of the fully gapped, momentum independent paired states of three-dimensional Dirac fermions. For this, we require the solution of fermion zero modes of the corresponding two-dimensional problem in the presence of a point vortex, in the plane perpendicular to the vortex line. Based on the spectral symmetry requirement for the existence of the zero mode, we identify the appropriate generalized Jackiw-Rossi Hamiltonians for different paired states. A four-dimensional generalized Jackiw-Rossi Hamiltonian possesses spectral symmetry with respect to an antiunitary operator, and gives rise to a single zero mode only for the {\em odd vorticity}, which is formally described by a $Z_2$ index. In the presence of generic perturbations such as chemical potential, Dirac mass, and Zeeman couplings, the associated two-dimensional problem for the odd parity topological superconducting state maps onto {\em two} copies of generalized Jackiw-Rossi Hamiltonian, and consequently an odd vortex binds two Majorana fermions. In contrast, there are no zero energy states for the topologically trivial $s$-wave superconductor in the presence of any chiral symmetry breaking perturbation in the particle-hole channel, such as regular Dirac mass. We show that the number of one-dimensional dispersive modes along the vortex line is also determined by the index of the associated two-dimensional problem. For an axial superfluid state in the presence of various perturbations, we discuss the consequences of the $Z_2$ index on the anomaly equations.