Majorana bound state of a Bogoliubov-de Gennes-Dirac Hamiltonian in arbitrary dimensions

TL;DR

This paper investigates Majorana zero-energy states in topological superconductors described by a BdG-Dirac Hamiltonian across arbitrary dimensions, establishing a link between wave functions, topological indices, and Chern numbers.

Contribution

It provides an explicit wave function solution, confirms the index theorem, and demonstrates the equivalence of topological invariants in a general dimensional setting.

Findings

Explicit Majorana wave function derived

Index theorem confirmed through analytical and topological indices

Chern number shown to reflect the number of zero-energy states

Abstract

We study a Majorana zero-energy state bound to a hedgehog-like point defect in a topological superconductor described by a Bogoliubov-de Gennes (BdG)-Dirac type effective Hamiltonian. We first give an explicit wave function of a Majorana state by solving the BdG equation directly, from which an analytical index can be obtained. Next, by calculating the corresponding topological index, we show a precise equivalence between both indices to confirm the index theorem. Finally, we apply this observation to reexamine the role of another topological invariant, i.e., the Chern number associated with the Berry curvature proposed in the study of protected zero modes along the lines of topological classification of insulators and superconductors. We show that the Chern number is equivalent to the topological index, implying that it indeed reflects the number of zero-energy states. Our theoretical model belongs to the BDI class from the viewpoint of symmetry, whereas the spatial dimension of the system is left arbitrary throughout the paper.