Counting Majorana bound states using complex momenta

TL;DR

This paper proves a formula to count Majorana bound states using complex momenta and applies it to various models, successfully mapping topological phases and linking complex solutions to localized Majorana wavefunctions.

Contribution

It provides a rigorous proof of a counting formula for Majorana bound states and demonstrates its effectiveness across multiple topological models.

Findings

Successfully maps topological phase diagrams.

Links complex momentum solutions to Majorana wavefunctions.

Identifies exceptional points in chiral symmetric systems.

Abstract

Recently, the connection between Majorana fermions bound to defects in arbitrary dimensions, and complex momentum roots of the vanishing determinant of the corresponding bulk Bogoliubov-de Gennes (BdG) Hamiltonian, has been established (EPL, 2015, $\textbf{110}$, 67005). Based on this understanding, a formula has been proposed to count the number ($n$) of the zero energy Majorana bound states, which is related to the topological phase of the system. In this paper, we provide a proof of the counting formula and we apply this formula to a variety of 1d and 2d models belonging to the classes BDI, DIII and D. We show that we can successfully chart out the topological phase diagrams. Studying these examples also enables us to explicitly observe the correspondence between these complex momentum solutions in the Fourier space, and the localized Majorana fermion wavefunctions in the position space. Finally, we corroborate the fact that for systems with a chiral symmetry, these solutions are the so-called "exceptional points", where two or more eigenvalues of the complexified Hamiltonian coalesce.