# A theorem for the existence of Majorana fermion modes in spin-orbit-coupled semiconductors

## TL;DR

This paper proves an index theorem confirming the existence of Majorana zero modes in spin-orbit-coupled semiconductors with proximity-induced superconductivity and Zeeman splitting, highlighting conditions for topological quantum phases.

## Contribution

It introduces a new index theorem for Majorana modes in semiconductors, extending the understanding of topological superconductivity in these systems.

## Key findings

- Majorana zero modes exist in certain vortex configurations.
- A topological phase transition occurs with increasing pairing potential.
- The theorem complements previous real-space solutions, confirming zero-energy Majorana states.

## Abstract

We prove an index theorem for the existence of Majorana zero modes in a semiconducting thin film with a sizable spin-orbit coupling when it is adjacent to an s-wave superconductor. The theorem, which is analogous to the Jackiw-Rebbi index theorem for the zero modes in mass domain walls in one-dimensional Dirac theory, applies to vortices with odd flux quantum in a semiconducting film in which s-wave superconductivity and a Zeeman splitting are induced by proximity effect. The momentum-space construction of the zero-mode solution presented here is complementary to the approximate real-space solution of the Bogoliubov-de Gennes equations at a vortex core [J. D. Sau et al., arXiv:0907.2239], proving the existence of non-degenerate zero-energy Majorana excitations and the resultant non-Abelian topological order in the semiconductor heterostructure. With increasing magnitude of the proximity-induced pairing potential, the non-Abelian superconducting state makes a topological quantum phase transition to an ordinary s-wave superconducting state which no topological order.