# Degeneracy lifting of Majorana bound states due to electron-phonon interactions

## TL;DR

This paper theoretically investigates how electron-phonon interactions influence the energy levels and stability of Majorana bound states in topological nanowires, revealing temperature-dependent broadening and splitting effects with potential resonant enhancements.

## Contribution

It introduces a detailed theoretical analysis of electron-phonon effects on Majorana bound states, including temperature dependence, system length scaling, and resonance phenomena due to phonon density of states singularities.

## Key findings

- Energy splitting remains exponentially small at zero temperature.
- Finite temperature causes level broadening independent of system length.
- Van Hove singularities enhance energy splitting and broadening.

## Abstract

We study theoretically how electron-phonon interaction affects the energies and level broadening (inverse lifetime) of Majorana bound states (MBSs) in a clean topological nanowire at low temperatures. At zero temperature, the energy splitting between the right and left MBSs remains exponentially small with increasing nanowire length $L$. At finite temperatures, however, the absorption of thermal phonons leads to the broadening of energy levels of the MBSs that does not decay with system length, and the coherent absorption/emission of phonons at opposite ends of the nanowire results in MBSs energy splitting that decays only as an inverse power-law in $L$. Both effects remain exponential in temperature. In the case of quantized transverse motion of phonons, the presence of Van Hove singularities in the phonon density of states causes additional resonant enhancement of both the energy splitting and the level broadening of the MBSs. This is the most favorable case to observe the phonon-induced energy splitting of MBSs as it becomes much larger than the broadening even if the topological nanowire is much longer than the coherence length. We also calculate the charge and spin associated with the energy splitting of the MBSs induced by phonons. We consider both a spinless low-energy continuum model, which we evaluate analytically, as well as a spinful lattice model for a Rashba nanowire, which we evaluate numerically.