# Fermion-parity switches of the ground state of Majorana billiards

## TL;DR

This paper investigates the fermion-parity switches in Majorana billiards, revealing how their density and statistics depend on geometry, magnetic field, and symmetries, and showing their universal behavior linked to random matrix theory.

## Contribution

It provides analytical formulas for fermion-parity switch density and demonstrates the universal statistical properties governed by random matrix ensembles.

## Key findings

- Fermion-parity switch density depends on billiard shape.
- Oscillations in switch density relate to classical periodic orbits.
- Switch spacings follow universal random matrix statistics.

## Abstract

Majorana billiards are finitely sized, arbitrarily shaped superconducting islands that host Majorana bound states. We study the fermion-parity switches of the ground state of Majorana billiards. In particular, we study the density and statistics of these fermion-parity switches as a function of applied magnetic field and chemical potential. We derive formulae that specify how the average density of fermion-parity switches depends on the geometrical shape the billiard. Moreover, we show how oscillations around this average value is determined by the classical periodic orbits of the billiard. Finally, we find that the statistics of the spacings of these fermion-parity switches are universal and are described by a random matrix ensemble, the choice of which depends on the antiunitary symmetries of the system in its normal state. We thus demonstrate that "one can hear (information about) the shape of a Majorana billiard" by investigating its "fermion-parity switch spectrum".