Can one hear the shape of a Majorana billiard?
Abstract: Topological insulators and superconductors have seen intense interst and shown exciting results and possibilities in the last decade, with applications in quantum computers, spintronics and quantum thermodynamics, among others. After a brief introduction to topological insulators and superconductors, I will discuss fermion parity switches in Majorana billiards, i.e. finitely sized, arbitrarily shaped superconducting islands that host Majorana fermions, where the superconductivity can either be inherent or induced via proximity effect, as a function of the external parameters of the Hamiltonian. In particular, I will talk about the density and statistics of these parity switches as a function of the applied magnetic field and chemical potential. I will review the formulae we derived that specify how the average density of these parity switches depends on the geometrical size of the billiard as well as its boundary. Moreover, I will examine how the oscillations around this average value is determined by the classical periodic orbits of the billiard. Finally, I will show our findings that the statistics of the spacings of these parity switches are universal and are described by an appropriate random matrix ensemble, the choice of which depends on the antiunitary symmetries of the system in its normal state. I will therefore demonstrate "one can hear (information about) the shape of a Majorana billiard" by investigating its "parity switch spectrum", alluding to Mark Kac's famous question about the relation of asymptotic spectral statistics to geometry.
 B. Pekerten, A. Teker, Ö. Bozat, M. Wimmer, and İ. Adagideli, Physical Review B 95, (2017)  A. M. Bozkurt, B. Pekerten, and İ. Adagideli, Physical Review B 97, (2018)  B. Pekerten, A. M. Bozkurt, and İ. Adagideli, arXiv 1812:11331, (2018)