Time-delay matrix, midgap spectral peak, and thermopower of an Andreev billiard

Abstract

We derive the statistics of the time-delay matrix (energy derivative of the scattering matrix) in an ensemble of superconducting quantum dots with chaotic scattering (Andreev billiards), coupled ballistically to M conducting modes (electron-hole modes in a normal metal or Majorana edge modes in a superconductor). As a first application we calculate the density of states ρ0 at the Fermi level. The ensemble average ⟨ρ0⟩=δ−10M[max(0,M+2α/β)]−1 deviates from the bulk value 1/δ0 by an amount depending on the Altland-Zirnbauer symmetry indices α,β. The divergent average for M=1,2 in symmetry class D (α=−1, β=1) originates from the midgap spectral peak of a closed quantum dot, but now no longer depends on the presence or absence of a Majorana zero mode. As a second application we calculate the probability distribution of the thermopower, contrasting the difference for paired and unpaired Majorana edge modes.