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

TL;DR

This paper analyzes the statistical properties of the time-delay matrix and density of states in chaotic superconducting quantum dots, revealing how symmetry classes affect midgap spectral peaks and thermopower distributions.

Contribution

It introduces a detailed statistical framework for the time-delay matrix in Andreev billiards, including new results on density of states and thermopower distributions across symmetry classes.

Findings

Average density of states depends on symmetry indices and deviates from bulk value.

Mid-gap spectral peak in class D quantum dots is independent of Majorana zero-modes.

Thermopower distribution varies between paired and unpaired Majorana edge modes.

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 $\rho_0$ at the Fermi level. The ensemble average $\langle\rho_0\rangle=\delta_0^{-1}M[\max(0,M+2\alpha/\beta)]^{-1}$ deviates from the bulk value $1/\delta_0$ by an amount depending on the Altland-Zirnbauer symmetry indices $\alpha,\beta$. The divergent average for $M=1,2$ in symmetry class D ($\alpha=-1$, $\beta=1$) originates from the mid-gap 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.