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 ρ 0 at the Fermi level. The ensemble average ρ 0 = δdeviates from the bulk value 1/δ 0 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.

Random-matrix theory of Majorana fermions and topological superconductors

“…Its eigenvalues D n > 0 are the delay times, and γ n ≡ 1/D n are the corresponding rates, each with the same degeneracy d T as the transmission eigenvalues. Marciani, Brouwer, and Beenakker (2014).…”

“…(79), where the transmission matrices t, t couple the N Majorana edge modes to the N electron-hole modes in the point contact, mediated by quasibound states in the quantum dot. In the circular ensembles the joint distribution of S 0 and D follows from the invariance (Brouwer, Frahm, and Beenakker, 1997;Marciani, Brouwer, and Beenakker, 2014)…”

“…For a random-matrix theory we assume that the Majorana modes are uniformly mixed with the modes in the point contact, via quasibound states in the quantum dot. Figure adapted from Marciani, Brouwer, and Beenakker (2014). V = −SδT , the Seebeck effect, with S the thermopower or Seebeck coefficient.…”

“…The distribution of the N = (N + N )/d T distinct delay times is given by (Marciani, Brouwer, and Beenakker, 2014)…”

Random matrix approaches to open quantum systems

“…This resembles a Wishart-Laguerre ensemble (13), but is directly expressed for Q S and supplemented with a determinantal factor. The joint distribution of rates λ n = 1/τ n is given by The same independence of S and Q S also occurs in the four classes with charge-conjugation symmetry at the symmetry point E = 0 (Marciani et al, 2014), where…”