Universal spectral statistics in Wigner-Dyson, chiral, and Andreev star graphs. II. Semiclassical approach

Abstract

A semiclassical approach to the universal ergodic spectral statistics in quantum star graphs is presented for all known ten symmetry classes of quantum systems. The approach is based on periodic orbit theory, the exact semiclassical trace formula for star graphs and on diagrammatic techniques. The appropriate spectral form factors are calculated upto one order beyond the diagonal and selfdual approximations. The results are in accordance with the corresponding random-matrix theories which supports a properly generalized Bohigas-Giannoni-Schmit conjecture.

Spectral correlations of individual quantum graphs

“…While all building blocks of semiclassical analysis on graphs are known [22,24], and a complete summation over all orbit pairs may be in reach, semiclassics on graphs is subject to the same limitations as in Hamiltonian systems. In particular, it is not clear how to extend its domain of applicability to times beyond the Heisenberg time.…”

On the spectral gap in Andreev graphs

“…Alternatively quantum graphs can be defined using the scattering formalism of Kottos and Smilansky [15]. Their approach has been used to construct Andreev graphs [12] (and other classes of quantum graphs that contain particle-hole symmetries). We will follow the latter approach and assume that the peripheral vertices of the star graphs have an interface to a superconductor such that Andreev reflections can occur with a certain probability.…”

“…No strong spectral gaps can be seen in such a regime. and time-reversal invariance then restrict the scattering matrix of the central vertex to the form [12] (3.5)…”

“…The bonds and the central vertex are assumed to be normalconducting. The electron-hole symmetry of the Bogoliubov-de Gennes equation 1 Andreev graphs with mixed signs for the Andreev scattering amplitude belong to a different universality class [12] -in an Andreev billiard this would correspond to having two interfaces to different superconductors whose pair potentials have a relative phase e −π = −1. No strong spectral gaps can be seen in such a regime.…”

“…However the effect of Andreev scattering on the spectrum is less strong and decays after a few mean level spacings away from Fermi energy (which can only be seen after sufficiently averaging over system parameters). Andreev scattering has been introduced to quantum maps [11] and quantum graphs [12]. The latter work investigated a class of Andreev graphs that corresponds to chaotic combined electron-hole dynamics with universal spectral statistics for non-standard symmetry classes.…”