ABSTRACT
In a series of two papers we investigate the universal spectral statistics of chaotic quantum systems in the ten known symmetry classes of quantum mechanics. In this first paper we focus on the construction of appropriate ensembles of star graphs in the ten symmetry classes. A generalization of the Bohigas-Giannoni-Schmit conjecture is given that covers all these symmetry classes. The conjecture is supported by numerical results that demonstrate the fidelity of the spectral statistics of star graphs to the corresponding Gaussian random-matrix theories.
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 up to one order beyond the diagonal and self-dual approximations. The results are in accordance with the corresponding random-matrix theories which supports a properly generalized Bohigas-Giannoni-Schmit conjecture.
ABSTRACT
The symmetry classification of complex quantum systems has recently been extended beyond the Wigner-Dyson classes. Several of the novel symmetry classes can be discussed naturally in the context of superconducting-normal hybrid systems such as Andreev billiards and graphs. In this paper, we give a semiclassical interpretation of their universal spectral form factors in the ergodic limit.
