Chaos on quantum graphs with Andreev scattering

The present thesis investigates the spectral statistics of superconducting-normalconducting hybrid systems. These hybrid systems are formed by a normal-metal non-integrable billiard being placed adjacent to a superconductor. The symmetry classification scheme of such systems due to Altland and Zirnbauer is at the basis of the thesis. For the mesoscopic systems described above, we give a semiclassical interpretation of the random-matrix theory prediction (by Altland and Zirnbauer) using periodic-orbit theory. Periodic-orbit theory links the quantum spectrum of a system with its classical periodic orbits. The model of choice for the treatment of the hybrid systems are quantum graphs. For an implementation of the hybrid character, the so-called Andreev scattering process is incorporated on the vertices of the graph. After an introduction to the concepts and methods used (chapter 1), a numerical treatise shows us how to generate an ensemble of graphs by appropriately choosing random scattering conditions at the vertices (chapter 2). The spectrum of these graphs coincides perfectly with the random-matrix theory predictions in the limit of large graphs. Models of Andreev graphs with symmetries of the classes C, CI, D, and DIII are formulated with the aid of Andreev star graphs (chapter 3). By the use of periodic-orbit theory, the short-time behaviour of the spectral form factor (the Fourier transform of the spectral density) is calculated semiclassically and shows excellent agreement with the predictions of Altland and Zirnbauer. All analytical calculations are supplemented with numerical results which are in perfect agreement with the analytical results and the random-matrix theory predictions. For symmetry classes C and CI, the approximation schemes developed with the help of quantum graphs have been carried over to the original physical system of Andreev billiards

Universal spectral statistics in Wigner-Dyson, chiral and Andreev star graphs I: construction and numerical results

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.Comment: 15 page

Universal spectral statistics in Wigner-Dyson, chiral and Andreev star graphs II: semiclassical approach

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 self-dual approximations. The results are in accordance with the corresponding random-matrix theories which supports a properly generalized Bohigas-Giannoni-Schmit conjecture.Comment: 15 Page

Universal spectral statistics of Andreev billiards: semiclassical approach

The classification of universality classes of random-matrix theory has recently been extended beyond the Wigner-Dyson ensembles. Several of the novel ensembles can be discussed naturally in the context of superconducting-normal hybrid systems. In this paper, we give a semiclassical interpretation of their spectral form factors for both quantum graphs and Andreev billiards.Comment: final improved version (to be published in Physical Review E), 6 pages, revtex

Spectral correlations of the massive QCD Dirac operator at finite temperature

We use the graded eigenvalue method, a variant of the supersymmetry technique, to compute the universal spectral correlations of the QCD Dirac operator in the presence of massive dynamical quarks. The calculation is done for the chiral Gaussian unitary ensemble of random matrix theory with an arbitrary Hermitian matrix added to the Dirac matrix. This case is of interest for schematic models of QCD at finite temperature.Comment: 19 pages, no figures, LaTeX (elsart.cls) minor changes, one reference adde