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Periodic-Orbit Theory of Universality in Quantum Chaos | | Sebastian Müller
; Stefan Heusler
; Petr Braun
; Fritz Haake
; Alexander Altland
; | | Date: |
23 Mar 2005 | | Subject: | Chaotic Dynamics; Mesoscopic Systems and Quantum Hall Effect | nlin.CD cond-mat.mes-hall | | Abstract: | We argue semiclassically, on the basis of Gutzwiller’s periodic-orbit theory, that full classical chaos is paralleled by quantum energy spectra with universal spectral statistics, in agreement with random-matrix theory. For dynamics from all three Wigner-Dyson symmetry classes, we calculate the small-time spectral form factor $K( au)$ as power series in the time $ au$. Each term $ au^n$ of that series is provided by specific families of pairs of periodic orbits. The contributing pairs are classified in terms of close self-encounters in phase space. The frequency of occurrence of self-encounters is calculated by invoking ergodicity. Combinatorial rules for building pairs involve non-trivial properties of permutations. We show our series to be equivalent to perturbative implementations of the non-linear sigma models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of orbit pairs are one-to-one with Feynman diagrams known from the sigma model. | | Source: | arXiv, nlin.CD/0503052 | | Services: | Forum | Review | PDF | Favorites | |
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