arXiv subjects

Sebastian Müller

Publications and source records attributed to Sebastian Müller.

95 records · Page 6Linked to original sources

arXiv2005

Periodic-Orbit Theory of Universality in Quantum Chaos

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 $K(τ)$ as power series in the time $τ$. Each term $τ^n$ 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.

nlin.CD↗

arXiv2005

Semiclassical Foundation of Universality in Quantum Chaos

We sketch the semiclassical core of a proof of the so-called Bohigas-Giannoni-Schmit conjecture: A dynamical system with full classical chaos has a quantum energy spectrum with universal fluctuations on the scale of the mean level spacing. We show how in the semiclassical limit all system specific properties fade away, leaving only ergodicity, hyperbolicity, and combinatorics as agents determining the contributions of pairs of classical periodic orbits to the quantum spectral form factor. The small-time form factor is thus reproduced semiclassically. Bridges between classical orbits and (the non-linear sigma model of) quantum field theory are built by revealing the contributing orbit pairs as topologically equivalent to Feynman diagrams.

nlin.CD↗

arXiv2003

Universal spectral form factor for chaotic dynamics

We consider the semiclassical limit of the spectral form factor $K(τ)$ of fully chaotic dynamics. Starting from the Gutzwiller type double sum over classical periodic orbits we set out to recover the universal behavior predicted by random-matrix theory, both for dynamics with and without time reversal invariance. For times smaller than half the Heisenberg time $T_H\propto \hbar^{-f+1}$, we extend the previously known $τ$-expansion to include the cubic term. Beyond confirming random-matrix behavior of individual spectra, the virtue of that extension is that the ``diagrammatic rules'' come in sight which determine the families of orbit pairs responsible for all orders of the $τ$-expansion.

nlin.CD↗

arXiv2003

Classical basis for quantum spectral fluctuations in hyperbolic systems

We reason in support of the universality of quantum spectral fluctuations in chaotic systems, starting from the pioneering work of Sieber and Richter who expressed the spectral form factor in terms of pairs of periodic orbits with self-crossings and avoided crossings. Dropping the restriction to uniformly hyperbolic dynamics, we show that for general hyperbolic two-freedom systems with time-reversal invariance the spectral form factor is faithful to random-matrix theory, up to quadratic order in time. We relate the action difference within the contributing pairs of orbits to properties of stable and unstable manifolds. In studying the effects of conjugate points, we show that almost self-retracing orbit loops do not contribute to the form factor. Our findings are substantiated by numerical evidence for the concrete example of two billiard systems.

nlin.CD↗

arXiv2002

Statistics of Self-Crossings and Avoided Crossings of Periodic Orbits in the Hadamard-Gutzwiller Model

Employing symbolic dynamics for geodesic motion on the tesselated pseudosphere, the so-called Hadamard-Gutzwiller model, we construct extremely long periodic orbits without compromising accuracy. We establish criteria for such long orbits to behave ergodically and to yield reliable statistics for self-crossings and avoided crossings. Self-encounters of periodic orbits are reflected in certain patterns within symbol sequences, and these allow for analytic treatment of the crossing statistics. In particular, the distributions of crossing angles and avoided-crossing widths thus come out as related by analytic continuation. Moreover, the action difference for Sieber-Richter pairs of orbits (one orbit has a self-crossing which the other narrowly avoids and otherwise the orbits look very nearly the same) results to all orders in the crossing angle. These findings may be helpful for extending the work of Sieber and Richter towards a fuller understanding of the classical basis of quantum spectral fluctuations.

nlin.CD↗