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Peter Braun
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The spectral fluctuations of quantum (or wave) systems with a chaotic classical (or ray) limit are mostly universal and faithful to random-matrix theory. Taking up ideas of Pechukas and Yukawa we show that equilibrium statistical mechanics for the fictitious gas of particles associated with the parametric motion of levels yields spectral fluctuations of the random-matrix type. Previously known clues to that goal are an appropriate equilibrium ensemble and a certain ergodicity of level dynamics. We here complete the reasoning by establishing a power law for the $hbar$ dependence of the mean parametric separation of avoided level crossings. Due to that law universal spectral fluctuations emerge as average behavior of a family of quantum dynamics drawn from a control parameter interval which becomes vanishingly small in the classical limit; the family thus corresponds to a single classical system. We also argue that classically integrable dynamics cannot produce universal spectral fluctuations since their level dynamics resembles a nearly ideal Pechukas-Yukawa gas.
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We suggest that random matrix theory applied to a classical action matrix can be used in classical physics to distinguish chaotic from non-chaotic behavior. We consider the 2-D stadium billiard system as well as the 2-D anharmonic and harmonic oscillator. By unfolding of the spectrum of such matrix we compute the level spacing distribution, the spectral auto-correlation and spectral rigidity. We observe Poissonian behavior in the integrable case and Wignerian behavior in the chaotic case. We present numerical evidence that the action matrix of the stadium billiard displays GOE behavior and give an explanation for it. The findings present evidence for universality of level fluctuations - known from quantum chaos - also to hold in classical physics.
Chaotic systems exhibit rich quantum dynamical behaviors ranging from dynamical localization to normal diffusion to ballistic motion. Dynamical localization and normal diffusion simulate electron motion in an impure crystal with a vanishing and finite conductivity, i.e., an Anderson insulator and a metal, respectively. Ballistic motion simulates a perfect crystal with diverging conductivity, i.e., a supermetal. We analytically find and numerically confirm that, for a large class of chaotic systems, the metal-supermetal dynamics crossover occurs and is universal, determined only by the systems symmetry. Furthermore, we show that the universality of this dynamics crossover is identical to that of eigenfunction and spectral fluctuations described by the random matrix theory.
We present experimental and numerical results for the long-range fluctuation properties in the spectra of quantum graphs with chaotic classical dynamics and preserved time-reversal invariance. Such systems are generally believed to provide an ideal basis for the experimental study of prob- lems originating from the field of quantum chaos and random matrix theory. Our objective is to demonstrate that this is true only for short-range fluctuation properties in the spectra, whereas the observation of deviations in the long-range fluctuations is typical for quantum graphs. This may be attributed to the unavoidable occurrence of short periodic orbits, which explore only the individual bonds forming a graph and thus do not sense the chaoticity of its dynamics. In order to corrobo- rate our supposition, we performed numerous experimental and corresponding numerical studies of long-range fluctuations in terms of the number variance and the power spectrum. Furthermore, we evaluated length spectra and compared them to semiclassical ones obtained from the exact trace formula for quantum graphs.
We present a semiclassical calculation of the generalized form factor which characterizes the fluctuations of matrix elements of the quantum operators in the eigenbasis of the Hamiltonian of a chaotic system. Our approach is based on some recently developed techniques for the spectral form factor of systems with hyperbolic and ergodic underlying classical dynamics and f=2 degrees of freedom, that allow us to go beyond the diagonal approximation. First we extend these techniques to systems with f>2. Then we use these results to calculate the generalized form factor. We show that the dependence on the rescaled time in units of the Heisenberg time is universal for both the spectral and the generalized form factor. Furthermore, we derive a relation between the generalized form factor and the classical time-correlation function of the Weyl symbols of the quantum operators.
While plenty of results have been obtained for single-particle quantum systems with chaotic dynamics through a semiclassical theory, much less is known about quantum chaos in the many-body setting. We contribute to recent efforts to make a semiclassical analysis of many-body systems feasible. This is nontrivial due to both the enormous density of states and the exponential proliferation of periodic orbits with the number of particles. As a model system we study kicked interacting spin chains employing semiclassical methods supplemented by a newly developed duality approach. We show that for this model the line between integrability and chaos becomes blurred. Due to the interaction structure the system features (non-isolated) manifolds of periodic orbits possessing highly correlated, collective dynamics. As with the invariant tori in integrable systems, their presence lead to significantly enhanced spectral fluctuations, which by order of magnitude lie in-between integrable and chaotic cases.
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