Universality in the spectra of strongly correlated systems

Parametric correlations versus fidelity decay: the symmetry breaking case

We provide formulas for fidelity decay and parametric energy correlations for random matrix ensembles where time-reversal invariance of the original Hamiltonian is broken by the perturbation. Like in the case of a symmetry conserving perturbation a simple relation between both quantities exists. Fidelity freeze is observed for systems with even and odd spin.

Parameter-dependent spectral statistics of chaotic quantum graphs: Neumann versus circular orthogonal ensemble boundary conditions

The parameter-dependent spectral statistics of totally connected quantum graphs with n = 4-30 vertices, such as the parametric velocities correlation functions and the distribution of curvatures, are studied. The inverse participation ratio (IPR), an important measure of localization effects, was also numerically investigated. In the calculations, we successfully used two different theoretical approaches. The first approach was based on the graphs' eigenenergies and wave functions calculations, while the second one used the eigenphases and the eigenvectors of the bond scattering matrix S(k). We considered graphs with Neumann and circular orthogonal ensemble (COE) boundary conditions. We show that in contrast to large Neumann graphs, for which the departure of many parameter-dependent spectral statistics from the random matrix theory (RMT) predictions is observed, for large COE graphs, the spectral statistics and IPR are in good agreement with the RMT predictions.

Departure of some parameter-dependent spectral statistics of irregular quantum graphs from random matrix theory predictions

Parameter-dependent statistical properties of spectra of totally connected irregular quantum graphs with Neumann boundary conditions are studied. The autocorrelation functions of level velocities c(x) and c[over ](omega,x) as well as the distributions of level curvatures and avoided crossing gaps are calculated. The numerical results are compared with the predictions of random matrix theory for Gaussian orthogonal ensemble (GOE) and for coupled GOE matrices. The application of coupled GOE matrices was justified by studying localization phenomena in graphs' wave functions Psi(x) using the inverse participation ratio and the amplitude distribution P(Psi(x)) .

Surprising relations between parametric level correlations and fidelity decay

Relations among fidelity, cross-form-factor (i.e., parametric level correlations), and level velocity correlations are found both by deriving a Ward identity in a two-matrix model and by comparing exact results, using supersymmetry techniques, in the framework of random matrix theory. A power law decay near Heisenberg time, as a function of the relevant parameter, is shown to be at the root of revivals recently discovered for fidelity decay. For cross-form-factors the revivals are illustrated by a numerical study of a multiply kicked Ising spin chain.

Parametric correlations of the energy levels of ray-splitting billiards

Parameter-dependent statistical properties of the spectra of ray-splitting billiards are studied experimentally and theoretically. The autocorrelation functions c(x) and c(omega,x) of level velocities as well as the generalized conductance C(0) are calculated for two different classically chaotic ray-splitting billiards. Experimentally a modified Sinai ray-splitting billiard is studied consisting of a thin microwave rectangular cavity with two quarter-circle-shaped Teflon inserts. The length of the cavity serves as the experimentally adjustable parameter. For the theoretical estimates of the parametric correlations we compute the quantum spectrum of a scaling triangular ray-splitting billiard. Our experimental and numerical results are compared with each other and with the predictions of random matrix theory.

Autocorrelation function of level velocities for ray-splitting billiards

We study experimentally and theoretically the autocorrelation function of level velocities c(x) and the generalized conductance C(0) for classically chaotic ray-splitting systems. Experimentally, a Sinai ray-splitting billiard was simulated by a thin microwave rectangular cavity with a quarter-circle Teflon insert. For the theoretical estimates of the autocorrelator c(x) and the conductance C(0) we made parameter-dependent quantum calculations of eigenenergies of an annular ray-splitting billiard. Our experimental and numerical results are compared to theoretical predictions of systems based on the Gaussian orthogonal ensemble in random matrix theory.

Universal level dynamics of complex systems

We study the evolution of the distribution of eigenvalues of a N x N matrix subject to a random perturbation drawn from (i) a generalized Gaussian ensemble and (ii) a non-Gaussian ensemble with a measure variable under the change of basis. It turns out that, in case (i), a redefiniton of the parameter governing the evolution leads to a Fokker-Planck equation similar to the one obtained when the perturbation is taken from a standard Gaussian ensemble (with invaraiant measure). This equivalence can therefore help us to obtain the correlations for various physically significant cases modeled by generalized Gaussian ensembles by using the already known correlations for standard Gaussian ensembles. For large N values, our results for both cases (i) and (ii) are similar to those obtained for the Wigner-Dyson gas as well as for the perturbation taken from a standard Gaussian ensemble. This seems to suggest the independence of evolution, in the thermodynamic limit, from the nature of perturbation involved as well as the initial conditions and therefore the universality of dynamics of the eigenvalues of complex systems.

Conductance fluctuations and quantum chaotic scattering in semiconductor microstructures

We review recent experiments on aperiodic conductance fluctuations in ballistic GaAs/AlGaAs microstructures in the shape of a stadium billiard and a circle with point-contact leads, measured at millikelvin temperatures. Much of the observed behavior can be analyzed within a semiclassical approach to quantum chaotic scattering. After a brief review of the Landauer-Buttiker formulation of coherent transport, a variety of novel experimental phenomena and comparisons to semiclassical theory are presented. In particular, we discuss quantum-enhanced backscattering, the power spectrum of conductance fluctuations, crossover to the high-magnetic-field and tunneling regimes, and an application allowing the rate of phase-randomizing scattering to be measured in chaotic ballistic microstructures.