Weak integrability breaking and level spacing distribution

Recently it was suggested that certain perturbations of integrable spin chains lead to a weak breaking of integrability in the sense that integrability is preserved at the first order in the coupling. Here we examine this claim using level spacing distribution. We find that the volume dependent crossover between integrable and chaotic level spacing statistics {which marks the onset of quantum chaotic behaviour, is markedly different for weak vs. strong breaking of integrability. In particular}, for the gapless case we find that the crossover coupling as a function of the volume LL scales with a 1/L^{2}1/L2 law for weak breaking as opposed to the 1/L^{3}1/L3 law previously found for the strong case.

The distribution of localization measures of chaotic eigenstates in the stadium billiard

The localization measures A (based on the information entropy) of localized chaotic eigenstates in the Poincaré-Husimi representation have a distribution on a compact interval [0;A0], which is well approximated by the beta distribution, based on our extensive numerical calculations. The system under study is the Bunimovich' stadium billiard, which is a classically ergodic system, also fully chaotic (positive Lyapunov exponent), but in the regime of a slightly distorted circle billiard (small shape parameter ") the diffusion in the momentum space is very slow. The parameter α = tH/tT , where tH and tT are the Heisenberg time and the classical transport time (diffusion time), respectively, is the important control parameter of the system, as in all quantum systems with the discrete energy spectrum. The measures A and their distributions have been calculated for a large number of ε and eigenenergies. The dependence of the standard deviation σ on α is analyzed, as well as on the spectral parameter β (level repulsion exponent of the relevant Brody level spacing distribution). The paper is a continuation of our recent paper (B. Batistić, Č. Lozej and M. Robnik, Nonlinear Phenomena in Complex Systems 21, 225 (2018)), where the spectral statistics and validity of the Brody level spacing distribution has been studied for the same system, namely the dependence of β and of the mean value < A > on α.

ROBUSTNESS OF STRONG CORRELATION BETWEEN EIGENVALUES AND DIAGONAL MATRIX ELEMENTS

In this paper we study robustness of strong correlation between eigenvalues and diagonal matrix elements sorted from the smallest to the largest, in the presence of large single-particle splittings for both realistic and schematic systems. We also study the nearest level spacing distribution obtained by the linear correlation.

Some Generic Properties of Level Spacing Distributions of 2D Real Random Matrices

We study the level spacing distribution P(S) of 2D real random matrices both symmetric as well as general, non-symmetric. In the general case we restrict ourselves to Gaussian distributed matrix elements, but different widths of the various matrix elements are admitted. The following results are obtained: An explicit exact formula for P(S) is derived and its behaviour close to S = 0 is studied analytically, showing that there is linear level repulsion, unless there are additional constraints for the probability distribution of the matrix elements. The constraint of having only positive or only negative but otherwise arbitrary non-diagonal elements leads to quadratic level repulsion with logarithmic corrections. These findings detail and extend our previous results already published in a preceding paper. For symmetric real 2D matrices also other, non-Gaussian statistical distributions are considered. In this case we show for arbitrary statistical distribution of the diagonal and non-diagonal elements that the level repulsion exponent ρ is always ρ = 1, provided the distribution function of the matrix elements is regular at zero value. If the distribution function of the matrix elements is a singular (but still integrable) power law near zero value of S, the level spacing distribution P(S) is a fractional exponent power law at small S. The tail of P(S) depends on further details of the matrix element statistics. We explicitly work out four cases: the uniform (box) distribution, the Cauchy-Lorentz distribution, the exponential distribution and, as an example for a singular distribution, the power law distribution for P(S) near zero value times an exponential tail.