Level statistics for nearly integrable systems

Mixing property of symmetrical polygonal billiards

The present work consists of a numerical study of the dynamics of irrational polygonal billiards. Our contribution reinforces the hypothesis that these systems can be strongly mixing, although never demonstrably chaotic, and discusses the role of rotational symmetries on the billiards boundaries. We introduce a biparametric polygonal billiard family with only C_{n} rotational symmetries. Initially, we calculate through the relative measure r(ℓ,θ;t) the phase space filling. This is done for some integer values of n and for a plane of parameters ℓ×θ. From the resulting phase diagram, we can identify the fully ergodic systems. The numerical evidence that symmetrical polygonal billiards can be strongly mixing is obtained by evaluating the position autocorrelation function Cor_{x}(t), which follows a power-law-type decay t^{-σ}. The strongly mixing property is indicated by σ=1. For odd, small values of n, the exponent σ≃1 is found. On the other hand, σ<1 (weakly mixing cases) for small, even values of n. Intermediate n values present σ≃1 independently of parity. For larger values of symmetry parameter n, the biparametric family tends to be a circular billiard (integrable case). For such values of n, we identified even less ergodic behavior at the pace at which n increases and σ decreases.

Entanglement in coupled kicked tops with chaotic dynamics

The entanglement of eigenstates in two coupled, classically chaotic kicked tops is studied in dependence of their interaction strength. The transition from the noninteracting and unentangled system toward full random matrix behavior is governed by a universal scaling parameter. Using suitable random matrix transition ensembles we express this transition parameter as a function of the subsystem sizes and the coupling strength for both unitary and orthogonal symmetry classes. The universality is confirmed for the level spacing statistics of the coupled kicked tops and a perturbative description is in good agreement with numerical results. The statistics of Schmidt eigenvalues and entanglement entropies of eigenstates is found to follow a universal scaling as well. Remarkably, this is not only the case for large subsystems of equal size but also if one of them is much smaller. For the entanglement entropies a perturbative description is obtained, which can be extended to large couplings and provides very good agreement with numerical results. Furthermore, the transition of the statistics of the entanglement spectrum toward the random matrix limit is demonstrated for different ratios of the subsystem sizes.

Comment on "Spectral analysis of deformed random networks"

We comment on the findings presented in [Phys. Rev. E 80, 046101 (2009)10.1103/PhysRevE.80.046101] concerning the spacing distribution of the spectrum of sparsely connected random networks. We point out that for clustered networks without any connection among them the spacing distribution is not described by Poisson but rather by a superposition of multiple Gaussian orthogonal ensemble (GOE) statistics. Therefore, the spacing distribution for a network having two dense subnetworks with few connections between the two subnetworks follows a transition from two-GOE to a GOE statistics and not as the article suggests a transition from Poisson statistics to the GOE.