Eigenstate entanglement between quantum chaotic subsystems: Universal transitions and power laws in the entanglement spectrum

Abstract

We derive universal entanglement entropy and Schmidt eigenvalue behaviors for the eigenstates of two quantum chaotic systems coupled with a weak interaction. The progression from a lack of entanglement in the noninteracting limit to the entanglement expected of fully randomized states in the opposite limit is governed by the single scaling transition parameter, Λ. The behaviors apply equally well to few-and many-body systems, e.g. interacting particles in quantum dots, spin chains, coupled quantum maps, and Floquet systems as long as their subsystems are quantum chaotic, and not localized in some manner. To calculate the generalized moments of the Schmidt eigenvalues in the perturbative regime, a regularized theory is applied, whose leading order behaviors depend on √ Λ. The marginal case of the 1/2 moment, which is related to the distance to closest maximally entangled state, is an exception having a √ Λ ln Λ leading order and a logarithmic dependence on subsystem size. A recursive embedding of the regularized perturbation theory gives a simple exponential behavior for the von Neumann entropy and the Havrda-Charvát-Tsallis entropies for increasing interaction strength, demonstrating a universal transition to nearly maximal entanglement. Moreover, the full probability densities of the Schmidt eigenvalues, i.e. the entanglement spectrum, show a transition from power laws and Lévy distribution in the weakly interacting regime to random matrix results for the strongly interacting regime. The predicted behaviors are tested on a pair of weakly interacting kicked rotors, which follow the universal behaviors extremely well.

Entanglement in coupled kicked tops with chaotic dynamics

“…In order to derive a perturbative expression for P (s) we closely follow the derivation in the CUE case given in Ref. [34] and adapt this to the COE transition ensemble. We choose the specific interaction for the coupled kicked tops given by Eq.…”

“…(17). The starting point is the perturbative expansion of the eigenphases ϕ jk of U according to [34]…”

“…The results for the CUE case have been obtained in Ref. [33,34]. The perturbative behavior of λ 1 and λ 2 can be determined following the steps in Sec.…”

“…Moreover a perturbation theoretical description for spectral statistics (consecutive level spacing distribution) and different measures of entanglement has been developed. For the entanglement entropies a recursively applied embedded perturbation theory describes the whole transition towards maximal entanglement [33,34]. Recently, a perturbative description of the time-dependence of entanglement entropies for initial product eigenstates was obtained which leads to a universal prediction after an appropriate rescaling of time [35].…”

“…Applying the recursive embedding of the regularized perturbation theory, following Ref. [34], leads to a description of the complete transition. Good agreement with numerical calculations for same as well as for different subsystem dimension is found.…”