Entanglement and localization transitions in eigenstates of interacting chaotic systems

The entanglement and localization in eigenstates of strongly chaotic subsystems are studied as a function of their interaction strength. Excellent measures for this purpose are the von Neumann entropy, Havrda-Charvát-Tsallis entropies, and the averaged inverse participation ratio. All the entropies are shown to follow a remarkably simple exponential form, which describes a universal and rapid transition to nearly maximal entanglement for increasing interaction strength. An unexpectedly exact relationship between the subsystem averaged inverse participation ratio and purity is derived that prescribes the transition in the localization as well.

Figure 1

The average eigenstate entropies $\langle S_k\rangle$ for von Neumann, and $k=2,3,4$ as a function of $\sqrt{\mathrm{\Lambda }}$. The lowest curve is for von Neumann, and the approach to asymptotic values is faster for larger $k$ values. The triangles are for ${\mathcal{U}}_{\text{RMT}}$ and the circles for ${\mathcal{U}}_{\text{KR}}$ with $N=50$. The lines correspond to Eq. (3). The inset is a magnification of the small-$\mathrm{\Lambda }$ region.