Page Curves for General Interacting Systems

TL;DR

This paper derives universal formulas for Renyi and von Neumann entanglement entropies in large interacting quantum systems, providing detailed Page curves applicable at infinite temperature and beyond.

Contribution

It presents detailed derivations of universal entanglement entropy formulas for cTPQ states, including the von Neumann Page curve, expanding on prior work with explicit formulas.

Findings

Universal formulas for Renyi entropies as functions of subsystem volume.

Explicit expression for the von Neumann Page curve.

Formulas applicable to infinite temperature and general interacting systems.

Abstract

We calculate in detail the Renyi entanglement entropies of cTPQ states as a function of subsystem volume, filling the details of our prior work [Nature Communications 9, 1635 (2018)], where the formulas were first presented. Working in a limit of large total volume, we find universal formulas for the Renyi entanglement entropies in a region where the subsystem volume is comparable to that of the total system. The formulas are applicable to the infinite temperature limit as well as general interacting systems. For example we find that the second Renyi entropy of cTPQ states in terms of subsystem volume is written universally up to two constants, $S_2(\ell)=-\ln K(\beta)+\ell\ln a(\beta)-\ln\left(1+a(\beta)^{-L+2\ell}\right)$, where $L$ is the total volume of the system and $a$ and $K$ are two undetermined constants. The uses of the formulas were already presented in our prior work and we mostly concentrate on the theoretical aspect of the formulas themselves. Aside from deriving the formulas for the Renyi Page curves, the expression for the von Neumann Page curve is also derived, which was not presented in our previous work.