Eigenstate capacity and Page curve in fermionic Gaussian states

TL;DR

This paper derives exact formulas for the entanglement capacity in fermionic Gaussian states and the SYK$_2$ model, revealing how it distinguishes between integrable and chaotic quantum systems.

Contribution

It provides the first exact finite-series expression for eigenstate capacity of entanglement in fermionic Gaussian states and analyzes its behavior in the SYK$_2$ model.

Findings

Average eigenstate CoE is independent of system size in the thermodynamic limit.

Leading volume-law coefficient approaches π²/8 - 1 for half-system partition.

Analytical results are confirmed by numerical computations.

Abstract

Capacity of entanglement (CoE), an information-theoretic measure of entanglement, defined as the variance of modular Hamiltonian, is known to capture the deviation from the maximal entanglement. We derive an exact expression for the average eigenstate CoE in fermionic Gaussian states as a finite series, valid for arbitrary bi-partition of the total system. Further, we consider the complex SYK$_2$ model in the thermodynamic limit and we obtain a closed-form expression of average CoE. In this limit, the variance of the average CoE becomes independent of the system size. Moreover, when the subsystem size is half of the total system, the leading volume-law coefficient approaches a value of $\pi^{2}/8 - 1$. We identify this as a distinguishing feature between integrable and quantum-chaotic systems. We confirm our analytical results by numerical computations.