Page curve for fermionic Gaussian states

Abstract

In a seminal paper, Page found the exact formula for the average entanglement entropy for a pure random state. We consider the analogous problem for the ensemble of pure fermionic Gaussian states, which plays a crucial role in the context of random free Hamiltonians. Using recent results from random matrix theory, we show that the average entanglement entropy of pure random fermionic Gaussian states in a subsystem of NA out of N degrees of freedom is given by (SA)G=(N-12)ψ(2N)+(14-NA)ψ(N)+(12+NA-N)ψ(2N-2NA)-14ψ(N-NA)-NA, where ψ is the digamma function. Its asymptotic behavior in the thermodynamic limit is given by (SA)G=N(log2-1)f+N(f-1)log(1-f)+12f+14log(1-f)+O(1/N), where f=NA/N≤1/2. Rema..