The Page Curve for Fermionic Gaussian States

TL;DR

This paper derives an exact formula for the average entanglement entropy of pure fermionic Gaussian states, revealing their asymptotic behavior and variance, and connecting to eigenstates of random quadratic Hamiltonians.

Contribution

It provides the first explicit formula for the average entanglement entropy of fermionic Gaussian states, extending Page's work to this important class of states.

Findings

Derived exact average entanglement entropy formula for fermionic Gaussian states.

Established asymptotic behavior of entropy in the thermodynamic limit.

Calculated the variance of entanglement entropy in the large system limit.

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 $N_A$ out of $N$ degrees of freedom is given by $\langle S_A\rangle_\mathrm{G}\!=\!(N\!-\!\tfrac{1}{2})\Psi(2N)\!+\!(\tfrac{1}{4}\!-\!N_A)\Psi(N)\!+\!(\tfrac{1}{2}\!+\!N_A\!-\!N)\Psi(2N\!-\!2N_A)\!-\!\tfrac{1}{4}\Psi(N\!-\!N_A)\!-\!N_A$, where $\Psi$ is the digamma function. Its asymptotic behavior in the thermodynamic limit is given by $\langle S_A\rangle_\mathrm{G}\!=\! N(\log 2-1)f+N(f-1)\log(1-f)+\tfrac{1}{2}f+\tfrac{1}{4}\log{(1-f)}\,+\,O(1/N)$, where $f=N_A/N$. Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by Lydzba, Rigol and Vidmar. Finally, we compute the variance in the thermodynamic limit, given by the constant $\lim_{N\to\infty}(\Delta S_A)^2_{\mathrm{G}}=\frac{1}{2}(f+f^2+\log(1-f))$.