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 ${N}_{A}$ out of $N$ degrees of freedom is given by ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{\mathrm{G}}=(N\ensuremath{-}\frac{1}{2})\mathrm{\ensuremath{\Psi}}(2N)+(\frac{1}{4}\ensuremath{-}{N}_{A})\mathrm{\ensuremath{\Psi}}(N)+(\frac{1}{2}+{N}_{A}\ensuremath{-}N)\mathrm{\ensuremath{\Psi}}(2N\ensuremath{-}2{N}_{A})\ensuremath{-}\frac{1}{4}\mathrm{\ensuremath{\Psi}}(N\ensuremath{-}{N}_{A})\ensuremath{-}{N}_{A}$, where $\mathrm{\ensuremath{\Psi}}$ is the digamma function. Its asymptotic behavior in the thermodynamic limit is given by ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{\mathrm{G}}=N(log2\ensuremath{-}1)f+N(f\ensuremath{-}1)log(1\ensuremath{-}f)+\frac{1}{2}f+\frac{1}{4}log(1\ensuremath{-}f)\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}O(1/N)$, where $f={N}_{A}/N\ensuremath{\le}1/2$. Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by \L{}yd\ifmmode \dot{z}\else \.{z}\fi{}ba, Rigol, and Vidmar. Finally, we compute the variance in the thermodynamic limit, given by the constant ${lim}_{N\ensuremath{\rightarrow}\ensuremath{\infty}}{(\mathrm{\ensuremath{\Delta}}{S}_{A})}_{\mathrm{G}}^{2}=\frac{1}{2}[f+{f}^{2}+log(1\ensuremath{-}f)]$.