Page curve for fermionic Gaussian states

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 N₀ out of N degrees of freedom is given by {S₀}₆= (N-12) (2N) + (14-N₀) (N) + (12+N₀-N) (2N-2N₀) -14 (N-N₀) -N₀, where is the digamma function. Its asymptotic behavior in the thermodynamic limit is given by {S₀}₆=N (log2-1) f+N (f-1) log (1-f) +12f+14log (1-f) 0. 16em{0ex}+0. 16em{0ex}O (1/N), where f=N₀/N1/2. Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by yd ż \. zba, Rigol, and Vidmar. Finally, we compute the variance in the thermodynamic limit, given by the constant lim₍ ({S₀) }₆^2=12f+f^2+log (1-f).