The Page Curve for Fermionic Gaussian States

• URL: http://arxiv.org/abs/2103.05416v2
• Date: Fri, 7 May 2021 05:36:05 GMT
• Title: The Page Curve for Fermionic Gaussian States
• Authors: Eugenio Bianchi, Lucas Hackl, Mario Kieburg
• Abstract summary: 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_Arangle_mathrmG!=! Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation. Finally, we compute the variance in the thermodynamic limit, given by the constant $lim_Ntoinfty(Delta S_A)2_mathrmG=
• Score: 1.0312968200748116
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\lim_{N\to\infty}(\Delta S_A)^2_{\mathrm{G}}=\frac{1}{2}(f+f^2+\log(1-f))$.

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