TY - JOUR

T1 - Page curve for fermionic Gaussian states

AU - Bianchi, Eugenio

AU - Hackl, Lucas

AU - Kieburg, Mario

N1 - Funding Information:
Acknowledgments. We would like to thank Pietro Donà, Peter Forrester, Marcos Rigol, and Lev Vidmar for inspiring discussions and comments on the manuscript. Special thanks goes to Lorenzo Piroli who pointed out an error in our analysis for the variance, which we subsequently corrected by deriving its exact form in the thermodynamic limit. L.H. gratefully acknowledges support by the Alexander von Humboldt Foundation. E.B. acknowledges support by the NSF via the Grant PHY-1806428 and by the John Templeton Foundation via the ID 61466 grant, as part of the “Quantum Information Structure of Spacetime (QISS)” project .
Publisher Copyright:
© 2021 American Physical Society.

PY - 2021/6/15

Y1 - 2021/6/15

N2 - 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. Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by Łydżba, Rigol, and Vidmar. Finally, we compute the variance in the thermodynamic limit, given by the constant limN→∞(ΔSA)G2=12[f+f2+log(1-f)].

AB - 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. Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by Łydżba, Rigol, and Vidmar. Finally, we compute the variance in the thermodynamic limit, given by the constant limN→∞(ΔSA)G2=12[f+f2+log(1-f)].

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U2 - 10.1103/PhysRevB.103.L241118

DO - 10.1103/PhysRevB.103.L241118

M3 - Article

AN - SCOPUS:85109022971

SN - 2469-9950

VL - 103

JO - Physical Review B

JF - Physical Review B

IS - 24

ER -