Journal
PHYSICAL REVIEW B
Volume 103, Issue 24, Pages -Publisher
AMER PHYSICAL SOC
DOI: 10.1103/PhysRevB.103.L241118
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Funding
- Alexander von Humboldt Foundation
- NSF [PHY-1806428]
- John Templeton Foundation as part of the Quantum Information Structure of Spacetime (QISS) project [61466]
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This study discusses the average entanglement entropy of pure fermionic Gaussian states in subsystems and derives its formula and asymptotic behavior. The results show that in the thermodynamic limit, the average entanglement entropy of these pure random fermionic Gaussian states shares consistency with the average over eigenstates of random quadratic Hamiltonians.
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 < S-A >(G) = ( N - 1/2)Psi(2N) + (1/4 - N-A)Psi(N) + (1/2 + N-A - N)Psi(2N - 2N(A)) - 1/4 Psi(N - N-A) - N-A, where Psi is the digamma function. Its asymptotic behavior in the thermodynamic limit is given by < S-A >(G) = N(log 2 - 1)f + N(f - 1) log(1 - f) +1/2f + 1/4log (1 - f) + O(1/N), where f = N-A/N <= 1/2. 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 ->infinity)(Delta S-A)(G)(2) = 1/2[f + f(2) + log(1 - f)].
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