Entanglement Entropy of Typical Eigenstates of Translationally Invariant Quadratic Hamiltonians

08/03/2017

Lev Vidmar, LFH, Eugenio Bianchi, Marcos Rigol

In a seminal paper [Phys.Rev.Lett. 71, 1291 (1993)], Page proved that the entanglement entropy of typical pure states is $S_{\rm typ}\simeq\ln{\cal D}_{\rm A} - (1/2) {\cal D}_{\rm A}^2/{\cal D}$ [/latex], for ${\cal D}_{\rm A}\leq\sqrt{\cal D}$ , where ${\cal D}_{\rm A}$ and ${\cal D}$ are the Hilbert space dimensions of the subsystem and the system, respectively. Typical pure states are hence (nearly) maximally entangled. We introduce new tools to compute the average entanglement entropy $\langle S\rangle$ of all eigenstates for quadratic fermionic Hamiltonians. With this, we prove that $S^+_1=\ln{\cal D}_{\rm A} - [1/(2\ln2)] (\ln{\cal D}_{\rm A})^2/\ln{\cal D}$ is an upper bound for translationally invariant quadratic Hamiltonians in any dimension. Consequently, typical eigenstates of such Hamiltonians are qualitatively different from typical pure states: they are not maximally entangled if the subsystem size is a finite fraction of the system size. We also prove that, in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal. We compare our predictions with numerical results for free fermions on a circle.

[10.1103/PhysRevLett.119.020601] | [arXiv:1703.02979]

Entanglement Entropy of Typical Eigenstates of Translationally Invariant Quadratic Hamiltonians

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