Entanglement Entropy of Typical Eigenstates of Translationally Invariant Quadratic Hamiltonians

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]