Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians

Lev Vidmar; Lucas Hackl; Eugenio Bianchi; Marcos Rigol

doi:10.1103/PhysRevLett.119.020601

paper

Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians

arXiv:1703.02979 · doi:10.1103/PhysRevLett.119.020601

Abstract

In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)], Page proved that the average entanglement entropy of subsystems of random pure states is $S_{ave} ≃ ln D_{A} - (1/2) D_{A}^{2} / D$ for $1 ≪ D_{A} \leq D$ , where $D_{A}$ and $D$ are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy $⟨ S ⟩$ of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models $ln D_{A} - (ln D_{A})^{2} / ln D \leq ⟨ S ⟩ \leq ln D_{A} - [1/ (2 ln 2)] (ln D_{A})^{2} / ln D$ . Consequently we prove that: (i) if the subsystem size is a finite fraction of the system size then $⟨ S ⟩ < ln D_{A}$ in the thermodynamic limit, i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal, i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.

4+6 pages, 3+2 figures, as published

Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians

Abstract

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