Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians

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 $S_{\rm ave}\simeq\ln{\cal D}{\rm A} - (1/2) {\cal D}{\rm A}^2/{\cal D}$for$1\ll{\cal D}{\rm A}\leq\sqrt{\cal D}$,where${\cal D}{\rm A}$and${\cal D}$ are the Hilbert space dimensions of the subsystem and the system, respectively. 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 $\ln{\cal D}{\rm A} - (\ln{\cal D}{\rm A})^2/\ln{\cal D} \leq \langle S \rangle \leq \ln{\cal D}{\rm A} - [1/(2\ln2)] (\ln{\cal D}{\rm A})^2/\ln{\cal D}$. Consequently we prove that: (i) if the subsystem size is a finite fraction of the system size then $⟨S⟩<\ln {\mathcal{D}}_{\mathrm{A}}$ 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.