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 S₀ₕ₄D₀- (1/2) D₀^2/D for 1D₀D, where D₀ 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 lnD₀- (lnD₀) ^2/lnD⟨S⟩D₀-1/ (2ln2) (lnD₀) ^2/lnD. Consequently, we prove that (i) if the subsystem size is a finite fraction of the system size, then ⟨S⟩<lnD₀ 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.