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 SₐᵥₑlnDA-(1/2)DA²/D for 1DA≤√D, where DA 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 $ln{D}A{-}(ln{D}A{)}²/lnD{≤}⟨S⟩{≤}ln{D}A{-}[1/(2ln2)](ln{D}A{)}²/lnD$. 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.