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-ave similar or equal to lnD(A)-(1/2)D-A(2)/D for 1 << D-A <= root 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 lnD(A)-(lnD(A))(2)/lnD <=< S ><= lnD(A)-[1/(2ln2)](2)/lnD. Consequently, we prove that (i) if the subsystem size is a finite fraction of the system size, then < S > < lnD(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.