Deviation from maximal entanglement for mid-spectrum eigenstates of local Hamiltonians

In a chain of $N$ spins governed by a local Hamiltonian, we consider eigenstates in a microcanonical ensemble in the middle of the energy spectrum. We prove that if the bandwidth of the ensemble is $C\ln^2N$ for a large constant $C$, then the average entanglement entropy of these eigenstates is bounded away from the maximum entropy. This result highlights the difference between the entanglement of mid-spectrum eigenstates of (chaotic) local Hamiltonians and that of random states, for the latter is nearly maximal. We also prove that the former is different from the thermodynamic entropy at the same energy.