Eigenstate Entanglement Entropy in Random Quadratic Hamiltonians

The eigenstate entanglement entropy has been recently shown to be a powerful tool to distinguish integrable from generic quantum-chaotic models. In integrable models, a unique feature of the average eigenstate entanglement entropy (over all Hamiltonian eigenstates) is that the volume-law coefficient depends on the subsystem fraction. Hence, it deviates from the maximal (subsystem fraction independent) value encountered in quantum-chaotic models. Using random matrix theory for quadratic Hamiltonians, we obtain a closed-form expression for the average eigenstate entanglement entropy as a function of the subsystem fraction. We test its correctness against numerical results for the quadratic Sachdev-Ye-Kitaev model. We also show that it describes the average entanglement entropy of eigenstates of the power-law random banded matrix model (in the delocalized regime), and that it is close but not the same as the result for quadratic models that exhibit localization in quasimomentum space.