Crossover Behavior of Entanglement Entropy for Energy Eigenstates of 1d and 2d Fermionic Systems

The entanglement entropy in ground states of typical condensed matter systems obeys the area law or a log-area law for critical systems. Subsystem entropies in random and thermal states obey a volume law. Here, we discuss the distribution of entanglement entropy in energy eigenstates of quasi-free fermionic systems as a function of energy and subsystem size. Numerical results are obtained with a Monte Carlo approach. We characterize the crossover behavior from the area or log-area law in the vicinity of the ground state and for small subsystem size to the volume law at higher energy and larger subsystem size. The coefficients of the volume law scaling can be matched to entropy densities in equilibrium thermal ensembles. For critical 1d systems at low energies, the universal crossover function matches the prediction from 1+1d conformal field theory for systems at nonzero temperatures. For 2d systems, we find a similar crossover behavior.