Entanglement Scaling Laws for Non-Interacting Fermions in Finite One-Dimensional Lattices

The structure of entanglement between two subsystems of a many-particle state contains useful information about many important physical characteristics. In particular, the entanglement entropy of gapped quantum states generally scales as a function of the surface area between the two subsystems, consistent with an informational holographic principle. Notably, the holographic scaling is known to be logarithmically violated for certain systems of non-interacting free fermions, a consequence of the gaplessness of the excitation spectrum. We numerically investigate the scaling of the bipartite entanglement entropy for non-interacting fermions on a one-dimensional lattice. A variety of entanglement measures, lattice fillings and lattice geometries are considered. Violation of the holographic scaling is apparent, even in small Bravais lattices where the excitation gap remains non-zero for all finite sizes. The numerical results are compared with calculations for infinite systems. The impact of lattice filling and a band gap on the entanglement scaling is also explored in this context.