Explicit Hamiltonians inducing volume law for entanglement entropy in fermionic lattices

We show how the area law for entanglement entropy may be violated by free fermions on a lattice, and we look for conditions leading to the emergence of a volume law. We give an explicit construction of the states with maximal entanglement entropy based on the fact that, once a bipartition of the lattice in two complementary sets A and A¯ is given, the states with maximal entanglement entropy (volume law) may be factored into Bell pairs (BPs) formed by two states with support on A and A¯. We then exhibit, for translational invariant fermionic systems on a lattice, a Hamiltonian whose ground state is such that it yields an exact volume law. As expected, the corresponding Fermi surface has a fractal topology. We also provide some examples of fermionic models for which the ground state may have an entanglement entropy SA between the area and the volume law, building an explicit example of a one-dimensional free fermion model where SA(L)Lβ, with β being intermediate between β=0 (area law) and β=1 (BP state inducing volume law). For this model, the dispersion relation has a "zigzag" structure leading to a fractal Fermi surface whose counting box dimension equals, for large lattices, β. Our analysis clearly relates the violation of the area law for the entanglement entropy of the ground state to the emergence of a nontrivial topology of the Fermi surface.