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Explicit Hamiltonians Inducing Volume Law for Entanglement Entropy in Fermionic Lattices

We show how the area law for the entanglement entropy may be violated by free fermions on a lattice and 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 $\bar{A}$ is given, the states with maximal entanglement entropy (volume law) may be factored into Bell-pairs (BP) formed by two states with support on $A$ and $\bar{A}$. We then exhibit, for translational invariant fermionic systems on a lattice, an Hamiltonian whose ground state is such to yield 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 $S_A$ between the area and the volume law, building an explicit example of a one-dimensional free fermion model where $S_A (L) \propto L^β$ 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 "zig-zag" 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 non-trivial topology of the Fermi surface.