Maximally Entangled Mode, Metal-Insulator Transition and Violation of Entanglement Area Law in Non-interacting Fermion Ground States

We study in this work the ground state entanglement properties of two models of non-interacting fermions moving in one-dimension (1D), namely random dimer model and power-law random banded model that exhibit metal-insulator transitions. We find that entanglement entropy grows either logarithmically or in a power-law fashion with subsystem size in the metallic phase or at metal-insulating critical point, thus violating the (1D version of) entanglement area law. No such violation is found in the insulating phase. We further find that characteristics of \emph{single fermion} states at the Fermi energy (which can \emph{not} be obtained from the many-fermion Slater determinant) is captured by the lowest energy single fermion mode of the \emph{entanglement} Hamiltonian; this is particularly true at the metal-insulator transition point. In addition, the inverse-participation ratio of the lowest energy single fermion mode of the {\em entanglement} Hamiltonian is proportional to that of the single fermion state at Fermi energy in all cases. Our results suggest entanglement is a powerful way to detect metal-insulator transitions, \emph{without} knowledge of the Hamiltonian of the system. Results on metal-insulator transition of 3D Anderson model will also be presented.