Quantum interference from sums over closed paths for electrons on a three-dimensional lattice in a magnetic field: total energy, magnetic moment, and orbital susceptibility

Abstract

We study quantum interference effects due to electron motion on a three-dimensional cubic lattice in a continuously-tunable magnetic field of arbitrary orientation and magnitude. These effects arise from the interference between magnetic phase factors associated with different electron closed paths. The sums of these phase factors, called lattice path-integrals, are ``many-loop" generalizations of the standard ``one-loop" Aharonov-Bohm-type argument. Our lattice path integral calculation enables us to obtain various important physical quantities through several different methods. The spirit of our approach follows Feynman's programme: to derive physical quantities in terms of ``sums over paths". From these lattice path-integrals we compute analytically, for several lengths of the electron path, the half-filled Fermi-sea ground-state energy of noninteracting spinless electrons in a cubic lattice. Our results are valid for any strength of the applied magnetic field in any direction. We also study in detail two experimentally important quantities: the magnetic moment and orbital susceptibility at half-filling, as well as the zero-field susceptibility as a function of the Fermi energy.