We study quantum interference effects due to electron motion on a three-dimensional cubic lattice in a continuously tunable magnetic field B 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, where the electron wave function picks up a phase factor ${\mathit{e}}^{\mathit{i}\mathrm{\Phi }}$ each time it travels around a closed loop enclosing a net flux Φ. 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 program: 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, ${\mathit{E}}_{\mathit{T}}$(B), of noninteracting spinless electrons in a cubic lattice. Our expressions for ${\mathit{E}}_{\mathit{T}}$ are valid for any strength of the applied magnetic field in any direction. Moreover, we provide an explicit derivation for the absolute minimum energy of the flux state. For various field orientations, we also study the quantum interference patterns and ${\mathit{E}}_{\mathit{T}}$(B) by exactly summing over ∼${10}^{29}$ closed paths in a cubic lattice, each one with its corresponding magnetic phase factor representing the net flux enclosed by each path. Furthermore, an expression for the total kinetic energy ${\mathit{E}}_{\mathit{T}}$(B,ν) for any electron filling ν close to one-half is obtained. We also study in detail two experimentally important quantities: the magnetic moment M(B) and orbital susceptibility χ(B) at half filling, as well as the zero-field susceptibility χ(μ) as a function of the Fermi energy μ. © 1996 The American Physical Society.

• Received 24 August 1995

DOI:https://doi.org/10.1103/PhysRevB.53.13374