Abelian gauge potentials on cubic lattices

The study of the properties of quantum particles in a periodic potential subjected to a magnetic field is an active area of research both in physics and mathematics, and it has been and is yet deeply investigated. In this chapter we discuss how to implement and describe tunable Abelian magnetic fields in a system of ultracold atoms in optical lattices. After reviewing two of the main experimental schemes for the physical realization of synthetic gauge potentials in ultracold setups, we study cubic lattice tight-bindingmodels with commensurate flux.We finally discuss applications of gauge potentials in one-dimensional rings.