  The theoretical treatment of homogeneous static magnetic fields in periodic
systems is challenging, as the corresponding vector potential breaks the
translational invariance of the Hamiltonian. Based on density operators and
perturbation theory, we propose, for insulators, a periodic framework for the
treatment of magnetic fields up to arbitrary order of perturbation, similar to
widely used schemes for electric fields. The second-order term delivers a new,
remarkably simple, formulation of the macroscopic orbital magnetic
susceptibility for periodic insulators. We validate the latter expression using
a tight-binding model, analytically from the present theory and numerically
from the large-size limit of a finite cluster, with excellent numerical
agreement.
