Orbital magnetism of coupled bands models

We develop a gauge-independent perturbation theory for the grand potential of itinerant electrons in two-dimensional tight-binding models in the presence of a perpendicular magnetic field. At first order in the field, we recover the result of the so-called {\it modern theory of orbital magnetization} and, at second order, deduce a new general formula for the orbital susceptibility. In the special case of two coupled bands, we relate the susceptibility to geometrical quantities such as the Berry curvature. Our results are applied to several two-band -- either gapless or gapped -- systems. We point out some surprising features in the orbital susceptibility -- such as in-gap diamagnetism or parabolic band edge paramagnetism -- coming from interband coupling. From that we draw general conclusions on the orbital magnetism of itinerant electrons in multi-band tight-binding models.