Berry curvature and quantum oscillation from multi-orbital coadjoint-orbit bosonization

We develop an effective field theory for a multi-orbital fermionic system using the method of coadjoint orbits for higher-dimensional bosonization. The dynamical bosonic fields are single-particle distribution functions defined on the phase space. We show that when projecting to a single band, Berry curvature effects naturally emerge. In particular, we consider the de Haas-van Alphen effect of a 2d Fermi surface, and show that the oscillation of orbital magnetization in an external field is offset by the Berry phase accumulated by the cyclotron around the Fermi surface. Beyond previously known results, we show that this phase shift holds even for interacting systems, in which the single-particle Berry phase is replaced by the static anomalous Hall conductance. Furthermore, we obtain the correction to the amplitudes of de Haas-van Alphen oscillations due to Berry curvature effects.