Magnetic-Field-Induced Geometric Response of Mean-Field Projectors: Streda Formula and Orbital Magnetization

We study the magnetic-field response of interacting electron systems within mean-field theory using perturbation theory. We show that the linear response of the mean-field density-matrix to a weak magnetic field is purely geometric: it depends only on wavefunction derivatives, the Berry connections linking the occupied and unoccupied subspaces, and is independent of the interaction potential and the quasiparticle dispersion. This leads to compact, gauge-invariant projector expressions for both the Středa formula and the formula for orbital magnetization. Our calculation explicitly elucidates the role of exchange and self-consistency in defining current vertices for orbital magnetization calculations. Our work establishes a direct connection between mean-field theory, quantum geometry and the non-interacting topological band theory.