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

TL;DR

The paper addresses how interacting electron systems in mean-field theory respond to a weak magnetic field. It shows that the linear response of the mean-field density matrix is geometric, determined solely by interband Berry connections between occupied and unoccupied subspaces, and is independent of the interaction potential and dispersion. Using this, it derives gauge-invariant projector expressions for the Středa formula and orbital magnetization, establishing a direct link between mean-field quasiparticles and noninteracting topological band theory. The results rely on a self-consistent vertex equation whose solution in the $q\to0$ limit reduces to the momentum-derivative of the mean-field Hamiltonian, consistent with Ward identities in conserving HF theory, thereby providing a unified geometric framework for mean-field systems.

Abstract

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.