Nonlinear response of Bloch electrons to a magnetic field: average density

Strong magnetic fields in crystals confine electrons to regions of space, e.g. in the quantum Hall effects. Here we ask the question, how does the density of electrons in a crystal change in the regime where the magnetic field is not so strong but beyond the linear response. We develop formalism to compute the average density of a clean crystal (of any space group) considering intraband and interband on equal footing in terms of unperturbed Bloch wave functions. To first order, we recover Streda-like formulas with an additional contribution from the orbital magnetic moments at the Fermi sea. If there is time reversal symmetry or the magnetic field becomes stronger, the second order term becomes important. To second order the expressions are more complex and naturally depend on geometric aspects of the wave functions such as the quantum metric tensor. Our expressions can be used to compute density changes of materials in realistic scenarios using large scale numerical codes, and, similar to the Streda formula, be used as a diagnostic tool for the existence (or lack thereof) of topological states.