Average density of Bloch electrons in a homogeneous magnetic field: A second-order response

We compute the average density of a three-dimensional multiband crystal of arbitrary symmetry, metal or insulator, to first and second order in a weak homogeneous magnetic field. To linear order and for insulators, the density follows the well-known Streda formula, but for metals there is an extra contribution from the orbital magnetic moments at the Fermi surface. To second order the average density depends on several microscopic processes. Among these, the quantum metric tensor plays an important role by generating a pseudo-magnetic moment resulting from the rotation of the Bloch wave functions in the complex projective plane. We also discuss the implications of our results for the volume and pressure. The method we develop is explicitly gauge invariant, considers intraband and interband processes on equal footing, accommodates relaxation processes, and can be readily extended to other observables.