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

TL;DR

This paper calculates the average density of Bloch electrons in a 3D crystal under a weak magnetic field, revealing new contributions at second order involving quantum geometry, with implications for material properties.

Contribution

It introduces a gauge-invariant method to compute second-order density responses, highlighting the role of quantum metric tensor and orbital magnetic moments.

Findings

First-order density follows Streda formula for insulators.

Second-order density depends on quantum metric tensor and microscopic processes.

Method applies to metals and insulators, considering intraband and interband effects.

Abstract

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.