A semiclassical approach to nonlinear transport and band geometry in solids

The semiclassical wave packet approach is a well-established method to study linear response transport in solids. It includes the effects of Berry curvature via the anomalous velocity and can be used to derive the quantized Hall conductance. However, this semiclassical formalism fails when considering nonlinear response or even linear response to a non-uniform external field. To go beyond uniform linear response, it is necessary to add corrections to the semiclassical approximations used in this formalism, but the issue is that it is not clear how this can be done systematically.

Here, we present a different semiclassical formalism based on the Wigner transformation which maps operators in the Hilbert space to functions defined over phase space. Within this formalism, ħ exists as a small parameter which can be used to do a systematic semiclassical expansion. To first order in ħ, this formalism coincides with the semiclassical wave packet formalism. In this work, we develop this formalism and expand to second order in ħ to study nonlinear transport from a semiclassical perspective. At second order in ħ, we generically find contributions from the quantum metric to transport current. We apply our formalism to understand how nonlinear transport probes the geometry of bands.