Finite Wavevector Electromagnetic Response in a Weyl Semimetal Model

Recent works have shown that linear and nonlinear optical conductivities of periodic crystals can encode topological and geometric attributes of the electronic bands. To date, work to extend these approaches to calculate responses to spatially inhomogeneous electric and magnetic fields has been limited to expansions around zero wavevector. Here we formulate a theory of linear and nonlinear conductivity for periodic systems at general wavevector. Crucially, our method does not make any assumptions on the form of the underlying Hamiltonian and ensures that current is conserved to all orders in the wavevector. To illustrate our approach, we apply this method to investigate the wavevector-dependent conductivity for a Weyl semimetal model. We examine the Hall response generated from this model, focusing on how the wavevector impacts the expression of topological properties in this response.