Inter-band Coherence Response to Electric Fields in Crystals

Abstract

In solid state conductors, linear response to a steady electric field is normally dominated by Bloch state occupation number changes that are correlated with group velocity and lead to a steady state current. However, for a number of important physical observables, the most important response even in conductors can be electric-field induced coherence between Bloch states in different bands, such as that responsible for screening in dielectrics. Examples include the anomalous and spin-Hall effects, spin torques, minimum conductivities and the chiral anomaly. In this paper we present a general quantum kinetic theory of linear response to an electric field which can be applied to solids with arbitrarily complicated band structures and includes the inter-band coherence response and the Bloch-state repopulation responses on an equal footing. We aim to enable extensive transport theory applications using computational packages based on maximally localized Wannier functions. The formalism is based on density-matrix equations of motion, on a Born approximation treatment of disorder, and on an expansion in scattering rate to leading non-trivial order. The quasiparticle bands are treated in a completely general manner that allows for arbitrary forms of the spin-orbit interaction and for the broken time reversal symmetry of magnetic conductors. We demonstrate that the inter-band response in conductors consists primarily of two terms: an intrinsic contribution due to the entire Fermi sea that captures the Berry curvature contribution to wave-packet dynamics, and an anomalous contribution caused by scattering that is sensitive to the presence of the Fermi surface. We solve for some electric-field response properties of simple model systems that are known to be dominated by interband coherence contributions and discuss an extensive list of complicated problems that cannot be solved analytically.