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In solid state conductors, linear response to a steady electric field is normally dominated by Bloch state occupation number changes. Recently it has been realized that, for a number of important physical observables, the dominant response is electric-field induced coherence between Bloch states in different bands. Examples include the anomalous and spin-Hall effects, spin torques in magnetic conductors, and the minimum conductivity and chiral anomaly in Weyl and Dirac semimetals. Here we first discuss the framework of 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 demonstrate that the inter-band response in conductors consists primarily of two terms: an intrinsic contribution due to the entire Fermi sea that captures, among other effects, 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. Next we discuss an important interband coherence effect on Hall transport. Classical charge transport, such as longitudinal and Hall currents in weak magnetic fields, is usually not affected by quantum phenomena. Yet relativistic quantum mechanics is at the heart of the spin-orbit interaction, which has been at the forefront of efforts to realize spin-based electronics, new phases of matter and topological quantum computing. In this work we demonstrate that quantum spin dynamics induced by the spin-orbit interaction are directly observable in classical charge transport. We determine the Hall coefficient RH of two-dimensional hole systems at low magnetic fields and show that it has a sizable spin-orbit contribution, which depends on the density p, is independent of temperature, is a strong function of the top gate electric field, and can reach 30% of the total. We provide a general method for extracting the spin-orbit parameter from magnetotransport data, applicable even at higher temperatures where Shubnikov-de Haas oscillations and weak antilocalisation are difficult to observe. Our work will enable experimentalists to measure spin-orbit parameters without requiring large magnetic fields, ultralow temperatures, or optical setups.
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