We present a general quantum kinetic theory of low-field magnetotransport in weakly disordered crystals that accounts fully for the interplay between electric-field induced interband coherence, Bloch-state scattering, and an external magnetic field. The quantum kinetic equation we derive for the Bloch-state density matrix naturally incorporates the momentum-space Berry phase effects whose influence on Bloch-state wavepacket dynamics is normally incorporated into transport theory in an ad hoc manner. The Berry phase correction to the momentum-space density of states in the presence of an external magnetic field implied by semiclassical wavepacket dynamics is captured by our theory as an intrinsic density-matrix response to a magnetic field. We propose a simple and general procedure for expanding the linear response of the Bloch-state density matrix to an electric field in powers of magnetic field. As an illustration, we apply our theory to magnetotransport in Weyl semimetals. We show that the chiral anomaly (positive magnetoconductivity quadratic in magnetic field) that appears when separate Fermi surface pockets surround distinct Weyl points survives only when intervalley scattering is very weak compared to intravalley scattering.

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. Recently it has been realized that, 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 in magnetic conductors, and the minimum conductivity and chiral anomaly in Weyl and Dirac semimetals. 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. One of the principal aims of our work is to enable extensive transport theory applications using computational packages constructed in terms of maximally localized Wannier functions. To this end we provide a complete correspondence between the Bloch and Wannier formulations of our theory. The formalism is based on densitymatrix equations of motion, on a Born approximation treatment of disorder, and on an expansion in scattering rate to leading non-trivial order. Our use of a Born approximation omits some physical effects and represents a compromise between comprehensiveness and practicality. 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, 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. To demonstrate the rich physics captured by our theory, we explicitly solve for some electric-field response properties of simple model systems that are known to be dominated by interband coherence contributions. At the same time we discuss an extensive list of complicated problems that cannot be solved analytically. Our goal is to stimulate progress in computational transport theory for electrons in crystals.

One of the intriguing properties characteristic to three-dimensional topological materials is the topological magnetoelectric phenomena arising from a topological term called the θ term. Such magnetoelectric phenomena are often termed the axion electrodynamics since the θ term has exactly the same form as the action describing the coupling between a hypothetical elementary particle, axion, and a photon. The axion was proposed about 40 years ago to solve the so-called strong CP problem in quantum chromodynamics and is now considered a candidate for dark matter. In this Tutorial, we overview theoretical and experimental studies on the axion electrodynamics in three-dimensional topological materials. Starting from the topological magnetoelectric effect in three-dimensional time-reversal invariant topological insulators, we describe the basic properties of static and dynamical axion insulators whose realizations require magnetic orderings. We also discuss the electromagnetic responses of Weyl semimetals with a focus on the chiral anomaly. We extend the concept of the axion electrodynamics in condensed matter to topological superconductors, whose responses to external fields can be described by a gravitational topological term analogous to the θ term.

We search for dynamical magnetoelectric phenomena in three-dimensional correlated systems with spin-orbit coupling. We focus on the antiferromagnetic insulator phases where the dynamical axion field is realized by the fluctuation of the antiferromagnetic order parameter. It is shown that the dynamical chiral magnetic effect, an alternating current generation by magnetic fields, emerges due to such time dependences of the order parameter as antiferromagnetic resonance. It is also shown that the anomalous Hall effect arises due to such spatial variations of the order parameter as antiferromagnetic domain walls. Our study indicates that spin excitations in antiferromagnetic insulators with spin-orbit coupling can result in nontrivial charge responses. Moreover, observing the chiral magnetic effect and anomalous Hall effect in our system is equivalent to detecting the dynamical axion field in condensed matter.

It has been suggested that certain antiferromagnetic topological insulators contain axion quasiparticles (AQs), and that such materials could be used to detect axion dark matter (DM). The AQ is a longitudinal antiferromagnetic spin fluctuation coupled to the electromagnetic Chern-Simons term, which, in the presence of an applied magnetic field, leads to mass mixing between the AQ and the electric field. The electromagnetic boundary conditions and transmission and reflection coefficients are computed. A model for including losses into this system is presented, and the resulting linewidth is computed. It is shown how transmission spectroscopy can be used to measure the resonant frequencies and damping coefficients of the material, and demonstrate conclusively the existence of the AQ. The dispersion relation and boundary conditions permit resonant conversion of axion DM into THz photons in a material volume that is independent of the resonant frequency, which is tuneable via an applied magnetic field. A parameter study for axion DM detection is performed, computing boost amplitudes and bandwidths using realistic material properties including loss. The proposal could allow for detection of axion DM in the mass range between 1 and 10 meV using current and near future technology.