The current density j B induced in a clean metal by a magnetic field B is formulated as the low-frequency limit of natural optical activity, or natural gyrotropy. Working with a multiband Pauli Hamiltonian, we obtain from the Kubo formula a simple expression for α gme ij = j B i /Bj in terms of the intrinsic magnetic moment (orbital plus spin) of the Bloch electrons on the Fermi surface. An alternate semiclassical derivation provides an intuitive picture of the effect, and takes into account the influence of scattering processes in dirty metals. This "gyrotropic magnetic effect" is fundamentally different from the chiral magnetic effect driven by the chiral anomaly and governed by the Berry curvature on the Fermi surface, and the two effects are compared for a minimal model of a Weyl semimetal. Like the Berry curvature, the intrinsic magnetic moment should be regarded as a basic ingredient in the Fermi-liquid description of transport in broken-symmetry metals. Introduction.-When a solid is placed in a static magnetic field the nature of the electronic ground state can change, leading to striking transport effects. A prime example is the integer quantum Hall effect in a quasi two-dimensional (2D) metal in a strong perpendicular field [1]. Novel magnetotransport effects have also been predicted to occur in 3D topological (Weyl) metals, such as an anomalous longitudinal magnetoresistence [2,3], and the chiral magnetic effect (CME), where an electric pulse E B induces a transient current j B [4]; both are related to the chiral anomaly that was originally discussed for Weyl fermions in particle physics [5,6]. In all these phenomena the role of the static B-field is to modify the equilibrium state, but an E-field is still required to put the electrons out of equilibrium and drive the current (since E = −Ȧ, the vector potential is time-dependent even for a static E-field).

We study nonlinear magneto-optical responses of metals by a semiclassical Boltzmann equation approach. We derive general formulas for linear and second order nonlinear optical effects in the presence of magnetic fields that include both Berry curvature and orbital magnetic moment. Applied to Weyl fermions, the semiclassical approach (i) captures the directional anisotropy of linear conductivity under magnetic field as a consequence of an anisotropic B 2 contribution, which may explain the low-field regime of recent experiments; (ii) predicts strong second harmonic generation proportional to B that is enhanced as the Fermi energy approaches the Weyl point, leading to large nonlinear Kerr rotation. Moreover, we show that the semiclassical formula for the circular photogalvanic effect arising from the Berry curvature dipole is reproduced by a full quantum calculation using a Floquet approach.

Several emergent phenomena and phases in solids arise from configurations of the electronic Berry phase in momentum space that are similar to gauge field configurations in real space such as magnetic monopoles. We show that the momentum-space analogue of the "axion electrodynamics" term E • B plays a fundamental role in a unified theory of Berry-phase contributions to optical gyrotropy in time-reversal invariant materials and the chiral magnetic effect. The Berry-phase mechanism predicts that the rotatory power along the optic axes of a crystal must sum to zero, a constraint beyond that stipulated by point group symmetry, but observed to high accuracy in classic experimental observations on α-quartz. Furthermore, the Berry mechanism provides a microscopic basis for the surface conductance at the interface between gyrotropic and nongyrotropic media.

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The spin wave spectrum of dense arrays of rectangular elements periodically arranged in a two-dimensional magnonic crystal with a complex unit cell and magnetized perpendicularly to the array plane has been characterized using broadband ferromagnetic resonance (FMR) spectroscopy. The crystal's unit cell consists of non-collinear orientations of constituting elongated rectangular elements. We found that only one mode is excited in the perpendicular-to-plane FMR in complete magnetic saturation. We also conducted out-of-plane angle resolved measurements of the FMR resonance field. We observe splitting of the singlet observed for the perfect perpendicular-to-plane orientation of the applied field into a doublet upon a tilt of the field from this orientation. The splitting of the singlet into a doublet is explained as an experimental evidence of dipole coupling of the elements on the arrays. Our experimental observations are in good agreement with the theory we developed to describe the magnetization dynamics on this periodic array.