Phase-Space Berry Phases in Chiral Magnets: Skyrmion Charge, Hall Effect, and Dynamics of Magnetic Skyrmions

The dynamics of electrons in solids is influenced by Berry phases in phase space (combined position and momentum space). Phase-space Berry phases lead to an effective force on the electrons, an anomalous contribution to the group velocity, and a correction to the density of states in phase space. In addition, Berry phases in position and in momentum space are related to topological winding numbers and can be used to characterize topologically distinct phases of matter. We study theoretically the effects of phase-space Berry phases in magnetic materials with weak spin-orbit coupling and a smoothly varying magnetization texture. Such magnetic textures appear generically in non-centrosymmetric magnetic materials with weak spin-orbit coupling due to a competition between the ferromagnetic exchange interaction and the weaker Dzyaloshinskii-Moriya interaction. In particular, the discovery of topologically stable whirls, so-called skyrmions, in the magnetization texture of these materials has attracted considerable attention due to prospects of applications in future magnetic storage devices. In part I of this thesis, we investigate the influence of phase-space Berry phases on the equilibrium properties of electrons in chiral magnets with weak spin-orbit coupling. We show that the strength of the Dzyaloshinskii-Moriya interaction in the long-wavelength limit can be calculated from Berry phases in mixed position/momentum space and that the same Berry phases lead to an electric charge of skyrmions in metallic chiral magnets. In insulators, the skyrmion charge of magnetic skyrmions turns out to be proportional to the topologically quantized second Chern number in phase space. This establishes a link between skyrmions in chiral magnets and the charged excitations in integer quantum Hall systems with small Zeeman splitting. In part II, we consider the Hall effect in the skyrmion lattice phase of chiral magnets in presence of spin-orbit coupling. It has been previously known that Berry phases in momentum space lead to the intrinsic part of the anomalous Hall effect, and that Berry phases in position space lead to an effective Lorentz force, resulting in the so-called topological Hall effect. By expanding the Kubo-Středa Formula for the Hall conductivity in gradients in position and momentum space, we show that the interplay between smooth magnetic textures and spin-orbit coupling leads to a previously disregarded contribution to the Hall effect, and we find a correction to the semiclassical formulation of the topological Hall effect. In part III, we study the influence of phase-space Berry phases on the dynamics of skyrmions in chiral magnets. Berry phases in mixed position/momentum space lead to a dissipationless momentum transfer from conduction electrons to skyrmions that is proportional to an applied electric field and independent of the (spin or electric) current. We further show that the electric charge of skyrmions, discussed in part I, influences the skyrmion motion only via hydrodynamic drag and ohmic friction in metals. In insulators, the quantized skyrmion charge couples directly to an applied electric field