-Anomalous Hall effect arises in systems with both spin-orbit coupling and magnetization. Generally, there are three mechanisms contributing to anomalous Hall conductivity: intrinsic, side jump, and skew scattering. The standard diagrammatic approach to the anomalous Hall effect is limited to computation of ladder diagrams. We demonstrate that this approach is insufficient. An important additional contribution comes from diagrams with a single pair of intersecting disorder lines. This contribution constitutes an inherent part of skew scattering on pairs of closely located defects and essentially modifies previously obtained results for anomalous Hall conductivity. We argue that this statement is general and applies to all models of anomalous Hall effect. We illustrate it by an explicit calculation for two-dimensional massive Dirac fermions with weak disorder. In this case, inclusion of the diagrams with crossed impurity lines reverses the sign of the skew scattering term and strongly suppresses the total Hall conductivity at high electron concentrations.Many ferromagnetic materials exhibit a finite Hall effect, i.e. transverse voltage in response to a current, without applying external magnetic field. This phenomenon is commonly referred to as the anomalous Hall effect (AHE) [1]. Two important ingredients of AHE are magnetization and spin-orbit interaction. The former breaks timereversal symmetry and exerts a force acting on electron spins while the latter couples the spins to orbital degrees of freedom thus giving rise to the transport effect.AHE can also occur as a result of valley or isospin polarization rather than ordinary ferromagnetism [2]. The spin-orbit coupling can also be of a more general form as it is, e.g. in graphene [3,4] where the role of spin is played by the sublattice index. An important part of the anomalous Hall signal originates in the Berry curvature thus being of a topological origin [5]. It is, therefore, natural that the discovery of materials like graphene and topological insulators [6,7], which are characterized by non-trivial Berry phase of quasiparticles, has considerably broadened the interest to AHE from both theory and experiment [8][9][10][11][12][13][14].Despite the long history [15][16][17] and high experimental relevance of AHE, its theoretical description is a challenging task often leading to confusions. In modern literature, two common approaches based on the Boltzmann kinetic equation and Kubo-Středa diagrammatic formalism are discussed. Boltzmann equation provides an intuitive quasiclassical approach to the effect [1,18] but requires an accurate account of several mechanisms of Hall conductivity: intrinsic, side-jump, and skew-scattering. Intrinsic AHE is attributed to topological properties of the band [19] and is thus independent of disorder. Skew scattering is due to the asymmetry in the impurity scattering crosssection and side jump refers to the transverse displacement of an electron being scattered. An alternative microscopic Kubo-Středa formalism is more...

We formulate a general microscopic approach to spin-orbit torques in thin ferromagnet/heavy-metal bilayers in linear response to electric current or electric field. The microscopic theory we develop avoids the notion of spin currents and spin-Hall effect. Instead, the torques are directly related to a local spin polarization of conduction electrons, which is computed from generalized Kubo-Středa formulas. A symmetry analysis provides a one-to-one correspondence between polarization susceptibility tensor components and different torque terms in the Landau-Lifshitz-Gilbert equation for magnetization dynamics. The spin-orbit torques arising from Rashba or Dresselhaus type of spin-orbit interaction are shown to have different symmetries. We analyze these spin-orbit torques microscopically for a generic electron model in the presence of an arbitrary smooth magnetic texture. For a model with spin-independent disorder we find a major cancellation of the torques. In this case the only remaining torque corresponds to the magnetization-independent Edelstein effect. Furthermore, our results are applied to analyze the dynamics of a skyrmion under the action of electric current.

Both longitudinal and anomalous Hall conductivity are computed in the model of two-dimensional Dirac fermions with a mass in the presence of arbitrary correlated weak disorder. The anomalous Hall conductivity is shown to be highly sensitive to the correlation properties of the random potential, such as the correlation length, while it remains independent of the integral disorder strength. This property extends beyond the Dirac model making the anomalous Hall effect an interesting tool to probe disorder correlations. arXiv:1710.09354v1 [cond-mat.dis-nn]

Dzyaloshinskii-Moriya interaction (DMI) is investigated in a 2D ferromagnet (FM) with spin-orbit interaction of Rashba type at finite temperatures. The FM is described in the continuum limit by an effective s-d model with arbitrary dependence of spin-orbit coupling (SOC) and kinetic energy of itinerant electrons on the absolute value of momentum. In the limit of weak SOC, we derive a general expression for the DMI constant D from a microscopic analysis of the electronic grand potential. We compare D with the exchange stiffness A and show that, to the leading order in small SOC strength α_{R}, the conventional relation D=(4mα_{R}/ℏ)A, in general, does not hold beyond the Bychkov-Rashba model. Moreover, in this model, both A and D vanish at zero temperature in the metal regime (i.e., when two spin sub-bands are partly occupied). For nonparabolic bands or nonlinear Rashba coupling, these coefficients are finite and acquire a nontrivial dependence on the chemical potential that demonstrates the possibility to control the size and chirality of magnetic textures by adjusting a gate voltage.

Skew scattering on rare impurity configurations is shown to dominate the anomalous Hall effect in a 2D Rashba ferromagnet. The mechanism originates in scattering on rare impurity pairs separated by distances of the order of the Fermi wavelength. The corresponding theoretical description goes beyond the conventional noncrossing approximation. The mechanism provides the only contribution to the anomalous Hall conductivity in the most relevant metallic regime and strongly modifies previously obtained results for lower energies in the leading order with respect to impurity strength.