On the Road to Connect Nonlinear Charged and Orbital Effects.*

A few years ago, it was theoretically proposed and experimentally demonstrated that nonmagnetic materials with nontrivial band topology support anomalous Hall effect at the second order in the electric field and in the absence of magnetic field. This nonlinear anomalous Hall effect emerges in noncentrosymmetric materials and was connected to the Berry curvature dipole of the ground state. In fact, the absence of a symmetry center in such materials unlocks the generation of nonequilibrium orbital magnetization, linear in the electric field. This effect is tagged the gyrotropic Edelstein effect. In this presentation, we investigate the intimate relation between the nonlinear anomalous Hall effect mechanism and gyrotropic Edelstein. As a matter of fact, these two effects are governed by the same crystalline symmetries and are both associated with the Berry curvature dipole. In this sense, the mathematical structure of these two quantities suggests that they are connected between each other, and must be a nontrivial function of the chemical potential. We address their corresponding behaviours in two different multiband model systems, remarking the fact that the spin-orbit coupling is not a necessary condition on both nonlinear Hall effect and gyrotropic Edelstein effect. From a theoretical point a view,  a formal relationship between the orbital and charge transports opens the possibility to unify the notions of the Berry curvature and the orbital moment, since in essence both objects are akin to an angular velocity of electrons in the adiabatic limit, when they are subjected to an external electric field.