Geometrical contributions to the Exchange interactions: From Equilibrium to Nonequilibrium

While the spin-spiral approach is a powerful method to calculate the exchange constants of realistic materials within density functional theory, it has the drawback that it does not explicitly express the exchange constants in terms of the electronic structure. In this talk we discuss how to express the exchange constants in terms of electronic structure properties, such as the mixed Berry curvature and the mixed quantum metric, which describe the geometrical properties of the electronic structure in mixed phase space [1]. While the mixed Berry curvature [2,3,4] plays a central role in the Dzyaloshinskii-Moriya interaction the symmetric exchange interaction involves additionally the quantum metric in mixed phase space [1]. Our expressions for the exchange constants bear a strong formal resemblance to Fukuyama's theory [5] of the orbital magnetic susceptibility, which can be expressed in terms of geometrical quantities as well [6]. In contrast to the spin-spiral approach, our formalism expresses the exchange constants directly in terms of the electronic structure information, which allows us to study the relationship to other effects and phenomena important in spintronics. For example, spin-transfer torque and spin-orbit torque [7,8] can be interpreted as nonequilibrium exchange interaction and nonequilibrium magnetic anisotropy. Consequently, the spin-orbit torque is given by the mixed Berry curvature. In first order of the spin-orbit interaction the Dzyaloshinskii-Moriya interaction is related to the ground-state spin current [9]. Thus, spin-currents excited by light are expected to lead to nonequilibrium DMI.References1. F. Freimuth, S. Blügel and Y. Mokrousov, arXiv:1701.08872.2. F. Freimuth, R. Bamler, Y. Mokrousov and A. Rosch, PRB 88, 214409 (2013).3. F. Freimuth, S. Blügel and Y. Mokrousov, JPCM 26, 104202 (2014).4. F. Freimuth, S. Blügel and Y. Mokrousov, JPCM 28, 316001 (2016).5. H. Fukuyama, Progress of Theoretical Physics 45, 704 (1971).6. Y. Gao, S. A. Yang, and Q. Niu, Phys. Rev. B 91, 214405 (2015).7. F. Freimuth, S. Blügel and Y. Mokrousov, PRB 92, 064415 (2015).8. F. Freimuth, S. Blügel and Y. Mokrousov, PRB 90, 174423 (2014).9. F. Freimuth, S. Blügel, and Y. Mokrousov, ArXiv eprints (2016), 1610.06541