Kubo formula for dc conductivity in the presence of spin-orbit coupling: a revision

I. A. Ado, M. Titov, Rembert A. Duine, Arne Brataas

Published 2023-09-12Version 1

We revise the Kubo formula for the electric dc conductivity in the presence of spin-orbit coupling (SOC). We discover that each velocity operator that enters this formula differs from $\partial H/\partial \boldsymbol p$, where $H$ is the Hamiltonian and $\boldsymbol p$ is the canonical momentum. Moreover, we find an additional contribution to the Hall dc conductivity from noncommuting coordinates that is missing in the conventional Kubo-Streda formula. We show that the widely used Rashba model does in fact provide a finite anomalous Hall dc conductivity in the metallic regime (in the noncrossing approximation). In addition to the Kubo formula, the Berry-phase theory of the anomalous Hall effect should also be revised for systems where the velocity operator differs from $\partial H/\partial \boldsymbol p$. While we focus on the Hall response of the charge current to the electric field, linear response theories of other SOC-related effects should be modified similarly.