Kubo formula for dc conductivity: generalization to systems with spin-orbit coupling

TL;DR

This paper revises the Kubo formula for dc conductivity in systems with spin-orbit coupling, revealing additional contributions and corrections to the velocity operator that impact the calculation of Hall conductivity.

Contribution

It introduces a generalized Kubo formula accounting for spin-orbit coupling effects, including noncommuting coordinates and modified velocity operators, improving accuracy in conductivity calculations.

Findings

Additional contribution from noncommuting coordinates to Hall conductivity

Velocity operator differs from ∂H/∂p in the presence of SOC

Rashba model yields finite anomalous Hall conductivity when corrections are included

Abstract

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. This contribution originates from the "electron-positron" matrix elements of the velocity and position operators. We argue that the widely used Rashba model does in fact provide a finite anomalous Hall dc conductivity in the metallic regime (in the noncrossing approximation) if SOC-corrections to the velocity and position operators are properly taken into account. While we focus on the response of the charge current to the electric field, linear response theories of other SOC-related effects should be modified similarly.