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We revise the Kubo formula for the electric dc conductivity in the presenceof spin-orbit coupling (SOC). We discover that each velocity operator thatenters this formula differs from $\partial H/\partial \boldsymbol p$, where $H$is the Hamiltonian and $\boldsymbol p$ is the canonical momentum. Moreover, wefind an additional contribution to the Hall dc conductivity from noncommutingcoordinates that is missing in the conventional Kubo-Streda formula. Thiscontribution originates from the "electron-positron" matrix elements of thevelocity and position operators. We argue that the widely used Rashba modeldoes in fact provide a finite anomalous Hall dc conductivity in the metallicregime (in the noncrossing approximation) if SOC-corrections to the velocityand position operators are properly taken into account. While we focus on theresponse of the charge current to the electric field, linear response theoriesof other SOC-related effects should be modified similarly.

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