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

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>∂</a:mi><a:mi>H</a:mi><a:mo>/</a:mo><a:mi>∂</a:mi><a:mi mathvariant="bold-italic">p</a:mi></a:mrow></a:math>, where <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>H</c:mi></c:math> is the Hamiltonian and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi mathvariant="bold-italic">p</d:mi></d:math> 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-Středa 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. Published by the American Physical Society 2024