Symmetrized decomposition of the Kubo-Bastin formula

The Smrcka-Streda version of Kubo's linear response formula is widely used in the literature to compute nonequilibrium transport properties of heterostructures. It is particularly useful for the evaluation of intrinsic transport properties associated with the Berry curvature of the Bloch states, such as anomalous and spin Hall currents as well as the dampinglike component of the spin-orbit torque. Here we demonstrate in a very general way that the widely used decomposition of the Kubo-Bastin formula introduced by Smrcka and Streda contains an overlap, which has lead to widespread confusion in the literature regarding the Fermi surface and Fermi sea contributions. To remedy this pathology, we propose a decomposition of the Kubo-Bastin formula based on the permutation properties of the correlation function and derive a set of formulas, without an overlap, that provides direct access to the transport effects of interest. We apply these formulas to selected cases and demonstrate that the Fermi sea and Fermi surface contributions can be uniquely addressed with our symmetrized approach.