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.

Figure 1

Energy dependence of (a) longitudinal conductivity, (b) fieldlike torque, (c) transverse conductivity, and (d) dampinglike torque in the two-dimensional magnetic Rashba gas. The solid red (blue) curve refers to the ${\mathcal{A}}_{\text{I}}$ (${\mathcal{A}}_{\text{II}}$) contribution, whereas the black curve is their sum ${\mathcal{A}}_{\text{I}}+{\mathcal{A}}_{\text{II}}$. The black (red) dots refer to ${\mathcal{A}}_{\mathrm{surf}}$ (${\mathcal{A}}_{\mathrm{sea}}$). The inset of (b) shows the band structure of the magnetic Rashba gas. The dashed horizontal line indicates the position of the avoided band crossing and the dotted line stands for the maximum energy taken in this calculation. The conductivity is in ${\mathrm{\Omega }}^{-1}{\text{m}}^{-1}$ and the spin conductivity is in $(\hslash /2e){\mathrm{\Omega }}^{-1}{\text{m}}^{-1}$.

Figure 2

Energy dependence of (a) longitudinal conductivity and (b) fieldlike torque in the multiorbital transition metal bilayer model. The solid red curve refers to the ${\mathcal{A}}_{\text{I}}$ and the black dots refer to ${\mathcal{A}}_{\mathrm{surf}}$. The conductivity is in ${\mathrm{\Omega }}^{-1}{\text{m}}^{-1}$ and the spin conductivity is in $(\hslash /2e){\mathrm{\Omega }}^{-1}{\text{m}}^{-1}$.

Figure 3

Energy dependence of (a) and (b) transverse conductivity and (c) and (d) dampinglike torque in the multiorbital transition metal bilayer model. The solid red (blue) curve refers to the ${\mathcal{A}}_{\text{I}}$ (${\mathcal{A}}_{\text{II}}$) contribution, the black curve is their sum ${\mathcal{A}}_{\text{I}}+{\mathcal{A}}_{\text{II}}$, and the red dots refer to ${\mathcal{A}}_{\mathrm{sea}}$. The conductivity is in ${\mathrm{\Omega }}^{-1}{\text{m}}^{-1}$ and the spin conductivity is in $(\hslash /2e){\mathrm{\Omega }}^{-1}{\text{m}}^{-1}$.

Figure 4

Angular dependence of (a) in-plane and (b) out-of-plane spin Hall effect in the noncollinear antiferromagnetic Kagome lattice model. The solid red (blue) curve refers to the ${\mathcal{A}}_{\text{I}}$ (${\mathcal{A}}_{\text{II}}$) contribution, the black curve is their sum ${\mathcal{A}}_{\text{I}}+{\mathcal{A}}_{\text{II}}$, and the black (red) dots refer to ${\mathcal{A}}_{\mathrm{surf}}$ (${\mathcal{A}}_{\mathrm{sea}}$). The inset displays the angle made by the applied electric field with respect to the crystal axes. The spin conductivity is in $(\hslash /2e){\mathrm{\Omega }}^{-1}$.