Gauge-invariant formulation of spin-current density-functional theory

Spin-currents and non-Abelian gauge potentials in electronic systems can be treated by spin-current density-functional theory, whose main input is the exchange-correlation (xc) energy expressed as a functional of spin-currents. Constructing a functional of spin-currents that is invariant under $\text{U}\left(1\right)\times \text{SU}\left(2\right)$ transformations is a long-standing challenge. We solve the problem by expressing the energy as a functional of a variable we call “invariant vorticity.” As an illustration we construct the xc energy functional for a two-dimensional electron gas with linear spin-orbit coupling and show that it is proportional to the fourth power of the spin-current.

Figure 1

Two admissible choices of paths connecting the origin to two points $\mathbf{r}={\mathbf{r}}_1$ and ${\mathbf{r}}_2$ for the evaluation of the path-ordered integral in Eq. (13). The solid path is a straight radial line, the dashed path proceeds along the $x$ axis first, and then along the $y$ axis.

Figure 2

Excess exchange-correlation energy $\Delta E_{xc}$ (filled circles) and excess exchange energy $\Delta E_x$ (empty rectangles) of a two-dimensional electron gas at Wigner-Seitz radius $r_s=1$ vs Rashba spin-orbit coupling constant $\overline{\alpha }=\alpha /v_{\text{F}}$ ($v_{\text{F}}=\sqrt{2}/r_s$ is the Fermi velocity) in units of $E_{xc}^0=-0.8$ and $E_x^0=-0.6$ (in the atomic units) respectively. Inset: same as in the main figure but in units of $E_{xc}^0{\overline{\alpha }}^4$ and $E_x^0{\overline{\alpha }}^4$, respectively for the exchange-correlation and exchange energies.

Figure 3

${\lambda }_{xc}$ of a two-dimensional electron gas with Rashba spin-orbit coupling vs Wigner-Seitz radius $r_s$, for varying Rashba coupling constant $\overline{\alpha }=\alpha /v_{\text{F}}$ $\left(v_{\text{F}}=\sqrt{2}/r_s\right)$. The limiting behavior at $\overline{\alpha }=0$ (solid line) is obtained by extrapolation.