Abstract
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 $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{SU}(2)$ 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.
Highlights
In the last few decades, density-functional theoryDFT ͑Ref. 1͒ has grown to be a widely used method for studying the ground-state properties of interacting many-electron systems and the range of its applications has been expanding
The basic variable of that theoryin addition to the usual particle and spin densitiesis the paramagnetic current density jpr, which has the following advantages upon the “physical current density” jr: ͑iit has no explicit dependence on the external vector potential—a property it shares with previous relativistic formulations[5,6] andiiit does not vanish in the limit of uniform density and magnetic field, which is vital to the construction of a local density approximationLDA
An interesting feature of the present construction is that the use of the paramagnetic spin-current as basic variables is mandatory: without them we could not define the link operators Lvrsee Eq ͑25͔͒, which are vital to the definition of the invariant vorticity
Summary
In the last few decades, density-functional theoryDFT ͑Ref. 1͒ has grown to be a widely used method for studying the ground-state properties of interacting many-electron systems and the range of its applications has been expanding. While this choice is not uniqueany gauge-invariant field which is in one-to-one correspondence to the vorticity would be in principle admissible, it was shown to be the most natural one, in the sense of leading to a local description of the effect of the magnetic field in the quasihomogeneous limit In extending their theory to spin polarized systems,[3] VR proved the analog of the Hohenberg-Kohn theorem for paramagnetic spin-currents, and observed that the xc energy Exc— a functional of paramagnetic spin-currents— should be invariant under local U1͒ ϫ SU2͒ gauge transformations. We will show below that the quantity N−1Jp, where Jp denotes the paramagnetic particle/spin-currents and N is the particle/spin densityagain, precise definitions will be given below, behaves under U1͒ ϫ SU2͒ gauge transformations precisely like the gauge potential A This suggests a path to constructing a gauge-invariant xc energy functional Exc. Let us introduce the “SU2͒-invariant vorticity”.
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