Gauge-Covariant Non-Abelian Density Functional Theory for Electronic Materials: An SU(2) Yang–Mills–Kohn–Sham Framework

Conventional density functional theory describes the ground-state propertiesof interacting electrons primarily through the scalar electron density n(r). Thisformulation becomes insufficient when the electronic system contains strong spin–orbit coupling, non-collinear magnetism, spin currents, orbital magnetism, or spatiallyvarying internal degrees of freedom.This paper proposes a generalized non-Abelian density functional theory in whichthe electronic state is represented by the setD(r) = {n(r), ji(r), ma(r), Jai(r)} , (1)where n is the particle density, jiis the Abelian particle-current density, mais theinternal spin density, and Jaiis the non-Abelian current density associated with the generators of SU(2).The theory introduces a matrix-valued gauge potentialAµ(r) = Aaµ(r)Ta , (2) with Ta = σa/2, and a non-Abelian field tensorFµν = ∂µAν − ∂νAµ − ig[Aµ, Aν]. (3)The proposed energy functional contains the non-interacting kinetic energy,Hartree energy, exchange-correlation energy, external electromagnetic coupling, Yang–Mills field energy, and a gauge-consistency constraint. The principal hypothesis isthat gauge-invariant combinations of ma, Jai , and Faµν can provide a systematicdescription of spin–orbit-coupled and non-collinear materials beyond conventionalspin-density functional theory