DFT–Yang–Mills Formulation of Noncollinear Magnetism in Strong-Exchange Materials

Noncollinear magnetism in strong-exchange materials is governed by the competition between isotropic exchange, anisotropic exchange, spin–orbit coupling, geometric frustration, orbital hybridization, and itinerant-electron motion. Conventional density functional theory can describe noncollinear magnetic states throughspin-density matrices and spinor Kohn–Sham equations. However, a gauge-theoreticalformulation provides a more unified description of spatially varying spin orientations, chiral exchange, spin transport, and emergent electromagnetic responses.This work proposes a DFT–Yang–Mills model in which the local spin frame isrepresented by an element of the non-Abelian group SU(2). A spatially varyingmagnetization generates an emergent gauge potential,Aµ(r) = −iU†(r)∂µU(r) = Aaµ(r)σa2. (1)where U(r) is a local spin rotation and σa are the Pauli matrices. The correspondingnon-Abelian field strength isFµν = ∂µAν − ∂νAµ − ig[Aµ, Aν]. (2)The proposed theory treats the gauge field as an emergent internal structure generated by local spin geometry, spin–orbit coupling, crystal symmetry, and electronichybridization. The model combines self-consistent noncollinear density functionaltheory with an effective Yang–Mills energy functional. This allows the exchangetensor, Dzyaloshinskii–Moriya interaction, spin stiffness, scalar spin chirality, andmagnetic topological charge to be described within a common mathematical framework.The main hypothesis is that exchange interactions in strong-exchange materialscan be expressed as the gauge-covariant electronic response to spatial variations ofthe local spin frame. The approach is applicable to spin spirals, frustrated antiferromagnets, chiral magnets, magnetic skyrmions, noncollinear superexchange, andmagnetoelastic coupling.