Topological Transitions Induced by Electron Coupling to an SU(2) Gauge Field: A Yang–Mills Framework for Quantum Materials

This paper develops a theoretical framework for studying topological transitions generated by the coupling between electrons and an emergent or externally engineered SU(2) gauge field. The electronic spinor is treated as a matter field transforming under a local non-Abelian symmetry, while spin–orbit coupling, magnetic textures, strain, interfacial exchange, and multiband hybridization generateeffective gauge potentials in spin space. In contrast to an Abelian U(1) field, anSU(2) field contains noncommuting components and therefore possesses a nonlinear field-strength contribution.The proposed formulation combines a gauge-covariant electronic Hamiltonian,a Yang–Mills action, a Higgs-like order-parameter sector, and topological responseterms. Topological transitions are characterized through the closing and reopening of the electronic gap, changes in Chern and Z2 invariants, Wilson-loop evolution,singularities of the non-Abelian Berry curvature, and changes in the second Cherncharacter in an extended parameter space.A set of new hypotheses is proposed in which the noncommutativity of the gaugepotential becomes an independent control parameter for topological engineering.The framework is relevant to spin–orbit-coupled semiconductors, topological insulators, magnetic textures, moiré materials, synthetic dimensions, photonic systems,ultracold atoms, and topological superconducting platforms.