Wilson Holonomy and Spectral Monodromy in Spin-Orbit Rings: Effective Gauge Connections and Loop Observables

Abstract: A spin-orbit Hamiltonian with an effective gauge structure carries two distinct loop objects that are routinely conflated: an energy-independent Wilson holonomy, which organizes interference and internal spin transport, and an energy-dependent monodromy, which quantizes the spectrum. We show that cleanly separating these objects supplies a precise, computable bridge between the loop/holonomy representation of gauge theories and condensed-matter spin-orbit transport. The construction maps a spin-orbit Hamiltonian to an effective $U(1)$ plus internal non-Abelian connection, reduces it to a first-order transport problem, and reads physical predictions from holonomy, monodromy, curvature, and eigenphase data. Two rings make the separation explicit. For a Dirac (graphene) ring with Rashba coupling and Aharonov-Bohm flux, the total holonomy factorizes exactly into a commuting $U(1)$ flux phase times an internal spin/pseudospin holonomy, and the spectrum follows from a holonomy-eigenvalue condition. For a Rashba-Dresselhaus ring, the internal $SU(2)$ transport is genuinely non-Abelian away from the $α=\pmβ$ pure-gauge locus, where curvature controls path ordering; spectral quantization then requires an explicit first-order reduction obtained by phase-space doubling of the second-order Schrödinger problem. A non-Abelian Stokes formulation and Magnus expansion serve as ordering diagnostics rather than spectral tools. Spin-network ideas enter only as historical geometric motivation, not as a dynamical import into spintronics.