2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/150891We present both the gauge theoretic description and the numerical calculations of the Berry phases with the real eigenstates, involving one with a many-body system as a background and the other with no such background. We demonstrate that for the former the sign of the Berry phase factor for a spin $\f{1}{2}$ particle (hole) coupled to a slow subsystem (phonon) depends on both the strength of electron correlations and the characteristics of the closed paths, unlike the cases for the latter.4 pages RevTex, uses epsf,twocolumn, 3 embedded postscript figures, submitted to PRL, acknowledgement appendedQuantum PhysicsCondensed Mattertext