Geometrical description of SU(2) Berry phases

The differential geometric background for the occurrence of SU(2) Berry phases in Fermi systems with time-reversal invariance is investigated. We explicitly construct the natural geometric structures on ${\mathrm{ssHssP}}^n$ (the quaternionic projective space) responsible for nontrivial holonomy. For the special case n=1, ${\mathrm{ssHssP}}^1$∼${\mathrm{ssS}}^4$, the connection that governs the time evolution corresponds to the SU(2) Yang-Mills instanton with self-dual curvature, giving the nonadiabatic generalization of the spin (3/2 case of Avron et al. [Phys. Rev. Lett. 61, 1329 (1988)].