Nonadiabatic noncyclic geometric phase and ensemble average spectrum of   conductance in disordered mesoscopic rings with spin-orbit coupling
  We generalize Yang's theory from the U(1) gauge field to the non-Abelian $U(1)\times SU(2)_{spin}$ gauge field. Based on this generalization and taking into account the geometric Pancharatnam phase as well as an effective Aharonov-Bohm (AB) phase in nonadiabatic noncyclic transport, we calculate the ensemble average Fourier spectrum of the conductance in disordered mesoscopic rings connected to two leads. Our theory can explain the experimental results reported by Morpurgo {\sl et al.} [Phys. Rev. Lett. {\bf 80}, 1050 (1998)] satisfactorily. In particular, we clarify that the experimentally observed splitting, as well as some structure on the sides of the main peak in the ensemble average Fourier spectrum, stem from the nonadiabatic noncyclic Pancharatnam phase and the effective AB phase, both being dependent on spin-orbit coupling. 
Shi-Liang Zhu and Z. D. Wang