Nonadiabatic noncyclic geometric phase of a spin-12particle subject to an arbitrary magnetic field

We derive a formula of the nonadiabatic noncyclic Pancharatnam phase for a quantum spin-$\frac{1}{2}$ particle subject to an arbitrary magnetic field. The formula is applied to three specific kinds of magneic fields. (i) For an orientated magnetic field, the Pancharatnam phase is derived exactly. (ii) For a rotating magnetic field, the evolution equation is solved analytically. The Aharonov-Anandan phase is obtained exactly and the Pancharatnam phase is computed numerically. (iii) We propose a kind of topological transition in one-dimensional mesoscopic ring subject to an in-plane magnetic field, and then address the nonadiabatic noncyclic effect on this phenomenon.