Oscillation of the tunnel splitting in nanospin systems within the particle mapping formalism

The oscillation of tunnel splitting in the biaxial spin system within magnetic field along the anisotropy axis is analyzed within the particle mapping approach, rather than in the (\theta-\phi) spin coherent-state representation. In our mapping procedure, the spin system is transformed into a particle moving in the restricted $S^1$ geometry whose wave function subjects to the boundary condition involving additional phase shift. We obtain the new topological phase that plays the same role as the Wess-Zumino action in spin coherent-state representation. Considering the interference of two possible trajectories, instanton and anti-instanton, we get the identical condition for the field at which tunneling is quenched, with the previous result within spin coherent-state representation.