Geometric phase and topological selection rules in spin tunneling problems

The topological interference for resonant tunneling in a single spin system with m-fold rotation symmetry is studied by the spin-coherent-state path integral approach under semiclassical approximation. It is found that the interference phase between different but symmetry-related tunneling paths is directly related to the quantization rule of the excited states. The interference phase can be geometrically interpreted as the area on the unit sphere enclosed by the tunneling paths and the quantized classical orbits, topological selection rules for both tunneling at excited states and resonant tunneling are thus found. The topological selection rule for resonant tunneling agrees with the experimental result on Fe-8 and the numerical results of the Mn-12 system.