Instantons and phase transition in quantum interference

In the biaxial nanospin system, quantum tunnelling of the magnetization vector oscillates with increasing magnetic field or temperature. When the number of instanons is equal to the number of anti-instantons, the annihilation of instanton-anti-instanton pair results in a vanished Berry curvature, it leads to the topological quenching point. When the number of instantons and anti-instantons is not equal, berry curvature is nonzero, this suggests that there are extra instantons that revive the tunnelling splitting. So the generation (annihilation) of instanton induced the revival (quenching) of tunnel splitting. This generation or annihilation is accompanied by different phase transitions which can be distinguished by (H/Hc)/((1-K2)/K1). When the system slides from the phase of H/Hc < 1 - K2/K1 into the phase of H/Hc > 1 - K2/K1, an instanton will be generated (annihilated), which revives (quenches) the quantum tunnelling. This conclusion also holds when phase transition occurs in the opposite direction, i.e., from H/Hc > 1 - K2/K1 to H/Hc < 1 - K2/K1. So there is a phase transition that occurs alternatively in the magnetic molecule system when the topological quenching of the tunnel splitting occurs quasi-periodically