Periodic Instanton and Phase Transition in Quantum Tunneling of Spin Systems

The quantum-classical transitions of the escape rates in a uniaxial spin model relevant to the molecular magnet Mn$_{12}$Ac and a biaxial anisotropic ferromagnetic particle are investigated by applying the periodic instanton method. The effective free energies are expanded around the top of the potential barrier in analogy to Landau theory of phase transitions. We show that the first-order transitions occur below the critical external magnetic field $h_x = 1/4$ for the uniaxial spin model and beyond the critical anisotropy constant ratio $\lambda = 1/2$ for the biaxial ferromagnetic grains, which are in good agreement with earlier works.