Quantum Tunneling and Quantum-Classical Transitions in Large Spin   Systems
  We have studied quantum tunneling of large spins in a biaxial spin system and in a single-axial spin system with a transverse magnetic field. The asymptotically exact eigenvalues and eigenstates of the spin systems are obtained by solving the Mathieu equation. When the height $4q$ of energy barrier formed by the anisotropy and the magnetic field exceeds some critical values $4q_{1T}(r)$, the ground states and the excited states split due to tunneling of large spins. We also have presented the phase diagram of these spin systems in the extremely weak spin-phonon coupling limit. The crossover from thermal to quantum regime is second order. With further decreasing temperature, the first order phase transition occurs from a thermally-assisted resonant tunneling to thermal regime as $q<q_{1T}(s)$. Our theory agrees with recent experimental observations well. 
Bambi Hu, De-gang Zhang, and Bo-zang Li (Center for Nonlinear Studies,   Hong Kong Baptist University)