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Macroscopic quantum tunneling and quantum-classical phase transitions of the escape rate in large spin systems

This article presents a review on the theoretical and the experimental developments on macroscopic quantum tunneling and phase transition of the escape rate in spin systems. We present the basic ideas with simplified calculations so that it is readable to both specialists and nonspecialists in this area of research. A brief derivation of the path integral formulation of quantum mechanics in its original form using the orthonormal position and momentum basis is reviewed. For spin systems such as single molecule magnets, the formulation of path integral requires the use of non-orthonormal spin coherent state in $(2s+1)$ dimensional Hilbert space, the coordinate independent and the coordinate dependent form of the spin coherent state path integral is derived. These two forms of spin coherent state path integral are applied to the tunneling of single molecule magnets through its magnetic anisotropy barrier. Most experimental and numerical results are presented. The suppression of tunneling for half-odd integer spin (spin-parity effect) at zero magnetic field is derived from both forms, which shows that this result (spin-parity effect) is independent of the coordinate. At nonzero magnetic field we present both the experimental and the theoretical results of the oscillation of tunneling splitting as a function of the applied magnetic field applied along the spin hard anisotropy axis direction. The experimental and the theoretical results of the tunneling in antiferromagnetic exchange coupled dimer model are also reviewed. As the spin coherent state path integral formalism is a semi-classical method, an alternative exact mapping of a spin system to a particle in a potential field (effective potential method) is derived. This effective potential method allows for the investigation of phase transition of the escape rate in spin systems.