Quantum-classical escape-rate transition of a biaxial spin system with a longitudinal field: A perturbative approach

The quantum-classical transition of the escape rate of the spin model $\mathcal{H}=-{\mathrm{DS}}_z^2-H_z{S}_z{+BS}_x^2$ is investigated by a perturbative approach with respect to B [D. A. Garanin, J. Phys A 24, L61 (1991)]. The transition is first order for $B<\mathrm{}{B}_c{(H}_z),$ the boundary line going to zero as $B_c/D\sim 1-H_z/\left(2SD\right)\mathrm{}$ in the strongly biased limit. The range of the first-order transition is thus larger than for the model $\mathcal{H}=-{\mathrm{DS}}_z^2-H_z{S}_z-H_x{S}_x$ studied earlier, where in the strongly biased case $H_{\mathrm{xc}}/\left(2SD\right)\mathrm{}\sim [1-H_z/(2SD)\mathrm{}{]}^{3/2}.$ The temperature of the quantum-classical transition, $T_0,$ behaves linearly in the strongly biased case for both models: $T_0\sim 2SD-H_z.$