Quantum-Classical Transition of the Escape Rate of a Uniaxial Spin System in an Arbitrarily Directed Field

TL;DR

This paper investigates the quantum-classical transition of the escape rate in a uniaxial spin system under arbitrary magnetic fields, revealing the order of transition and critical field values relevant for experiments.

Contribution

It provides a detailed analysis of the transition order and critical field boundaries for quantum escape rates in a large-spin model with arbitrary field orientation.

Findings

First-order transition with discontinuous dΓ/dT below phase boundary.

Second-order transition above the phase boundary.

Critical field h_{xc} depends on bias and is approximately 0.2345 δ^{3/2} in the biased limit.

Abstract

The escape rate \Gamma of the large-spin model described by the Hamiltonian H = -DS_z^2 - H_zS_z - H_xS_x is investigated with the help of the mapping onto a particle moving in a double-well potential U(x). The transition-state method yields $\Gamma$ in the moderate-damping case as a Boltzmann average of the quantum transition probabilities. We have shown that the transition from the classical to quantum regimes with lowering temperature is of the first order (d\Gamma/dT discontinuous at the transition temperature T_0) for h_x below the phase boundary line h_x=h_{xc}(h_z), where h_{x,z}\equiv H_{x,z}/(2SD), and of the second order above this line. In the unbiased case (H_z=0) the result is h_{xc}(0)=1/4, i.e., one fourth of the metastability boundary h_{xm}=1, at which the barrier disappears. In the strongly biased limit \delta\equiv 1-h_z << 1, one has h_{xc} \cong (2/3)^{3/4}(\sqrt{3}-\sqrt{2})\delta^{3/2}\cong 0.2345 \delta^{3/2}, which is about one half of the boundary value h_{xm} \cong (2\delta/3)^{3/2} \cong 0.5443 \delta^{3/2}.The latter case is relevant for experiments on small magnetic particles, where the barrier should be lowered to achieve measurable quantum escape rates.