A usual method for introducing finite temperature effects in condensed matter physics is to introduce a thermal noise in the equations of motion; hence the physically meaningful quantities are the moments of the statistical distribution of the dynamical variables. For systems described in terms of position and momentum, such as in molecular dynamics, the temperature can be understood as the fluctuations around the equilibrium. A fundamental issue with this approach is that when going towards zero temperature, one can, in principle, compute both position and momenta to arbitrary precision, hence violating the Heisenberg uncertainty principle. However, using path integrals, one can recover an effective classical system with correct quantum expectation values at low-temperature. These methods are known as Path-Integral Molecular Dynamics.
We build a similar approach for a quantum spin in a magnetic field, which, while being a classical approach, can account for effects due to the underlying spin quantization. We begin by expressing the spin algebra in the spin coherent states basis, thus enabling the transition from a discrete to a continuous quantum description. We then rewrite its quantum mechanics in the framework of Path Integrals, hence yielding a partition function as an integral rather than a sum. We approximate the resulting partition function in the high-temperature limit by recasting the quantum matrix elements into an effective exponential form, thus deriving an effective, equivalent classical Hamiltonian. In this form, the quantized nature of spin takes the form of an effective anisotropy term added to the applied field. We then proceed to solve this effective classical model by implementing an effective field derived from this classical Hamiltonian into an atomistic model from which we derive thermodynamic observables and show that our effective model produces accurate quantum expectation values within its relevant temperature range.
