Quantum-classical correspondence for spins at finite temperatures with application to Monte Carlo simulations

TL;DR

The paper addresses how to faithfully map the thermodynamics of quantum spins at finite temperature onto a classical description. It proves that in the large-$S$ limit the quantum partition function $Z_Q(S)$ equals the classical partition function $Z_C(S_C)$ with $S_C^2=S(S+1)$ up to corrections of order $1/S_C^2$, and that residual quantum effects form a systematic expansion in $1/[S(S+1)]$. The authors derive the effective classical Hamiltonian parameters $J_C=J S(S+1)$, $H_C=g\mu_B H\sqrt{S(S+1)}$, and a quantum-renormalized anisotropy $D_C=D[S(S+1)-3/4]$, enabling classical Monte Carlo (CMC) simulations to predict transition temperatures. They validate the approach by computing $T_c$ for a suite of materials (e.g., MnF$_2$, MnTe, FePS$_3$, CrI$_3$) and demonstrating good agreement with experimental data, thereby providing a rigorous basis for using classical simulations to study realistic quantum spin systems at finite temperatures. The work also discusses limitations near and below the ordering temperature and in metals, outlining how the framework can guide parameter selection and quantify quantum corrections in practical materials modeling.

Abstract

We consider quantum-to-classical mapping for an arbitrary system of interacting spins at finite temperatures. We prove that, in the large-$S$ limit, the asymptotic form of the partition function coincides with that of a classical model for spins of length $S_C=\sqrt{S(S+1)}$. Quantum corrections to the leading term form a series in powers of $1/[S(S+1)]$. This representation provides a rigorous basis for classical modeling of realistic magnetic Hamiltonians. As an application, the classical Monte Carlo simulations are performed to compute transition temperatures for several topical materials with known interaction parameters, including MnF$_2$, MnTe, Rb$_2$MnF$_4$, MnPSe$_3$, FePS$_3$, FePSe$_3$, CoPS$_3$, CrSBr, and CrI$_3$. The resulting transition temperatures show good agreement with experimental data.

Figures (1)

• Figure 1: Fourth-order cumulants versus temperature for the spin models of MnF$_2$ (a) and FePSe$_3$ (b). Phase transition temperatures are indicated by dashed lines. If not shown, the error bars are smaller than the symbol sizes.