Monte Carlo study of the classical antiferromagnetic $J_1$-$J_2$-$J_3$ Heisenberg model on a simple cubic lattice

TL;DR

This work investigates the thermodynamics of the classical $J_1$-$J_2$-$J_3$ Heisenberg model on a simple cubic lattice to understand frustration-driven phase behavior. It employs extensive Monte Carlo simulations with a heat-bath algorithm and Binder cumulant analysis, comparing results to mean-field and Tyablikov approximations across broad exchange ratios. A key finding is a minimum in the Neel temperature at $J_2^{(c)}=J_3+1/4$, with the frustration ratio $f=|\theta|/T_N$ increasing with $J_3$; Binder analysis reveals both second- and first-order transitions depending on the ordering wave vector, and a critical-like $ u$ for certain regimes. The results highlight the crucial role of fluctuations in frustrated magnets and bear relevance to antiferromagnetic perovskites such as CaMnO$_3$ and HgMnO$_3$, informing interpretation of experimental thermodynamics and guiding theoretical approaches beyond mean-field theory.

Abstract

An extensive Monte Carlo study of the classical Heisenberg model on a simple cubic lattice with antiferromagnetic exchange interactions $J_n$ between the first, second, and third neighbors is performed in a broad region of $J_2 / J_1$, $J_3 / J_1$ ratios, and temperature. The character of the phase transitions is analyzed via the Binder cumulant method. The Neel temperature $T_{\mathrm{N}}$ and the frustration parameter (the ratio $f= |θ|/T_{\mathrm{N}}$, $θ$ being the Curie-Weiss temperature) are calculated. A comparison with the Tyablikov approximation is carried out. The strength of the frustration effects is explored. Possible applications to antiferromagnetic perovskites, such as CaMnO$_3$ and HgMnO$_3$, are discussed.

Figures (9)

• Figure 1: Illustration of the exchange interaction parameters ($J_1, J_2, J_3$) used in the model calculations.
• Figure 2: Average absolute value of the total $\mathbf{Q}_1$-vector magnetic moment $| \mathbf{M}_{\mathbf{Q}_1} |$ as a function of temperature for $J_2 = 0.391$, $J_3 = 0.05$ and $L = 10$ (left panel); its derivative on temperature (middle panel). Temperature dependence of the specific heat for the same parameters (right panel).
• Figure 3: The Neel temperature as a function of $J_2$ calculated in the Monte Carlo simulation for $J_3 = 0.05$ and lattice sizes $L = 10$, 20, 30, 40 (empty markers connected with solid lines) and compared with the Tyablikov approximation, $L = \infty$ (bottom dotted line) and the mean field theory (upper dash-dotted line). Arrows point to Neel temperatures for $J_2$ = 0.95 (point A), 0.393 (point B), 0.3 = $J_2^{(\mathrm{c})}$ (point C), and 0.31 (point D).
• Figure 4: Temperature dependence of the Binder cumulant $U_{\mathbf{Q}_0}$ in the vicinity of the point A ($J_2 = 0.095$, $J_3 = 0.05$) on Fig. \ref{['tn_L10-40']} for different lattice sizes $L$. $T_{\mathrm{cross}}$ = 1.2146. The inset shows $U_{\mathbf{Q}_0}$ as a function of $x = (T/T_{\mathrm{cross}}-1)L^{1/\nu}$ with $\nu=0.7$.
• Figure 5: Temperature dependence of the Binder cumulant $U_{\mathbf{Q}_1}$ in the vicinity of the point B ($J_2 = 0.393$, $J_3 = 0.05$) on Fig. \ref{['tn_L10-40']} for different lattice sizes $L$. The inset shows the positions of the maximums of $U_{\mathbf{Q}_1}$ as a function of $L$.