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

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.