The Phase Transition of Square Lattice Antiferromagnets at Finite Temperature

The magnetic properties of the two-dimensional $J_1-J_2$ model with both exchanges $J_1$ and $J_2$ being antiferromagnetic and a single-ion anisotropy at nonzero temperature are investigated. As $J_2\mathrm{/}{J}_1<1\mathrm{/}2$ ($>1\mathrm{/}2$), only the N'{e}el (collinear) state exists. When $J_2\mathrm{/}{J}_1=1\mathrm{/}2$, both the N'{e}el and collinear states can exist and have the same N'{e}el temperature. The calculated free energies show that there can occur a phase transition between the two states below the N'{e}el point when the single-ion anisotropy is strong enough. It is a first-order transition at nonzero temperature. It is possible that the doping in real materials can modify the ratio of $J_2\mathrm{/}{J}_1$ to reach $1\mathrm{/}2$ so as to implement the phase transition.