Phase transitions in a frustrated Ising antiferromagnet on a square lattice

The phase transitions occurring in the frustrated Ising square antiferromagnet with first- $\left(J_1<0\right)$ and second-nearest-neighbor $\left(J_2<0\right)$ interactions are studied within the framework of the effective-field theory with correlations based on the different cluster sizes and for a wide range of $R=J_2/\left|J_1\right|$. Despite the simplicity of the model, it has proved difficult to precisely determine the order of the phase transitions. In contrast to the previous effective-field study, we have found a first-order transition line in the region close to $R=-0.5$ not only between the superantiferromagnetic and paramagnetic $\left(R<-0.5\right)$ but also between antiferromagnetic and paramagnetic $\left(R>-0.5\right)$ phases.

Figure 1

Ground-state configurations of the $J_1\text{−}{J}_2$ Ising model on the square lattice showing (a) aniferromagnetic and (b) superantiferromagnetic states for the two-spin cluster approximation defined by the Hamiltonian Eqs. (3) and (11), respectively. Two sublattices are marked by black and white circles.

Figure 2

Phase diagram in the coupling-temperature plane for the $J_1\text{−}{J}_2$ Ising model on the square lattice based on the two-spin cluster. Solid and dashed lines indicate second- and first-order transitions, respectively, while the black circles denote the position of a tricritical point. $\mathrm{AF}$ and $\mathrm{SAF}$ are the antiferromagnetic and superantiferromagnetic ordered phases, respectively, and $P$ is the paramagnetic phase. The inset shows phase diagram when the two-spin cluster is chosen in the vertical direction.

Figure 3

Phase diagram in the coupling-temperature plane for the $J_1\text{−}{J}_2$ Ising model on the square lattice based on the nine-spin cluster. Solid and dashed lines indicate second- and first-order transitions, respectively, while the black circles denote the position of a tricritical point. $\mathrm{AF}$ and $\mathrm{SAF}$ are the antiferromagnetic and superantiferromagnetic ordered phases, respectively, and $P$ is the paramagnetic phase.

Figure 4

Two options of the six-spin cluster with spins $s_i,s_j,s_k,s_l,s_m$, and $s_n$ for the superantiferromagnetic arrangement on the square lattice: (a) in the horizontal direction and (b) in the vertical direction. Two sublattices are marked by black and white circles. Each cluster has 10 nn and 14 nnn spins.

Figure 5

The same as described in the legend of Fig. 2, but for the six-spin cluster.

Figure 6

Temperature dependencies of the order parameters of $m_{\mathrm{AF}}$ in (a) and $m_{\mathrm{SAF}}$ in (b) for the $J_1\text{−}{J}_2$ Ising model on the square lattice based on the nine-spin cluster, when the frustration parameter $R$ is changed. The dashed lines indicate the first-order transitions.