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Cluster mean-field theory study of $J_{1}-J_{2}$ Heisenberg model on a square lattice

We study the spin-1/2 $J_{1}$-$J_{2}$ Heisenberg model on a square lattice using the cluster mean-field theory. We find a rapid convergence of phase boundaries with increasing cluster size. By extrapolating the cluster size $L$ to infinity, we obtain accurate phase boundaries $J_{2}^{c1} \approx 0.42$ (between the N$\acute{e}$el antiferromagnetic phase and nonmagnetic phase), and $J_{2}^{c2} \approx 0.59$ (between nonmagnetic phase and the collinear antiferromagnetic phase). The transitions are identified unambiguously as second order at $J_{2}^{c1}$ and first order at $J_{2}^{c2}$. At finite temperature, we present a complete phase diagram with stable, meta-stable and unstable states near $J_{2}^{c2}$, being relevant to that of the anisotropic $J_1-J_2$ model. The uniform as well as staggered magnetic susceptibilities are also discussed.