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We find two magnetically ordered phases, one with N\\' eel order, for $\\alpha \\leq 0.4$, and another with collinear order, for $\\alpha\\geq 0.6$, separated by a nonmagnetic region, for $0.4\\leq \\alpha \\leq 0.6$, where a gapped spin liquid is found. The transition at $\\alpha=0.4$ is of the second order while the one at $\\alpha=0.6$ is of the first order and the spin gaps cross at $\\alpha=0.5$. 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B. Silva Neto, T. P. Cysne","submitted_at":"2015-04-06T17:42:45Z","abstract_excerpt":"We obtain the complete phase diagram of the antiferromagnetic $J_{1}$-$J_{2}$ model, $0\\leq \\alpha = J_2/J1 \\leq 1$, within the framework of the $O(N)$ nonlinear sigma model. We find two magnetically ordered phases, one with N\\' eel order, for $\\alpha \\leq 0.4$, and another with collinear order, for $\\alpha\\geq 0.6$, separated by a nonmagnetic region, for $0.4\\leq \\alpha \\leq 0.6$, where a gapped spin liquid is found. The transition at $\\alpha=0.4$ is of the second order while the one at $\\alpha=0.6$ is of the first order and the spin gaps cross at $\\alpha=0.5$. 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