Magnetic quantum phase transitions of the antiferromagnetic J₁-J₂ Heisenberg model

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 $\alpha=0.5$. Our results are exact at $N\rightarrow\infty$ and agree with numerical results from different methods.