# Frustrated Ising model

Consider a 2D Ising model with nearest neighbour, and second nearest neighbour interactions

$$\mathcal{H}= -J_1\sum_{\langle ij\rangle}\sigma_i \sigma_j-J_2\sum_{\langle\langle ik\rangle\rangle}\sigma_i \sigma_k$$

where $$\sigma =\pm 1$$. And $$|J_1|=|J_2|$$

For $$J_1>0$$ and $$J_2<0$$ the system is frustrated since $$J_1$$ prefers ferromagnetic ordering but $$J_2$$ prefers antiferromagnetic ordering. How do I calculate which state minimizes the energy?

I was thinking one could try different combinations and see which arrangement minimizes the "frustration", but maybe there's a better way? Seems like a lot of work

Would really appreciate some input

The SAF phase consists of alternating rows of $$+$$ and $$-$$ spins in one direction (which can be horizontal or vertical, giving an additional degeneracy on top of the usual up-down spin degeneracy).
The relevant energies per spin are \begin{align*} E_\text{F}/N &= -2J_1 -2J_2 \\ E_\text{AF}/N &= 2J_1 -2J_2 \\ E_\text{SAF}/N &= 2J_2 \end{align*} The lines along which SAF becomes energetically equal to either AF or F correspond to $$J_2=-\frac{1}{2}|J_1|$$. So the case in which you are interested, $$J_1>0$$ and $$J_2=-J_1$$, lies well within the SAF region.