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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

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This model (assuming a square lattice) has been studied in the literature for a while. K Binder and DP Landau Phys Rev B, 21, 1941 (1980) discuss the three possibilities for the ground state: ferromagnetic (F), antiferromagnetic (AF), and superantiferromagnetic (SAF). Basically one considers the likely structures and picks the one with the lowest energy. They also conducted Monte Carlo simulations to determine the phase behaviour at higher temperatures, and never saw any more complicated structures. There's more discussion in J Oitmaa J Phys A, 14, 1159 (1981), and I drew the following diagram based on Fig 1 from that paper.

enter image description here

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.

There may be discussions of this model in standard texts, but I don't have any of those to hand, sorry, and the papers I cited both require a subscription. There is a connection with the anisotropic next-nearest-neighbour Ising or ANNNI model (which I saw mentioned in a comment on your earlier question), and that model is of interest both experimentally and theoretically.

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  • $\begingroup$ Thank you so much! I don't think it's been mentioned in our course book, and unfortunateley, I had my exam yesterday. But I'm happy to finally understand this. $\endgroup$ – user183359 Oct 23 '18 at 10:06
  • $\begingroup$ That's fine; feel free to accept the answer at some point if you wish to do so (the checkmark on the left). $\endgroup$ – user197851 Oct 23 '18 at 10:11

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