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Good evening! I'm at the beginning of my study about the Ising model and it has been proposed to me this problem: Find all periodic ground-state configuration for the following one-dimensional Ising model of interacting spins, $s_i=+1$ or $ -1 $ $$ H=-\sum_{i\in\mathbb{Z}} (s_{i}s_{i+1}-s_{i}s_{i+2}-4s_{i}s_{i+3}) $$ Actually, I don't know how to procede. I know that the ground states, by definition, are the configuration of minimal energy. But then, is this possible to compute these configuration for this specific model in an explicit way? I hope that someone could give me a hint, because I think that I'm a little bit confused.. Thank you very much!

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The state with minimal energy will be an alteration of +1 and -1 spins. The reason is that the 1-4 interactions will dominate the rest. Each spin in your system will interact with 6 others. Even though 1-2 (neighbours) and 1-3 interactions will be unfavourable in the alternating sequence, two favourable 1-4 interactions with the weight -4 will have more impact.

You can probably make this more rigorous by showing that flipping a single spin or creating a domain with different alternation will require positive energy.

Hope this helps.

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