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I am working on spin-1 Ising model and I am new in this field. it seems that spin-1 Ising model in one dimension can be exactly solved by transfer matrix similar with spin 1/2 Ising model, am I right on this? also I like to know whether spin-1 Ising model is exactly solvable in two-dimension?
The Hamiltonian is :H$=\sum_{<ij>}S_i^zS_j^z$ with $S_z$=diag(1,0,-1).

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  1. Yes, this can be solved with a transfer matrix similar to the spin-1/2 Ising model - this is true pretty much for any classical model.

  2. I doubt this can be solved exactly in 2D - at least, I don't think anyone has tried, and there's very few models in 2D which can be solved exactly. Certainly, a mapping to free fermions as for spin-1/2 seems to fail. But no one can know for sure (unless someone knows that it has been solved exactly) - it is always possible that some model can be solved exactly but either no-one has tried, or despite trying people have not yet figured out how to solve it.

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