# Simulating the Ising Model, but with three states instead of two

Recall the homogeneous Ising energy of a configuration σ in the absence of a magnetic field is given by the Hamiltonian function

$$H(\sigma) = -\sum_{\langle i~j\rangle} \sigma_i \sigma_j ,$$

where the sum is over pairs of adjacent spins. Recall the notation $$⟨ij⟩$$ indicates that sites $$i$$ and $$j$$ are nearest neighbors. The configuration probability is given by the Boltzmann distribution with inverse temperature $$β ≥ 0:$$

$$P(\sigma) = \frac{e^{-\beta H(\sigma)}}{Z}$$ where $$Z$$ is the partition function. When simulating the Ising model, the standard method is to use the Metropolis algorithm which makes use of an acceptance criteria with probabilities drawn from distribution $$P(\sigma)$$.

My question is, what should the the acceptance criteria be if $$\sigma\in\{-1,0,1\}$$ instead of the usual two spin state $$\sigma\in\{-1,1\}$$ ?

I don't remember the details, wasn't it that the acceptance probability depends on the changes in the energy? something like $$P_{acceptance}= \min\{1,e^{-\beta\Delta E}\}$$ . If this is the case, you will only have to select a spin at random, assign a new value in [-1,0,1] at random, and then accept the change with the probability $$P_{acceptance}$$.