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

Question

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 $\sigma\in\{-1,1\}$ ?

Answer 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 $P_{acceptance}$.

I recommend the book by R.Toral and P. Colet "Stochastic numerical methods: an introduction for students and scientists" for details and good code examples.