What is the fundamental difference between the Ising and Potts models?

Question

What I understand is that the only difference between the Ising and Potts models is that Ising has two types of spins, and Potts has n types of spins.

However, I am wondering if the Hamiltonian (energy formula) is the same or different.

Are the Hamiltonianas the same or different?

What other differences do they have?

Answer 1

Fundamentally, there is not much of a difference. A Potts model is a kind of generalized Ising model that includes more spin states. The Hamiltonians can be written with similar forms.

The Ising model can be written as

$$H(\vec\sigma) = -\sum_{\langle i,j\rangle} J_{i,j}\sigma_i \sigma_j -\mu \sum_i \sigma_i,\tag{1}$$

where $\langle i,j \rangle$ denotes nearest neighbor pairs, and $\sigma_i \in \{-1, +1\}$ are the lattice spin states. $J_{i,j}$ is a nearest neighbor interaction energy that can be spin-pair dependent, and $\mu$ is an on-site energy term. Alternatively, if you use spin values of $\sigma_i \in \{0, 1\}$ we call this a lattice gas model. The model parameters can be updated to give equivalent results with either choice of spin states.

The $Q$-state Potts model can be written similarly:

$$H(\vec\sigma) = -\sum_{\langle i,j\rangle} J_{i,j}\delta_{\sigma_i,\sigma_j} - \sum_i \mu_i\sigma_i \tag{2}$$

Here $\sigma_i \in \{0, 1, 2, \dots, Q\}$ are the $Q$ spin states and $\delta_{\sigma_i,\sigma_j}$ is the Kronecker delta.

There are also other extensions of the Ising model that include more than just nearest neighbor and on-site interactions. In my opinion, the most general model (often found in alloy modeling literature) is referred to as the “cluster expansion”. This model accounts for many-body interactions in the system and can be expressed as:

$$H(\vec\sigma) = J_0 + \sum_i J_i \sigma_i + \sum_{i,j} J_{i,j}\sigma_i \sigma_j + \sum_{i,j,k} J_{i,j,k}\sigma_i\sigma_j\sigma_k + \dots \tag{3}$$

Unlike the Ising model and Potts model shown above, the sums are not restricted to nearest neighbors, but run over all possible pairs, triplets, quadruplets, etc. of sites in the spin lattice. Sanchez, Ducastelle, and Gratias have shown that, provided an orthonormal set of spin functions (such as discrete Chebyshev polynomials), one can construct a complete and orthonormal basis set of cluster functions (products of spin functions), that can provide a formally exact description of the energy of a spin-lattice system. Because the cluster expansion is infinite in nature, it must be truncated for practical work. Clearly, the Ising model is a special case of the more general cluster expansion.