Definition of the Ashkin-Teller model There seems to be two different definitions of the so-called "Ashkin-Teller model", and I'm not sure which one is the one assumed by physicists (or which one is of more interest).
First, in wikipedia, the model seems to be defined as the 4-state clock model. Very briefly, if we consider the "spins" to the following four angles $\sigma\in \{0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\}$, then the Hamiltonian is given as
$$ H = -\sum_{\langle ij \rangle} \cos(\theta_j - \theta_i)$$
meaning that neighboring spins prefer to be aligned, then perpendicular, then anti-aligned, in that order. This is the definition that makes more intuitive sense to me, perhaps due to its correspondence with discrete gauge models.
On the other hand (for instance in page 333 of Grimmet's RCM book), the model is defined as
$$H = +\sum_{\langle ij \rangle} \delta(\theta_i, \theta_j) $$
where the interaction $\delta$ produces $0$ if the two spins are aligned, $J_1$ if the two spins are anti-aligned, and $J_2$ if the two spins are perpendicular. And the assumption is $J_2 > J_1 > 0$, meaning that the spins prefer to be aligned, then anti-aligned, then perpendicular, different from the above behavior. (One can check the above definition is equivalent to setting $J_1=2$ and $J_2=1$)
My questions are:

*

*Which definition is the one normally assumed by physicists?

*Which model is mathematically more interesting (I guess this is a broad question)? For example, one can show the first model can be considered two indepedent replicas of the Ising model, which yields fairly interesting graphical representations such as this and this. For the second model, one can prove the existence of a double phase-transition using a graphical representation and standard percolation arguments (see also page 333 of Grimmet's book).

 A: The model whose Hamiltonian is
$$H=-J\sum_{(i,j)}\cos\Big({2\pi\over q}(\sigma_i-\sigma_j)\Big),
   \quad\sigma_i=0,\ldots, q-1$$
is the q-state clock model. It was introduced by Potts in 1952. In the same paper, Potts also introduced the Hamiltonian
$$H=-J\sum_{(i,j)}\delta_{\sigma_i,\sigma_j},
   \quad\sigma_i=0,\ldots, q-1$$
which is now called the Potts model. The Ashkin-Teller model is defined by
$$H=-J_1\sum_{(i,j)} \sigma_i\sigma_j
 -J_2\sum_{(i,j)} \tau_i\tau_j
 -K\sum_{(i,j)} \sigma_i\sigma_j\tau_i\tau_j,\quad
  \sigma_i=\pm 1,\tau_i=\pm 1$$
and corresponds to two Ising models coupled by their local energy. The special case $J_1=J_2=K$ is equivalent to the 4-state clock model.
These are the definitions assumed by physicists. The Potts model allows for the interesting Fortuin-Kasteleyn representation but the Ashkin-Teller may be the more interesting to mathematicians since it is related to an 8-vertex model.
A: In their original paper, Ashkin and Teller considered the following model: at each vertex of the lattice, there is a spin taking one of 4 possible values, denoted by A, B, C and D. The interaction energy between neighboring spins could take 4 possible values: $\epsilon$ (for pairs of type AA, BB, CC or DD), $\epsilon'$ (for AB or CD), $\epsilon''$ (for AC or BD) and $\epsilon'''$ (for AD or BC).
It was later realized (the first reference I am aware of is this one) that this model could be rewritten in terms of Ising spins, by placing two Ising spins at each vertex of the lattice, say $\sigma_i$ and $\tau_i$, setting ${\rm A}=(+,+)$, ${\rm B}=(+,-)$, ${\rm C}=(-,+)$ and ${\rm D}=(-,-)$, and considering the Hamiltonian
$$
- \sum_{i\sim j} \bigl( J_\sigma \sigma_i\sigma_j + J_\tau \tau_i\tau_j + J_{\sigma\tau} \sigma_i\sigma_j\tau_i\tau_j + C \bigr).
$$
One easily checks that this yields the same energies as before if we choose
\begin{align*}
J_\sigma &= \tfrac14(-\epsilon - \epsilon' + \epsilon'' + \epsilon'''),
\quad
&J_\tau &= \tfrac14(-\epsilon + \epsilon' - \epsilon'' + \epsilon'''),
\\
J_{\sigma\tau} &= \tfrac14(-\epsilon + \epsilon' + \epsilon'' - \epsilon'''),
\quad
&C &= \tfrac14(-\epsilon - \epsilon' - \epsilon'' - \epsilon''').
\end{align*}
(Of course, the constant $C$ plays no role as it does not affect the Gibbs measure).
As far as I am concerned, this is the model expected by physicists when speaking about the Ashkin-Teller model. Sometimes, though, people have particular cases in mind, maybe because, in their paper, Ashkin and Teller already quickly focused on the special case $\epsilon'=\epsilon''=\epsilon'''$, or equivalently $J_\sigma=J_\tau=J_{\sigma\tau}$, which is equivalent to the 4-state Potts model (note, however, that Potts would only introduce his model 8 years after the paper by Ashkin and Teller).
