How to define the order parameter of the q-state Potts model? The order parameter of Ising model can be defined as $m=\frac{N_1-N_2}{N}$, if $N$ is the total number of lattice points, $N_1$ and $N_2$ is the number of lattice points spin up and down respectively, $N=N_1+N_2$.
But I am not able to write down the order parameter of the q-state Potts model. Any help will be appreciated!
 A: First one needs to gain a deeper understanding of the order parameter of the Ising model. The magnetization $m$ can be written as $m=(+1)n_1+(-1)n_2$, where $n_p=N_p/N$ ($p=1,2$) is the number density of each spin. So what do the coefficients $(\pm1)$ stand for? They are two possible magnetizations of a single Ising spin, which form the representation of the $\mathbb{Z}_2$ group. So the Ising spin is also known as the $\mathbb{Z}_2$ spin.
Now it is straight forward to extend the formulation to the $q$-state Potts model. Simply replace the number 2 by $q$, i.e. to consider the Potts spin as the $\mathbb{Z}_q$ spin, whose "magnetization" must be taken from the representation of the $\mathbb{Z}_q$ group, which are $q$th roots of unity $e^{2\pi ip/q}$ ($p=1,2,\cdots,q$). Given the representation of the Potts spin, it is easy to write down the order parameter
$$m=\sum_{p=1}^qe^{2\pi i\frac{p}{q}}n_p,$$
with $n_p=N_p/N$ (let $N=\sum_pN_p$) being the number density of the $p$th type of the Potts spin. Note that the order parameter is complex in general. In the disordered phase, all types of Potts spin appear with equal probability, i.e. $n_1=n_2=\cdots=n_q$, and in this case, we do have $m=0$ due to the cancellation of the phase, which is consistent with the idea of the order parameter: "a quantity that is zero in the disordered phase and non-zero in the ordered phase".
A: The order parameter for the q-state Potts model has a couple different choices found in the literature. 
The standard $q$-state Potts model, given by 
$\mathcal{H} = J\sum_{\langle i j \rangle} \delta_{ij}$
has an order parameter that can be defined in $q-1$ dimensions, where $q$ unit vectors define a hypertetrahedron in $q-1$ dimensions [1,2]. This is rather difficult to describe in a stack exchange post, but the wikipedia page on simplexes can help [3].
An alternative definition of the order parameter, in a system with $N$ spins and $n_i$ is the number of spins in state $i \in \{1,2,\dotsc,q\}$:
$m = \frac{\frac{q}{N}\max(n_1,n_2,\dotsc,n_q)-1}{q-1}$
can be found in the literature.
Alternatively, if you are looking at the planar Potts model, also known as the clock model, then the choice given by Everett applies. The clock model Hamiltonian is given by: 
$\mathcal{H} = J \sum_{\langle i j \rangle} \cos(\theta_i - \theta_j)$
where $\theta_i = (2\pi/q)i$ and $i \in \{1,2,\dotsc,q\}$.
[1] https://link.aps.org/pdf/10.1103/RevModPhys.54.235
[2] https://iopscience.iop.org/article/10.1088/0305-4470/8/9/019/meta 
[3] https://en.wikipedia.org/wiki/Simplex
