What is the order parameter of 2D generalized $XY$ model?

Question

I'm now studying the phase transition of 2D generalized XY model. This model considered here has a mixture with ferromagnetic and nematic-like interactions,

$$mathcal{H}=-sum_{langle i jrangle}left[Delta cos left(theta_{i}-theta_{j}right)+(1-Delta) cos left(q theta_{i}-q theta_{j}right)right],$$
where the the sum is over the nearest neighbors spins on a square lattice, $0 leq Delta leq 1$, and $q$ is a positive integer. This model has three phases, paramagnetic(P), Ferromagnetic(F) and Nematic(N).

I am especially studying $q=3$ case. According to this paper (https://arxiv.org/abs/1401.4442), the transition between N-P and F-P are Kosterlitz-Thouless universality class and N-F belong to the 3-states Potts universality class. The order parameter is defined as
 $m_{k}=frac{1}{L^{2}}left|sum_{i} exp left(i k theta_{i}right)right|$ , $ k= 1$ for 3-states Potts universality class and $k = 3$ for Kosterlitz-Thouless universality class. I can't understand why the order parameter is defined as $m_{k}$. Plese teach me.

Fraguh · Answer

A way to characterize order in the XY model is through the spin-spin correlation function

$$begin{equation}
chi(vec{r}) = langlevec{S}(vec{R}+vec{r}) cdot vec{S}(vec{R})rangle = langle cos(theta(vec{r}) - theta(vec{0}) rangle = langle e^{i( theta(vec{r}) - theta(vec{0}))}  - i sin(theta(vec{r}) - theta(vec{0}) rangle
end{equation}$$

where I've used translational invariance to define $vec{R}=0$ by the self consistent relation $theta(vec{0}) = langle theta rangle $.

Then the sine term in $chi$ vanishes upon averaging and we can write

begin{equation}
chi(vec{r}) = langle e^{i delta theta(vec{r})} rangle  = e^{frac{-1}{2}langle delta theta^2(vec{r})rangle}
end{equation} where $delta theta = theta - langle theta rangle$ and the last equality follows if we consider $delta theta$ to be a gaussian random variable.

I am unsure of the motivation for an order parameter $m_k$ with $k$ dependence. At the moment I am working on a similar exercise with hamiltonian

begin{equation}
H(theta)=-Ksum_{ineq j}cos(theta^i - theta^j) - h_psum_{i}cos(p theta^i)
end{equation}

and, using the order parameter defined above, I have identified a third temperature scale associated with $p$ that may lie above or below the Kosterlitz-Thouless transition depending on the value of $p$.

What is the order parameter of 2D generalized $XY$ model?

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