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

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,

$$\mathcal{H}=-\sum_{\langle i j\rangle}\left[\Delta \cos \left(\theta_{i}-\theta_{j}\right)+(1-\Delta) \cos \left(q \theta_{i}-q \theta_{j}\right)\right],$$ 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.

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

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

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

$$\begin{equation} H(\theta)=-K\sum_{i\neq j}\cos(\theta^i - \theta^j) - h_p\sum_{i}\cos(p \theta^i) \end{equation}$$

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 $$p$$.

• Thank you for your answer. I have ever see (but not study) this order parameter. So,I can understand all you answered. N-F phase transitoion in my model belongs to 3-state potts model universality and the order parameter m_{k} is similar to the magnetization of potts model. Do you know any ralations? – Ryo Feb 4 '19 at 7:13