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 $p$.

  • $\begingroup$ 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? $\endgroup$
    – Ryo
    Feb 4 '19 at 7:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.