## XY Model

The XY model (sometimes called the planar model) consists of spins on a lattice where each spin can take on any angle $\theta_i$ between $0$ and $2\pi$. The energy is given by $$E = -J \sum_{\lt ij \gt} \cos{(\theta_i - \theta_j)},$$ where $J$ is the interaction energy that sets the scale for the energy. It has been shown that the magnetization vanishes for all nonzero temperatures $T$, and thus it was thought that there was no phase transition. However, Kosterlitz and Thouless showed that there is a phase transition, $T_{\rm KT}$ such that below $T_{\rm KT}$ the spin-spin correlation function falls off as a power law and above $T_{\rm KT}$ this correlation falls off exponentially. The phase transition is caused by the unbinding of pairs of vortices, which are an example of topological defects.

The program simulates the XY model, calculates various thermodynamic quantities, and shows the location of the vortices.

Problem: Simulations of the two-dimensional $XY$ (planar) model

In this model the spins are located on a $d$-dimensional lattice, but are restricted to point in any direction in the plane. The interaction between two nearest neighbor spins is given by $-J {\bf s}_1 \cdot {\bf s}_2$, where ${\bf s}_1$ and ${\bf s}_2$ are two unit spin vectors.

One of the interesting features of the $XY$ model in two dimensions is that the mean magnetization $\langle m\rangle = 0$ for all nonzero temperatures, but there is a phase transition at a nonzero temperature $T_{\rm KT}$ known as the Kosterlitz-Thouless transition. (J. Michael Kosterlitz and David J. Thouless received the 2016 Nobel Prize in Physics for discovering and explaining this phase transition, which led to finding similar phase transitions in many other systems and the recognition of the importance of topological defects such as the vortices found in the planar model. F. Duncan M. Haldane shared the prize that year for related work.) For $T \leq T_{\rm KT}$ the spin-spin correlation $C(r)$ decreases as a power law; for $T > T_{\rm KT}$, $C(r)$ decreases exponentially. The power law decay of $C(r)$ for $T \leq T_{\rm KT}$ implies that every temperature below $T_{\rm KT}$ acts as a critical point.

Program XYModel uses the Metropolis algorithm to simulate the $XY$ model in two dimensions. In this case a spin is chosen at random and rotated by a random angle up to a maximum value $\delta$.

1. Rewrite the interaction $-J {\bf s}_i \cdot {\bf s}_j$ between nearest neighbor spins $i$ and $j$ in a simpler form by substituting $s_{i,x} = \cos\theta_i$ and $s_{i,y} = \sin \theta_i$, where the phase $\theta_i$ is measured from the horizontal axis in the counter-clockwise direction. Show that the result is $-J \cos(\theta_i - \theta_j)$.
2. An interesting feature of the $XY$ model is the existence of vortices and antivortices. A vortex is a region of the lattice where the spins rotate by at least $2\pi$ as you trace a closed path. Run the simulation with the default parameters and observe the locations of the vortices. Follow the arrows as they turn around a vortex. A vortex is indicated by a square box. What is the difference between a positive (blue) and negative (yellow) vortex? Does a vortex ever appear isolated? Count the number of positive vortices and negative vortices. Is the number the same at all times?
3. Click the New button, change the temperature to 0.2, set the initial configuration to random, and run the simulation. You should see quenched-in vortices which don't change with time. Are there an equal number of positive and negative vortices? Are there isolated vortices whose centers are more than a lattice spacing apart?
4. Click the New button and set the initial configuration to ordered and the temperature to 0.2. Also set steps per display to 100 so that the simulation will run much faster. Run the simulation for at least $1000\,$mcs to equilibrate and $10,000\,$mcs to collect data, and record your estimates of the energy, specific heat, vorticity, and the susceptibility. Repeat for temperatures from 0.3 to 1.5 in steps of 0.1. Plot the energy and specific heat versus the temperature. (The susceptibility diverges for all temperatures below the transition, which occurs near $T=0.9$. The location of the specific heat peak is different from the transition temperature. The program computes $\chi=(1/NT^2)\langle M^2\rangle$ instead of the usual expression, because $\langle M\rangle=0$ in the thermodynamic limit.) Is the vorticity (the mean number density of vortices) a smooth function of the temperature?
5. Look at configurations showing the vortices near the Kosterlitz-Thouless transition at $T=T_{\rm KT}\approx0.9$. Is there any evidence that the positive vortices are moving away from the negative vortices? The Kosterlitz-Thouless transition is due to this unbinding of vortex pairs.

## Resource

Problem 9.21 in Statistical and Thermal Physics: With Computer Applications, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).

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