XY Model in Two Dimensions

Setup

Each site of a 2D square lattice carries a continuous angle \(\theta_i \in [0, 2\pi)\), with ferromagnetic nearest-neighbour coupling. The XY model Hamiltonian is $$ H = -J \sum_{\langle ij \rangle} \cos(\theta_i - \theta_j). $$ There is no long-range order at finite \(T\), but the model undergoes a BKT transition at $$ T_{\mathrm{BKT}} \approx 0.893\, J. $$ Below it, vortex–antivortex pairs are bound and correlations decay as a power law; above it, free vortices proliferate and correlations decay exponentially.

Method

The angles evolve by overdamped Langevin dynamics (Model A), $$ \dot{\theta}_i = -\Gamma J \sum_{j \in \mathrm{nn}(i)} \sin(\theta_i - \theta_j) + \eta_i(t), $$ with white noise satisfying $$ \langle \eta_i(t)\eta_j(t') \rangle = 2\Gamma T\, \delta_{ij}\delta(t-t'). $$ The integrator is the Euler–Maruyama scheme, with \(\xi_i \sim \mathcal{N}(0,1)\) sampled per site each step. Defaults \(J=\Gamma=1\); periodic boundaries.

Interaction

Left-click inserts a vortex (\(+1\)); right-click inserts an antivortex (\(-1\)).

With periodic boundary conditions, the net topological charge must remain zero, so inserting a single defect generally creates compensating defects elsewhere.

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