XY Model — Vortex Unbinding

WHAT IS THIS

Take a flat lattice where every site holds a single arrow, free to point in any direction, and let neighbouring arrows prefer to align while heat jiggles them. In two dimensions this system can never truly order — waves of gentle disagreement always destroy perfect alignment. Yet it still has a phase transition, one carried entirely by topology: point defects where the direction winds a full turn (vortices and antivortices), which are born and die only in pairs. Below the Kosterlitz-Thouless temperature they hold each other in bound pairs; above it they unbind into a swarm and erase the field's memory.

This study runs the 2D XY model — a checkerboard Metropolis Monte Carlo on continuous angles — in real time on the GPU, and images the field the way a liquid-crystal film looks between crossed polarizers.

SAME ENGINE, A DIFFERENT TOPOLOGY

Ising decides by domains. The XY model decides by defects.

PARAMETERS EXPLORED

Each image below records its exact parameter set.

PROCESS — PARAMETER SWEEPS

The quench as a strip — a hot plasma of tens of thousands of defects dropped cold, annihilating in pairs frame by frame until only a handful survive. Coarsening slows as it goes; the Monte-Carlo clock is ramped to keep the on-screen rate steady.

SIGNATURE — ORDER WITHOUT ORDER

A transition with no broken symmetry — topology alone decides it.

Mermin and Wagner proved a two-dimensional field of directions can never truly order — and yet Berezinskii, Kosterlitz and Thouless found a transition hiding there anyway. Below T_KT the correlations die slowly, as a power law; above it, exponentially fast. The difference is not symmetry but topology: whether vortex-antivortex pairs hold each other bound.

This engine was checked against that signature before any image was kept: the measured correlation function crosses from power law to exponential at the transition, the vortex density explodes 540-fold, and the total topological charge stays exactly zero — every vortex born with its antivortex, every death a pair annihilation.

COLOUR = CROSSED POLARIZERS

A field of directions is exactly what a thin liquid-crystal film shows under a polarizing microscope: dark brushes where the director lines up with a polarizer axis (intensity ∝ sin²2θ), filled between with birefringence interference colours, and defects wearing four-armed brush crosses. The palette here is an artistic take on that interference sequence — indigo, steel, champagne, straw gold, orchid.

Defect cores are tinted by their topological charge — a quantity of the model itself — vortices warm, antivortices cool.

The colours are a visual interpretation of the angle field, not spectroscopic measurements.

REFERENCES

1. V. L. Berezinskii. "Destruction of Long-range Order in One-dimensional and Two-dimensional Systems having a Continuous Symmetry Group I. Classical Systems." Soviet Physics JETP, vol.32, 493-500 (1971).
2. J. M. Kosterlitz, D. J. Thouless. "Ordering, metastability and phase transitions in two-dimensional systems." Journal of Physics C: Solid State Physics, vol.6, 1181-1203 (1973).
3. J. M. Kosterlitz. "The critical properties of the two-dimensional xy model." Journal of Physics C: Solid State Physics, vol.7, 1046-1060 (1974).
4. N. D. Mermin, H. Wagner. "Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models." Physical Review Letters, vol.17, 1133-1136 (1966).
5. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller. "Equation of State Calculations by Fast Computing Machines." The Journal of Chemical Physics, vol.21, 1087-1092 (1953).