STUDY #17 · 2026 · IN OBSERVATION

XY Model — Vortex Unbinding

A model-driven visual study of topological defects in a two-dimensional field of directions.

MOVING IMAGE — DEFECT ANNIHILATION quench T 1.25 → 0.28 · pairs only · schlieren

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.

the resolved field — a long anneal, few survivors quilting the weave T 0.25 · 6500 sweeps · 44 defects · Σq 0 · schlieren

Motif XY model / Kosterlitz-Thouless transition / vortex-antivortex pairs / topological defects / schlieren texture / Monte Carlo

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer. The visual output was selected through parameter exploration.

Observation A lattice of continuous directions is held at a temperature. True order is forbidden in two dimensions, yet one dial still crosses a transition: topological defects — vortices and antivortices, born and dying only in pairs — stay bound below T_KT and unbind into a plasma above it. Quenched cold, thousands of defects annihilate in pairs until a few survivors quilt the flowing weave.

Reference J. M. Kosterlitz & D. J. Thouless, "Ordering, metastability and phase transitions in two-dimensional systems," Journal of Physics C, vol.6, 1181-1203 (1973); V. L. Berezinskii, Soviet Physics JETP, vol.32, 493-500 (1971).

Tools Python / NumPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

SAME ENGINE, A DIFFERENT TOPOLOGY

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

Study #10 — Ising Study #17 — XY

The spin two choices — up or down a continuous direction — any angle at all

What carries the transition areas — magnetic domains growing and freezing points — vortex pairs binding and unbinding

The symmetry broken below Tc — one phase wins never broken — topology alone crosses the line

What annihilates domain walls, absorbed as one phase spreads vortex and antivortex, dying only in pairs

PARAMETERS EXPLORED

param meaning effect on the image

T temperature (T_KT ≈ 0.893) the one dial: smooth weave → breathing bound pairs → unbinding → defect plasma

quench a hot start dropped to low temperature thousands of defects annihilate in pairs — the film's time axis

h external field breaks the U(1) symmetry: combs the field flat and sweeps the vortices out

J coupling (fixed 1) the stiffness of alignment — the spin-wave rigidity

δ proposal width (numerical) fixes the Monte-Carlo clock rate of the footage

Each image below records its exact parameter set.

SELECTED STILLS — 6

the resolved field — few survivors quilt the weave T 0.25 · 6500 sweeps · 44 defects · schlieren

the free-defect plasma — order's memory erased T 1.15 · 16166 defects · m 0.011 · schlieren

just above the line — pairs beginning to let go T 0.95 · 3862 defects · m 0.53 · schlieren

bound pairs breathing in the weave T 0.7 · 336 defects · m 0.084 · schlieren

combed flat — a field breaking the symmetry from outside T 0.7 · h 0.25 · m 0.85 · schlieren

the same field in the superfluid palette T 0.25 · 6500 sweeps · superfluid

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.

the annihilation strip T 1.25 → 0.28 · defects 21940 → 20

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.

the proof of the invisible transition — power law below, exponential above, defects exploding at T_KT G(r) ~ r^(−0.089) ⇄ e^(−r/ξ) · ρ_v × 540 · Σq = 0

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 plasma between crossed polarizers — warm and cool cores in a dark weave T 1.15 · vortex warm / antivortex cool · schlieren

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

REFERENCES

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).
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).
J. M. Kosterlitz, "The critical properties of the two-dimensional xy model," Journal of Physics C: Solid State Physics, vol.7, 1046-1060 (1974).
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).
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).