The Physical Seam · compact angles under heat
When Pairs Let Go
Place a vortex pair, measure its quantized winding, then drive the Kosterlitz RG equations to find vortex unbinding, the universal stiffness jump, and the essential singularity without pretending the cutoff model predicts a lattice transition exactly.
Every arrow below is only an angle. There is no place for a conventional magnetization to hide at nonzero temperature in the infinite short-range two-dimensional XY model. Yet a loop can remember what is inside it, and that integer survives every smooth rearrangement that does not drag a core across the loop.
Put the loop somewhere decisive
Choose one core, both, or empty space. The field is rebuilt on a by lattice. Each edge difference is wrapped before the loop sum is taken, so a full turn cannot disappear behind the branch cut.
Click empty field to place a pair. Drag either core or drag inside the white loop. Pointer positions snap to half-lattice coordinates so a core never sits on a sampled site.
Keyboard-operable precision controls
A loop around one core returns one signed turn. Around both, the charges cancel. This establishes quantized winding, but no more. It does not establish a thermodynamic transition.
Now make the dismissal fail
The leading dilute-vortex flow has two running variables. K = J/(kBT) is the bare dimensionless stiffness, and y = exp(−μ/(kBT)) is the bare fugacity set by the core energy.
chosen trajectorynumerical separatrixπK = 2
The escape scale
Above the separatrix, the integration stops when y = . Calling that scale ℓ*, the instrument sets ξ/a = exp(ℓ*). It samples temperatures and fits the computed points to ln(ξ/a) = c + b/√t.
The flow has two destinations. On the cold side, fugacity falls toward zero and the long-distance stiffness remains nonzero. On the hot side, fugacity eventually runs to order one, where this dilute approximation announces its own breakdown. The critical trajectory ends at πK = , so the renormalized stiffness approaches Kren = . Combining the same limit with η = 1/(2πKren) gives η(Tc) = .
For a neutral helium-4 film, converting the dimensionless jump to areal mass density gives ρs(Tc−)/Tc = kg m−2 K−1. This is a zero-frequency, long-wavelength limit using the helium-4 atomic mass fixed in the check. It is not the total density and not a three-dimensional density.
The tempting one-vortex balance uses the bare stiffness and gives kBT/J = . The square-lattice simulation benchmark transcribed as βKTJ = instead implies kBTKT/J = . Their mismatch is the point: the universal statement concerns the renormalized long-distance stiffness, not the microscopic bare J.
The check
The browser and the downloadable verifier independently rebuild the quantities above. The shipped page engine uses these explicit choices:
| phase field | angle evaluations |
|---|---|
| plaquettes | wrapped four-edge sums |
| wrap convention | (−π, π], tolerance |
| RG integrator | RK4, Δℓ = , at most steps |
| temperature shooting | bisections on [, ] |
| escape convention | stop at y = , set ξ/a = exp(ℓ*) |
| essential fit | equally spaced t values from to , unweighted ordinary least squares |
| maximum RG work | RK4 steps per sweep |
- The vortex cores are point defects, snapped to half-lattice coordinates. The displayed arrows sample their analytic atan2 field. The loop is a grid-aligned square, and display rounding occurs only after the raw wrapped sum is divided by 2π.
- The RG equations are the leading dilute Coulomb-gas truncation. The free core energy is chosen by the reader. The shooting criterion is the finite-cutoff condition K(ℓmax) = 2/π, not an exact microscopic solution.
- The fit amplitude b and intercept c are nonuniversal. The threshold y = , the ultraviolet length a, the finite ℓmax, and the chosen fit window all change them. The essential form, not the fitted amplitude, is the robust claim here.
- The helium conversion fixes m = kg, kB = J K−1, and ℏ = J s. Paired fermions require the pair mass. Finite frequency, finite size, and substrate nonuniformity round an experimental onset.
- The displayed Monte Carlo benchmark is a source transcription. Its reciprocal is recomputed live. It is model-specific, unlike 2/π and η = 1/4.
Run the independent checker with node research/kosterlitz-thouless/verify-kosterlitz-thouless.mjs. It executes this page's own engine, compares it with a separate implementation, and deliberately mutates the calculation to show the checks can fail.
The lattice number is still numerical
The universal endpoint is not the open question. The nearest unresolved edge is microscopic: no exact transition temperature is known for the nearest-neighbor square-lattice XY model, and difficult logarithmic corrections make numerical error budgets method-sensitive. Published values include from Hasenbusch in 2005, from Hsieh, Kao, and Sandvik in 2013 after including a subleading logarithmic correction, and from Jha's 2020 tensor calculation. The latter two quoted intervals do not overlap. That is not permission to average them.
As of 2026-08-01, this page therefore uses Hasenbusch only as a named benchmark and makes no exact-lattice claim. The live flow cannot adjudicate the discrepancy because its core fugacity and ultraviolet matching are free inputs, precisely the microscopic information the universal equations discard.
Primary records: Hasenbusch 2005, Hsieh, Kao, and Sandvik 2013, and Jha 2020. The historical trail begins with Berezinskii's 1970 Russian work, published in English in 1971, followed by Kosterlitz and Thouless in 1973. The jump prediction is Nelson and Kosterlitz, 1977.