Correlation-length singularity at the XY transition
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For the ordinary nearest-neighbor cosine XY model on the square lattice, does √(βc−β)log ξ(β) approach a positive finite constant as β rises to βc from below? The correlation length uses inverse axis mass after the infinite-volume limit.
Why it matters and what counts as progress
Why it matters: The limit is the precise essential-singularity scaling predicted by Berezinskii–Kosterlitz–Thouless theory for this model.
What counts as progress: A proved finite positive limit under this normalization and order of limits, or a correction to the scaling; results for Villain or Coulomb-gas variants require separate transfer.
How the problem developed
- 1974 · Renormalization-group result [1]
J. M. Kosterlitz Found that the two-dimensional square-lattice XY correlation length diverges faster than any power near the transition, anticipating the Berezinskii–Kosterlitz–Thouless essential singularity.
- 5 Oct 2026 · Reported theorem [2]
OpenAI The preprint reports the essential singularity for the nearest-neighbor cosine XY model on the square lattice, with the thermodynamic limit taken before the separation limit.
AI contribution
Essential Singularity of the Correlation Length in the Planar XY Model
For the nearest-neighbor cosine XY model on the square lattice, the manuscript claims sqrt(βc−β) log ξ(β) tends to a finite positive model-specific constant as β rises to βc from below. Here ξ is defined from the inverse axis mass after the infinite-volume and separation limits in that order.
Sources
Website timeline
Publication dates and editorial changes.
OpenAI reports the XY-model correlation-length essential singularity