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Diffusion-warm sampling of the XY model enables fast thermalization at scale

NeurIPS 2026 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Hamiltonian Monte Carlo · 0 citations (OpenAlex)

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Trains a temperature-conditioned, rotation-invariant score-based diffusion model on 20x20 XY-model samples and uses its outputs as warm starts for Wolff MCMC, cutting thermalization time roughly tenfold on larger lattices.

Problem

Sampling equilibrium configurations of the 2D XY model with continuous spins is hampered by very long MCMC thermalization times at low temperature, and ML samplers trained on small lattices fail to generalize to larger sizes.

Approach

Use a temperature-conditioned (classifier-free guidance) score-based diffusion model on the torus, trained on small 20x20 lattices, to generate warm-start configurations for much larger lattices, followed by a few Wolff cluster updates to correct the physics.

Key results

• Diffusion-warmed MCMC reduces thermalization time by roughly an order of magnitude versus random-start Wolff MCMC across tested sizes and temperatures (about 10x at beta=1.5).
• For a 50x50 system at beta=1, about 30 Wolff updates on diffusion samples make C(r) match 1000 fully thermalized MCMC samples.
• Diffusion samples alone underestimate helicity modulus and low-temperature correlations for L>20; 30 Wolff steps correct these observables.

Surprising findings

• Naive size extrapolation of the diffusion model without MCMC correction fails to reproduce long-range correlations and spin waves, and tiling 20x20 samples performs worse than the diffusion model.

Limitations

• Scaling of total diffusion-plus-MCMC runtime with system size is not established; authors say further study is needed.
• Diffusion samples alone do not reproduce long-wavelength spin waves and long-range correlations at larger L; MCMC correction is required.
• Evaluation uses only 1000 samples per setting and a limited set of temperatures and sizes; hyperparameter tuning was limited.

Experimental setup

Datasets: XY model Wolff-MCMC samples (20x20, 11 temperatures)

Benchmarks: XY model long-run Wolff MCMC reference observables (energy, magnetization, helicity modulus, heat capacity) on L=20 to 6

Metrics: thermalization time (MCMC steps and wall-clock) spin-spin correlation C(r) relative error of energy, magnetization, helicity modulus, h

Baselines: Wolff MCMC from random initialization random tiling of 20x20 samples

Builds on: Wolff cluster algorithm score-based SDE diffusion classifier-free guidance torsional diffusion

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