Lattice Configuration Generation with a Self-Learning Diffusion Model

We show that a diffusion sampler for lattice-field configurations can be trained without preparing training data by an external Monte Carlo calculation. Starting from exactly sampled configurations at $\beta=0$, we construct a self-bootstrap sampler, SLDiffusion, in which periodic Gaussian proposals with a fixed learned score are Metropolis--Hastings corrected, at each $\beta$, against the same physical target at every noise level, and only replay configurations from the resulting chain are used to train the score at the next stage. In the two-dimensional compact XY model, self-training proceeds from $\beta=0.30$ to $0.50$ at $L=4$. At $\beta=0.5$, the energy and vortex densities for $L=4,6,8,12$ agree with independent Hybrid Monte Carlo calculations within $1.35\sigma$. Volume-native retraining at $L=8$ and $12$ improves both the proposal displacement and autocorrelation. The integrated autocorrelation times of the energy and vortex densities remain below two for all volumes studied. These results demonstrate that a Metropolis-corrected diffusion sampler can be self-trained without configurations drawn in advance from the target coupling.