Diffusion models for the $O(3)$ nonlinear sigma model

Daniel Schubring

cond-mat.stat-mechOct 29, 2026

Diffusion models for the $O(3)$ nonlinear sigma model

Authors: Daniel Schubring

Organizations: Department of Physics, City College of New York, CUNY New York, NY 10031, USA

Abstract

A score-based diffusion model is trained on the two-dimensional classical Heisenberg model, the lattice representation of the $O(3)$ nonlinear sigma model. Neural networks are trained on three different coarse-grainings of the same training set to implement a multiscale sampling scheme that modestly improves both the accuracy and computational cost of direct sampling on the finest lattice. Stein (or Schwinger-Dyson) diagnostics for the two-point correlation function are introduced to assess the quality of the learned score without relying on reverse diffusion sampling. These diagnostics are also incorporated into an additional training loss. We discuss several technical considerations relevant to implementing diffusion models on $S^2$ . These include a consideration of lifting the network input to $S^3$ , as in the $CP^1$ representation, and developing a boundary-layer approximation to the short-time spherical heat kernel that extends its usefulness to larger diffusion times.

Figures & tables

Figure 1: A two-point correlation function for an

O(3)

model coarse-grained from a

64^{2}

lattice (

\beta=1.5

) to a

16^{2}

lattice (black). One-parameter renormalizations of

\beta

\beta^{\prime}

, fitting either the correlation length (blue) or the energy per site (red), are compared to the generated configurations from a diffusion model (purple). See also Table 1 .

Method	$C(1)$	$m^{-1}$	$\chi_{m}$
Equilibrium	$0.278183(6)$	$2.6551(7)$	$11.350(3)$
$\mathrm{DM}_{16}$ (1600 evals)	$0.2781(2)$	$2.68(2)$	$11.39(8)$
$\mathrm{DM}_{16}$ (200 evals)	$0.2794(3)$	$2.59(2)$	$11.04(8)$
$\beta^{\prime}=0.768$	$\emph{0.2782}$	$1.06$	$4.269$
$\beta^{\prime}=1.14$	$0.4452$	$\emph{2.66}$	$16.41$

Table 1: Comparison of observables on the coarse-grained

16^{2}

lattice with samples from both the diffusion model and

O(3)

models at effective couplings

\beta^{\prime}

. The observables used to determine each

\beta^{\prime}

are indicated in italics.

Figure 2: Distributions of the topological charge

Q

(left) and the number of lattice dislocations

N_{\mathrm{dis}}

(right) for the two-dimensional

O(3)

model at

\beta=1.5

on a

64^{2}

lattice. The equilibrium heat-bath reference (black line) is compared with

10^{4}

configurations generated by a diffusion model using 800 score evaluations (blue histogram). Systematic errors in the configurations generated by the model are largely corrected after a cheap refinement of four heat-bath sweeps (purple error bars).

Figure 3: Editing short-distance structure with diffusion models. Configurations of classical spins are visualized by mapping spin orientations to saturated colors. An image of Seurat’s painting A Sunday Afternoon on the Island of La Grande Jatte is first converted to saturated colors and then to a corresponding spin configuration. A diffusion model trained on the $O(3)$ model at $\beta=1.5$ is then used to modify the short-distance structure in a manner consistent with the local statistics of the equilibrium field theory. At the upper right is an enlarged $64^{2}$ sample of the face of the man wearing a top hat. At the lower right is a genuine equilibrium $64^{2}$ configuration for comparison. The edited painting has an energy per site of $-1.201$ and a density of $2.40\times 10^{-3}$ topological lattice dislocations, compared with $-1.203$ and $2.26\times 10^{-3}$ in the equilibrium theory, respectively.

Figure 4: Comparison of approximations to the scalar score

\partial_{z}\log K(\theta,\tau)

, where

z=\cos\theta

. The spectral expansion with

\ell_{\max}=10

is used at large

\tau

, while the MPSD and boundary-layer approximations are used at small

\tau

. These approximations are shown at the switching time

\tau=0.137

, alongside a higher-order spectral expansion (

\ell_{\max}=15

Figure 5: Two-point correlation function

C(r)

on the

64^{2}

lattice. Solid lines show equilibrium correlations at diffusion times

u=0

0.5

, and

0.75

, while circles show the corresponding Stein estimates ( 40 ). Diamonds show correlations measured from configurations generated by the diffusion model using 800 score evaluations.

Lattice	$u$	Method	$C(1)$	$C(2)$	$C(4)$	$C(8)$
$64^{2}$	$0$	Equilibrium	0.601683(2)	0.433864(4)	0.278183(6)	0.147033(8)
		$\mathrm{DM}_{A}$ (800 evals)	0.60031(8)	0.43188(9)	0.2767(1)	0.1473(2)
		$\mathrm{DM}_{A}$ (200 evals)	0.59803(7)	0.4269(1)	0.2700(1)	0.1405(1)
		$\text{Stein}_{A}$	0.60429	0.44230	0.28220	0.14989
		$\text{Stein}_{B}$	0.60087	0.44004	0.28040	0.14901
		$\text{Stein}_{C}$	0.59751	0.43700	0.27888	0.14815

Table 2: Comparison of correlations in the equilibrium reference ensemble, samples from the diffusion models, and Stein diagnostic estimates of the correlation function. On the

64^{2}

lattice, three models were used. Model

A

was determined as the best checkpoint for sampling and Models

B

and

C

were trained with and without a

C(1)

Stein residual loss term.

Hardware	Method	it/s	$\mathrm{DM}_{64}$ evals
RTX PRO 6000 Blackwell	$\mathrm{HB}_{64}$	31.03	0.0393
	$\mathrm{DM}_{16}$	12.16	0.100
	$\mathrm{DM}_{32}$	3.19	0.382
	$\mathrm{DM}_{64}$	1.22	1.000
A100-SXM4-40GB	$\mathrm{HB}_{64}$	23.8	0.0580
	$\mathrm{DM}_{16}$	11.3	0.122

Table 3: Computational costs of heat bath sweeps and diffusion model score evaluations on two GPUs. The final column gives the cost of each operation in equivalent

\mathrm{DM}_{64}

score evaluations, calculated from the measured iteration rates.

Figure 6: Illustration of various super-resolution schemes. The operator $\mathcal{P}$ denotes prolongation to a lattice with twice the linear size, while $\mathcal{L}$ denotes Langevin correction with the coarse spins held fixed. The operators $\mathcal{F}$ and $\mathcal{R}$ denote forward and reverse diffusion, respectively. The shaded region corresponds to diffusion time $u=u_{\mathrm{noise}}=0.5$ , while the unshaded region corresponds to $u=0$ . In method A, all prolongation and refinement steps are performed at $u=0$ , following the initial generation of a $16^{2}$ configuration by reverse diffusion. In methods $B_{1}$ and $B_{2}$ , the configuration is first prolonged to $64^{2}$ at $u=0$ , then forward diffused to $u_{\mathrm{noise}}$ . Method $B_{1}$ uses the remaining score evaluations entirely for reverse diffusion to $u=0$ , whereas $B_{2}$ divides them between Langevin correction at $u_{\mathrm{noise}}$ and reverse diffusion. In method $C$ , the initial $16^{2}$ configuration is reverse diffused only to $u_{\mathrm{noise}}$ , where prolongation and refinement are performed. The final stages then follow method $B_{2}$ .

Method	$C(1)$	$C(2)$	$C(4)$	$C(8)$	$C(16)$	Evals
$\boldsymbol{32^{2}}$ lattice, $\boldsymbol{u=0}$
Ref.	0.433864(4)	0.278183(6)	0.147033(8)	0.05464(1)	0.01848(2)	—
AB	0.4316(2)	0.2772(2)	0.1450(2)	0.0530(3)	0.0172(5)	724
Direct	0.4233(1)	0.2690(2)	0.1398(2)	0.0508(3)	0.0165(3)	800
Interp.	0.4035(3)	0.2590(3)	0.1368(3)	0.0495(4)	0.0160(4)	124
$\boldsymbol{32^{2}}$ lattice, $\boldsymbol{u=0.5}$

Table 4: Comparison of

C(r)

obtained from the reference ensemble (Ref.), direct diffusion model sampling, and various super-resolution methods (see Fig. 6 ) including Gaussian interpolation without final DM correction (Interp.). Computational costs are expressed in equivalent score evaluations (Evals) of

\mathrm{DM}_{32}

for the

32^{2}

lattice and

\mathrm{DM}_{64}

for the

64^{2}

lattice, using the A100 timings reported in Table 3 .

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Equilibration under heat-bath Monte Carlo on the

64^{2}

lattice, starting from random (hot), uniform (cold), and diffusion-generated configurations. The four panels show the nearest-neighbor correlation

C(1)

, long-distance correlation

C(32)

, magnetic susceptibility

\chi_{m}

, and topological susceptibility

\chi_{t}

. Diffusion-generated configurations are obtained using

\mathrm{DM}_{16}

with RK2 or mixed SDE sampling, followed by prolongation and restricted heat-bath refinement. For these curves, zero sweeps corresponds to the beginning of unrestricted heat-bath evolution, after prolongation and restricted refinement have been completed.

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