Point-Group Equivariant Neural Networks for Local Dynamics in Lattice Systems

Yunhao Fan; Gia-Wei Chern

cond-mat.str-elJan 7, 2026

Point-Group Equivariant Neural Networks for Local Dynamics in Lattice Systems

Authors: Yunhao Fan, Gia-Wei Chern

Organizations: Department of Physics, University of Virginia, Charlottesville, VA 22904, USA

Abstract

We develop point-group equivariant neural networks for learning local mappings used in dynamical simulations of lattice systems. The same network acts on a finite neighborhood around each site, respecting lattice translations through shared parameters. The local configuration is decomposed into features that transform according to irreducible representations of the point group, and hidden features retain this organization throughout the network. Equivariant linear maps and symmetry-allowed tensor products then learn nonlinear spatial correlations while preserving the required transformation laws. This approach avoids constructing a fixed set of hand-crafted invariant descriptors, whose selection requires choices about which local correlations to retain. It also allows the complexity of the learned mapping to grow without enlarging its spatial cutoff. We demonstrate the construction by learning the forces governing adiabatic dynamics in the square-lattice Holstein model. Comparisons with exact diagonalization test the accuracy of the predicted forces and the evolution of spatial correlations. Force evaluation scales linearly with system size, allowing the trained model to simulate charge-density-wave ordering on large lattices. These simulations reproduce anomalously slow coarsening of the charge-density-wave domains, illustrating the use of the learned local mapping for collective dynamics.

Figures & tables

Figure 1: Point-group equivariant neural-network framework for local lattice force fields, illustrated with a square-lattice environment. The scalar fields in a finite neighborhood $\mathcal{C}_{i}$ are decomposed into irreducible representations (IRs) of its point group, $D_{4}$ for the example shown. Hidden features retain their prescribed transformation laws, and the invariant sector supplies the scalar force prediction. Equivariant updates combine two pathways: linear mixing of features within an IR sector and tensor-product coupling followed by projection onto a target IR using Clebsch–Gordan coefficients. The resulting features are linearly mixed within each target sector and passed through equivariant nonlinear activations. The same construction applies to other lattice geometries with their corresponding point groups and representation data.

Figure 2: Square-lattice Holstein model used to demonstrate the local point-group ENN force field. Itinerant electrons couple to scalar variables $Q_{i}$ describing local $A_{1}$ -type structural distortions. Rotations and reflections permute these site variables, while their spatial arrangement supplies nontrivial $D_{4}$ representations to the ENN.

Layer	IR Channels $A_{1}\oplus A_{2}\oplus B_{1}\oplus B_{2}\oplus E$	Nonlinearity
Input	$9\oplus 3\oplus 6\oplus 5\oplus 11$	–
Expanded 1	$128\oplus 115\oplus 130\oplus 123\oplus 264$	–
Hidden 1	$32\oplus 32\oplus 32\oplus 32\oplus 32$	ReLU
Expanded 2	$2512\oplus 2576\oplus 2576\oplus 2576\oplus 4128$	–
Hidden 2	$8\oplus 8\oplus 8\oplus 8\oplus 8$	ReLU
Expanded 3	$148\oplus 164\oplus 164\oplus 164\oplus 264$	–

Table 1: Architecture of the ENN for the 2D Holstein model. Entries give channel multiplicities in the indicated IR order; each

E

channel has two components. Hidden layers use ReLU for the amplitude activation

\sigma

in Eq. ( 9 ).

Figure 3: (a) Scatter plot comparing the forces predicted by the ENN, $F_{\mathrm{ML}}$ , with the exact forces $F_{\mathrm{exact}}$ obtained from exact diagonalization. Blue and orange symbols denote training and test configurations, respectively. (b) Histogram of the force prediction error $\delta=F_{\mathrm{ML}}-F_{\mathrm{exact}}$ for the test set, showing a narrow, approximately symmetric distribution centered around zero.

Figure 4: Equal-time lattice correlation functions $C_{ij}=\langle Q_{i}Q_{j}\rangle$ as a function of the site separation $r_{ij}$ following a thermal quench to $T=0.1$ , shown at three representative times: (a) $n_{\rm step}=200$ , (b) $n_{\rm step}=1000$ , and (c) $n_{\rm step}=9000$ . Blue circles denote results obtained from Langevin dynamics using ENN-predicted forces, while red triangles correspond to exact diagonalization (ED)–based Langevin simulations. Each data point is averaged over 30 independent runs to reduce statistical fluctuations.

Figure 5: Coarsening dynamics of the CDW phase in the Holstein model from ENN-based Langevin dynamics simulations. (a)–(c) Real-space snapshots of the CDW order-parameter field (top) and the corresponding ensemble-averaged structure factors (bottom) at representative times following a thermal quench. (d) Scaled structure factor $S(q,t)/L^{2}(t)$ versus $qL(t)$ at different times, showing approximate data collapse over the displayed time window. (e) Characteristic length scale $L(t)$ extracted from the structure factor as a function of time for three temperatures, showing slow growth with fitted apparent exponents smaller than the Allen–Cahn value $1/2$ .

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

IR	$e$	$2C_{4}$	$C_{2}$	$2\sigma_{v}$	$2\sigma_{d}$	Basis examples
$A_{1}$	1	1	1	1	1	$1$ , $x^{2}+y^{2}$
$A_{2}$	1	1	1	$-1$	$-1$	$xy(x^{2}-y^{2})$
$B_{1}$	1	$-1$	1	1	$-1$	$x^{2}-y^{2}$
$B_{2}$	1	$-1$	1	$-1$	1	$xy$
$E$	2	0	$-2$	0	0	$(x,y)$

Appendix

Table 2: Character table for the planar

D_{4}

action. Here

e

is the identity,

\sigma_{v}

denotes reflection about a coordinate axis, and

\sigma_{d}

denotes reflection about a diagonal.

$\Gamma$	$C^{\Gamma}_{xx}$	$C^{\Gamma}_{yy}$	$C^{\Gamma}_{xy}$	$C^{\Gamma}_{yx}$
$A_{1}$	$+1$	$+1$	$0$	$0$
$A_{2}$	$0$	$0$	$+1$	$-1$
$B_{1}$	$+1$	$-1$	$0$	$0$
$B_{2}$	$0$	$0$	$+1$	$+1$

Appendix

Table 3: Nonzero Clebsch–Gordan coefficients

C^{\Gamma}_{ij}

for the decomposition

E\otimes E\rightarrow\Gamma

in the point group

D_{4}\simeq C_{4v}

, defined by

f^{(\Gamma)}=\sum_{i,j=x,y}C^{\Gamma}_{ij}f_{i}^{(E,1)}f_{j}^{(E,2)}

	$A_{1}$	$A_{2}$	$B_{1}$	$B_{2}$	$E$
$A_{1}$	$A_{1}$	$A_{2}$	$B_{1}$	$B_{2}$	$E$
$A_{2}$		$A_{1}$	$B_{2}$	$B_{1}$	$E$
$B_{1}$			$A_{1}$	$A_{2}$	$E$
$B_{2}$				$A_{1}$	$E$
$E$					$A_{1}\oplus A_{2}\oplus B_{1}\oplus B_{2}$

Appendix

Table 4: Direct products of irreducible representations of the

D_{4}

point group.

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