Equivariant Neural Networks for Force-Field Models of Lattice Systems

TL;DR

This work addresses the challenge of modeling lattice-system dynamics with symmetry-aware, scalable force fields. It introduces symmetry-preserving equivariant neural networks (ENNs) that operate on IR-decomposed local environments to predict on-site forces while exactly respecting discrete lattice symmetries. Applying the framework to the Holstein model, the authors demonstrate accurate local force predictions and enable large-scale simulations that reveal anomalously slow, temperature-dependent CDW coarsening and dynamical scaling, unobserved in smaller-scale studies. The approach offers a general, data-efficient pathway to connect microscopic electronic processes with mesoscale dynamical phenomena in strongly correlated lattice systems, with potential applicability to a broad class of electron–lattice and electron–spin models.

Abstract

Machine-learning (ML) force fields enable large-scale simulations with near-first-principles accuracy at substantially reduced computational cost. Recent work has extended ML force-field approaches to adiabatic dynamical simulations of condensed-matter lattice models with coupled electronic and structural or magnetic degrees of freedom. However, most existing formulations rely on hand-crafted, symmetry-aware descriptors, whose construction is often system-specific and can hinder generality and transferability across different lattice Hamiltonians. Here we introduce a symmetry-preserving framework based on equivariant neural networks (ENNs) that provides a general, data-driven mapping from local configurations of dynamical variables to the associated on-site forces in a lattice Hamiltonian. In contrast to ENN architectures developed for molecular systems -- where continuous Euclidean symmetries dominate -- our approach aims to embed the discrete point-group and internal symmetries intrinsic to lattice models directly into the neural-network representation of the force field. As a proof of principle, we construct an ENN-based force-field model for the adiabatic dynamics of the Holstein Hamiltonian on a square lattice, a canonical system for electron-lattice physics. The resulting ML-enabled large-scale dynamical simulations faithfully capture mesoscale evolution of the symmetry-breaking phase, illustrating the utility of lattice-equivariant architectures for linking microscopic electronic processes to emergent dynamical behavior in condensed-matter lattice systems.

Figures (6)

• Figure 1: Schematic of the scalable ML force-field framework based on an equivariant neural network (ENN) for condensed-matter lattice systems. The ML model maps the local classical field configurations $\mathcal{C}_i$ in the neighborhood of site $i$ to the corresponding local energy $\epsilon_i$ or force vector $\bm{\mathcal{F}}_i$. The classical variables in $\mathcal{C}i$ are first decomposed into symmetry-adapted basis functions $\bm f^{(\Gamma, r)}$ associated with the irreducible representations (IRs) of the lattice symmetry group---$D_4$ (or $C_{4v}$) for the square lattice case shown here. These IR-adapted components serve as inputs to the ENN, in which every hidden unit and output channel transforms according to a specified IR, ensuring symmetry-consistent predictions.
• Figure 2: Implementation of an equivariant neural network (ENN) for point-group symmetries relevant to condensed-matter lattice systems. Each node $x^{(\Gamma, r)}_k$ transforms as a component of the symmetry-adapted basis of the irreducible representation (IR) $\Gamma$ of the point group. Panel (a) depicts the forward propagation of node features from one layer $\bm x^{(\Gamma, r)}$ to the next $\bm y^{(\Gamma, r)}$ via two distinct channels. The first channel, shown in panel (b), is a fully connected transformation acting within a single IR sector. The second channel, illustrated in panel (c), mixes features belonging to different IRs: symmetry-preserving tensor products of prior-layer features are combined using Clebsch-Gordan coefficients to form intermediate nodes $\tilde{\bm x}^{(\Gamma, r)}$. These IR products follow the multiplication rules summarized in panel (d). The resulting intermediate features are then propagated to the next layer through another fully connected transformation, completing the equivariant update.
• Figure 3: Schematic of the Holstein model, illustrating itinerant electrons coupled to scalar dynamical variables $Q_i$ that represent local $A_1$-type structural distortions at each lattice site.
• Figure 4: (a) Scatter plot comparing the force components 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 5: 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.