Graph neural network force fields for adiabatic dynamics of lattice Hamiltonians

TL;DR

A GNN-based force-field framework for the adiabatic dynamics of lattice Hamiltonians is developed and demonstrated and it is established as an elegant and efficient architecture for symmetry-aware, large-scale dynamical simulations of correlated lattice systems.

Abstract

Scalable and symmetry-consistent force-field models are essential for extending quantum-accurate simulations to large spatiotemporal scales. While descriptor-based neural networks can incorporate lattice symmetries through carefully engineered features, we show that graph neural networks (GNNs) provide a conceptually simpler and more unified alternative in which discrete lattice translation and point-group symmetries are enforced directly through local message passing and weight sharing. We develop a GNN-based force-field framework for the adiabatic dynamics of lattice Hamiltonians and demonstrate it for the semiclassical Holstein model. Trained on exact-diagonalization data, the GNN achieves high force accuracy, strict linear scaling with system size, and direct transferability to large lattices. Enabled by this scalability, we perform large-scale Langevin simulations of charge-density-wave ordering following thermal quenches, revealing dynamical scaling and anomalously slow sub--Allen--Cahn coarsening. These results establish GNNs as an elegant and efficient architecture for symmetry-aware, large-scale dynamical simulations of correlated lattice systems.

Figures (7)

• Figure 1: Schematic of the BP-type force-field architecture for the semiclassical Holstein model. Starting from the input lattice distortion configuration $\{Q_i\}$ (left), the local environment within a cutoff radius $r_c$ around a target site $i$ is first processed through a symmetry-adapted descriptor. The descriptor maps the local configuration to a set of generalized coordinates $\{G_m\}$, constructed such that all eight symmetry-related configurations generated by the $D_4$ point-group operations of the square lattice yield identical descriptor values. These symmetry-invariant descriptors $\{G_m\}$ are then fed into a fully connected neural network, which outputs a single scalar $F_i$ corresponding to the force acting on site $i$.
• Figure 2: Force prediction benchmark for the descriptor+MLP model. (a) Parity plot comparing the machine-learning predicted force $F_{\mathrm{ML}}$ with the exact diagonalization result $F_{\mathrm{ED}}$ for both training (blue) and test (orange) datasets. The near-perfect alignment along the diagonal demonstrates excellent predictive accuracy and generalization. (b) Distribution of the force prediction error $\delta = F_{\mathrm{ML}} - F_{\mathrm{ED}}$. The error histogram is sharply peaked around zero, indicating unbiased predictions with small variance and high numerical fidelity.
• Figure 3: Schematic architecture of the graph neural network (GNN) for learning the adiabatic dynamics of the Holstein model. Each layer of the network corresponds to a lattice representation of the physical system. The input layer consists of the on-site lattice distortion configuration ${Q_i}$ defined on lattice sites. At each hidden layer, the lattice sites carry multi-channel scalar node features $V^{(\ell)}_{i,\alpha}$ that encode progressively more nonlocal information through successive message-passing operations. The output layer produces the predicted forces $F_i$ acting on each lattice site. The message-passing update at each layer is illustrated schematically as a single-layer fully connected neural network acting on the local neighborhood of a site, combining same-site contributions ($W^{(\mathrm{self})}$) and neighbor contributions ($W^{(\mathrm{NN})}$) with shared parameters across the lattice. This locality-preserving and weight-sharing structure ensures translation-equivariant predictions and scalability to large lattice sizes.
• Figure 4: Force prediction benchmark for the GNN model. (a) Parity plot comparing the machine-learning predicted forces $F_{\mathrm{ML}}$ with the exact-diagonalization reference values $F_{\mathrm{ED}}$ for both training (blue) and test (orange) datasets. The close alignment of data points along the diagonal indicates high predictive accuracy and good generalization. (b) Distribution of the force prediction error $\delta = F_{\mathrm{ML}} - F_{\mathrm{ED}}$. The error histogram is narrowly centered around zero, demonstrating unbiased predictions with small variance and high numerical accuracy.
• Figure 5: Force prediction benchmark for the energy-based GNN model. (a) Parity plot of machine-learning forces $F_{\mathrm{ML}}$---obtained from automatic differentiation of the predicted energy---versus exact-diagonalization values $F_{\mathrm{ED}}$ for training (blue) and test (orange) data. The close alignment along the diagonal indicates high accuracy and good generalization. (b) Distribution of the force error $\delta = F_{\mathrm{ML}} - F_{\mathrm{ED}}$, sharply centered at zero, demonstrating unbiased and low-variance predictions.