Machine Learning Modeling of Charge-Density-Wave Recovery After Laser Melting

TL;DR

This work tackles nonequilibrium CDW dynamics in a laser-driven Holstein model by separating lattice forces into a slow, quasi-adiabatic component and a fast electronic bath term. A graph neural network learns the adiabatic force as a local, time-dependent functional of the lattice, enabling linear-scaling simulations, while a minimal Langevin bath captures residual nonadiabatic effects during recovery. The combined ML force-field and bath model reproduce long-time CDW recovery and real-space domain patterns with high fidelity, offering a scalable route to driven correlated materials beyond direct nonadiabatic methods. The approach generalizes to other driven electron–phonon and correlated systems, potentially enabling practical simulations of ultrafast dynamics on large lattices.

Abstract

We investigate the nonequilibrium dynamics of a laser-pumped two-dimensional spinless Holstein model within a semiclassical framework, focusing on the melting and recovery of long-range charge-density-wave order. Accurately describing this process requires fully nonadiabatic electron-lattice dynamics, which is computationally demanding due to the need to resolve fast electronic motion over long time scales. By analyzing the structure of the lattice force during nonequilibrium evolution, we show that the force naturally separates into a smooth quasi-adiabatic component and a residual bath-like contribution associated with fast electronic fluctuations. The quasi-adiabatic component depends only on the instantaneous local lattice configuration and can be efficiently learned using machine-learning techniques, while a minimal Langevin description of the bath term captures the essential features of the recovery dynamics. Combining these elements enables efficient and scalable simulations of long-time nonequilibrium dynamics on large lattices, providing a practical route to access driven correlated systems beyond the reach of direct nonadiabatic approaches.

Figures (6)

• Figure 1: Pump-induced melting and recovery of the lattice order parameter. Colored curves show individual stochastic realizations of order parameter, with blue (red) indicating positive (negative) long-time values. The black solid line denotes the ensemble-averaged order parameter $O(t)$, while the green solid line shows the ensemble-averaged spatial fluctuation $\sigma(t)$. The left panel shows early-time dynamics ($t/\tau_0 \leq 20$), highlighting the transient suppression of charge order after excitation, and the right panel displays the long-time recovery dynamics.
• Figure 2: Time evolution of the local density $n_i(t)$ for two representative sites. (a) Early-time dynamics ($t/\tau_0 \leq 15$), showing the comparison between the non-adiabatic evolution $n_i^{\mathrm{nad}}(t)$ (faint blue and red colors) and the adiabatic reference $n_i^{\mathrm{ad}}(t)$ (solid red and blue). (b) Late-time dynamics ($400 \leq t/\tau_0 \leq 415$). Insets in both panels show the distribution of $\chi_i = n_i^{\mathrm{ad}}(t) - n_i^{\mathrm{nad}}(t)$ within the corresponding time windows.
• Figure 3: Schematic of the graph neural network (GNN) framework for learning the adiabatic lattice force. (Left) Snapshot of lattice distortions $\{Q_i\}$ in the two-dimensional Holstein model following excitation by a short laser pulse, illustrating a nonequilibrium distorted configuration. The set of on-site distortions $\{Q_i\}$ serves as the input to the machine-learning model. (Middle) Graph neural network architecture, where lattice sites are represented as nodes and local neighborhoods are encoded through message-passing layers. Multiple hidden layers iteratively aggregate information from nearby sites, capturing the local environment dependence implied by the locality principle. (Right) GNN output corresponding to the predicted adiabatic on-site force $F_i^{\rm ad}$ acting on each lattice site. The model thus provides a linear-scaling, symmetry-aware mapping from instantaneous lattice configurations to the adiabatic force field.
• Figure 4: Benchmark of the machine-learning force prediction against the exact adiabatic force. (a) Scatter plot comparing the ML-predicted force $F^{\rm ML}$ with the adiabatic force $F^{\rm ad}$ obtained from the reference calculation. The close clustering of data points along the diagonal indicates excellent agreement over the full force range. (b) Histogram of the prediction error $\Delta F = F^{\rm ML} - F^{\rm ad}$, demonstrating a narrow, approximately symmetric error distribution centered near zero.
• Figure 5: Recovery dynamics of the order parameter following photoexcitation. The exact nonadiabatic evolution (black solid line) is ensemble averaged over 30 independent initial conditions, with the shaded band indicating the corresponding standard deviation. The ML-based dynamics (red solid line) are averaged over the same number of realizations. The close agreement between the two curves, together with the overlap within the fluctuation window, demonstrates the accuracy of the learned force model combined with the effective bath description.