Machine learning nonequilibrium phase transitions in charge-density wave insulators

TL;DR

The paper addresses the computational bottleneck of nonequilibrium electronic forces in voltage-driven CDW insulators by developing a neural-network force-field that directly predicts instantaneous local forces from the local lattice environment in the Holstein model. Under adiabatic lattice dynamics, electronic forces follow $F_i = g \langle \hat{n}_i \rangle - k Q_i - \kappa \sum_{j \in \mathcal{N}(i)} Q_j$, with the nonequilibrium electron density from NEGF, and the lattice evolves via $dQ_i/dt = -(1/\gamma) F_i + \eta_i(t)$. The model employs symmetry-adapted descriptors and a deep network trained on NEGF data, achieving close agreement with NEGF-BD in reproducing domain-wall propagation during the CDW-to-metal transition on a $28\times 30$ lattice, while offering orders-of-magnitude speedups. This work demonstrates that ML force-field frameworks can be extended to nonequilibrium, nonconservative electronic forces and points toward future use of equivariant networks for vector or tensor force fields in driven quantum materials.

Abstract

Nonequilibrium electronic forces play a central role in voltage-driven phase transitions but are notoriously expensive to evaluate in dynamical simulations. Here we develop a machine learning framework for adiabatic lattice dynamics coupled to nonequilibrium electrons, and demonstrate it for a gating induced insulator to metal transition out of a charge density wave state in the Holstein model. Although exact electronic forces can be obtained from nonequilibrium Green's function (NEGF) calculations, their high computational cost renders long time dynamical simulations prohibitively expensive. By exploiting the locality of the electronic response, we train a neural network to directly predict instantaneous local electronic forces from the lattice configuration, thereby bypassing repeated NEGF calculations during time evolution. When combined with Brownian dynamics, the resulting machine learning force field quantitatively reproduces domain wall motion and nonequilibrium phase transition dynamics obtained from full NEGF simulations, while achieving orders of magnitude gains in computational efficiency. Our results establish direct force learning as an efficient and accurate approach for simulating nonequilibrium lattice dynamics in driven quantum materials.

Figures (4)

• Figure 1: Machine-learning framework for adiabatic dynamics of a voltage-driven Holstein system. (a) Schematic of the gated Holstein setup: a central square-lattice Holstein system is coupled to two electrodes, with the right electrode acting as a substrate and the left electrode providing electrostatic gating. An applied voltage $V$ drives the system out of equilibrium, while local lattice distortions $Q_i$ evolve adiabatically in response to the electronic degrees of freedom. The dashed region indicates the local neighborhood $\mathcal{C}_i$ of a reference site-$i$ used as input to the ML model. (b) Overview of the ML force-prediction architecture, which generalizes the Behler-Parrinello framework to driven lattice systems. The neighborhood lattice distortions are first decomposed into symmetry-adapted components and processed into descriptor variables that are invariant under the on-site lattice point-group symmetry (here the mirror group $C_s$). These invariant features are then fed into a fully connected neural network, whose output directly yields the local force $F_i$ acting on the lattice degree of freedom at site-$i$.
• Figure 2: Benchmark of ML-predicted electronic forces against NEGF results. (a) Scatter plot comparing ML-predicted forces with reference NEGF forces for the training (blue) and test (orange) datasets. Forces are rescaled to the interval $[-1,1]$ for comparison. (b) Normalized distribution of the prediction error $\delta = F_{\mathrm{NEGF}} - F_{\mathrm{ML}}$ for the test dataset, illustrating the accuracy and bias of the ML force predictions.
• Figure 3: Comparison of metallic-phase expansion obtained from NEGF-BD and ML-BD simulations under identical driving conditions. Panels (a) and (b) show snapshots of the on-site electron density $n_i$ on a $28\times30$ lattice region at representative time steps, demonstrating close quantitative and qualitative agreement between the two approaches in capturing the nonequilibrium expansion of the metallic phase.
• Figure 4: Time evolution of the CDW-metal interface position $\xi$ obtained from Brownian-dynamics quench simulations. Yellow triangles represent results from NEGF-driven dynamics, while green circles correspond to ML-driven simulations. The close quantitative agreement between the two trajectories demonstrates that the ML force field accurately reproduces the nonequilibrium electronic driving forces governing domain-wall propagation.