Kinetics of Peierls dimerization transition: Machine learning force-field approach

TL;DR

The study addresses the computational bottleneck in simulating nonequilibrium CDW dynamics driven by Peierls instability by developing a symmetry-aware Behler–Parrinello–type ML force field. By decomposing energy into local contributions and using group-theoretical bispectrum descriptors, the approach achieves $O(N)$ scaling while preserving electronic–lattice fidelity under the adiabatic approximation. Large-scale ML–Langevin simulations reveal a two-stage coarsening: an initial accelerated regime with $L(t) \sim t^{0.7}$ due to anisotropic domain-wall motion, and a late-time $L(t) \sim t^{1/2}$ Allen–Cahn regime; the ML results quantitatively reproduce ED benchmarks. This framework enables quantitative mesoscale modeling of complex electron–phonon dynamics and can be extended to more intricate orders and photoinduced transitions, offering a scalable path to multi-scale condensed-matter simulations.

Abstract

We present a machine learning (ML) force-field framework for simulating the non-equilibrium dynamics of charge-density-wave (CDW) order driven by the Peierls instability. Since the Peierls distortion arises from the coupling between lattice displacements and itinerant electrons, evaluating the adiabatic forces during time evolution is computationally intensive, particularly for large systems. To overcome this bottleneck, we develop a generalized Behler-Parrinello neural-network architecture -- originally formulated for ab initio molecular dynamics -- to accurately and efficiently predict forces from local structural environments. Using the locality of electronic responses, the resulting ML force field achieves linear scaling efficiency while maintaining quantitative accuracy. Large-scale dynamical simulations using this framework uncover a two-stage coarsening behavior of CDW domains: an early-time regime characterized by a power-law growth $L \sim t^α$ with an effective exponent $α\approx 0.7$, followed by a crossover to the Allen-Cahn scaling $L \sim \sqrt{t}$ at late times. The enhanced early-time coarsening is attributed to anisotropic domain-wall motion arising from electron-mediated directional interactions. This work demonstrates the promise of ML-based force fields for multiscale dynamical modeling of condensed-matter lattice models.

Figures (9)

• Figure 1: Schematic illustration of the four degenerate charge-density-wave (CDW) ground states of the Su–Schrieffer–Heeger (SSH) model on a square lattice. Colored edges denote the short, dimerized bonds corresponding to regions of enhanced charge density. The arrow colors indicate distinct domain types that emerge during the coarsening process, as shown in Fig. \ref{['fig:MLlargesimulation']}.
• Figure 2: Schematic diagram of the Behler-Parrinello architecture for ML force-field model of the Peierls system. The model comprises two main components: (i) a lattice descriptor and (ii) a multilayer neural network. The descriptor preserves the $D_4$ point-group symmetry of the lattice, with symmetry-adapted features $\{\eta_i^{(\Gamma,r)}\}$ constructed from the irreducible representations (IRs) of the local lattice environment $\mathcal{C}_i$. These features are input to the neural network to predict the local energy $\epsilon_i$, and the total energy $E$ is obtained by summing over all lattice sites. Atomic forces $\mathbf F _i$ are then computed as derivatives of the total energy with respect to the local lattice displacements $\mathbf{u}_i$.
• Figure 3: Benchmark of the ML force-field model against the exact force on the lattice. Panels (a) and (b) compare the ML-predicted forces $F_{\mathrm{ML}}$ with the exact forces $F_{\mathrm{eexact}}$ along the $x$ and $y$ directions, respectively. Panels (c) and (d) show the corresponding error distributions $\delta = F_{\mathrm{ML}} - F_{\mathrm{exact}}$ in each direction. The standard deviations of the prediction errors are $\sigma_x = 0.002$ and $\sigma_y = 0.003$.
• Figure 4: Comparison of time-dependent correlation functions $C(r,t)$ obtained from Langevin simulations using the ML force-field model and the exact-diagonalization (ED) method. The correlation function is evaluated as the average of $C^{xx}_{ij}$ and $C^{yy}_{ij}$ for site pairs $(i,j)$ separated by distance $r = r_{ij}$ along the diagonal direction. Each curve represents an ensemble average over 100 independent simulations on a $50\times50$ lattice with randomized initial conditions.
• Figure 5: Snapshots from ML-Langevin simulations on a $300\times300$ lattice showing the temporal evolution of the phase field $\theta_i$, defined via the local CDW order parameter $\bm{\phi}_i = (\phi^x_i, \phi^y_i)$ [Eq. (\ref{['eq:local_CDW_order']})]. The phase $\theta_i = \arctan(\phi^y_i / \phi^x_i)$ varies continuously from $0$ to $2\pi$, with $\theta_i = 0, \pi/2, \pi$, and $3\pi/2$ corresponding to the four degenerate ground states illustrated in Fig. \ref{['fig:4pipi']}. The bottom panels display vector plots of $\bm{\phi}_i$ within the white boxed regions of the corresponding top panels. At early times, the system exhibits diagonal domain structures that give rise to anomalous coarsening behavior. At later stages, the evolution crosses over to the conventional Allen-Cahn coarsening regime characterized by curvature-driven domain growth.