Kinetics of orbital ordering in cooperative Jahn-Teller models: Machine-learning enabled large-scale simulations

TL;DR

The study tackles the kinetics of orbital ordering in cooperative Jahn-Teller systems relevant to CMR manganites by developing a symmetry-aware, locality-based neural-network force field to predict electronic forces driving adiabatic JT dynamics. The model adopts $E_{\mathrm{ML}}=\sum_i \epsilon_i$ with site energies $\epsilon_i=\epsilon(\mathcal{C}_i)$ and computes conservative forces via $\mathbf F_i=-\partial E_{\mathrm{ML}}/\partial \mathcal{Q}_i$, trained on exact diagonalization data for $30\times30$ lattices. Large-scale Langevin simulations on $100\times100$ lattices reveal a two-stage coarsening of the $C$-type orbital/JT order, with an initial rapid growth followed by late-stage freezing tied to nearly straight domain walls and interfacial anisotropy. This work demonstrates a scalable framework for multi-scale modeling of correlated electron systems and sets the stage for incorporating spin dynamics and Hubbard interactions in future BP-type ML force fields for CMR materials, enabling more comprehensive simulations of their rich phase behavior.

Abstract

We present a scalable machine learning (ML) force-field model for the adiabatic dynamics of cooperative Jahn-Teller (JT) systems. Large scale dynamical simulations of the JT model also shed light on the orbital ordering dynamics in colossal magnetoresistance manganites. The JT effect in these materials describes the distortion of local oxygen octahedra driven by a coupling to the orbital degrees of freedom of $e_g$ electrons. An effective electron-mediated interaction between the local JT modes leads to a structural transition and the emergence of long-range orbital order at low temperatures. Assuming the principle of locality, a deep-learning neural-network model is developed to accurately and efficiently predict the electron-induced forces that drive the dynamical evolution of JT phonons. A group-theoretical method is utilized to develop a descriptor that incorporates the combined orbital and lattice symmetry into the ML model. Large-scale Langevin dynamics simulations, enabled by the ML force-field models, are performed to investigate the coarsening dynamics of the composite JT distortion and orbital order after a thermal quench. The late-stage coarsening of orbital domains exhibits pronounced freezing behaviors which are likely related to the unusual morphology of the domain structures. Our work highlights a promising avenue for multi-scale dynamical modeling of correlated electron systems.

Figures (11)

• Figure 1: Schematic diagram of the vibronic modes for the MnO6 octahedron: (a) the symmetry-preserving breathing mode, (b) and (c) the symmetry-breaking JT modes. In terms of oxygen displacements, the coordinates of these normal modes are: $Q^{A_1} = (-X_1 + X_2 - X_3 + X_4 - X_5 + X_6)/\sqrt{6}$, $Q^x = (-X_1+X_2 +X_3 - X_4)/2$, and $Q^z = (-X_1 + X_2 - X_3 + X_4 + 2 X_5 - 2 X_6)/\sqrt{12}$, where $X_i$ denotes the $x$ coordinates of the $i$-th oxygen, and so on.
• Figure 2: Schematic diagram of (a) $C$-type orbital order, and (b) the concomitant antiferro-distortive JT order. Also shown in panel (b) is the vector representation of the orbital/JT order. The arrows represent either the doublet vector $\bm Q^E$ or the expectation value of the orbital pseudo-spin $\langle \hat{\bm\tau} \rangle$. These two vectors are related to each other via Eq. (\ref{['eq:tau-Q']}) in the ground state.
• Figure 3: Schematic diagram of the ML force-field model for the cooperative JT models. A lattice descriptor transforms the neighborhood distortion configuration $\mathcal{C}_i$ into effective coordinates $\{ G_m \}$ which are then fed into a fully connected neural network (NN). The output node of the NN corresponds to the local site-energy $\epsilon_i$. The combination of the descriptor and the NN provides an approximation for the universal function $\varepsilon(\cdot)$. The corresponding total potential energy $E$ is obtained from the summation of the local energies. Automatic differentiation is employed to compute the derivatives $\partial E_{\rm ML} / \partial \bm{\mathcal{Q}}_i$ for the effective forces acting on the breathing and JT modes.
• Figure 4: Benchmark of the ML force-field models for JT model with a filling fraction $f = 0.49$. Panels (a)--(c) on the left show the ML predicted forces $F^{\rm ML}$ versus the ground truth $F^{\rm ED}$ obtained from exact diagonalization method for the three vibronic modes of the octahedron. These forces are normalized by the electron-phonon coupling constant, which is set to $\lambda = 1.25$. The corresponding histograms of the prediction errors are shown in panels (d)--(f). Similar results are also obtained for the ML models for the half-filling case.
• Figure 5: Comparison of time-dependent correlation functions $C^{xx}(r, t)$ defined in Eq. (\ref{['eq:Cxx']}) obtained from Langevin simulations with the ML force-field model and the ED method. The correlation functions were obtained from 40 independent thermal quench simulations on a $30\times 30$ lattice with a filling fraction $f = 0.49$.