Enhanced coarsening of charge density waves induced by electron correlation: Machine-learning enabled large-scale dynamical simulations

TL;DR

The study tackles non-equilibrium phase ordering in strongly correlated electron systems, where conventional theories struggle to capture coupled electronic and order-parameter dynamics at large scales. It introduces a symmetry-aware neural-network force-field that predicts local lattice forces in the adiabatic Hubbard-Holstein framework, enabling linear-scaling Langevin dynamics and large-scale CDW simulations. The authors demonstrate that electron correlation can enhance CDW coarsening via a disorder-screening mechanism tied to self-energy renormalization, with late-time dynamics obeying $C(r,t)=f(r/L(t))$ and $L(t)\sim t^{1/2}$ in certain regimes. This work provides a scalable route for multi-scale modeling of correlated electron dynamics and offers new physical insight into how electron-electron interactions reshape non-equilibrium CDW growth.

Abstract

The phase ordering kinetics of emergent orders in correlated electron systems is a fundamental topic in non-equilibrium physics, yet it remains largely unexplored. The intricate interplay between quasiparticles and emergent order-parameter fields could lead to unusual coarsening dynamics that is beyond the standard theories. However, accurate treatment of both quasiparticles and collective degrees of freedom is a multi-scale challenge in dynamical simulations of correlated electrons. Here we leverage modern machine learning (ML) methods to achieve a linear-scaling algorithm for simulating the coarsening of charge density waves (CDWs), one of the fundamental symmetry breaking phases in functional electron materials. We demonstrate our approach on the square-lattice Hubbard-Holstein model and uncover an intriguing enhancement of CDW coarsening which is related to the screening of on-site potential by electron-electron interactions. Our study provides fresh insights into the role of electron correlations in non-equilibrium dynamics and underscores the promise of ML force-field approaches for advancing multi-scale dynamical modeling of correlated electron systems.

Figures (4)

• Figure 1: Machine learning (ML) force-field model for the adiabatic Hubbard-Holstein model. a Schematic diagram of the ML model for computing the force acting on the lattice degree of freedom $Q$ at site $\mathbf{r}_i$. The local lattice configuration $\mathcal{C}_i=\left\{Q_0,Q_1,\dots,Q_n\right\}$ is selected by the cutoff radius $R_c$. The size of disk represents the relative intensity of $Q$. The selected local lattice configuration is mapped into the symmetry-invariant descriptors $\left\{G_0,G_1,\dots,G_n\right\}$ as the input to the neural network using the irreducible representation of the point group $D_4$. The output of the neural network produces the force $F_i$. b- e: Benchmark for the forces computed from the ML force-field model for adiabatic dynamics of Hubbard-Holstein model. Panels b and d plot the ML predicted force $F_{\mathrm{ML}}$ vs the forces computed from the Gutzwiller approximation $F_{\mathrm{GA}}$ for Hubbard $U/t_{\mathrm{nn}}=0.6$ and $U/t_{\mathrm{nn}}=0.9$, respectively. The corresponding histograms, c and e, show the spread of the error $\delta F=F_{\mathrm{ML}}-F_{\mathrm{GA}}$. The standard deviations of the errors are $\sigma = 0.00577$ and 0.00370, respectively.
• Figure 2: Comparison of lattice correlation function $\langle Q_i Q_j\rangle$ obtained from Langevin simulations with the ML force-field model and the Gutswiller approximation (GA). Thermal quench simulations of a $14\times 14$ system at two different Hubbard $U/t_{\mathrm{nn}}=0.6$ ( a- c) and $U/t_{\mathrm{nn}}=0.9$ ( d- f) were carried out to produce these correlation functions at various time $t$ after the quench.
• Figure 3: Snapshots of local CDW parameter $\phi_i$ at various time after a thermal quench of the Holstein model for Hubbard $U/t_{\mathrm{nn}}=0.6$ ( a- c) and $U/t_{\mathrm{nn}}=0.9$ ( d- f). An initial random configuration is suddenly quenched to a temperature $k_B T/t_{\mathrm{nn}}= 0.001$ at time $t = 0$. The ML adiabatic Langevin dynamics is used to simulate the relaxation of the system toward equilibrium. The red and blue regions correspond to two types of CDW domains related by the $\mathbb{Z}_2$ symmetry, respectively.
• Figure 4: a The evolution of the characteristic length $L(t)$ of the CDW domain with respect to time for five Hubbard $U$ values ranging from $U/t_{\mathrm{nn}}=0.3$ to $U/t_{\mathrm{nn}}=1.5$. The upper three dashed lines show the growth of CDW domain recovers the Allen-Cahn growth law $L(t)\sim t^{1/2}$ (dashed line) when $U/t_{\mathrm{nn}}\geq 0.9$. b- e Collapse of the correlation functions onto to single curve after rescaling $r/L(t)$ for $U/t_{\mathrm{nn}}=0.6$, $0.9$, $1.2$ and $1.5$.