Anomalous coarsening and nonlinear diffusion of kinks in an one-dimensional quasi-classical Holstein model

TL;DR

The paper addresses anomalously slow CDW coarsening in a one-dimensional Holstein model at half-filling, where domain growth is mediated by topological kinks. It combines full-model simulations with a quasi-classical limit (no electron hopping) to isolate the electronic constraints on kink dynamics, revealing a temperature-dependent departure from the classic diffusion-law growth. The authors show that cooperative two-kink hopping, required to conserve electron number, leads to a density-dependent diffusion $D\sim D_0\rho^a$ and a coarsening law $\rho\sim t^{-1/(2+a)}$, i.e., $z=2+a$, with $a$ increasing as temperature decreases. The mechanism persists across canonical and grand-canonical ensembles, offering a transparent framework for constrained defect dynamics with broad implications for phase ordering in the full Holstein model and related electron–phonon systems.

Abstract

We study the phase-ordering dynamics of a quasi-classical Holstein model. At half-filling, the zero-temperature ground state is a commensurate charge-density-wave (CDW) with alternating occupied and empty sites. This quasi-classical formulation enables us to isolate the role of electrons in coarsening dynamics. Following a thermal quench, CDW domains grow through the diffusion and annihilation of kinks -- topological defects separating the two symmetry-related CDW orders. While standard diffusive dynamics predicts domain sizes scaling as the square root of time, our large-scale simulations reveal a slower power-law growth with a temperature-dependent exponent. We trace this anomalous behavior to a cooperative kink hopping arising from Fermi-Dirac statistics of electrons and quasi-conservation of electron numbers. The correlated-hopping of kinks in turn gives rise to an effective diffusion coefficient that depends on the kink density. These results uncover a new mechanism for slow coarsening and carry implications for phase-ordering in the full Holstein model and related electron-phonon systems.

Figures (9)

• Figure 1: CDW order in 1D and kinks. The electron density modulation is described by a scalar order parameter $\Delta$ as $n_i = \overline{n} + \Delta (-1)^i$. The two degenerate ground states in panels (a) and (b) correspond to opposite signs of $\Delta$. In an inhomogeneous CDW states, domains of opposite CDW order are connected by topological defects, known as kinks, as illustrated in panels (c) and (d).
• Figure 2: Density of kinks $\rho$ versus time of the 1D semi-classical Holstein model with various hopping amplitudes $t_{\rm nn}$ at the temperature $T = 0.032$.
• Figure 3: Configuration snapshots of the quasi-classical Holstein model following a thermal quench, showing the electron occupation number $n_i$ (top panels) and the corresponding local lattice distortion $Q_i$ (bottom panels) at three representative times: (a) $t = 1$, (b) $t=200$, and (c) $t=10000$.
• Figure 4: Spatial correlation function of the lattice displacement field, $C(r,t)$, evaluated at several representative times following the thermal quench for two temperatures. Here $r = r_{ij}$ is the distance between a pair of sites-$i$ and $j$. Panels (a) and (b) show results for $T=0.048$ and $T=0.08$, respectively.
• Figure 5: Density of kinks $\rho$ versus time of the 1D quasi-classical Holstein model at varying quench temperatures. The dashed lines correspond to power-law decay $1/t^{1/(2+a)}$, with the nonlinear parameter $a = 1.1$, 1.35, 1.7, 2 for temperatures $T = 0.08$, 0.064, 0.048, 0.032 (from bottom to top). Simulations are performed on a chain of length $N = 10000$.