Suppressed coarsening after an interaction quench in the Holstein chain

TL;DR

We address how coarsening-like ordering unfolds after an interaction quench in an isolated quantum–classical system, focusing on the one-dimensional semiclassical Holstein model under energy-conserving Ehrenfest dynamics. Using large-scale simulations at half filling, we identify three dynamical regimes (nonequilibrium metal, quasi-coarsening, and arrested CDW) and show that the intermediate regime exhibits slow, scale-invariant domain growth with $L(t)\sim t^{1/3}$ and $n(t)\sim t^{-1/3}$, driven by kink motion that is diffusive but intermittently scattered by the electronic background. To explain this, we develop an effective kink-kinetics model with unbiased diffusion and stochastic annihilation or elastic scattering upon encounters, reproducing the observed scaling and highlighting the role of an internal bath in reshaping defect dynamics without dissipation. These results reveal a distinct coarsening mechanism in isolated hybrid systems and point to a broader class of prethermal, energy-conserving dynamics where internal quantum degrees of freedom qualitatively modify defect kinetics.

Abstract

We investigate the nonequilibrium dynamics induced by an interaction quench in the semiclassical Holstein model within the Ehrenfest nonadiabatic framework, which describes an isolated hybrid quantum-classical system with strictly conserved total energy. Focusing on the half-filled case, where the equilibrium ground state exhibits commensurate charge-density-wave (CDW) order for any nonzero coupling, we identify three distinct post-quench dynamical regimes as a function of the final electron-phonon coupling: a nonequilibrium metallic state without CDW order, an intermediate regime characterized by slow scale-invariant ordering dynamics, and a frozen CDW state with arrested coarsening and immobile kinks. We analyze the intermediate regime in detail and uncover an unconventional algebraic decay of the kink density, $n \sim t^{-1/3}$, distinct from both ballistic annihilation and diffusive coarsening in classical dissipative systems. We show that this anomalous exponent arises from the hybrid nature of the dynamics: while the lattice evolves deterministically, the electronic degrees of freedom act as an effective internal bath that induces diffusive kink motion without energy dissipation. An effective reaction-diffusion description, incorporating both annihilation and elastic scattering of kinks, quantitatively accounts for the observed scaling behavior. Our results reveal a distinct coarsening mechanism in isolated hybrid systems, demonstrating how internal quantum dynamics can qualitatively reshape defect kinetics far from equilibrium.

Figures (6)

• Figure 1: Dynamical phase diagram of the post-quench Holstein model. Shown is the spatiotemporal evolution of the local CDW order parameter $\phi_i(t)$ (top row) following an interaction quench at fixed adiabatic parameter $r=0.3$, for representative final electron–phonon couplings $\lambda$. The corresponding lattice displacements $Q_i(t)$ within the dashed regions are displayed in the bottom row. Three qualitatively distinct dynamical regimes are identified. Left: a nonequilibrium metallic regime ($\lambda < \lambda_{c1}$), in which CDW correlations remain short-ranged and fluctuate around zero, with no stable domain formation. Middle: a quasi-coarsening regime ($\lambda_{c1} < \lambda < \lambda_{c2}$), characterized by the nucleation of CDW domains separated by mobile kinks whose dynamics leads to slow, reaction-limited coarsening. Right: an arrested CDW regime ($\lambda > \lambda_{c2}$), where strong local CDW order develops rapidly, but kink motion is progressively suppressed at late times, resulting in frozen domain walls and dynamically arrested coarsening. From our numerical simulations, we estimate the critical couplings as $\lambda_{c1} \approx 0.4$ and $\lambda_{c2} \approx 1.0$.
• Figure 2: Nonequilibrium steady-state distributions in the weak-coupling metallic phase. (a) Late-time momentum-resolved electronic occupation $f_{\rm neq}(k)$, obtained from the single-particle density matrix in the nonadiabatic simulations. This distribution serves as the nonequilibrium analogue of the Fermi-Dirac function and retains a sharp Fermi surface, consistent with a metallic state. (b) Late-time distributions of the lattice displacement $Q$ and conjugate momentum $P$, which remain narrowly peaked and approximately Gaussian, indicating weak lattice fluctuations and the absence of CDW order.
• Figure 3: (a) Equal-time correlation function $C(r,t)$ of the local CDW order parameter, defined in Eq. (\ref{['eq:corr-func']}), shown at representative times after the quench. Correlations progressively extend to larger length scales as time evolves. The dashed line indicates one half of the on-site value $C(0,t)$. (b) Time evolution of the correlation length $L(t)$, defined implicitly by $C(L,t)=\tfrac{1}{2}C(0,t)$. At late times, this characteristic length displays an algebraic growth $L(t)\sim t^{1/3}$, substantially slower than the diffusive coarsening expected for random-walk dynamics.
• Figure 4: Kink decay in the quasi-coarsening regime. Time evolution of the kink density $n(t)$ following the interaction quench. At late times, $n(t)$ exhibits an algebraic decay $n(t)\sim t^{-1/3}$ (dashed line). Over the full time window, this decay tracks the growth of the characteristic length scale via $L(t)\sim 1/n(t)$, indicating that the post-quench dynamics in the quasi-coarsening regime are governed by kink motion and interaction-driven annihilation.
• Figure 5: Space–time evolution of CDW domains and kink trajectories. Representative space–time configurations of the local CDW order parameter in the quasi-coarsening regime, illustrating the dynamics of kinks (domain walls) separating CDW domains of opposite phase (indicated by contrasting colors). The kink trajectories are visible as the interfaces between domains. Magenta crosses mark kink-kink annihilation events, while pairs of black circles with outward-pointing arrows denote scattering events in which kinks collide but do not annihilate. The coexistence of annihilation and nonannihilating scattering highlights the reaction-limited nature of the coarsening dynamics.