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arXiv:2512.07744

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

Jang, Ho, Yang, Yang, Chern, Gia-Wei

Published Dec 8, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2512.07744

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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.
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∼t−1/3n \sim t^{-1/3}n∼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.
The Holstein Hamiltonian describes fermions hopping on a lattice and interacting locally with dispersionless phonon degrees of freedom. In the low-density limit, dressed quasiparticles, polarons and bipolarons, propagate with an effective mass. At higher densities, pairs can condense into a low-temperature superconducting phase and, at or near commensurate filling on a bipartite lattice, to charge density wave (CDW) order. CDW formation breaks a discrete symmetry and hence occurs via a second-order (Ising) transition and therefore at a finite Tcdw{T}_{\mathrm{cdw}}Tcdw​ in two dimensions. Quantum Monte Carlo calculations have determined Tcdw{T}_{\mathrm{cdw}}Tcdw​ for a variety of geometries, including square, honeycomb, and Lieb lattices. The superconducting transition, on the other hand, in d=2d=2d=2 is in the Kosterlitz-Thouless universality class and is much less well characterized. In this paper we determine Tsc{T}_{\mathrm{sc}}Tsc​ for the square lattice for several values of the density $\ensuremath{\rho}$ and phonon frequency ${\ensuremath{\omega}}_{0}$. We find that quasilong-range order sets in at ${T}_{\mathrm{sc}}\ensuremath{\lesssim}t/20$, where ttt is the near-neighbor hopping amplitude, consistent with previous rough estimates from simulations which extrapolated to only the temperatures we reach from considerably higher TTT. We also show evidence of a discontinuous evolution of the density as the CDW transition is approached at half filling.
We present a pseudospin formulation for the post-quench dynamics of charge-density-wave (CDW) order in the half-filled spinless Holstein model on a square lattice, assuming spatially homogeneous evolution. This Anderson pseudospin description captures the coherent nonequilibrium dynamics of the coupled electron-lattice system. Numerical simulations reveal three distinct dynamical regimes of the CDW order parameter following a quench-locked oscillations, Landau-damped dynamics, and overdamped relaxation-closely paralleling quench dynamics in BCS superconductors and other electronically driven symmetry-breaking phases. Crucially, however, the presence of dynamical lattice degrees of freedom leads to qualitatively different long-time behavior. In particular, while the oscillation amplitude is reduced in the damped regimes, CDW oscillations do not fully decay but instead persist indefinitely due to feedback from the lattice field. We further show that these persistent oscillations are characterized by a nonequilibrium electronic distribution, which provides an intuitive understanding of both their amplitude and the renormalization of the oscillation frequency relative to the bare Holstein phonon frequency. Our results highlight the essential role of lattice dynamics in nonequilibrium ordered phases and establish a clear distinction between electron-lattice-driven CDW dynamics and their purely electronic counterparts.
We use determinant quantum Monte Carlo to study the half-filled `bond-Holstein' model on a square lattice. We find that the model exhibits a charge-density-wave (CDW) phase transition with a critical temperature TcdwT_\mathrm{cdw}Tcdw​ considerably higher than that of the canonical `site-Holstein' model. Using a finite-size scaling analysis of the charge structure factor ScdwS_{\rm cdw}Scdw​, we obtain TcdwT_\mathrm{cdw}Tcdw​ to greater than one percent accuracy. At the same time, local observables also show clear signatures consistent with the transition temperatures inferred from our scaling analysis. We attribute the enhanced CDW tendencies to a phonon-mediated nearest-neighbor electron repulsion that is directly proportional to the dimensionless electron-phonon coupling λλλ in the atomic (t→0t\rightarrow 0t→0) limit. This behavior contrasts with the site-Holstein case, where the same limit yields only an on-site attraction. We supplement our analysis with results from several unsupervised machine learning methods, which not only confirm our estimates of TcdwT_\mathrm{cdw}Tcdw​ but also provide insight into the high-temperature crossover between a metallic and bipolaron liquid regime.
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.
We study the Holstein model of spinless fermions, which at half filling exhibits a quantum phase transition from a metallic Tomonaga-Luttinger liquid phase to an insulating charge-density-wave (CDW) phase at a critical electron-phonon coupling strength. In our work, we focus on the real-time evolution starting from two different types of initial states that are CDW ordered: (i) ideal CDW states with and without additional phonons in the system and (ii) correlated ground states in the CDW phase. We identify the mechanism for CDW melting in the ensuing real-time dynamics and show that it strongly depends on the type of initial state. We focus on the far-from-equilibrium regime and emphasize the role of electron-phonon coupling rather than dominant electronic correlations, thus complementing a previous study of photoinduced CDW melting [H. Hashimoto and S. Ishihara, Phys. Rev. B 96, 035154 (2017)]. The numerical simulations are performed by means of matrix-product-state based methods with a local basis optimization (LBO). Within these techniques, one rotates the local (bosonic) Hilbert spaces adaptively into an optimized basis that can then be truncated while still maintaining a high precision. In this work, we extend the time-evolving block decimation (TEBD) algorithm with LBO, previously applied to single-polaron dynamics, to a half-filled system. We demonstrate that in some parameter regimes, a conventional TEBD method without LBO would fail. Furthermore, we introduce and use a ground-state density-matrix renormalization group method for electron-phonon systems using local basis optimization. In our examples, we account for up to Mph=40{M}_{\mathrm{ph}}=40Mph​=40 bare phonons per site by working with O(10)O(10)O(10) optimal phonon modes.