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arXiv:2312.13727 · Physical review. B./Physical review. B

Pattern formation in charge density wave states after a quantum quench

Lingyu Yang, Yang Yang, Gia-Wei Chern

Published May 9, 2024• 6 citations
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Abstract

We study postquench dynamics of charge density wave (CDW) order in the square-lattice $t\text{\ensuremath{-}}V$ model. The ground state of this system at half-filling is characterized by a checkerboard modulation of particle density. A generalized self-consistent mean-field method, based on the time-dependent variational principle, is employed to describe the dynamical evolution of the CDW states. Assuming a homogeneous CDW order throughout the quench process, the time-dependent mean-field approach is reduced to the Anderson pseudospin method. Quench simulations based on the Bloch equation for pseudospins produce three canonical behaviors of order-parameter dynamics: phase-locked persistent oscillation, Landau-damped oscillation, and dynamical vanishing of the CDW order. We further develop an efficient real-space von Neumann equation method to incorporate dynamical inhomogeneity into simulations of quantum quenches. Our large-scale simulations uncover complex pattern formations in the postquench CDW states, especially in the strong quench regime. The emergent spatial textures are characterized by super density modulations on top of the short-period checkerboard CDW order. Our demonstration of pattern formation in quenched CDW states, described by a simple broken ${Z}_{2}$ symmetry, underscores the importance of dynamical inhomogeneity in quantum quenches of many-body systems with more complex orders.

DOI
10.1103/physrevb.109.195133

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We investigate the out-of-equilibrium dynamics of a photoexcited charge-density-wave (CDW) state in the square-lattice Holstein model, in a setup similar to a pump-probe experiment. At half-filling, the ground state of this system is characterized by a checkerboard modulation of particle densities, accompanied by a concomitant lattice distortion. An efficient real-space dynamics method integrating the von Neumann equation for electron density matrix and Newton equation for classical lattice dynamics is developed to simulate the dynamical evolution of a photoexcited Holstein system. We find that the energy injected by a short pump pulse results in the reduction of the CDW order and the generation of coherent phonons. At strong photoexcitations, while the CDW order is melted in the sense that the time-averaged order parameter vanishes, a dynamical CDW state is self-sustained by the strong coherent phonon oscillations. Our large-scale simulations further uncover a dynamical regime of intermediate fluence where complex pattern formation is induced by the pump pulse. The emergent spatial textures are characterized by super density modulations on top of the short-range checkerboard CDW order. Our demonstration of pattern formation highlights the significance of dynamical inhomogeneity in quantum many-body systems in pump-probe experiments.
We investigate the out-of-equilibrium dynamics of a photo-excited charge-density-wave (CDW) state in the square-lattice Holstein model, in a setup similar to a pump-probe experiment. At half-filling, the ground state of this system is characterized by a checkerboard modulation of particle densities, accompanied by a concomitant lattice distortion. An efficient real-space dynamics method integrating the von~Neumann equation for electron density matrix and Newton equation for classical lattice dynamics is developed to simulate the dynamical evolution of a photo-excited Holstein system. We find that the energy injected by a short pump pulse results in the reduction of the CDW order and the generation of coherent phonons. At strong photoexcitations, while the CDW order is melted in the sense that the time-averaged order parameter vanishes, a dynamical CDW state is self-sustained by the strong coherent phonon oscillations. Our large-scale simulations further uncover a dynamical regime of intermediate fluence where complex pattern formation is induced by the pump pulse. The emergent spatial textures are characterized by super density modulations on top of the short-range checkerboard CDW order. Our demonstration of pattern formation highlights the significance of dynamical inhomogeneity in quantum many-body systems in pump-probe experiments.
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.
By combining the Baeriswyl wave function with equilibrium and time-dependent variational principles, we develop a nonequilibrium formalism to study quantum quenches for two-dimensional spinless fermions with nearest-neighbor hopping and repulsion. The variational ground-state energy, the charge-density wave (CDW) order parameter, and the short-time dynamics agree convincingly with the results of numerically exact simulations. We find that, depending on the initial and final interaction strength, the quenched system either exhibits oscillatory behavior or relaxes to a time-independent steady state. The time-averaged expectation value of the CDW order parameter rises sharply when crossing from the steady-state regime to the oscillating regime, indicating that the system, being nonintegrable, shows signs of thermalization with an effective temperature above or below the equilibrium critical temperature, respectively.
We investigate the formation of charge and spin ordering by starting from a noninteracting state and studying how it evolves in time under a Hamiltonian with finite electronic interactions. We consider the one-dimensional, half-filled extended Hubbard model, which we solve within time-dependent density matrix renormalization group. By employing linear finite-time quenches in on-site and nearest-neighbor interactions, we find the existence of impulse, intermediate, and adiabatic regimes of time evolution. For the quenches we analyze, we observe that the adiabatic regime is reached with distinct ramping timescales depending on whether the charge density wave (CDW) or the spin density wave (SDW) is formed. The former needs to be slower than the latter to prevent entangled excited states from being accessed during the quench. More interestingly, in the intermediate regime, we observe an enhancement of the entanglement entropy with respect to its initial value, which precedes the formation of the CDW ordering; a similar enhancement is not seen in the quench toward SDW. Our findings also show that the breaking of the system integrability, by turning on the nearest-neighbor interactions, does not give rise to significant changes in the nonequilibrium behavior within the adiabatic approximation.
We investigate the quench dynamics in the dipolar Bose-Hubbard model (DBHM) in one dimension. The boson hopping is constrained by dipole conservation and shows fractonic dynamics. The ground states at large Hubbard interaction U are Mott insulators at integer filling and a period-2 charge density wave (CDW) at half-integer filling. We focus on Mott-to-Mott and CDW-to-CDW quenches and find that dipole correlation spreading shows the light-cone behavior with the Lieb-Robinson (LR) velocity proportional to the dipole kinetic energy J and the square of the density in the case of Mott quench at integer filling. An effective model for postquench dynamics is constructed under the dilute-dipole approximation and fits the numerical results well. For the CDW quench, we observe a much reduced LR velocity of order J 2 / U and additional periodic features in the time direction. The emergence of a CDW ground state and the reduced LR velocity at half-integer filling can both be understood by careful application of the second-order perturbation theory. The oscillatory behavior arises from quantum scars in the quadrupole sector of the spectrum and is captured by a PXP-like model that we derive by projecting the DBHM to the quadrupolar sector of the Hilbert space. Published by the American Physical Society 2024