Charge-density wave phases in quantum materials stem from the complex interplay of electronic and lattice degrees of freedom. Nowadays, various time-resolved spectroscopy techniques allow to actively manipulate such phases and monitor their dynamics in real time. Modeling such nonequilibrium dynamics theoretically is a great challenge and exact methods can usually only treat a small number of atoms and finitely many phonons. We approach the melting of charge-density waves in a Holstein model after a sudden switch-on of the electronic hopping from two perspectives. We prove that in the noninteracting and in the strong-coupling limit, the CDW order parameter on high-dimensional hypercubic lattices obeys a factorization relation for long times, such that its dynamics can be reduced to the one-dimensional case. Secondly, we present numerical results from semiclassical techniques based on the truncated Wigner approximation for two spatial dimensions. A comparison with exact data obtained for a Holstein chain shows that a semiclassical treatment of both the electrons and phonons is required in order to correctly describe the phononic dynamics. This is confirmed, in addition, for a quench in the electron-phonon coupling strength.
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 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 bare phonons per site by working with
O(10) optimal phonon modes.
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 in two dimensions. Quantum Monte Carlo calculations have determined
Tcdw for a variety of geometries, including square, honeycomb, and Lieb lattices. The superconducting transition, on the other hand, in
d=2 is in the Kosterlitz-Thouless universality class and is much less well characterized. In this paper we determine
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
t is the near-neighbor hopping amplitude, consistent with previous rough estimates from simulations which extrapolated to only the temperatures we reach from considerably higher
T. We also show evidence of a discontinuous evolution of the density as the CDW transition is approached at half filling.
The formation of charge-density-wave order in Dirac fermion systems via electron-phonon coupling represents a significant topic in condensed matter physics. In this work, we investigate this phenomenon within the Holstein model on the honeycomb lattice, with a specific focus on the effect of disorder. While the interplay between electron-electron interactions and disorder has long been a central theme in the field, recent attention has increasingly turned to the combined influence of disorder and electron-phonon coupling. Using determinant quantum Monte Carlo simulations, we concentrate on the phase transitions of charge-density-wave order on the honeycomb lattice. Disorder is introduced through the random hopping of electrons in the system, which can localize electrons via the Anderson effect. Our primary result is that disorder suppresses the charge-density-wave phase, and the interplay between disorder and electron-phonon interactions extends the phase area. We also determine the transition temperature
βc to the ordered phase as a function of the electron-phonon coupling. Additionally, we observed a suppression of electron kinetic energy and dc conductivity under disorder, highlighting the role of Anderson localization in the degradation of electronic transport. These findings offer significant theoretical insight into the stability and critical phenomena of correlated phases in disordered two-dimensional systems.
The quench dynamics of the Hubbard model in tilted and harmonic potentials is discussed within the semiclassical picture. Applying the fermionic truncated Wigner approximation (fTWA), the dynamics of imbalances for charge and spin degrees of freedom is analyzed and its time evolution is compared with the exact simulations in a one-dimensional lattice. Quench from charge or spin density waves is considered. We show that introduction of harmonic and spin-dependent linear potentials sufficiently validates fTWA for longer times. Such an improvement of fTWA is also obtained for the higher-order correlations in terms of quantum Fisher information for charge and spin channels. This allows us to discuss the dynamics of larger system sizes and connect our discussion to the recently introduced Stark many-body localization. In particular, we focus on a finite two-dimensional system and show that at intermediate linear potential strength, the addition of a harmonic potential and spin dependence of the tilt results in subdiffusive dynamics, similar to that of disordered systems. Moreover, for specific values of harmonic potential, we observed phase separation of ergodic and nonergodic regions in real space. The latter fact is especially important for ultracold-atom experiments in which harmonic confinement can be easily imposed, causing a significant change in relaxation times for different lattice locations.
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
Tcdw considerably higher than that of the canonical `site-Holstein' model. Using a finite-size scaling analysis of the charge structure factor
Scdw, we obtain
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
λ in the atomic (
t→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
Tcdw but also provide insight into the high-temperature crossover between a metallic and bipolaron liquid regime.