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arXiv:1202.4684 · Physical Review B

Lattice and surface effects in the out-of-equilibrium dynamics of the Hubbard model

Patrice André, Marco Schirò, Michele Fabrizio

Published May 10, 2012• 16 citations
arXivHTMLPDF

Abstract

We study, by means of the time-dependent Gutzwiller approximation, the out-of-equilibrium dynamics of a half-filled Hubbard-Holstein model of correlated electrons interacting with local phonons. Inspired by pump-probe experiments, where intense light pulses selectively induce optical excitations that trigger a transient out-of-equilibrium dynamics, here we inject energy in the Hubbard bands by a nonequilibrium population of empty and doubly-occupied sites. We first consider the case of a global perturbation, acting over the whole sample, and find evidence of a mean-field dynamical transition where the lattice gets strongly distorted above a certain energy threshold, despite the weak strength of the electron-phonon coupling by comparison with the Hubbard repulsion. Next, we address a slab geometry for a correlated heterostructure and study the relaxation dynamics across the system when the perturbation acts locally on the first layer. While for weak deviations from equilibrium the excited surface is able to relax by transferring its excess energy to the bulk, for large deviations, the excess energy stays instead concentrated in the surface layer. This self-trapping occurs both in the absence as well as in the presence of electron-phonon coupling. Phonons actually enforce the trapping by distorting at the surface.

DOI
10.1103/physrevb.85.205118

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We study, by means of the time-dependent Gutzwiller approximation, the out-of-equilibrium dynamics of a half-filled Hubbard-Holstein model of correlated electrons interacting with local phonons. Inspired by pump-probe experiments, where intense light pulses selectively induce optical excitations that trigger a transient out-of-equilibrium dynamics, here we inject energy in the Hubbard bands by a nonequilibrium population of empty and doubly-occupied sites. We first consider the case of a global perturbation, acting over the whole sample, and find evidence of a mean-field dynamical transition where the lattice gets strongly distorted above a certain energy threshold, despite the weak strength of the electron-phonon coupling by comparison with the Hubbard repulsion. Next, we address a slab geometry for a correlated heterostructure and study the relaxation dynamics across the system when the perturbation acts locally on the first layer. While for weak deviations from equilibrium the excited surface is able to relax by transferring its excess energy to the bulk, for large deviations, the excess energy stays instead concentrated in the surface layer. This self-trapping occurs both in the absence as well as in the presence of electron-phonon coupling. Phonons actually enforce the trapping by distorting at the surface.
The spectral response and physical features of the 2D Hubbard-Holstein model are calculated both in equilibrium at zero and low chemical dopings, and after an ultrashort powerful light pulse, in undoped systems. At equilibrium and at strong charge-lattice couplings, the optical conductivity reveals a three-peak structure in agreement with experimental observations. After an ultrashort pulse and at nonzero electron-phonon interaction, phonon and spin subsystems oscillate with the phonon period T(ph)≈80 fs. The decay time of the phonon oscillations is about 150-200 fs, similar to the relaxation time of the charge system. We propose a criterion for observing these oscillations in high T(c) compounds: the time span of the pump light pulse τ(pump) has to be shorter than the phonon oscillation period T(ph).
We study quenches of the interaction and electron-phonon coupling parameter in the Hubbard-Holstein model, using nonequilibrium dynamical mean field theory. The calculations are based on a generalized Lang-Firsov scheme for time-dependent interactions or externally driven phonons, and an approximate strong-coupling impurity solver. The interaction quench calculations reveal the phonon-assisted decay of excess doublons, while the quenches of the electron-phonon coupling lead to persistent oscillations of the phonons and to a phonon-enhanced doublon production.
We investigate the nonequilibrium dynamics of the weak-coupling Hubbard-Holstein model after a sudden switch-on of the electron-phonon interaction within nonequilibrium dynamical mean-field theory (DMFT). Using the self-consistent Migdal approximation for the electron-phonon coupling together with second-order perturbation theory for the electron-electron interaction, we show that the relaxation dynamics exhibits a crossover between electron-dominated and phonon-dominated regimes, extending to finite Hubbard interaction the scenario previously identified in the Holstein model. To investigate the microscopic buildup of the thermal state, we analyze the dynamics within the Step-by-Step DMFT framework. In the plane of real time and DMFT iteration number, thermalization is marked by a sharp propagating front. This front appears in electronic observables already for weak quenches within the simulated time window, whereas the phononic sector exhibits a visible front only at sufficiently strong coupling. Thus, at weak coupling the local dispersionless phonons show a delayed onset of front formation, while near and beyond the crossover the front develops on comparable timescales in both the electronic and phononic sectors. Whenever both fronts are resolved, they propagate with the same velocity, showing that thermalization spreads coherently through the coupled electron-phonon system.
We study equilibrium and nonequilibrium properties of electron-phonon systems described by the Hubbard-Holstein model using dynamical mean-field theory. In equilibrium, we benchmark the results for impurity solvers based on the one-crossing approximation and slave-rotor approximation against non-perturbative numerical renormalization group reference data. We also examine how well the low-energy properties of the electron-boson coupled systems can be reproduced by an effective static electron-electron interaction. The one-crossing and slave-rotor approximations are then used to simulate insulator-to-metal transitions induced by a sudden switch-on of the electron-phonon interaction. The slave-rotor results suggest the existence of a critical electron-phonon coupling above which the system is transiently trapped in a nonthermal metallic state with coherent quasiparticles. The same quench protocol in the one-crossing approximation results in a bad metallic state.
We study the isotope effects arising from the coupling of correlated electrons with dispersionless phonons by considering the Hubbard-Holstein model at half-filling within the dynamical mean-field theory. In particular we calculate the isotope effects on the quasiparticle spectral weight ZZZ, the renormalized phonon frequency, and the static charge and spin susceptibilities. In the weakly correlated regime $U∕t\ensuremath{\lesssim}1.5$, where UUU is the Hubbard repulsion and ttt is the bare electron half-bandwidth, the physical properties are qualitatively similar to those characterizing the Holstein model in the absence of Coulomb repulsion, where the bipolaronic binding takes place at large electron-phonon coupling and it is reflected in divergent isotope responses. On the contrary in the strongly correlated regime $U∕t\ensuremath{\gtrsim}1.5$, where the bipolaronic metal-insulator transition becomes of first order, the isotope effects are bounded, suggesting that the first-order transition is likely driven by an electronic mechanism, rather then by a lattice instability. These results point out how the isotope responses are extremely sensitive to phase boundaries and they may be used to characterize the competition between the electron-phonon coupling and the Hubbard repulsion.
Within the time-dependent Gutzwiller approximation applied to one-dimensional Holstein and Su-Schrieffer-Heeger-Hubbard models, we study the influence of electron correlations on the phonon self-energy. For the local Holstein coupling, we find that the phonon-frequency renormalization gets weakened upon increasing the on-site interaction UUU for all momenta. In contrast, correlations can enhance the phonon-frequency shift for small wave vectors in the Su-Schrieffer-Heeger-Hubbard model. Moreover, the time-dependent Gutzwiller approximation applied to the latter model provides a mechanism which leads to phonon-frequency corrections at intermediate momenta due to the coupling with double-occupancy fluctuations. Both models display a shift of the nesting induced to a q=0q=0q=0 instability when the on-site interaction becomes sufficiently strong and thus establishing phase separation as a generic phenomenon of strongly correlated electron-phonon coupled systems.