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arXiv:0904.0976 · Physical Review Letters

Thermalization after an Interaction Quench in the Hubbard Model

Martin Eckstein, Marcus Kollar, Philipp Werner

Published Jul 29, 2009• 460 citations
arXivHTMLPDF

Abstract

We use nonequilibrium dynamical mean-field theory to study the time evolution of the fermionic Hubbard model after an interaction quench. Both in the weak-coupling and in the strong-coupling regime the system is trapped in quasistationary states on intermediate time scales. These two regimes are separated by a sharp crossover at U(c)dyn=0.8 in units of the bandwidth, where fast thermalization occurs. Our results indicate a dynamical phase transition which should be observable in experiments on trapped fermionic atoms.

DOI
10.1103/physrevlett.103.056403

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We use nonequilibrium dynamical mean-field theory to study the time evolution of the fermionic Hubbard model after an interaction quench. Both in the weak-coupling and in the strong-coupling regime the system is trapped in quasistationary states on intermediate time scales. These two regimes are separated by a sharp crossover at U(c)dyn=0.8 in units of the bandwidth, where fast thermalization occurs. Our results indicate a dynamical phase transition which should be observable in experiments on trapped fermionic atoms.
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 the nonequilibrium dynamics in the fermionic Hubbard model after a sudden change of the interaction strength. To this scope, we introduce a time-dependent variational approach in the spirit of the Gutzwiller ansatz. At the saddle-point approximation, we find at half filling a sharp transition between two different regimes of small and large coherent oscillations, separated by a critical line of quenches where the system is found to relax. Any finite doping washes out the transition, leaving aside just a sharp crossover. In order to investigate the role of quantum fluctuations, we map the model onto an auxiliary quantum Ising model in a transverse field coupled to free fermionic quasiparticles. Remarkably, the Gutzwiller approximation turns out to correspond to the mean-field decoupling of this model in the limit of infinite coordination lattices. The advantage is that we can go beyond mean field and include Gaussian fluctuations around the non-equilibrium mean-field dynamics. Unlike at equilibrium, we find that quantum fluctuations become massless and eventually unstable before the mean-field dynamical critical line, which suggests they could even alter qualitatively the mean-field scenario.
We study the crossover between the sudden quench limit and the adiabatic dynamics of superconducting states in the attractive Hubbard model. We focus on the dynamics induced by the change of the attractive interaction during a finite ramp time. The ramp time is varied in order to track the evolution of the dynamical phase diagram from the sudden quench to the equilibrium limit. Two different dynamical regimes are realized for quenches towards weak and strong coupling interactions. At weak coupling the dynamics depends only on the energy injected into the system, whereas a dynamics retaining memory of the initial state takes place at strong coupling. We show that this is related to a sharp transition between a weak and a strong coupling quench dynamical regime, which defines the boundaries beyond which a dynamics independent from the initial state is recovered. Comparing the dynamics in the superconducting and nonsuperconducting phases, we argue that this is due to the lack of an adiabatic connection to the equilibrium ground state for nonequilibrium superconducting states in the strong coupling quench regime.
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The generic nonequilibrium evolution of a strongly interacting fermionic system is studied. For strong quenches, a collective collapse-and-revival phenomenon is found to extend over the whole Brillouin zone. A qualitatively distinct behavior occurs for weak quenches where only weak wiggling occurs. Surprisingly, no evidence for prethermalization is found in the weak-coupling regime. In both regimes, indications for relaxation beyond oscillatory or power-law behavior are found and used to estimate relaxation rates without resorting to a probabilistic ansatz. The relaxation appears to be fastest for intermediate values of the quenched interaction.