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arXiv:1307.3438 · Physical Review B

Interaction quenches in the two-dimensional fermionic Hubbard model

Simone A. Hamerla, Götz Silvester Uhrig

Published Mar 13, 2014• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.89.104301

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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.
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.
In the last years the impressive progress on the experimental side led to a variety of new experiments allowing to address systems out of equilibrium.In this way the behavior of such systems far from equilibrium is no longer a purely theoretical issue but indeed observable.New experimental techniques, like particles trapped in optical lattices, render a realization of quantum systems with nearly arbitrary system parameters possible and provide a possibility to study their time evolution.Systems out of equilibrium are characterized by the fact, that these systems are in highly excited states giving rise to totally new fascinating properties.In the present thesis one-and two-dimensional fermionic Hubbard models out of equilibrium are discussed.The system is taken out of equilibrium by a so-called interaction quench.At the beginning the system is prepared in the groundstate of the non-interacting Hamiltonian.At a time t the interaction between the fermions is suddenly turned on so that the time evolution is governed by the whole, interacting Hamiltonian.Hence the system is prepared in the groundstate of one Hamiltonian but evolves according to a di_erent Hamiltonian.Consequently the system ends up in a highly excited state.To describe such a system a method based on an expansion of the Heisenberg equations of motion to highest order possible is developed in this thesis.This method provides an exact description of the time evolution on short and intermediate time scales after the quench.As the method reveal exact results and does not rely on any perturbative assumption, a study of arbitrarily large interaction strengths is possible.Besides, the method is one of the few methods capable of two-dimensional systems.In the following the method used in this thesis is explained and advantages and disadvantages of the approach are thematized.For this purpose the results of the developed iterated equation of motion approach are compared to results obtained in calculation up to second order in the interaction.Then the dynamics of the onedimensional model is discussed with a focus on the relation of the results derived by the iterated equations of motion approach to results obtained by bosonization theory, the behavior for strong interactions and the dynamical transition from the weak to the strong quench regime.Furthermore, the Hubbard model is studied on a two-dimensional square lattice.This model is fundamentally di_erent from the one-dimensional model: In contrast to the one-dimensional model the two-dimensional model is not integrable allowing a true relaxation of the system.For this system a calculation up to second order in the interaction is performed and compared to the results of the iterated equation of motion approach.Besides, the time evolution of the momentum distribution and the influence of doping on the dynamics is studied.Moreover, a first estimate for relaxation times is provided without relying on the assumption of a mixed state.
We investigate the time evolution of correlations in the Bose-Hubbard model following a quench from the superfluid to the Mott insulator. For large values of the final interaction strength the system approaches a distinctly nonequilibrium steady state that bears strong memory of the initial conditions. In contrast, when the final interaction strength is comparable to the hopping, the correlations are rather well approximated by those at thermal equilibrium. The existence of two distinct nonequilibrium regimes is surprising given the nonintegrability of the Bose-Hubbard model. We relate this phenomenon to the role of quasiparticle interactions in the Mott insulator.
Thermalization in strongly correlated fermionic systems remains a central open problem in quantum many-body physics. In this work, we investigate the real-time dynamics and the approach to thermalization in the two-dimensional Hubbard model, a paradigmatic framework for correlated electrons, relevant to high-temperature superconductivity and ultracold quantum simulation. Focusing on the half-filled square lattice, we monitor the time evolution of the double occupancy following a quench in the on-site interaction UUU, and assess whether its long-time value is captured by a canonical thermal ensemble. We employ time-dependent variational Monte Carlo methods combined with transformer-based Neural-Network Quantum States to accurately describe the nonequilibrium dynamics of fermions, especially for the behavior at long times, thereby accessing regimes that were previously inaccessible to numerical simulations. Our results reveal two dynamical behaviors: for weak to intermediate interactions, the double occupancy rapidly approaches the thermal prediction, consistent with ergodic evolution; beyond a critical interaction UCU_{C}UC​, the dynamics remains distinct from the thermal expectation on the timescales investigated, revealing signatures of a prethermal plateau delaying fast relaxation. These results establish numerical simulation as a powerful tool to probe nonequilibrium quantum phenomena in correlated fermionic matter.
The study of the interplay between interaction-induced correlations and nonequilibrium initial conditions in many-body systems has recently attracted a lot of attention. New experimental techniques provide high control over many-body systems and systematic access to their nonequilibrium regime. In many systems the nonequilibrium properties of a Fermi liquid can be relevant. A first approach to their understanding is the main content of this thesis. At the beginning, nonequilibrium phenomena, Landau's theory of a Fermi liquid, the Hubbard model, time evolution in quantum mechanics, related experiments, and the flow equation method are reviewed. The key observation of this thesis, namely a characteristic mismatch of expectation values in equilibrium and nonequilibrium, is first illustrated for the squeezed oscillator. For this one-particle model Hamiltonian the perturbative approach can be compared with the exact solution and it is shown that the mismatch holds even beyond perturbation theory. Afterwards these observations are generalized to a larger class of one-particle models. Then the nonequilibrium behavior of a Fermi liquid is examined by analyzing the Fermi liquid phase of the Hubbard model in more than one dimension. After a sudden switch-on of a weak two-particle interaction to the noninteracting Fermi gas the relaxation of the many-body system is observed. For this purpose, the flow equation transformation is implemented for the Hubbard Hamiltonian. Then the discussion of the momentum distribution function and of the kinetic energy displays a three-step relaxation behavior of the Fermi liquid from the initial perturbation until thermalization is reached. Firstly, the sudden switch inserts excitation energy into the system which drives the following dynamics. Then a rapid initial build-up of correlations is caused by initial dephasing and leads to the establishment of a quasiparticle picture. By that the system enters into a quasi-stationary state which can be long-lasting for weak interaction. This state shows prethermalization of the kinetic energy which already has relaxed to its final values. However, the momentum distribution still resembles a zero temperature Fermi liquid. Its later relaxation on a second time scale is caused by a residual two-particle interaction which allows for scattering processes and can be described by a quantum Boltzmann equation. The physical origin of this delayed relaxation can be traced back to an interplay of translational invariance of the Hubbard model and the Pauli principle for fermions; together they restrict the phase space for scattering events. Comparing with similar work I conjecture on the generic nature of the findings made for the quenched Fermi liquid in other many-body systems. Finally I point out to the potential relevance of the delayed relaxation for the observation of further nonequilibrium phenomena, for instance in BCS systems. In order to extend the study of sudden switching to arbitrary switching processes the calculation is repeated using the Keldysh perturbation theory. First evaluations for a linear increase of the interaction strength are given. Extensions of this work are suggested and may motivate further research.
We study the relaxation dynamics of interacting one-dimensional fermions with band curvature after a weak quench in the interaction parameter at zero temperature. Our model lies within the class of interacting Luttinger liquids, where the harmonic Luttinger theory is extended by a weak-integrability-breaking phonon scattering term. In order to solve for the nonequilibrium time evolution, we use quantum kinetic equations exploiting the resonant but subleading character of the phonon interaction term. The interplay between phonon scattering and the quadratic Luttinger theory leads to the emergence of three distinct spatiotemporal regimes for the fermionic real-space correlation function. It features the crossover from a prequench to a prethermal state, finally evolving towards a thermal state on increasing length and time scales. The characteristic algebraically decaying real-space correlations in the prethermalized regime become modulated by an amplitude that is decaying in time according to a stretched exponential as an effect of the interactions. The asymptotic thermalization dynamics is governed by energy transport over large distances from the thermalized to the nonthermalized regions via macroscopic, dynamical slow modes. This is revealed in an algebraic decay of the system's effective temperature. The numerical value of the associated exponent agrees with the dynamical critical exponent of the Kardar-Parisi-Zhang universality class. We also discuss a criterion for the applicability of this theory away from the integrable limit of noninteracting fermions.