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arXiv:1804.07547 · Physical Review A

Strong quenches in the one-dimensional Fermi-Hubbard model

Philip Bleicker, Götz Silvester Uhrig

Published Sep 10, 2018• 7 citations
arXivHTMLPDF

Abstract

The one-dimensional Fermi-Hubbard model is used as test bed for strong global parameter quenches. With the aid of iterated equations of motion in combination with a suitable scalar product for operators, we describe the dynamics and the long-term behavior in particular of the system after interaction quenches. This becomes possible because the employed approximation allows for oscillatory dynamics avoiding spurious divergences. The infinite-time behavior is captured by an analytical approach based on stationary phases; no numerical averages over long times need to be computed. We study the most relevant frequencies in the dynamics after the quench and find that the local interaction $U$ as well as the bandwidth $W$ dominate. In contrast to former studies, a crossover instead of a sharp dynamical transition depending on the strength of the quench is identified. For weak quenches, the bandwidth is more important while for strong quenches the local interaction $U$ dominates.

DOI
10.1103/physreva.98.033602

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The one-dimensional Fermi-Hubbard model is used as test bed for strong global parameter quenches. With the aid of iterated equations of motion in combination with a suitable scalar product for operators, we describe the dynamics and the long-term behavior in particular of the system after interaction quenches. This becomes possible because the employed approximation allows for oscillatory dynamics avoiding spurious divergences. The infinite-time behavior is captured by an analytical approach based on stationary phases; no numerical averages over long times need to be computed. We study the most relevant frequencies in the dynamics after the quench and find that the local interaction UUU as well as the bandwidth WWW dominate. In contrast to former studies, a crossover instead of a sharp dynamical transition depending on the strength of the quench is identified. For weak quenches, the bandwidth is more important while for strong quenches the local interaction UUU dominates.
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.
Motivated by recent experiments, we study the relaxation dynamics and thermalization in the one-dimensional Bose-Hubbard model induced by a global interaction quench. Specifically, we start from an initial state that has exactly one boson per site and is the ground state of a system with infinitely strong repulsive interactions at unit filling. Using exact diagonalization and the density-matrix renormalization-group method, we compute the time dependence of such observables as the multiple occupancy and the momentum distribution function. Typically, the relaxation to stationary values occurs over just a few tunneling times. The stationary values are identical to the so-called diagonal ensemble on the system sizes accessible to our numerical methods and we further observe that the microcanonical ensemble describes the time averages of many observables reasonably well for small and intermediate interaction strength. The expectation values of observables in the canonical ensemble agree quantitatively with the time averages obtained from the quench at small interaction strengths, and qualitatively provide a good description even in parameter regimes where the microcanonical ensemble is not applicable due to finite-size effects. We discuss our numerical results in the framework of the eigenstate thermalization hypothesis. Moreover, we also observe that the diagonal and the canonical ensembles are practically identical for our initial conditions already on the level of their respective energy distributions for small interaction strengths. Finally, we discuss implications of our results for the interpretation of a recent sudden expansion experiment [Phys. Rev. Lett. 110, 205301 (2013)], in which the same interaction quench was realized.
Using the time-dependent density matrix renormalization-group method and exact diagonalization, we study the nonequilibrium dynamics of the one-dimensional Fermi-Hubbard model following a quantum quench or a ramp of the on-site interaction strength. We are particularly interested in the nonequilibrium evolution of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) correlations, which, at finite spin polarizations and for attractive interactions, are the dominant two-body correlations in the ground state. For quenches from the noninteracting to the attractive regime, we investigate the dynamical emergence of FFLO correlations and their signatures in the pair quasi--momentum distribution function. We observe that the postquench double occupancy exhibits a maximum as the interaction strength becomes of the order of the bandwidth. Finally, we study quenches and ramps from attractive to repulsive interactions, which imprint FFLO correlations onto repulsively bound pairs. We show that a quite short ramp time is sufficient to wipe out the characteristic FFLO features in the postquench pair momentum distribution functions.
We study a nonequilibrium dynamics of a one-dimensional spin-imbalanced Fermi-Hubbard model following a quantum quench of on-site interaction, realizable, for example, in Feshbach-resonant atomic Fermi gases. We focus on the post-quench evolution starting from the initial BCS and Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) ground states and analyze the corresponding spin-singlet, spin-triplet, density-density, and magnetization-magnetization correlation functions. We find that beyond a light-cone crossover time, rich post-quench dynamics leads to thermalized and pre-thermalized stationary states that display strong dependence on the initial ground state. For initially gapped BCS state, the long-time stationary state resembles thermalization with the effective temperature set by the initial value of the Hubbard interaction. In contrast, while the initial gapless FFLO state reaches a stationary pre-thermalized form, it remains far from equilibrium. We suggest that such post-quench dynamics can be used as a fingerprint for identification and study of the FFLO phase.
After a quantum quench, a sudden change of parameters, generic many-particle quantum systems are expected to equilibrate. A few collisions of quasiparticles are usually sufficient to establish approximately local equilibrium. Reaching global equilibrium is, however, much more difficult as conserved quantities have to be transported for long distances to build up a pattern of fluctuations characteristic for equilibrium. Here we investigate the quantum quench of the one-dimensional bosonic Hubbard model from infinite to finite interaction strength UUU using semiclassical methods for weak and exact diagonalization for strong quenches. Equilibrium is approached only slowly, as ${t}^{\ensuremath{-}1/2}$ with subleading corrections proportional to ${t}^{\ensuremath{-}3/4}$, consistent with predictions from hydrodynamics. We show that these long-time tails determine the relaxation of a wide range of physical observables.
We study the time evolution of two fermionic one-dimensional models (spinless fermions with nearest-neighbor repulsion and the Hubbard model) exposed to an interaction quench for short and moderate times. The method used to calculate the time dependence is a semi-numerical approach based on the Heisenberg equation of motion. We compare the results of this approach with the results obtained by bosonization implying power law behavior. Indeed, we find that power laws describe our results well, but our results raise the issue of which exponents occur. For spinless fermions, it seems that the Tomonaga–Luttinger parameters work well, which also describe the equilibrium low-energy physics. But for the Hubbard model this is not the case. Instead, we find that exponents from the bosonization around the initial state work well. Finally, we discuss what can be expected for the long-time behavior.