In this thesis, the main goal is to study the kinetic dynamics based on Hubbard model in dimensions d>2. We analyse rigorously some basic properties of the dynamics in the following scaling limit. As the coupling constant converges to zero, we rescale the time variable to capture the slow dynamics: we let coupling constant go to zero, time to infinity in such a way that the product of square of the coupling constant and time has a finite limit. The main ingredient of the kinetic limit is that, combining some basic ideas of scattering theory (long time scale) and perturbation theory (small coupling parameter), it automatically selects from the microscopic dynamics some dominating terms. This gives rise to a new description of time evolution (given by quantum Boltzmann equation), whose form can already be postulated from the second order perturbative expansion. The limit equation is expected to capture nontrivial information about the original system. Besides the weak coupling dynamics, we also discuss two complementary results in related Fermion systems.
In this thesis, the main goal is to study the kinetic dynamics based on Hubbard model in dimensions d>2. We analyse rigorously some basic properties of the dynamics in the following scaling limit. As the coupling constant converges to zero, we rescale the time variable to capture the slow dynamics: we let coupling constant go to zero, time to infinity in such a way that the product of square of the coupling constant and time has a finite limit. The main ingredient of the kinetic limit is that, combining some basic ideas of scattering theory (long time scale) and perturbation theory (small coupling parameter), it automatically selects from the microscopic dynamics some dominating terms. This gives rise to a new description of time evolution (given by quantum Boltzmann equation), whose form can already be postulated from the second order perturbative expansion. The limit equation is expected to capture nontrivial information about the original system. Besides the weak coupling dynamics, we also discuss two complementary results in related Fermion systems.
After a sudden disruption, weakly interacting quantum systems first relax to a prethermalized state that can be described by perturbation theory and a generalized Gibbs ensemble. Using these properties of the prethermalized state we perturbatively derive a kinetic equation which becomes a quantum Boltzmann equation in the scaling limit of vanishing interaction. Applying this to interaction quenches in the fermionic Hubbard model we find that the momentum distribution relaxes to the thermal prediction of statistical mechanics. For not too large interaction, this two-stage scenario provides a quantitative understanding of the time evolution leading from the initial pure via a metastable prethermal to the final thermal state.
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.
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.
Abstract For a transverse-field Ising chain with weak long-range interactions we develop a perturbative scheme, based on quantum kinetic equations, around the integrable nearest-neighbour model. We introduce, discuss, and benchmark several truncations of the time evolution equations up to eighth order in the Jordan–Wigner fermionic operators. The resulting set of differential equations can be solved for lattices with O(102) sites and facilitates the computation of spin expectation values and correlation functions to high accuracy, at least for moderate timescales. We use this scheme to study the relaxation dynamics of the model, involving prethermalisation and thermalisation. The techniques developed here can be generalised to other spin models with weak integrability-breaking terms.
Based on the references [P1–P6], we investigate in this cumulative PhD thesis the dynamics and transport in closed and open interacting spin chain models. We focus on (i) the connection between the open and the closed system dynamics [P1, P2], (ii) the influence of various system parameters on the transport behavior in closed and open systems [P3, P5], and in addtion (iii) the dynamics and transport in closed models [P4, P6]. As for (i), we explore the connection of the dynamics in open systems, modeled by the Lindblad equation, and the correlation functions in the closed system [P1]. Building on this connection for magnetization transport, we study whether the time evolution of the open quantum system can be described in terms of classical correlation functions generated by the Hamiltonian equations of motion for real vectors [P2]. Second, we study in (ii) the influence of different parameters on the diffusion constants in closed and open systems [P3, P5]. In Ref. [P3], we study the impact of integrability-breaking perturbations in both closed and open spin chain systems. While we find a continuous change of the diffusion constant with perturbation strength in the closed system, in the open system, in the weak-perturbation limit, we indicate the possibility of a diverging diffusion constant. Also, we investigate the influence of the system-bath coupling on the transport behavior in boundary-driven open systems [P5]. We observe that the diffusion constants depend on the system-bath coupling and that in the nonequilibrium steady state, they inevitably retain finite-size effects, limiting the computation of long-time transport coefficients. Additional for (iii), in Ref. [P4], we first examine the impact of perturbations on the dynamics in closed spin chain models. We employ projector-operator techniques to understand, on perturbative grounds, the nontrivial damping of the standard correlation function. Second, we also discuss a dynamical quantum typicality approach for studying the energy transport in bipartite systems at low temperatures [P6].
This thesis is about the derivation of effective mean field equations and their next-order corrections starting from nonrelativistic many-body quantum theory. Mean field equations provide an approximate ansatz for the description of interacting many-particle systems. In this ansatz the interaction between the particles is replaced by a self-consistent external potential leading to an effective one-body description of the many-particle system. Next-order corrections provide an approximation which goes one step further and tries to capture also subleading effects that are not resolved by the mean field ansatz. We present mathematical proofs for the validity of such effective theories for different models that are motivated, e.g., from the theory of ultracold atoms (the bosonic Hartree equation and the corresponding Bogoliubov theory) and from plasma physics (the motion of a tracer particle through a degenerate and dense electron gas). Starting from a many-body Schrödinger equation, our goal is to show that the solutions converge in a particular limit to the solutions to an effective mean field equation and its next-order corrections. After a short introduction and a summary in Chapter one, we present the main part of this work in three self-contained chapters. In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.e., for a coupling constant g_N=1/N. It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective one-particle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of Bose-Einstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single one-particle quantum state. Our emphasis lies in the description of the few particles that fluctuate around the Bose-Einstein condensate. We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the N-particle space. This result allows us to prove several other assertions. Among other things, it is used to derive the Hartree equation with optimal speed of convergence 1/N for initial states that are close to ground states of interacting systems and also to prove convergence of the N-particle solution towards a time evolution obtained from the Bogoliubov Hamiltonian on Fock space. Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Here, we derive a novel estimate for low energy eigenfunctions stating that the probability for finding l particles out of their total number N not in the condensate is exponentially small in the number l. Using this bound, we then prove that the ground state wave function of the microscopic model satisfies certain quasifree type properties. The exponential decay is moreover used to provide an alternative proof for the validity of Bogoliubov's approximation for the low-lying energy eigenvalues. Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energy-momentum dispersion relation which is linear for small momentum. The linearity of the effective dispersion relation is an essential ingredient for the explanation of the superfluid character of the Bose gas. In Chapter four we study the time evolution of a single particle coupled through a pair potential to a dense and homogeneous ideal Fermi gas in two spatial dimensions. This type of model is well known in plasma physics where it is used to describe the energy loss of ions moving through a dense and degenerate electron gas. We analyze the model for a coupling parameter g=1 and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift. To leading order, this energy shift is given by the spatially homogeneous mean field potential produced by the gas particles, whereas at next-to-leading order, one has to consider an additional correction to the mean field energy which is due to so-called recollision processes.