The recent developments in experimental techniques such as pump-probe spectroscopy, new technologies in building two dimensional materials, and the advent of highly tunable systems in the form of ultracold optical lattices, trapped ions and array of atoms have opened a new path to investigate the dynamics of strongly correlated system out of equilibrium, which would not be possible with the conventional methods.These new developments not only provided a powerful tool to study some of the most fundamental questions in the nonequilibrium regime of quantum systems but also made it possible to explore new phenomena which has not been observed before.Motivated by some of these experiments, in this thesis, we study a wide range of strongly correlated systems out of equilibrium.We will use a combination of theoretical and computational methods such as sum rules, nonequilibrium Green's function, nonequilibrium dynamical mean-eld theory (DMFT), and an exact solution of the non-interacting model to address a wide spectrum of systems out of equilibrium.First, we have generalized a formalism for the nth derivative of a time-dependent operator in the Heisenberg representation and employ it to the spectral sum rules in which we obtain a qualitative understanding of the pump eect on electron-phonon coupling in high T c superconductors.Then, by using the nonequilibrium DMFT for the Falicov-Kimball model, we obtain the current-voltage prole of a multilayer device which consists of a single barrier region (usually insulator plane) connected to a number of metallic leads in both sides.To improve the conservation of current and lling 6.12 ∆ ln (t) for the XX chain with dierent initial states . . . . . . . . . .6.13 ∆ ln (t) for the XY chain with various γ and h parameters . . . . . . .6.14 |C ln | 2 , |F ln | 2 , and ∆ ln (t) for the XY chain . . . . . . . . . . . . . . .6.15 The continuous blue curve is the function -Ai 2 (-X) . . . . . . . . .6.16 The evolution of the entanglement entropy in the XY chain . . . . . .6.17 v e max (2) and v e max (1) for dierent values of a and h . . . . . . . . . . .6.18 Π(l, L)αl for periodic system with total length L = 2000 . . . . . .6.19 Π(l, L)αl for open system with total length L = 2000 . . . . . . . .6.20 The coecient of the logarithmic term for two congurations . . . . .6.21 Π(l, L)αl for periodic system with total length L = 300 with respect to l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.22 Values of f (x), α min and α max with respect to x . . . . . . . . . . . .6.23 The contributions of dierent ranks k in the Shannon information for two sizes l = 14 and 26 . . . . . . . . . . . . . . . . . . . . . . . . . .6.24 The error E(x m , l) in the evaluation of Shannon information coming from the truncation at the rank k = x m l . . . . . . . . . . . . . . . .6.25 The contributions of dierent ranks k in the Shannon information . .6.26 The evolution of logarithmic formation probability of dierent congurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.27 The evolution of Shannon information of a subsystem with dierent sizes201 6.28 The mutual information between a pair of dimers . . . . . . . . . . .6.29 The evolution of mutual information between two adjacent subsystems in two dierent cases . . . . . . . . . . . . . . . . . . . . . . . . . . .
The recent developments in experimental techniques such as pump-probe spectroscopy, new technologies in building two dimensional materials, and the advent of highly tunable systems in the form of ultracold optical lattices, trapped ions and array of atoms have opened a new path to investigate the dynamics of strongly correlated system out of equilibrium, which would not be possible with the conventional methods.These new developments not only provided a powerful tool to study some of the most fundamental questions in the nonequilibrium regime of quantum systems but also made it possible to explore new phenomena which has not been observed before.Motivated by some of these experiments, in this thesis, we study a wide range of strongly correlated systems out of equilibrium.We will use a combination of theoretical and computational methods such as sum rules, nonequilibrium Green's function, nonequilibrium dynamical mean-eld theory (DMFT), and an exact solution of the non-interacting model to address a wide spectrum of systems out of equilibrium.First, we have generalized a formalism for the nth derivative of a time-dependent operator in the Heisenberg representation and employ it to the spectral sum rules in which we obtain a qualitative understanding of the pump eect on electron-phonon coupling in high T c superconductors.Then, by using the nonequilibrium DMFT for the Falicov-Kimball model, we obtain the current-voltage prole of a multilayer device which consists of a single barrier region (usually insulator plane) connected to a number of metallic leads in both sides.To improve the conservation of current and lling 6.12 ∆ ln (t) for the XX chain with dierent initial states . . . . . . . . . .6.13 ∆ ln (t) for the XY chain with various γ and h parameters . . . . . . .6.14 |C ln | 2 , |F ln | 2 , and ∆ ln (t) for the XY chain . . . . . . . . . . . . . . .6.15 The continuous blue curve is the function -Ai 2 (-X) . . . . . . . . .6.16 The evolution of the entanglement entropy in the XY chain . . . . . .6.17 v e max (2) and v e max (1) for dierent values of a and h . . . . . . . . . . .6.18 Π(l, L)αl for periodic system with total length L = 2000 . . . . . .6.19 Π(l, L)αl for open system with total length L = 2000 . . . . . . . .6.20 The coecient of the logarithmic term for two congurations . . . . .6.21 Π(l, L)αl for periodic system with total length L = 300 with respect to l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.22 Values of f (x), α min and α max with respect to x . . . . . . . . . . . .6.23 The contributions of dierent ranks k in the Shannon information for two sizes l = 14 and 26 . . . . . . . . . . . . . . . . . . . . . . . . . .6.24 The error E(x m , l) in the evaluation of Shannon information coming from the truncation at the rank k = x m l . . . . . . . . . . . . . . . .6.25 The contributions of dierent ranks k in the Shannon information . .6.26 The evolution of logarithmic formation probability of dierent congurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.27 The evolution of Shannon information of a subsystem with dierent sizes201 6.28 The mutual information between a pair of dimers . . . . . . . . . . .6.29 The evolution of mutual information between two adjacent subsystems in two dierent cases . . . . . . . . . . . . . . . . . . . . . . . . . . .
In this thesis, we describe methods for solving quantum impurity models out of equilibrium, and consider several applications of these methods. Quantum impurity models describe systems in which a small subsystem (the impurity) is coupled to a large external environment (the bath). These models are useful in describing a wide variety of physical systems, including quantum dots, absorbed adatoms, and molecular junctions. They are also important for the study of lattice models, such as the Hubbard model, where they appear as auxiliary problems within the framework of dynamical mean field theory (DMFT). Many interesting phenomena in these systems, such as transport through a quantum dot, or optical pumping of a Mott insulator, occur away from equilibrium. Theoretical description of these phenomena requires methods that are capable of handling both real time evolution and strong correlation effects presenting a formidable challenge. In this thesis, we describe several methods for the solution of non-equilibrium quantum impurity models which meet this challenge. These methods are based on an expansion in the coupling between the impurity and the bath known as the hybridization expansion. Chapter 1 gives an introduction to the problems addressed in this thesis. Chapter 2 provides a brief overview of Keldysh formalism for nonequilibrium problems. Chapter 3 describes the bare hybridization expansion for the Anderson impurity model as well as its bold extensions. In particular we derive the non-crossing approximation (NCA), the one-crossing approximation (OCA), the inchworm expansion, and a bold expansion for the impurity Green's function. Chapter 4 describes the implementation of a quantum Monte Carlo algorithm based on the inchworm expansion. The rest of the thesis describes applications of these algorithms to problems of physical interest. Chapters 5 and 6 apply the methods previously described to understanding the Kondo effect in quantum dot systems. The first chapter examines the behavior of the Kondo effect under an applied voltage bias, a classic non-equilibrium strongly-correlated problem. The second chapter examines different ways of measuring the cloud of electrons which screen the impurity spin in the Kondo regime. Chapters 7 and 8 apply the same methods to the Hubbard model using the Dynamical mean-field theory (DMFT) mapping. The first chapter studies the Hubbard model in equilibrium using the numerically exact inchworm quantum Monte Carlo method. The second chapter studies the possibility of dynamic switching between the metal and Mott-insulator states of the Hubbard model using the one-crossing approximation (OCA). Finally chapter 9 gives some overall conclusions and suggests future directions for the development of hybridization expansion methods.
The study of nonequilibrium phenomena in correlated lattice systems has developed into one of the most active and exciting branches of condensed matter physics. This research field provides rich new insights that could not be obtained from the study of equilibrium situations, and the theoretical understanding of the physics often requires the development of new concepts and methods. On the experimental side, ultrafast pump-probe spectroscopies enable studies of excitation and relaxation phenomena in correlated electron systems, while ultracold atoms in optical lattices provide a new way to control and measure the time evolution of interacting lattice systems with a vastly different characteristic time scale compared to electron systems. A theoretical description of these phenomena is challenging because, first, the quantum-mechanical time evolution of many-body systems out of equilibrium must be computed and second, strong-correlation effects which can be of a nonperturbative nature must be addressed. This review discusses the nonequilibrium extension of the dynamical mean field theory (DMFT), which treats quantum fluctuations in the time domain and works directly in the thermodynamic limit. The method reduces the complexity of the calculation via a mapping to a self-consistent impurity problem, which becomes exact in infinite dimensions. Particular emphasis is placed on a detailed derivation of the formalism, and on a discussion of numerical techniques, which enable solutions of the effective nonequilibrium DMFT impurity problem. Insights gained into the properties of the infinite-dimensional Hubbard model under strong nonequilibrium conditions are summarized. These examples illustrate the current ability of the theoretical framework to reproduce and understand fundamental nonequilibrium phenomena, such as the dielectric breakdown of Mott insulators, photodoping, and collapse-and-revival oscillations in quenched systems. Furthermore, remarkable novel phenomena have been predicted by the nonequilibrium DMFT simulations of correlated lattice systems, including dynamical phase transitions and field-induced repulsion-to-attraction conversions.
Strongly correlated systems, i.e., quantum materials for which the interactions between its constituents are strong, are good candidates for the development of applications based on quantum-mechanical principles, such as quantum computers. Two paradigmatic models of strongly correlated systems are heavy-fermionic systems and one-dimensional spin-12 systems, with and without quenched disorder. In the past decade, improvement in computational methods and a vast enhancement in computational power has made it possible to study these systems in a a non-perturbative manner. In this thesis we present state-of-the-art numerical methods to investigate the properties of strongly correlated systems, and we apply these methods to solve a couple of selected problems in quantum condensed matter theory. We start by revisiting the phase diagram of the Falicov-Kimball model on the square lattice which can be considered as a heavy-fermionic systems. This model describes an interplay between conduction electrons and heavy electrons and reveals several distinct metal-insulator phase transitions. Using a lattice Monte-Carlo method, we study the transport properties of the model. Our analysis describes the role of temperature and interaction strength on the metal-insulator phase transitions in the Falicov-Kimball model. The second part of the thesis investigate the spatial structure of the entanglement in ground and thermal statesof the transverse-field Ising chain. We use the logarithmic negativity as a measure for the entanglement between two disjoint blocks. We investigate how logarithmic negativity depends on the spatial separation between two blocks, which can be viewed as the entanglement analog of a spatial correlation function. We find sharp entanglement thresholds at a critical distance beyond which the logarithmic negativity vanishes exactly and thus the two blocks become unentangled. Our results hold even in the presence of long-ranged quantum correlations, i.e., at the system’s quantum critical point. Using Time-Evolving Block Decimation (TEBD), we explore this feature as a function of temperature and size of the two blocks. We present a simple model to describe our numerical observations. In the last part of this thesis, we introduce an order parameter for a many-body localized spin-glass (MBL-SG) phase. We show that many-body localized spin-glass order can also be detected from two-site reduced density matrices, which we use to construct an eigenstate spin-glass order parameter. We find that this eigenstate spin-glass order parameter captures spin-glass phases in random Ising chains, both in many-body eigenstates as well as in the nonequilibrium dynamics, from a local in time measurement. We discuss how our results can be used to observe MBL-SG order within current experiments in Rydberg atoms and trapped ion systems.
Strongly correlated quantum systems display a myriad of complex, fascinating behaviour. The competition between charge, spin, orbital, or lattice degrees of freedom and their respective energy scales gives rise to rich and complex emergent phases. More recently, the non-equilibrium properties of correlated quantum many-body systems have come into the focus of active research, with remarkable advances. While the advent of tensor network based simulations on the theoretical side has allowed access to the full time-evolution of the quantum state as it traverses the Hilbert space, the experimental progress in the ultracold atom community has allowed physicists to enter the paradigm of analog quantum simulation. Their tunability and isolation from the environment makes them ideal platforms to emulate and study open problems in condensed matter physics and go beyond the simulational capabilities of state of the art classical simulations. In this thesis, we explore the dynamics of correlated Fermi gases. We develop and present different facets of applicability of radiofrequency modulation techniques to drive the system out of equilibrium. We begin by considering a homogeneous, three-dimensional Fermi gas in the BCS-BEC crossover and develop an off-resonant radiofrequency transfer scheme to excite the Higgs mode of the superfluid. In a subsequent study we investigate the tunability of the coherence between Cooper pairs by tuning the duration of interaction ramps of the internal interaction strength. These works highlight the possibility to externally tune and stabilise complex quantum many-body states away from equilibrium. Using quasi-exact matrix product state simulations, we study the response of an attractively interacting, one-dimensional Fermi-Hubbard model to weak radiofrequency perturbations. We reveal the emergence of two distinct dynamical regimes in the time evolution and are able to relate it back to the underlying excitation spectrum of the system. Finally, we explore the possibility to perform quantum quenches through radiofrequency π-pulses, achieving nearly complete population transfer between different interacting fermionic states. We reveal non-trivial dynamical effects in the pair correlation of the final state, as well as the excitation of a collective trap mode of the system.
The main topic of this thesis is the study of many-body effects in strongly correlated one- or quasi one-dimensional condensed matter systems. These systems are characterized by strong quantum and thermal fluctuations, which make mean-field methods fail and demand for a fully numerical approach. Fortunately, a numerical method exist, which allows to treat unusually large one -dimensional system at very high precision. This method is the density-matrix renormalization group method (DMRG), introduced by Steve White in 1992. Originally limited to the study of static problems, time-dependent DMRG has been developed allowing one to investigate non-equilibrium phenomena in quantum mechanics. In this thesis I present the solution of three conceptionally different problems, which have been addressed using mostly the Krylov-subspace version of the time-dependent DMRG. My findings are directly relevant to recent experiments with ultracold atoms, also carried out at LMU in the group of Prof. Bloch. The first project aims the ultimate goal of atoms in optical lattices, namely, the possibility to act as a quantum simulator of more complicated condensed matter system. The underline idea is to simulate a magnetic model using ultracold bosonic atoms of two different hyperfine states in an optical superlattice. The system, which is captured by a two-species Bose-Hubbard model, realizes in a certain parameter range the physics of a spin-1/2 Heisenberg chain, where the spin exchange constant is given by second order processes. Tuning of the superlattice parameters allows one to controlling the effect of fast first order processes versus the slower second order ones. The analysis is motivated by recent experiments, %\cite{Folling2007,Trotzky2008} where coherent two-particle dynamics with ultracold bosonic atoms in isolated double wells were detected. My project investigates the coherent many-particle dynamics, which takes place after coupling the double well. I provide the theoretical background for the next step, the observation of coherent many-particle dynamics after coupling the double wells. The tunability between the Bose-Hubbard model and the Heisenberg model in this setup could be used to study experimentally the differences in equilibration processes for non-integrable and Bethe ansatz integrable models. It turns out that the relaxation in the Heisenberg model is connected to a phase averaging effect, which is in contrast to the typical scattering driven thermalization in nonintegrable models In the second project I study a many-body generalization of the original Landau-Zener formula. This formula gives the transition probability between the two states of a quantum mechanical two-level system, where the offset between the two levels is varying linearly in time. In a recent experiment this framework has been extended to a many-body system consisting of pairwise tunnel-coupled one-dimensional Bose liquids. It was found that the tunnel coupling between the tubes and the intertube interactions strongly modify the original Landau-Zener picture. After a introduction to the two-level and the three-level Landau-Zener problem I present my own results for the quantum dynamics of the microscopic model and the comparison to the experimental results. I have calculated both Landau-Zener sweeps as well as the time-evolution after sudden quenches of the energy offset. A major finding is that for sufficiently large initial density quenches can be efficiently used to create quasi-thermal states of arbitrary temperatures. The third project is more mathematical and connects the fields of quantum computation and of quantum information. Here, the main purpose is to analyse systematically the effects of decoherence on maximally entangled multi-partite states, which arise typically during quantum computation processes. The bigger the number of entangled qubits the more fragile is its entanglement under the influence decoherence. As starting point I consider first two entangled qubits, whereby one qubit interacts with an arbitrary environment. For this particular case I have derived a factorization law for the disentanglement. Next, I calculate the decrease of entanglement of two , three and four entangled qubits, general W- and general GHZ-state, coupled to a global spin-1/2 bath or several independent spin-1/2 baths , one for each qubit. Although there is no appropriate entanglement measure for three and more qubits, it turns out that this decrease is directly related to the increase of entanglement between the central system and the bath. This implies the formation of a much bigger multipartite entangled network. Thus, using the von Neumann entropy and the Wootters concurrence, I derive a simple upper bound for the bath-induced entanglement breaking power of the initially maximally entangled multi-partite states.
In this thesis, strongly correlated quantum many-body systems in equilibrium and in out-of-equilibrium situations are investigated. This is done by applying and developing well established numerical methods. The focus of the thesis lies in the development and application of the density matrix renormalization group method (DMRG) to quantum many-body systems out of equilbrium. In this thesis, in addition to the DMRG, methods for the exact diagonalization of the Hamiltonian of the system, like the Lanczos- or the Jacobi-Davidson method, are treated. An extension of the Lanczos method makes it possible to treat the time evolution of strongly correlated quantum systems with an accuracy comparable to machine precision. This method is the basis for a possible extension of the DMRG for the treatment of systems out of equilibrium, the so-called adaptive time-dependent DMRG ("adaptive t-DMRG"). A second variant of the adaptive t-DMRG uses the Suzuki-Trotter decomposition of the time-evolution operator. An error analysis demonstrates that both methods, for a suitable choice of control parameters, have errors < 1% at the end of the time evolution in complicated quantities like, e.g., the momentum distribution function. We apply these numerical methods to investigate the quantum critical behavior of a variant of the Hubbard model and to treat two non-equilibrium situations of current interest. Extensive use of the DMRG makes it possible to clarify the quantum-critical behavior of the ionic Hubbard model; in particular, the numerical results demonstrate quite clearly that the correct scenario has two critical points; at the first critical point, only the charge degrees of freedom are critical, while at the second one only the spin degrees of freedom are critical. Quite surprisingly, the Mott-insulator phase in the strong-coupling regime shows a divergent electrical susceptibility. Next, the dynamics of a system of so-called soft-core bosons is treated. The particles initially are trapped in a deep box-potential and the dynamics is investigated after releasing them from this trapping potential. Similar to exact results obtained for so-called hard-core bosons, (quasi-)coherent matter waves emerge, demonstrating the possibility of realizing an atom laser in experiments on optical lattices. We find that the wave vector of the emerging matter wave can be tuned by changing the strength of the interaction between the particles. The second non-equilibrium situation is a so-called "quantum-quench" of a system of strongly correlated fermions, i.e., the evolution of the system after suddenly changing an intrinsic parameter like, e.g., the interaction strength between the particles, is investigated. In this case, we are mainly interested in the long-time behavior; in particular, the question arises whether, relying on Boltzmann's ergodic hypothesis, relaxation to a thermal state is obtained. We find that, in general, non-thermal final states are reached, which can be described by generalized Gibbs-Boltzmann ensembles.