An artificial quantum dot which is attached to two leads exhibits a non-equilibrium Kondo effect due to the presence of a finite source-drain voltage. The construction of a theory which provides numerically computable expectation values of physical observables for this non-perturbative quantum effect within an appropriate non-equilibrium quantum statistical ensemble is an unsolved problem. The present thesis investigates an imaginary-time approach by Han and Heary [1] by means of its formal structure and computational feasibility. The theory maps the non-equilibrium system to an infinite set of auxiliary effective-equilibrium systems which can be examined with conventional techniques. An analytic continuation is necessary to reobtain non-equilibrium expectation values from effective-equilibrium ones. It gives rise to an inverse problem. We show that continuous-time quantum Monte-Carlo (CT-QMC) methods provide a numerically controlled solution of the set of auxiliary systems, in particular for parameter values which are deep inside the non-equilibrium Kondo regime. For static local observables on the quantum dot we provide reasonable numerical estimates with clear signatures of non-equilibrium Kondo physics, using a standard MaxEnt approach to the inverse problem of analytic continuation. In order to compute local dynamical observables and transport characteristics, a theoretical framework based on the function theory of several complex variables is developed. It defines the inverse problem for the MaxEnt procedure. Resulting predictions for spectral functions and I-V characteristics in the weak- and intermediate-coupling regime are in good agreement with other methods. [1] J. E. Han, R. J. Heary, Phys. Rev. Lett. 99, 236808 (2007)
An artificial quantum dot which is attached to two leads exhibits a non-equilibrium Kondo effect due to the presence of a finite source-drain voltage. The construction of a theory which provides numerically computable expectation values of physical observables for this non-perturbative quantum effect within an appropriate non-equilibrium quantum statistical ensemble is an unsolved problem. The present thesis investigates an imaginary-time approach by Han and Heary [1] by means of its formal structure and computational feasibility. The theory maps the non-equilibrium system to an infinite set of auxiliary effective-equilibrium systems which can be examined with conventional techniques. An analytic continuation is necessary to reobtain non-equilibrium expectation values from effective-equilibrium ones. It gives rise to an inverse problem. We show that continuous-time quantum Monte-Carlo (CT-QMC) methods provide a numerically controlled solution of the set of auxiliary systems, in particular for parameter values which are deep inside the non-equilibrium Kondo regime. For static local observables on the quantum dot we provide reasonable numerical estimates with clear signatures of non-equilibrium Kondo physics, using a standard MaxEnt approach to the inverse problem of analytic continuation. In order to compute local dynamical observables and transport characteristics, a theoretical framework based on the function theory of several complex variables is developed. It defines the inverse problem for the MaxEnt procedure. Resulting predictions for spectral functions and I-V characteristics in the weak- and intermediate-coupling regime are in good agreement with other methods. [1] J. E. Han, R. J. Heary, Phys. Rev. Lett. 99, 236808 (2007)
Correlations are a central ingredient of modern condensed matter physics. With the advent of mesoscopic physics, it became possible to study correlated systems in non-equilibrium situations, which can be reached by e.g. driving a constant current through a quantum dot. A basic question is how correlations on the one hand and a non-equilibrium current on the other hand influence each other. Here this interplay is studied for the model system of a quantum dot in the Kondo regime using the real time renormalization group (RTRG) method. This method is a functional renormalization group (RG) method allowing a non-perturbative description of correlated systems in non-equilibrium. For the Kondo quantum dot the method is generalized to the case of double vertices. The RTRG method not only provides equations for the calculation of the renormalized coupling between the quantum dot spin and the reservoir electrons, but also equations for the calculation of the non-equilibrium decoherence rate and the non-equilibrium current.The RG equations are derived to second order. This approximation is expected to be valid as long as the quantum dot does not enter the strong coupling regime, i.e. the effective coupling strength between the quantum dot spin and the electrons does not become too large. It is found that this is the case if the bias voltage applied to the dot or the temperature is much larger than the Kondo temperature.It is shown that the RG equations lead to coupling functions depending on the energy of the incident and the scattered electron. This dependence is simplified to the dependence on the average energy of the incident and the scattered electron in the weak coupling regime.A closer inspection of the RG equations for the couplings shows that they not only contain terms leading to a logarithmic increase of the coupling strength but also subleading terms which are unimportant in the weak coupling regime. The RG equations are solved analytically for the case where the subleading terms are neglected. It is found that the coupling functions develop resonances at energies associated with the chemical potentials of the reservoirs. The height of these resonances is determined by the decoherence rate. However, for bias voltages much larger than the Kondo temperature these resonances do not affect the conductance for a two terminal setup, so the conductance does not provide a measure for the decoherence rate in this case. It is found that the conductance for the two terminal setup is a universal function of voltage/Kondo temperature and temperature/Kondo temperature in the weak coupling regime. This function is calculated analytically in the limits of vanishing voltage or temperature. Furthermore, it is found that for zero temperature the decoherence rate is proportional to the non-equilibrium current flowing through the system. This supports the physical picture that a non-equilibrium current is associated with a decoherence rate, which cuts the logarithmic scaling of the coupling strength.Finally, the RG equations are solved numerically for zero temperature including the subleading corrections found before. As expected, the subleading corrections have no impact on the decoherence rate and the conductance in the weak coupling regime. On the other hand, it is found that they lead to strong deviations from the results of the analytic leading logarithmic approximation if the bias voltage approaches the Kondo temperature. Generally, it is found that the subleading corrections lead to a saturation of the decoherence rate and the conductance to some finite value. However, for the numerical solution a numerical regularization parameter has to be introduced. It turns out that the saturation value found depends on its value, so no statements on the precise value of the decoherence rate and the current can be made in this regime.
Within the imaginary-time theory for nonequilibrium in quantum dot systems the calculation of dynamical quantities like Green's functions is possible via a suitable quantum Monte Carlo algorithm. The challenging task is to analytically continue the imaginary-time data for both complex voltage and complex frequency onto the real variables. To this end a function-theoretical description of dynamical observables is introduced and discussed within the framework of the mathematical theory of several complex variables. We construct a feasible maximum-entropy algorithm for the analytical continuation by imposing a continuity assumption on the analytic structure and provide results for spectral functions in stationary nonequilibrium and current-voltage characteristics for different values of the dot charging energy.
This work is motivated by the fact that the investigation of non-equilibrium phenomena in strongly correlated electron systems has developed into one of the most active and exciting branches of condensed matter physics as it provides rich new insights that could not be obtained from the study of equilibrium situations. However, a theoretical description of those phenomena is missing. Therefore, in this thesis, we develop a numerical method that can be used to study two minimal models -- the Hubbard model and the Anderson impurity model with general parameter range and time dependence. We begin by introducing the theoretical framework and the general features of the Hubbard model. We then describe the dynamical mean field theory (DMFT), which was first invented by Georges in 1992. It provides a feasible way to approach strongly correlated electron systems and reduces the complexity of the calculations via a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. We employ the non-equilibrium extension of DMFT and map the Hubbard model to the single impurity Anderson model (SIAM). Since the fundamental component of the DMFT method is a solver of the single impurity Anderson model, we continue with a description of the formalism to study the real-time dynamics of the impurity model staring at its thermal equilibrium state. We utilize the non-equilibrium strong-coupling perturbation theory and derive semi-analytical approximation methods such as the non-crossing approximation (NCA) and the one-crossing approximation (OCA). We then use the Quantum Monte-Carlo method (QMC) as a numerically exact method and present proper measurements of local observables, current and Green's functions. We perform simulations of the current after a quantum quench from equilibrium by rapidly applying a bias voltage in a wide range of initial temperatures. The current exhibits short equilibrium times and saturates upon the decrease of temperature at all times, indicating Kondo behavior both in the transient regime and in the steady state. However, this bare QMC solver suffers from a dynamical sign problem for long time propagations. To overcome the limitations of this bare treatment, we introduce the ``Inchworm algorithm'', based on iteratively reusing the information obtained in previous steps to extend the propagation to longer times and stabilize the calculations. We show that this algorithm greatly reduces the required order for each simulation and re-scales the exponential challenge to quadratic in time. We introduce a method to compute Green's functions, spectral functions, and currents for inchworm Monte Carlo and show how systematic error assessments in real time can be obtained. We illustrate the capabilities of the algorithm with a study of the behavior of quantum impurities after an instantaneous voltage quench from a thermal equilibrium state. We conclude with the applications of the unbiased inchworm impurity solver to DMFT calculations. We employ the methods for a study of the one-band paramagnetic Hubbard model on the Bethe lattice in equilibrium, where the DMFT approximation becomes exact. We begin with a brief introduction of the Mott metal insulator phase diagram. We present the results of both real time Green's functions and spectral functions from our nonequilibrium calculations. We observe the metal-insulator crossover as the on-site interaction is increased and the formation of a quasi-particle peak as the temperature is lowered. We also illustrate the convergence of our algorithms in different aspects.
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.
In this thesis, we use a functional quantum field theoretical approach to investigate the non-equilibrium time evolution of an Anderson quantum dot with the main focus on the Kondo regime. We employ a real-time Keldysh path integral formulation to find an effective action. From the two-particle irreducible effective action, we derive, from the variational principle, the exact real-time Kadanoff–Baym equations of motion for the full propagator. We study these dynamic equations for the single impurity Anderson model, which decribes a quantum dot coupled to two finite-temperature leads. We take the tunnelling to the leads into account exactly. In order to solve the Kadanoff–Baym equations numerically we have to approximate them. For this purpose, we make a non-perturbative approximation by summing an infinite number of Feynman diagrams in the direct (s)-, particle-particle (t)-, particle-hole (u)-, and stu-channels. The aim of our investigation is to analyse the non-equilibrium realtime evolution of the quantum dot after a hybridisation and interaction quench into its stationary state. The main focus is on the narrowing of the Kondo resonance and the formation of the Hubbard side bands. Following on from this, the main achievement is the transient, as well as the stationary electrical current, through the quantum dot and the investigation of the dependence on temperature and magnetic field. We compare our results with other methods, such as functional renormalisation group and iterative sum of path integrals, and find a very good agreement in most situations.
In a recent work [17] Han and Heary introduced a formalism approaching steady-state quantum transport through mesoscopic structures which maps the non-equilibrium problem onto a family of auxiliary equilibrium quantum impurity systems by introducing imaginary voltages. We apply continuous-time quantum Monte-Carlo solvers to obtain precise and unbiased imaginary-time data for these auxiliary models. Physical observables are obtained by an analytical continuation in both Matsubara frequency and complexified voltage using a Maximum Entropy Method (MEM). The MEM is introduced by means of a kernel function compatible with the analytical structure of the theory. While it remains a yet challenging task to obtain reliable spectral functions, this unbiased procedure seems to indicate that the formalism yields results which are compatible with those of other methods.