Dynamical mean-field theory (DMFT) is a widely used, nonperturbative computational framework for strongly correlated electron systems. This thesis extends DMFT in two directions: treating disorder and higher-order correlators away from equilibrium, and using near-term quantum computers as impurity solvers. In the first part, we develop two methodological advances for the nonequilibrium DMFT treatment of disordered, interacting systems. The first overcomes the central computational bottleneck of the recently developed nonequilibrium DMFT plus coherent potential approximation (DMFT+CPA) by replacing its explicit numerical disorder averaging with an exact analytical integration, exploiting a normalization property of the Schwinger-Keldysh contour; applied to a disorder-and-interaction quench of the Anderson-Hubbard model, this reveals that weak random disorder promotes thermalization. The second demonstrates that the out-of-time-order correlator (OTOC) can be successfully computed within the existing DMFT+CPA framework, by generalizing the Schwinger-Keldysh contour to a double-folded structure; as a proof-of-principle demonstration, weak disorder is found to accelerate the OTOC's decay in the half-filled Anderson-Hubbard model in the intermediate-coupling regime. In the second part, we explore the use of near-term quantum computers as DMFT impurity solvers. A stable real-time DMFT iteration scheme, designed for compatibility with quantum hardware, is first developed and validated classically against the metal-to-insulator transition. A symmetry-adapted variational quantum eigensolver (VQE) is then benchmarked as the Anderson impurity solver for the converged DMFT Hamiltonian, and is found to reproduce ground-state energies accurately and to extract the single-particle Green's function reliably at intermediate to strong coupling, while accurate ground-state energies are shown not to guarantee accurate dynamics at weak coupling, a key challenge for extending such approaches to correlated metals. Together, we demonstrate that DMFT can be methodologically extended to study disordered and interacting systems away from equilibrium, and to incorporate near-term quantum hardware as a genuinely new computational resource to enhance its capabilities.
Dynamical mean-field theory (DMFT) is a widely used, nonperturbative computational framework for strongly correlated electron systems. This thesis extends DMFT in two directions: treating disorder and higher-order correlators away from equilibrium, and using near-term quantum computers as impurity solvers. In the first part, we develop two methodological advances for the nonequilibrium DMFT treatment of disordered, interacting systems. The first overcomes the central computational bottleneck of the recently developed nonequilibrium DMFT plus coherent potential approximation (DMFT+CPA) by replacing its explicit numerical disorder averaging with an exact analytical integration, exploiting a normalization property of the Schwinger-Keldysh contour; applied to a disorder-and-interaction quench of the Anderson-Hubbard model, this reveals that weak random disorder promotes thermalization. The second demonstrates that the out-of-time-order correlator (OTOC) can be successfully computed within the existing DMFT+CPA framework, by generalizing the Schwinger-Keldysh contour to a double-folded structure; as a proof-of-principle demonstration, weak disorder is found to accelerate the OTOC's decay in the half-filled Anderson-Hubbard model in the intermediate-coupling regime. In the second part, we explore the use of near-term quantum computers as DMFT impurity solvers. A stable real-time DMFT iteration scheme, designed for compatibility with quantum hardware, is first developed and validated classically against the metal-to-insulator transition. A symmetry-adapted variational quantum eigensolver (VQE) is then benchmarked as the Anderson impurity solver for the converged DMFT Hamiltonian, and is found to reproduce ground-state energies accurately and to extract the single-particle Green's function reliably at intermediate to strong coupling, while accurate ground-state energies are shown not to guarantee accurate dynamics at weak coupling, a key challenge for extending such approaches to correlated metals. Together, we demonstrate that DMFT can be methodologically extended to study disordered and interacting systems away from equilibrium, and to incorporate near-term quantum hardware as a genuinely new computational resource to enhance its capabilities.
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.
Correlated electron systems exhibit a variety of intriguing emergent phenomena, including Mott physics, pseudogap behavior, and unconventional superconductivity. One of the most widely used numerical approaches for investigating these systems is dynamical mean-field theory (DMFT). This dissertation addresses two related problems: how to develop better quantum impurity solvers for DMFT and how to use DMFT results to gain a better understanding of correlated electron systems. The first part of this dissertation develops numerical methods for solving quantum impurity problems, which constitute the central computational task in DMFT calculations. In Chap. 4, we develop a method that combines a tensor-train numerical integration scheme with the inchworm hybridization expansion for quantum impurity problems. Benchmarks on spinless, single-orbital, and multi-orbital impurity models demonstrate the convergence of the Green’s functions to exact or reference results and identify the regimes in which tensor-train compression of Feynman diagrams and the inchworm expansion itself are effective. In Chap. 5, we introduce the concept of counterterms within the hybridization expansion and apply it to a bare hybridization expansion solver that relies on a sum-of-exponentials integration technique. Tests on spinless, multi-orbital, and realistic embedding impurity problems show that counterterms can significantly reduce the required perturbative order and improve the efficiency of hybridization-expansion calculations. The second part of this dissertation applies the cluster extension of DMFT to the single-band Hubbard model. In Chap. 6, we introduce a fluctuation decomposition for the self-energy that eliminates the Fierz ambiguity encountered in conventional fluctuation diagnostics of the Hubbard model. The resulting analysis shows that spin fluctuations make an important contribution to the pseudogap self-energy, while multi-boson contributions are also necessary for a complete description. In Chap. 7, we analyze the spin-fermion coupling and its role in d-wave pairing. Our calculations show that, for antinodal fermions in the pseudogap regime, their coupling to antiferromagnetic spin fluctuations is strongly enhanced relative to the non-interacting limit. A modified Bethe–Salpeter equation analysis further shows that this enhanced spin–fermion coupling boosts the spin-fluctuation-induced contribution to the d-wave pairing susceptibility.
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 developments of quantum computing algorithms and experiments for atomic scale simulations have largely focused on quantum chemistry for molecules, while their application in condensed matter systems is scarcely explored. Here we present a quantum algorithm to perform dynamical mean field theory (DMFT) calculations for condensed matter systems on currently available quantum computers, and demonstrate it on two quantum hardware platforms. DMFT is required to properly describe the large class of materials with strongly correlated electrons. The computationally challenging part arises from solving the effective problem of an interacting impurity coupled to a bath, which scales exponentially with system size on conventional computers. An exponential speedup is expected on quantum computers, but the algorithms proposed so far are based on real time evolution of the wavefunction, which requires high-depth circuits and hence very low noise levels in the quantum hardware. Here we propose an alternative approach, which uses the variational quantum eigensolver (VQE) method for ground and excited states to obtain the needed quantities as part of an exact diagonalization impurity solver. We present the algorithm for a two site DMFT system, which we benchmark using simulations on conventional computers as well as experiments on superconducting and trapped ion qubits, demonstrating that this method is suitable for running DMFT calculations on currently available quantum hardware.
Strong electronic correlations are the root of many fascinating phenomena of condensed matter physics. However, they also make calculations notoriously difficult. This thesis is concerned with the development and improvement of computational methods for strongly correlated electron systems. After a brief recapitulation of the Green’s function formalism,which is the theoretical basis of most such methods, we discuss the analytical continuation from imaginary to real frequencies. Especially the maximum entropy method is explained in detail and an implementation there of is presented.The remainder of this thesis deals with methods to compute Green’s functions on thei maginary frequency axis. First, the Anderson impurity model (AIM) is considered. It can be solved numerically – a task for which several methods exist. We focus on continuous-time quantum Monte Carlo with worm sampling, through which various kinds of correlation functions can be calculated. On this basis we derive and implement symmetric improved estimators that allow for the computation of the self-energy and vertex functions where the level of stochastic noise is largely reduced with respect to conventional estimators.As a next step the dynamical mean-field theory (DMFT) is introduced as a method to capture local correlations in the Hubbard model (HM) by self-consistently mapping it onto the AIM. We study the convergence behavior of this iterative procedure and the improvement of convergence due to mixing. Then, we compare electron scattering rates from DMFT in the weak-coupling regime of the square-lattice HM to results from Boltzmann scattering theory.While local correlations account for many interesting phenomena, such as the Mott-Hubbard metal-insulator transition, they are not sufficient to correctly describe, e.g., the physics of phase transitions in two- or three-dimensional systems. It is thus the final step in this thesis to also include nonlocal correlations by means of the dynamical vertex approximation (DΓA). For this we develop a new self-consistency scheme, which is then applied to several systems: We study the square-lattice HM, where we compare to benchmark methods in the weak-coupling regime, but also present results for intermediate and strong coupling. Furthermore, a two-band model with doping and a three-band Wannier model for strontium vanadate are studied. As an application of our new self-consistent DΓA, we study the effects of geometrical frustration in the kagome lattice, where we also compare our results to determinant quantum Monte Carlo.
We discuss a nonequilibrium dynamical mean-field framework for simulating inhomogeneous Hubbard models with local disorders. Our approach treats electron interactions and disorders on equal footing, by considering only local dynamical fluctuations. The theory reduces to nonequilibrium dynamical mean-field theory in the presence of only electron-electron interactions and to the coherent potential approximation in noninteracting systems with disorders. Both time-dependent and steady-state problems are treated by implementing the theory on the three-branch Kadanoff-Baym contour and two-branch Schwinger-Keldysh contour, respectively. Benchmarks on a eight-site cube show that the method yields rather accurate spectral functions in both the weakly and strongly interacting regimes. In a cubic lattice, we demonstrate energy conservation after an interaction quench and thermalization after just a few hopping times in both clean and disordered systems. As an application, we study transport through a serial double quantum-dot sandwiched between two leads, focusing on the current and dot occupations after a voltage quench.