In this thesis, we use the Dual Fermion approximation (DF) to study the Hubbard model on the 2D square lattice by using two-particle quantities such as susceptibilities and full vertex functions. In the first chapter we give an introduction to the strongly correlated systems including their definitions and features. Then, in the next chapter, to solve our model we present some numerical methods. The numerical methods that we used for our study are the dynamical mean-field theory (DMFT), the continuous-time auxiliary field impurity solver (CTAUX), Fluctuation diagnostics and Dual Fermion approximation (DF). The DF method provides high-resolution results for single-particle quantities such as the density of states (DOS) and Self Energy as well two particles spin susceptibility and vertex functions. With these results, we can present high-resolution results of the fluctuation diagnostics method with minimal computational expense. We examine the full Q-vector dependence of the extended self-energy at the Nodal and Antinodal k-points and provide interpretation for the claim that spin fluctuations are responsible for the observed metal-to-insulator behaviour.
In this thesis, we use the Dual Fermion approximation (DF) to study the Hubbard model on the 2D square lattice by using two-particle quantities such as susceptibilities and full vertex functions. In the first chapter we give an introduction to the strongly correlated systems including their definitions and features. Then, in the next chapter, to solve our model we present some numerical methods. The numerical methods that we used for our study are the dynamical mean-field theory (DMFT), the continuous-time auxiliary field impurity solver (CTAUX), Fluctuation diagnostics and Dual Fermion approximation (DF). The DF method provides high-resolution results for single-particle quantities such as the density of states (DOS) and Self Energy as well two particles spin susceptibility and vertex functions. With these results, we can present high-resolution results of the fluctuation diagnostics method with minimal computational expense. We examine the full Q-vector dependence of the extended self-energy at the Nodal and Antinodal k-points and provide interpretation for the claim that spin fluctuations are responsible for the observed metal-to-insulator behaviour.
In this thesis, we use the dynamical cluster approximation to study strongly correlated electron systems, especially from the angle of two-particle quantities, such as dynamical susceptibilities and vertex functions. The thesis starts with an introduction to the strongly correlated systems, including their definitions, prominent features, applications, difficulties in explaining them theoretically and some numerical approaches developed. The following section is an introduction to one family of strongly correlated systems we focus on in this thesis, the high temperature cuprates. The salient features in their general phase diagrams are described and discussed. Then an overview of the model we used to study high Tc cuprates is provided, with its limitations and extensions. To solve this model, the numerical method we employ for our study is the dynamical mean-field theory, the dynamical cluster approximation and the continuous time auxiliary field impurity solver. the last part of Chap.1 contains brief derivations for these algorithms. In Chap. 2, we apply dynamical cluster approximation to solve the one-band 2D Hub- bard model. The physical quantities of interest are two-particle quantitites, such as the dynamical susceptibility, irreducible vertex functions and full vertex functions. In this chapter, we describe how to obtain these susceptibilities via linear response theory and write down a detailed example for superconducting susceptibility. Finally we show how to calculate two particle quantities within DCA and obtain phase boundary with them. Chap. 3 is based on one of our publications. In this work, we specifically address the problem of optimizing the superconducting transition temperature in the 2D Hubbard model by analyzing wide regions of parameter space in density, interaction, and second- nearest-neighbor hopping strength.We mainly focus on d-wave superconductivity but show results of other symmetries in the last section. Chap. 4 follows another publication of ours. We study the temperature and doping evolution of NMR response in the normal state of the 2D Hubbard model using cluster dynamical mean-field theory. We simulate the Knight shift, the spin-echo decay rate and the spin-lattice relaxation time, and compare them to the cuprates experimental results. The last chapter extends the calculation of the NMR response to the superconducting state with the Nambu formalism. We show the detailed formulas and diagrams to calculate two-particle quantities, including the Dyson-Schwinger equation of motion.
This thesis will describe efforts to enhance our ability to simulate the 2D Hubbard model. Chapter 2 provides a background on the main computational techniques used throughout the work, including quantum Monte Carlo (QMC), dynamical mean field theory (DMFT), the dynamical cluster approximation (DCA), the continuous time auxiliary field algorithm (CTAUX), and the Maximum Entropy Method (MEM) for numeric analytic continuation. Chapter 3 presents new work on applying Twisted Boundary Conditions to the DCA framework. This method is applied in a effort to access thermodynamic (i.e. large system) information about the Hubbard model without the immense computational expense required to simulate large lattices directly. Chapter 4 describes efforts to study the extended 2D Hubbard model, which re-introduces non-local interactions between electrons that drive the formation of charge ordered phases. The work includes a thorough analysis of the phase diagram of the model away from half-filling, as well as analysis of the effect of non-local interactions on anti-ferromagnetic fluctuations and competition between the charge order and AFM states. Chapter 5 includes a derivation of the dual fermions diagrammatic expansion and the dual fermions ladder approximation used to compute corrections to single site DMFT calculations. The method is applied to the 2D Hubbard model in order to study the evolution of its spectral function, Fermi surface, and momentum dependent mass renormalization as the temperature, interaction strength, and doping are changed. The great advantage of this method is that we are able to access arbitrary momenta throughout the Brillouin zone, in contrast to the limitation to a few cluster momenta that characterize cluster methods like DCA.
In this thesis, we tackle various problems in strongly correlated electron systems, which can be addressed properly via the non-perturbative dynamical mean field theory (DMFT) approach using the continuous time quantum Monte Carlo method as an impurity solver. First, we revisit the old Nagaoka ferromagnetism problem in the U=oo Hubbard model and study the stability of the ferromagnetic state as a function of the temperature, the doping level, and the next-nearest-neighbor hopping t'. We then address the nature of the Mott transition in the two-dimensional Hubbard model at half-filling using cluster DMFT. Cluster DMFT can incorporate the short-range correlations beyond DMFT by extending the spatial range in which correlations are treated exactly to a finite cluster size. The non-local correlations reduce substantially the critical interaction U and modify the shape of the transition lines in the phase diagram. We then concentrate on the calculation of two-particle response functions from the ab initio perspective by means of computing the one-particle excitation spectrum using the combination of the density functional theory (DFT) and DMFT and extracting the two-particle irreducible vertex function from a local two-particle Green's function computed within DMFT. In particular, we derive the equations for calculating the magnetic/charge susceptibility and the pairing susceptibility in superconductivity. This approach is applied to the Hubbard model and the periodic Anderson model and we determine the phase diagram of magnetism and superconductivity in these models. We show that the superconducting phase is indeed stable near the magnetic phase where the pairing interaction mediated by spin fluctuations is dominantly enhanced. The non-local correlation effect to superconductivity is also discussed using the dual fermion approach and the dynamical vertex approximation. We finally apply the vertex function approach within DFT+DMFT to a Fe-based superconductor, BaFe2As2, and compute the dynamical magnetic susceptibility in this material. Our calculation results show a good agreement with the magnetic excitation spectra observed in a neutron scattering experiment. The response function calculation method derived in this thesis can capture both a localized and an itinerant nature of collective excitations in strongly correlated electron systems.
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.
We develop a codebase for performing dynamical mean field theory (DMFT) and dual fermion (DF) calculations on the Hubbard model, by modifying, extending, and interfacing several pre-existing open-source codes. This is used to examine the Hubbard model on a face-centered cubic (fcc) lattice, to obtain numerically-exact results at the level of DMFT and investigate the effect of non-local electron correlations within the dual fermion approach. Our code centres around a modified implementation of the CTHYB quantum Monte Carlo technique used as an impurity solver. We discuss the modifications made, and validate our code by comparing its results to known results in the literature. The DF calculations are obtained by using the CTHYB results as inputs to an extension of the OpenDF code, which we likewise validate by comparison of our results to those of previous authors. We study the magnetic phase diagram, magnetic susceptibilities, and single-particle dynamics of the Hubbard model on the fcc lattice. Within DMFT a large FM region is found above half-filling, the phase boundary of which is located precisely by susceptibility and magnetisation calculations. A large AFM phase is found around and below half-filling with contributions from instabilities towards the X and W points. Single-particle spectra are obtained using the Maximum Entropy method for a wide range of temperatures and fillings demonstrating how the spectra change as the FM phase boundary is crossed. The non-local correlations taken into account by the DF approach reduce the critical temperatures of both the FM and AFM regions. Corrections in the FM region are small and strongly suggest that DMFT is able to capture the relevant physics in this region of the phase diagram. The code developed as part of this project opens the door for further investigations of the Hubbard model on the fcc, and other lattices, across wide ranges of parameter space.
We apply a recently introduced hybridization-flow functional renormalization group scheme for Anderson-like impurity models as impurity solver in a dynamical mean field theory (DMFT) approach to lattice Hubbard models. We present how this scheme is capable of reproducing metallic and insulating solutions of the lattice model. Our setup also offers a numerically rather inexpensive method to calculate two-particle correlation functions. For the paramagnetic Hubbard-model on the Bethe lattice in infinite dimensions we calculate the local two-particle-vertex for the metallic and the insulating phase. Then we go to a two-site cluster-DMFT-scheme for the two-dimensional Hubbard-model that includes short-range antiferromagnetic fluctuations and obtain the local and non-local two-particle-vertex-functions. We discuss the rich frequency structures of these vertices and compare with the vertex in the single-site solution.