We investigate the electronic structure of the strongly correlated electron systems with an orbital degeneracy within the framework of the iterated-perturbation theory (IPT) (1, 2) of the dynamical mean fleld method. We consider the multi-orbital Hubbard model in the limit of inflnite lattice spatial dimensions. Our motivation comes from recent studies of the orbital efiects in strongly correlated elctron systems such as tran- sition metal oxides, strontium ruthenates, manganities, and organic metals. Almost all of these materials consist of valence bands coming from d shells, where intra- and inter- orbital Coulomb interactions are equally important. The important part of the paper is a detailed discussion on the problem of the analytical continuation of the complex frequency Matsubara Grenn function from the Matsubara points to the real frequency axis. The analytical continuation is a nessessary step in the numerical calculations if we are interested in calculating the physical quantities such as the density of states or transport quantities.
We investigate the electronic structure of the strongly correlated electron systems with an orbital degeneracy within the framework of the iterated-perturbation theory (IPT) (1, 2) of the dynamical mean fleld method. We consider the multi-orbital Hubbard model in the limit of inflnite lattice spatial dimensions. Our motivation comes from recent studies of the orbital efiects in strongly correlated elctron systems such as tran- sition metal oxides, strontium ruthenates, manganities, and organic metals. Almost all of these materials consist of valence bands coming from d shells, where intra- and inter- orbital Coulomb interactions are equally important. The important part of the paper is a detailed discussion on the problem of the analytical continuation of the complex frequency Matsubara Grenn function from the Matsubara points to the real frequency axis. The analytical continuation is a nessessary step in the numerical calculations if we are interested in calculating the physical quantities such as the density of states or transport quantities.
In this thesis we consider three different models for strongly correlated electrons, namely a multi-band Hubbard model as well as the spinless Falicov-Kimball model, both with a semi-elliptical density of states in the limit of infinite dimensions d, and the attractive Hubbard model on a square lattice in d=2.\n\nIn the first part, we study a two-band Hubbard model with unequal bandwidths and anisotropic Hund's rule coupling (J_z-model) in the limit of infinite dimensions within the dynamical mean-field theory (DMFT). Here, the DMFT impurity problem is solved with the use of quantum Monte Carlo (QMC) simulations. Our main result is that the J_z-model describes the occurrence of an orbital-selective Mott transition (OSMT), in contrast to earlier findings. We investigate the model with a high-precision DMFT \nalgorithm, which was developed as part of this thesis and which supplements QMC with a high-frequency expansion of the self-energy.\nThe main advantage of this scheme is the extraordinary accuracy of the numerical solutions, which can be obtained already with moderate computational effort, so that studies of multi-orbital systems within the DMFT+QMC are strongly improved. We also found that a suitably defined\nFalicov-Kimball (FK) model exhibits an OSMT, revealing the close connection of the Falicov-Kimball physics to the J_z-model in the OSM phase.\n\nIn the second part of this thesis we study the attractive Hubbard model in two spatial dimensions within second-order self-consistent perturbation theory.\nThis model is considered on a square lattice at finite doping and at low temperatures. Our main result is that the predictions of first-order perturbation theory (Hartree-Fock approximation) are renormalized by a factor of the order of unity even at arbitrarily weak interaction (U->0). The \nrenormalization factor q can be evaluated as a function of the filling n for 0<n<1. In the limit n->0, the q-factor vanishes, signaling the divergence of self-consistent perturbation theory in this limit. Thus we present the first asymptotically exact results at weak-coupling for the negative-U Hubbard model in d=2 at finite doping.
Predicting the physical properties of strongly correlated electron systems (SCES) requires the combined Density Functional Theory and Dynamical Mean-Field Theory (DFT+DMFT) framework. However, the computational bottle-neck lies in the impurity solver. This work focuses on rigorously benchmarking the Multi-Orbital Iterated Perturbation Theory (MO-IPT) impurity solver against the numerically exact Continuous-Time Quantum Monte Carlo (CTQMC) method, specifically through a detailed study of the three-orbital Hubbard model. The two-orbital model serves as a canonical system to test performance under realistic multi-band conditions. Our systematically derived MO-IPT overcomes the critical limitations of CTQMC, including exponential scaling with orbital number, the need for analytic continuation, and the severe fermionic sign problem. By demonstrating high accuracy across varying interaction strengths and temperatures, we validate MO-IPT as an inexpensive, yet reliable, replacement solver essential for future material-specific investigations.
We review the dynamical mean-field theory of strongly correlated electron systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact for models of correlated electrons in the limit of large lattice coordination (or infinite spatial dimensions). It extends the standard mean-field construction from classical statistical mechanics to quantum problems. We discuss the physical ideas underlying this theory and its mathematical derivation. Various analytic and numerical techniques that have been developed recently in order to analyze and solve the dynamical mean-field equations are reviewed and compared to each other. The method can be used for the determination of phase diagrams (by comparing the stability of various types of long-range order), and the calculation of thermodynamic properties, one-particle Green's functions, and response functions. We review in detail the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to experiments on three-dimensional transition-metal oxides. We present an overview of the rapidly developing field of applications of this method to other systems. The present limitations of the approach, and possible extensions of the formalism are finally discussed. Computer programs for the numerical implementation of this method are also provided with this article.
For a long time strong electronic correlations in metals have mainly been associated with Mottness, the proximity to a Mott metal-insulator transition (MIT), where large Coulomb interactions induce the localization of charges. However, triggered by the discovery of the iron-based superconductors about ten years ago, it was realized that multi-orbital materials with only moderate Coulomb but sizeable Hund’s rule interactions – so-called Hund metals – allow for a distinct screening mechanism towards strong correlations: Hundness. Here, Hund’s rule constrains the spin rather than the charge dynamics. This discovery led to a vividly debated fundamental issue in the field of strongly correlated condensed matter systems, which is the main topic of the present thesis: what is the origin of strong correlations in the normal phase of Hund metals, Mottness or Hundness? And what are their decisive fingerprints? The goal of this dissertation is twofold. First, we present and advance our method: the numerical renormalization group (NRG) as viable real-frequency multi-band impurity solver for dynamical mean-field theory (DMFT), a common approach to tackle strongly correlated systems. Second, we apply DMFT+NRG to shed light on the Hund-metal problem raised above. In the first part of this thesis we present our state-of-the-art NRG solver, which offers direct access to data with unprecedented real-frequency spectral resolution at arbitrarily low energies and temperatures in contrast to commonly used Quantum Monte Carlo solvers. It is based on matrix product states and exploits non-abelian symmetries to reduce numerical costs. In the case of orbital symmetry, this allows us to treat multi-band models with more than two bands, and thus to tackle the Hund-metal problem for the first time with NRG. For multi-band models without orbital symmetry, an “interleaved” scheme of NRG (iNRG) was recently developed, dramatically increasing the numerical efficiency. Remarkably, the accuracy of iNRG is comparable to standard NRG, as we reveal in a detailed study. This finding establishes iNRG as a promising DMFT solver for material-specific model simulations. In the second part of this thesis we study a minimal toy model for Hund metals with DMFT+NRG, the orbital-symmetric three-band Hubbard-Hund model (3HHM) close to a lattice filling of 1/3. Our major insight is “spin-orbital separation” (SOS), a Hund’s-ruleinduced two-stage Kondo-type screening process, in which orbital screening occurs at much higher energies than spin screening. In Hund metals, i.e. far from a MIT phase boundary, SOS thus causes large electron masses by strongly reducing the coherence scale below which a Fermi liquid is formed. Further, it opens up a broad incoherent and strongly particle-hole asymmetric intermediate energy regime that reaches up to bare excitation scales. This SOS regime shows fractional power-law behavior and is characterized by resilient “Hund quasiparticles” with itinerant orbital degrees of freedom coupled non-trivially to quasi-free large spins. At zero temperature, the local density of states exhibits a two-tier quasiparticle peak on top of a broad incoherent background. In contrast, in Mott-correlated metals, i.e. close to the MIT phase boundary, the SOS regime becomes negligibly small and the Hubbard bands are well separated. These findings lead to distinct signatures of Hundness and Mottness in the temperature dependence of ARPES spectra, static local susceptibilities, resistivity, thermopower and entropy, many of which were also found in realistic simulations of the archetypal Hund- and Mott-correlated materials, Sr2RuO4 and V2O3. In summary, we provide evidence that and elucidate how Hundness evokes strong correlation effects in Hund metals. This might help to better interpret experimental results and guide superconducting theories.
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.
The dynamical mean-field theory (DMFT) maps a correlated lattice problem onto an impurity problem of a single correlated site coupled to an uncorrelated bath. Most implementations solve the DMFT equations using quantum Monte-Carlo sampling on the imaginary time and frequency (Matsubara) axis. We will here review alternative methods using exact diagonalization, i.e., representing the many-body ground state of the impurity as a sum over Slater determinants and calculating Green’s functions using iterative Lanczos procedures. The advantage being that these methods have no sign problem, can handle involved multi-orbital Hamiltonians (low crystal symmetry, spin-orbit coupling) and – when working completely on the real axis – do not need a mathematically ill-posed analytical continuation. The disadvantage of traditional implementations of exact diagonalization has been the exponential scaling of the calculation problem as a function of number of bath discretization points. In the last part we will review how recent advances in exact diagonalization can evade the exponential barrier thereby increasing the number of bath discretization points to reach the thermodynamic limit.