We develop a method for calculating the self-energy of a quantum impurity coupled to a continuous bath by stochastically generating a distribution of finite Anderson models that are solved by exact diagonalization, using the noninteracting local spectral function as a probability distribution for the sampling. The method enables calculation of the full analytic self-energy and single-particle Green's function in the complex frequency plane, without analytic continuation, and can be used for both finite and zero temperature at arbitrary fillings. Results are in good agreement with imaginary frequency data from continuous-time quantum Monte Carlo calculations for the single-impurity Anderson model and the two-orbital Hubbard model within dynamical mean-field theory (DMFT) as well as real frequency data for self-energy of the single-band Hubbard model within DMFT using the numerical renormalization group. The method should be applicable to a wide range of quantum impurity models and particularly useful when high-precision real frequency results are sought.
We develop a method for calculating the self-energy of a quantum impurity coupled to a continuous bath by stochastically generating a distribution of finite Anderson models that are solved by exact diagonalization, using the noninteracting local spectral function as a probability distribution for the sampling. The method enables calculation of the full analytic self-energy and single-particle Green's function in the complex frequency plane, without analytic continuation, and can be used for both finite and zero temperature at arbitrary fillings. Results are in good agreement with imaginary frequency data from continuous-time quantum Monte Carlo calculations for the single-impurity Anderson model and the two-orbital Hubbard model within dynamical mean-field theory (DMFT) as well as real frequency data for self-energy of the single-band Hubbard model within DMFT using the numerical renormalization group. The method should be applicable to a wide range of quantum impurity models and particularly useful when high-precision real frequency results are sought.
We propose a method for estimating smooth real-frequency self-energy in the dynamical mean-field theory with the finite-temperature exact diagonalization (DMFT-ED). One of the benefits of DMFT-ED calculations is that one can obtain real-frequency spectra without a numerical analytic continuation. However, these spectra are spiky and strongly depend on the way to discretize a continuous bath (e.g., the number of the bath sites). The present scheme is based on a recently proposed compact representation of imaginary-time Green's functions, the intermediate representation (IR). The projection onto the IR basis acts as a noise filter for the discretization errors in the self-energy. This enables to extract the physically relevant part from the noisy self-energy. We demonstrate the method for single-site DMFT calculations of the single-band Hubbard model. We also show the results can be further improved by numerical analytic continuation.
We introduce an alternative approach for obtaining smooth finite-temperature spectral functions of quantum impurity models using the numerical renormalization group (NRG) technique. It is based on calculating first the Green's function on the imaginary-frequency axis, followed by an analytic continuation to the real-frequency axis using Pad\'e approximants. The arbitrariness in choosing a suitable kernel in the conventional broadening approach is thereby removed and, furthermore, we find that the Pad\'e method is able to resolve fine details in spectral functions with less artifacts on the scale of $\ensuremath{\omega}\ensuremath{\sim}T$. We discuss the convergence properties with respect to the NRG calculation parameters (discretization $\ensuremath{\Lambda}$, z-averaging, truncation cutoff) and the number of Matsubara points taken into account in the analytic continuation. We test the technique on the single-impurity Anderson model and the Hubbard model (within the dynamical mean-field theory). For the Anderson impurity model, we discuss the shape of the Kondo resonance and its temperature dependence. For the Hubbard model, we discuss the inner structure of the Hubbard bands in metallic and insulating solutions at half-filling, as well as in the doped Mott insulator. Based on these test cases, we conclude that the Pad\'e approximant approach provides improved results for spectral functions at low-frequency scales of $\ensuremath{\omega}\ensuremath{\lesssim}T$ and that it is capable of resolving sharp spectral features also at high frequencies.
We here present how a self-consistent solution of the dynamical mean field theory equations can be obtained using exact diagonalization of an Anderson impurity model with accuracies comparable to those found using renormalization group or quantum Monte Carlo methods. We show how one can solve a correlated quantum impurity coupled to several hundred uncorrelated bath sites, using a restricted active basis set. The number of bath sites determines the resolution of the obtained spectral function, which consists of peaks roughly spaced by the band width divided by the number of bath sites. The self-consistency cycles are fully performed on the real frequency axis and expressed as numerical stable matrix operations. The same impurity solver has been used on Ligand Field and finite size cluster calculations and is capable of treating involved Hamiltonians including the full rotational invariant Coulomb interaction, spin-orbit coupling and low symmetry crystal-fields. The proposed method allows for the calculation of a variety of correlation functions at little extra cost.
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.
An essential role for the description of correlated quantum many-particle systems is played by impurity problems. These consist of a small number of localized orbitals, where the electrons are subject to a Coulomb-interaction, coupled to noninteracting bath degrees of freedom. Impurity models are used for the description of magnetic impurities in metals, of quantum points in nanostructures, and in the context of the dynamical mean field theory (DMFT). For their solution a large number of numerical methods exists. We mention here quantum Monte Carlo methods, the numerical renormalization group, or the exact diagonalization method. In this thesis we introduce a new approach to impurity problems, which is based on the functional renormalization group (fRG). In contrast to the methods mentioned above this approach is approximative, but it can be applied with comparatively lower numerical effort. The aim of this thesis is to examine the advantages and limits of this method. An important role in this connection is played by the single impurity Anderson model (SIAM), which is already understood very well with other methods and is seen as a prototype for models with strong local correlations. The spin fluctuations of this model are governed by the so-called Kondo energy scale and a central question of this thesis is whether this scale can be reproduced with our approach. We begin with a short introduction to the Green's function formalism for correlated quantum many particle systems. Then the fRG flow equations for the one-particle irreducible vertex functions are deduced, and several approximation schemes are discussed. In chapter 4 we introduce the fRG scheme used in this thesis. At first the SIAM is mapped to a semi-infinite chain, in which the interacting orbital is given by the first site of this chain. The system is then subdivided into two parts. The first part, which is called 'core', consists of the interacting orbital and the first L bath sites. The remaining bath sites form the second part. At the beginning both parts are decoupled from each other such that the core can be solved exactly. Starting with this exact solution the coupling between the core and the remaining bath sites is switched on slowly in a renormalization group flow. This way it is possible to calculate the one-particle and two-particle correlation functions on the correlated site. We call this flow scheme 'hybridization flow'. The flow equations are formulated in an effective theory on the first bath site outside the core, which turns out to be advantageous compared to other implementations of the hybridization flow. In chapter 5 this hybridization flow method is applied to the SIAM with semi-elliptic bath density of states. We discuss the differences which arise for different core-sizes (L = 0,1,2,3) and for different truncations of the fRG flow equations. In the cases L=0 and L=2 the local Fermi liquid properties of the SIAM and the associated Kondo scale can not be reproduced in both approximation schemes. For the core sizes L=1 and L=3 the ground state of the core is already a spin singlet state in the beginning of the flow and the Fermi liquid properties are reproduced. The Kondo energy scale, which determines the width of the central quasi-particle peak and the size of the local spin-susceptibility, is however not accurately resolved. This is partly due to the finite temperature used in our implementation such that for larger interaction strength the temperature is above the Kondo energy scale. The application of the hybridization flow method to quantum impurity problems in the context of the DMFT is discussed in chapter 6. In this approach lattice models with a local interaction, like the Hubbard model, are mapped to an effective impurity problem for a single lattice site embedded in a dynamical mean field representing the influence of the other electrons. This scheme is exact in infinitely many dimensions and a nonperturbative approximation method for finite-dimensional systems. In particular the Mott metal-insulator transition in the Hubbard model is successfully described by this method. One class of quantities, which is easily accessible from the hybridization flow scheme, is given by the local two-particle correlation functions. These play an essential role in non-local extensions of the DMFT. Therefore we mainly focus on these quantities. At first we apply our flow scheme to the Hubbard model on the Bethe lattice in infinite dimensions and calculate the local one-particle irreducible vertex functions in the insulating and the metallic phase in good agreement with previous calculations, which use the exact diagonalization method as impurity solver. After this we extend our flow scheme to a cluster DMFT method, which includes short-range antiferromagnetic correlations and calculate again the local and non-local vertex functions for the two-dimensional Hubbard model. We compare these to the results of the single-site DMFT.
The self-energy method for quantum impurity models expresses the correlation part of the self-energy in terms of the ratio of two Green's functions and allows for a more accurate calculation of equilibrium spectral functions than is possible directly from the one-particle Green's function [Bulla et al., J. Phys.: Condens. Matter 10, 8365 (1998)], for example, within the numerical renormalization group method. In addition, the self-energy itself is a central quantity required in the dynamical mean field theory of strongly correlated lattice models. Here, we show how to generalize the self-energy method to the time-dependent situation for the prototype model of strong correlations, the Anderson impurity model. We use the equation-of-motion method to obtain closed expressions for the local Green's function in terms of a time-dependent correlation self-energy, with the latter being given as a ratio of a one-particle time-dependent Green's function and a higher-order correlation function. We benchmark this self-energy approach to time-dependent spectral functions against the direct approach within the time-dependent numerical renormalization group method. The self-energy approach improves the accuracy of time-dependent spectral function calculations, and the closed-form expressions for the Green's function allow for a clear picture of the time evolution of spectral features at the different characteristic time scales. The self-energy approach is of potential interest also for other quantum impurity solvers for real-time evolution, including time-dependent density matrix renormalization group and continuous-time quantum Monte Carlo techniques.