Nonequilibrium dynamical mean-field theory (DMFT) solves correlated lattice models by obtaining their local correlation functions from an effective model consisting of a single impurity in a self-consistently determined bath. The recently developed mapping of this impurity problem from the Keldysh time contour onto a time-dependent single-impurity Anderson model (SIAM) [C. Gramsch et al., Phys. Rev. B 88, 235106 (2013)] allows one to use wave-function-based methods in the context of nonequilibrium DMFT. Within this mapping, long times in the DMFT simulation become accessible by an increasing number of bath orbitals, which requires efficient representations of the time-dependent SIAM wave function. These can be achieved by the multiconfiguration time-dependent Hartree (MCTDH) method and its multilayer extensions. We find that MCTDH outperforms exact diagonalization for large baths in which the latter approach is still within reach and allows for the calculation of SIAMs beyond the system size accessible by exact diagonalization. Moreover, we illustrate the computation of the self-consistent two-time impurity Green's function within the MCTDH second quantization representation.
Nonequilibrium dynamical mean-field theory (DMFT) solves correlated lattice models by obtaining their local correlation functions from an effective model consisting of a single impurity in a self-consistently determined bath. The recently developed mapping of this impurity problem from the Keldysh time contour onto a time-dependent single-impurity Anderson model (SIAM) [C. Gramsch et al., Phys. Rev. B 88, 235106 (2013)] allows one to use wave-function-based methods in the context of nonequilibrium DMFT. Within this mapping, long times in the DMFT simulation become accessible by an increasing number of bath orbitals, which requires efficient representations of the time-dependent SIAM wave function. These can be achieved by the multiconfiguration time-dependent Hartree (MCTDH) method and its multilayer extensions. We find that MCTDH outperforms exact diagonalization for large baths in which the latter approach is still within reach and allows for the calculation of SIAMs beyond the system size accessible by exact diagonalization. Moreover, we illustrate the computation of the self-consistent two-time impurity Green's function within the MCTDH second quantization representation.
We derive an exact mapping from the action of nonequilibrium dynamical mean-field theory (DMFT) to a single-impurity Anderson model (SIAM) with time-dependent parameters, which can be solved numerically by exact diagonalization. The representability of the nonequilibrium DMFT action by a SIAM is established as a rather general property of nonequilibrium Green functions. We also obtain the nonequilibrium DMFT equations using the cavity method alone. We show how to numerically obtain the SIAM parameters using Cholesky or eigenvector matrix decompositions. As an application, we use a Krylov-based time propagation method to investigate the Hubbard model in which the hopping is switched on, starting from the atomic limit. Possible future developments are discussed.
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 improved relaxation approach of Meyer and co-workers is extended by employing the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) theory. Within this approach the trial wave function is expressed by a recursive, multilayered expansion that is the same as in a normal ML-MCTDH simulation. Calculation of an energy eigenstate then proceeds by iteratively diagonalizing the Boltzmann operator in the top layer representation using a Lanczos/Arnoldi method while relaxing the single particle functions of all layers using the ML-MCTDH imaginary time propagation. Numerical iteration is stopped until the relative energy and/or density of the target state meet the convergence criteria. The application of this multilayer improved relaxation method is illustrated by computing the energy splitting for the lowest pair of eigenstates as well as the spin observables and entanglement entropy of the spin-boson model. A large number of degrees of freedom, up to 10,000 bath modes, can be treated, and thus the appropriate phase transition at a critical system-bath coupling strength may be predicted via appropriate extrapolations.
The multi-layer multi-configurational time-dependent Hartree (MCTDH) approach is an efficient method to study quantum dynamics in real and imaginary time. The present work explores its potential to describe quantum fluids. The multi-layer MCTDH approach in second quantization representation is used to study lattice models beyond one dimension at finite temperatures. A scheme to map the lattice sites onto the MCTDH tree representation for multi-dimensional lattice models is proposed. A statistical sampling scheme previously used in MCTDH calculations is adapted to facilitate an efficient description of the thermal ensemble. As example, a two-dimensional hard-core Bose-Hubbard model is studied considering up to 64 × 64 lattice sites. The single particle function basis set size required to obtain converged results is found to not increase with the lattice size. The numerical results properly simulate the finite temperature Berezinskii-Kosterlitz-Thouless phase transition.
The multilayer multiconfigurational time-dependent Hartree (MCTDH) in optimized second quantization representation (oSQR) approach combines the tensor contraction scheme of the multilayer MCTDH approach with the use of an optimized time-dependent orbital basis. Extending the original work on the subject [U. Manthe and T. Weike, J. Chem. Phys. 146, 064117 (2017)], here MCTDH-oSQR propagation in imaginary time and properties related to particle number conservation are studied. Differences between the orbital equation of motion in real and imaginary time are highlighted and a new gauge operator, which facilitates efficient imaginary time propagation, is introduced. Studying Bose-Hubbard models, particle number conservation in MCTDH-oSQR calculations is investigated in detail. Interesting properties of the single-particle functions used in the multilayer MCTDH representation are identified. Based on these results, a tensor contraction scheme, which explicitly utilizes particle number conservation, is suggested.