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arXiv:cond-mat/0311291 · Journal of Physics Condensed Matter

Density-matrix renormalization group approach to quantum impurity problems

Satoshi Nishimoto, Eric Jeckelmann

Published Jan 16, 2004• 54 citations
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Abstract

A dynamic density-matrix renormalization group approach to the spectral properties of quantum impurity problems is presented. The method is demonstrated on the spectral density of the flat-band symmetric single-impurity Anderson model. We show that this approach provides the impurity spectral density for all frequencies and coupling strengths. In particular, Hubbard satellites at high energy can be obtained with a good resolution. The main difficulties are the necessary discretization of the host band hybridized with the impurity and the resolution of sharp spectral features such as the Abrikosov–Suhl resonance.

DOI
10.1088/0953-8984/16/4/010

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A dynamic density-matrix renormalization group approach to the spectral properties of quantum impurity problems is presented. The method is demonstrated on the spectral density of the flat-band symmetric single-impurity Anderson model. We show that this approach provides the impurity spectral density for all frequencies and coupling strengths. In particular, Hubbard satellites at high energy can be obtained with a good resolution. The main difficulties are the necessary discretization of the host band hybridized with the impurity and the resolution of sharp spectral features such as the Abrikosov–Suhl resonance.
In this thesis we investigated, implemented, gauged, optimized, and applied a numerical approach given by the dynamic density-matrix renormalization (D-DMRG). The dynamic density-matrix renormalization is the extension of standard DMRG to the calculation of dynamic quantities. D-DMRG provides valuable numerical information on dynamic correlations by computing convolutions of the corresponding spectral densities. The dynamics at zero temperature is determined by computing the expectation values in the local propagator. This can be realized by targeting not only at the ground state and the excited state, but also at the resolvent applied to the excited state. This additional targeted state is called the correction vector. The calculation of the frequency-dependent correction vector is numerically extremely demanding due to the inversion of an almost singular non-hermitian matrix. We compared the performance of several iterative solvers for linear equation systems and managed to stabilize the inversion problem of the D-DMRG by using optimized algorithms. The main limitation of the correction vector D-DMRG is that one cannot obtain data for purely real frequencies but only for frequencies with a certain imaginary part. Hence the extraction of the behavior at purely real frequencies is one of the main problems to be solved in using the D-DMRG. We illustrated how and to which extent such data can be deconvolved to retrieve the wanted spectral densities. We discussed and compared various algorithms to achieve this extraction and presented a non-linear approach from the family of maximum entropy methods. This approach provides a continuous, positive ansatz for the wanted spectral density with the least bias (LB). Even relative abrupt changes of the spectral density can be reproduced satisfactorily. In the vicinity of singularities spurious oscillations occur. The least-bias ansatz can be made more robust towards small numerical inaccuracies and finite-size effects by including besides the entropy functional a xi-functional in the functional to be minimized. We investigated the dynamic propagator of the symmetric single impurity Anderson model by D-DMRG. The Abrikosov-Suhl resonance (ASR) was calculated to gauge the D-DMRG and to demonstrate that features at low energies can be resolved. The low energy scale of the SIAM was reproduced over two orders of magnitude. The Hubbard satellites were investigated. From the broadened D-DMRG raw data we extracted the width and the shifts from the atomic positions. By means of the LB extraction we were able to address the line shape of the satellites. It was found that the Hubbard satellites are strongly asymmetric, the peak is very pronounced and the maximum value is very high. The Hubbard model on the Bethe lattice with infinite coordination number was investigated by means of the dynamic mean-field theory (DMFT) using the D-DMRG as impurity solver. A high-resolution investigation of the electron spectra close to the metal-to-insulator transition in dynamic mean-field theory was presented. The all-numerical, consistent confirmation of a smooth transition at zero temperature is provided.
In this work we outline a simple and numerically inexpensive approach to describe the spectral features of the single-impurity Anderson model. The method combines aspects of the density-matrix embedding theory (DMET) approach with a spectral broadening approach inspired by those used in numerical renormalization-group methods. At zero temperature for a wide range of UUU, the spectral function produced by this approach is found to be in good agreement with general expectations as well as more advanced and complex numerical methods such as DMRG-based schemes. The theory developed here is simply transferable to more complex impurity problems.
This article focuses on the calculation of spectral functions for single- and multi-impurity models using the density matrix renormalization group (DMRG). To calculate spectral functions in DMRG, the correction vector method is presently the most widely used approach. One, however, always obtains Lorentzian convoluted spectral functions, which in applications like the dynamical mean-field theory can lead to wrong results. In order to overcome this restriction, we use chain decompositions in which the resulting effective Hamiltonian can be diagonalized completely to calculate a discrete ``peak'' spectrum. We show that this peak spectrum is a very good approximation to a deconvolution of the correction vector spectral function. Calculating this deconvoluted spectrum directly from the DMRG basis set and operators is the most natural approach, because it uses only information from the system itself. Having calculated this excitation spectrum, one can use an arbitrary broadening to obtain a smooth spectral function or directly analyze the excitations. As a nontrivial test, we apply this method to obtain spectral functions for a model of three coupled Anderson impurities. Although we are focusing in this article on impurity models, the proposed method for calculating the peak spectrum can be easily adapted to usual lattice models.
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 $\ensuremath{\Lambda}$, zzz-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.
The density matrix renormalization group method is introduced in energy space to study Anderson impurity models. The method allows for calculations in the thermodynamic limit and is advantageous for studying not only the dynamical properties, but also the quantum entanglement of the ground state at the vicinity of an impurity quantum phase transition. This method is applied to obtain numerically exactly the ground-state phase diagram of the single-impurity Anderson model on the honeycomb lattice at half-filling. The calculation of local static quantities shows that the phase diagram contains two distinct phases, the local moment (LM) phase and the asymmetric strong coupling (ASC) phase, but no Kondo screening phase. These results are supported by the local spin and charge excitation spectra, which exhibit qualitatively different behavior in these two phases and also reveal the existence of the valence fluctuating point at the phase boundary. For comparison, we also study the low-energy effective pseudogap Anderson model using the method introduced here. Although the high-energy excitations are obviously different, we find that the ground-state phase diagram and the asymptotically low-energy excitations are in good quantitative agreement with those for the single-impurity Anderson model on the honeycomb lattice, thus providing a quantitative justification for the previous studies based on low-energy approximate approaches. Furthermore, we find that the lowest entanglement level is doubly degenerate for the LM phase, whereas it is singlet for the ASC phase and is accidentally threefold degenerate at the valence fluctuating point. This should be contrasted with the degeneracy of the energy spectrum because the ground state is found to be always singlet. Our results therefore clearly demonstrate that the low-lying entanglement spectrum can be used to determine with high accuracy the phase boundary of the impurity quantum phase transition.