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arXiv:2204.03165 · Physical review. B./Physical review. B

Root- N Krylov-space correction vectors for spectral functions with the density matrix renormalization group

Alberto Nocera, G. Alvarez

Published Nov 3, 2022• 15 citations
arXivHTMLPDF

Abstract

We propose a method to compute spectral functions of generic Hamiltonians using the density matrix renormalization group (DMRG) algorithm directly in the frequency domain, based on a modified Krylov-space decomposition to compute the correction vectors. Our approach entails the calculation of the root-$N$ ($N=2$ is the standard square root) of the Hamiltonian propagator using Krylov-space decomposition and repeating this procedure $N$ times to obtain the actual correction vector. We show that our method greatly alleviates the burden of keeping a large bond dimension at large target frequencies, a problem found with conventional correction-vector DMRG, whereas achieving better computational performance at large $N$. We apply our method to spin and charge spectral functions of $t\text{\ensuremath{-}}J$ and Hubbard models in the challenging two-leg ladder geometry and provide evidence that the root-$N$ approach reaches a much improved spectral resolution compared to the conventional correction vector.

DOI
10.1103/physrevb.106.205106

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We propose a method to compute spectral functions of generic Hamiltonians using the density matrix renormalization group (DMRG) algorithm directly in the frequency domain, based on a modified Krylov-space decomposition to compute the correction vectors. Our approach entails the calculation of the root-NNN (N=2N=2N=2 is the standard square root) of the Hamiltonian propagator using Krylov-space decomposition and repeating this procedure NNN times to obtain the actual correction vector. We show that our method greatly alleviates the burden of keeping a large bond dimension at large target frequencies, a problem found with conventional correction-vector DMRG, whereas achieving better computational performance at large NNN. We apply our method to spin and charge spectral functions of t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J and Hubbard models in the challenging two-leg ladder geometry and provide evidence that the root-NNN approach reaches a much improved spectral resolution compared to the conventional correction vector.
Frequency-dependent correlations, such as the spectral function and the dynamical structure factor, help illustrate condensed matter experiments. Within the density matrix renormalization group (DMRG) framework, an accurate method for calculating spectral functions directly in frequency is the correction-vector method. The correction vector can be computed by solving a linear equation or by minimizing a functional. This paper proposes an alternative to calculate the correction vector: to use the Krylov-space approach. This paper then studies the accuracy and performance of the Krylov-space approach, when applied to the Heisenberg, the t-J, and the Hubbard models. The cases studied indicate that the Krylov-space approach can be more accurate and efficient than the conjugate gradient, and that the error of the former integrates best when a Krylov-space decomposition is also used for ground state DMRG.
Frequency-dependent correlations, such as the spectral function and the dynamical structure factor, help understand condensed matter experiments. Within the DMRG framework, an accurate method for calculating spectral functions directly in frequency is the correction-vector method. The correction-vector can be computed by solving a linear equation or by minimizing a functional. This paper proposes an alternative to calculate the correction vector: to use the Krylov-space approach. This paper then studies the accuracy and performance of the Krylov-space approach, when applied to the Heisenberg, the t-J, and the Hubbard models. The cases studied indicate that Krylov-space approach can be more accurate and efficient than conjugate gradient, and that the error of the former integrates best when a Krylov-space decomposition is also used for ground state DMRG.
Current widely-used approaches to calculate spectral functions using the density-matrix renormalization group in frequency space either necessarily include an artificial broadening (correction-vector method), have limited resolution (time-domain density-matrix renormalization group with Fourier transform method), or are limited to low-energy properties or single dominant modes (original continued fraction method). Here we propose an adaptive Lanczos-vector method to calculate the coefficients of a continued fraction expansion of the spectral function iteratively. We show that one can obtain a very accurate representation of the spectral function very efficiently, and that one can also directly extract the spectral weights and poles for the discrete system.
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.
Current widely used approaches to calculate spectral functions using the density-matrix renormalization group in frequency space either necessarily include an artificial broadening (correction-vector method), have limited resolution (time-domain density-matrix renormalization group with Fourier transform method), or are limited to low-energy properties or single dominant modes (original continued fraction method). Here we propose an adaptive Lanczos-vector method to calculate the coefficients of a continued fraction expansion of the spectral function iteratively. We show that one can obtain a very accurate representation of the spectral function very efficiently, and that one can also directly extract the spectral weights and poles for the discrete system.
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.