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arXiv:1607.03538 · Physical review. E

Spectral functions with the density matrix renormalization group: Krylov-space approach for correction vectors

Alberto Nocera, Gonzalo Álvarez

Published Nov 21, 2016• 91 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.94.053308

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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.
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 $t\text{\ensuremath{-}}J$ 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.
The density matrix renormalization group (DMRG) method allows for very precise calculations of ground state properties in low-dimensional strongly correlated systems. We investigate two methods to expand the DMRG to calculations of dynamical properties. In the Lanczos vector method the DMRG basis is optimized to represent Lanczos vectors, which are then used to calculate the spectra. This method is fast and relatively easy to implement, but the accuracy at higher frequencies is limited. Alternatively, one can optimize the basis to represent a correction vector for a particular frequency. The correction vectors can be used to calculate the dynamical correlation functions at these frequencies with high accuracy. By separately calculating correction vectors at different frequencies, the dynamical correlation functions can be interpolated and pieced together from these results. For systems with open boundaries we discuss how to construct operators for specific wave vectors using filter functions.
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.
We present a tangent-space Krylov method for efficient computation of zero-temperature real-frequency spectral functions on top of ground-state matrix product states obtained from the density matrix renormalization group. It relies on projecting the Hamiltonian to the tangent space of the ground-state matrix product states, where resolvents can be efficiently represented using Krylov space techniques. This enables direct computations of excitation energies and the corresponding spectral weights. We demonstrate the accuracy and efficiency of the tangent-space Krylov approach by showcasing spectral data for various models. These include the 1D Haldane-Shastry and Heisenberg models as benchmarks. As an interesting application, we study the Hubbard model on a cylinder at half filling, augmented by a density-assisted hopping (DAH) term. We find that DAH leads to particle-hole asymmetric single-particle mobilities and lifetimes in the resulting Mott insulator, and identify the responsible scattering processes. Furthermore, we find that DAH influences the dispersion of Green’s function zeros beyond its range, which has a frustrating effect on the Mott insulator studied here.
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.