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arXiv:1609.01248 · Physical review. B./Physical review. B

Adaptive broadening to improve spectral resolution in the numerical renormalization group

Seung‐Sup B. Lee, Andreas Weichselbaum

Published Dec 12, 2016• 42 citations
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Abstract

We propose an adaptive scheme of broadening the discrete spectral data from numerical renormalization group (NRG) calculations to improve the resolution of dynamical properties at finite energies. While the conventional scheme overbroadens narrow features at large frequency by broadening discrete weights with constant width in log-frequency, our scheme broadens each discrete contribution individually based on its sensitivity to a $z$-shift in the logarithmic discretization intervals. We demonstrate that the adaptive broadening better resolves various features in noninteracting and interacting models at comparable computational cost. The resolution enhancement is more significant for coarser discretization as typically required in multiband calculations. At low frequency below the energy scale of temperature, the discrete NRG data necessarily needs to be broadened on a linear scale. Here we provide a method that minimizes transition artifacts in between these broadening kernels.

DOI
10.1103/physrevb.94.235127

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We propose an adaptive scheme of broadening the discrete spectral data from numerical renormalization group (NRG) calculations to improve the resolution of dynamical properties at finite energies. While the conventional scheme overbroadens narrow features at large frequency by broadening discrete weights with constant width in log-frequency, our scheme broadens each discrete contribution individually based on its sensitivity to a zzz-shift in the logarithmic discretization intervals. We demonstrate that the adaptive broadening better resolves various features in noninteracting and interacting models at comparable computational cost. The resolution enhancement is more significant for coarser discretization as typically required in multiband calculations. At low frequency below the energy scale of temperature, the discrete NRG data necessarily needs to be broadened on a linear scale. Here we provide a method that minimizes transition artifacts in between these broadening kernels.
Numerical renormalization-group (NRG) calculations of quantum impurity models, based on a logarithmic discretization in energy of electronic or bosonic Hamiltonians, provide a powerful tool to describe physics involving widely separated energy scales, as typically encountered in nanostructures and strongly correlated materials. This main advantage of the NRG was however considered a drawback for resolving sharp spectral features at finite energy, such as dissipative atomic peaks. Surprisingly, we find a bunching of many-body levels in NRG spectra near dissipative resonances, and exploit this by combining the widely used Oliveira's zzz trick, using an averaging over few discrete NRG spectra, with an optimized frequency-dependent broadening parameter $b(\ensuremath{\omega})$. This strategy offers a tremendous gain in computational power and extracts all the needed information from the raw NRG data without a priori knowledge of the various energy scales at play. As an application we investigate with high precision the crossover from coherent to incoherent dynamics in the spin boson model.
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 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.
In the numerical renormalization-group (NRG) calculations of spectral functions of quantum impurity models, the results are always affected by discretization and truncation errors. The discretization errors can be alleviated by averaging over different discretization meshes (``zzz-averaging''), but since each partial calculation is performed for a finite discrete system, there are always some residual discretization and finite-size errors. The truncation errors affect the energies of the states and result in the displacement of the delta-peak spectral contributions from their correct positions. The two types of errors are interrelated: for coarser discretization, the discretization errors increase, but the truncation errors decrease since the separation of energy scales is enhanced. In this work, it is shown that by calculating a series of spectral functions for a range of the total number of states kept in the NRG truncation, it is possible to estimate the errors and determine the error bars for spectral functions, which is important when making accurate comparison to the results obtained by other methods and for determining the errors in the extracted quantities (such as peak positions, heights, and widths). The closely related problem of spectral broadening is also discussed: it is shown that the overbroadening contorts the results without, surprisingly, reducing the variance of the curves. It is thus important to determine the results in the limit of zero broadening. The method is applied to determine the error bounds for the Kondo peak splitting in an external magnetic field. For moderately strong fields, the results are consistent with the Bethe ansatz study by Moore and Wen [Phys. Rev. Lett. 85, 1722 (2000)]. We also discuss the regime of large $U/\ensuremath{\Gamma}$ ratio. It is shown that in the strong-field limit, a spectral step is observed in the spectrum precisely at the Zeeman frequency until the field becomes so strong that the step merges with the atomic spectral peak.
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.
The numerical renormalization group (NRG) is rephrased as a variational method with the cost function given by the sum of all the energies of the effective low-energy Hamiltonian. This allows us to systematically improve the spectrum obtained by NRG through sweeping. The ensuing algorithm has a lot of similarities to the density matrix renormalization group (DMRG) when targeting many states, and this synergy of NRG and DMRG combines the best of both worlds and extends their applicability. We illustrate this approach with simulations of a quantum spin chain and a single impurity Anderson model where the accuracy of the effective eigenstates is greatly enhanced as compared to the NRG, especially in the transition to the continuum limit.