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arXiv:1906.05050 · Physical review. B./Physical review. B

Self-energy Padé approach for analytic continuation: Application to the zero-gap Kondo lattice model

Annamária Kiss

Published Dec 16, 2019• 5 citations
arXivHTMLPDF

Abstract

The self-energy Pad\'e approach is discussed as an analytic continuation of numerical methods that evaluate correlation functions in imaginary time. Instead of direct analytic continuation of the correlation functions, the Pad\'e method is applied for the self-energy in this approach, which is then used for deriving the Green's functions at real energies. We demonstrate that the self-energy Pad\'e approach is more stable and robust against statistical errors compared to the direct way. The characteristics and success of the self-energy Pad\'e approach are analyzed by actual calculations for the illustrative examples of the noninteracting Anderson lattice and interacting Hubbard model. A zero-gap Kondo lattice model with linearly vanishing conduction electron density of states at the Fermi level is also studied by using the self-energy Pad\'e approach as an analytic continuation. We investigate the properties of the Kondo insulating state including the dependence of the insulating gap on the Kondo coupling and coherence effects. Furthermore, we identify two energy scales from dynamic and thermodynamic quantities that are associated as a direct and an indirect gap in a band hybridization picture.

DOI
10.1103/physrevb.100.214417

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The self-energy Pad'e approach is discussed as an analytic continuation of numerical methods that evaluate correlation functions in imaginary time. Instead of direct analytic continuation of the correlation functions, the Pad'e method is applied for the self-energy in this approach, which is then used for deriving the Green's functions at real energies. We demonstrate that the self-energy Pad'e approach is more stable and robust against statistical errors compared to the direct way. The characteristics and success of the self-energy Pad'e approach are analyzed by actual calculations for the illustrative examples of the noninteracting Anderson lattice and interacting Hubbard model. A zero-gap Kondo lattice model with linearly vanishing conduction electron density of states at the Fermi level is also studied by using the self-energy Pad'e approach as an analytic continuation. We investigate the properties of the Kondo insulating state including the dependence of the insulating gap on the Kondo coupling and coherence effects. Furthermore, we identify two energy scales from dynamic and thermodynamic quantities that are associated as a direct and an indirect gap in a band hybridization picture.
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 ill-posed analytic continuation problem for Green's functions and self-energies is investigated by revisiting the Pad\'e approximants technique. We propose to remedy the well-known problems of the Pad\'e approximants by performing an average of several continuations, obtained by varying the number of fitted input points and Pad\'e coefficients independently. The suggested approach is then applied to several test cases, including Sm and Pr atomic self-energies, the Green's functions of the Hubbard model for a Bethe lattice and of the Haldane model for a nanoribbon, as well as two special test functions. The sensitivity to numerical noise and the dependence on the precision of the numerical libraries are analyzed in detail. The present approach is compared to a number of other techniques, i.e., the nonnegative least-squares method, the nonnegative Tikhonov method, and the maximum entropy method, and is shown to perform well for the chosen test cases. This conclusion holds even when the noise on the input data is increased to reach values typical for quantum Monte Carlo simulations. The ability of the algorithm to resolve fine structures is finally illustrated for two relevant test functions.
We present a new algorithm to analytically continue the self-energy of quantum many-body systems from Matsubara frequencies to the real axis. The method allows straightforward, unambiguous computation of electronic spectra for lattice models of strongly correlated systems from self-energy data that has been collected with state-of-the-art continuous time solvers within dynamical mean-field simulations. Using well-known analytical properties of the self-energy, the analytic continuation is cast into a constrained minimization problem that can be formulated as a quadratic programmable optimization with linear constraints. The algorithm is validated against exactly solvable finite-size problems, showing that all features of the spectral function near the Fermi level are very well reproduced and coarse features are reproduced for all energies. The method is applied to two well-known lattice problems, the two-dimensional Hubbard model at half filling, where the momentum dependence of the gap formation is studied, as well as a multiband model of NiO, for which the spectral function can be directly compared to experiment. Agreement with published results is very good.
The self-energy method for quantum impurity models expresses the correlation part of the self-energy in terms of the ratio of two Green's functions and allows for a more accurate calculation of equilibrium spectral functions than is possible directly from the one-particle Green's function [Bulla et al., J. Phys.: Condens. Matter 10, 8365 (1998)], for example, within the numerical renormalization group method. In addition, the self-energy itself is a central quantity required in the dynamical mean field theory of strongly correlated lattice models. Here, we show how to generalize the self-energy method to the time-dependent situation for the prototype model of strong correlations, the Anderson impurity model. We use the equation-of-motion method to obtain closed expressions for the local Green's function in terms of a time-dependent correlation self-energy, with the latter being given as a ratio of a one-particle time-dependent Green's function and a higher-order correlation function. We benchmark this self-energy approach to time-dependent spectral functions against the direct approach within the time-dependent numerical renormalization group method. The self-energy approach improves the accuracy of time-dependent spectral function calculations, and the closed-form expressions for the Green's function allow for a clear picture of the time evolution of spectral features at the different characteristic time scales. The self-energy approach is of potential interest also for other quantum impurity solvers for real-time evolution, including time-dependent density matrix renormalization group and continuous-time quantum Monte Carlo techniques.
Direct analytic continuation of the self-energy is used to determine the effect of antiferromagnetic ordering on the spectral function and optical conductivity of a Mott insulator. Comparison of several methods shows that the most robust estimation of the gap value is obtained by use of the real part of the continued self-energy in the quasiparticle equation within the single-site dynamical mean-field theory of the two-dimensional square lattice Hubbard model, where, for a UUU that is slightly greater than the Mott critical value, antiferromagnetism increases the gap by about 80%.
Analytic continuation (AC) from the imaginary-time Green's function to the spectral function is a crucial process for numerical studies of the dynamical properties of quantum many-body systems. This process, however, is an ill-posed problem; that is, the obtained spectrum is unstable against the noise of the Green's function. Though several numerical methods have been developed, each of them has its own advantages and disadvantages. The sparse modeling (SpM) AC method, for example, is robust against the noise of the Green's function but suffers from unphysical oscillations in the low-energy region. We propose a method that combines the SpM AC with the Pad\'e approximation. This combination, called SpM-Pad\'e, inherits robustness against noise from SpM and low-energy accuracy from Pad\'e, compensating for the disadvantages of each. We demonstrate that the SpM-Pad\'e method yields low-variance and low-biased results with almost the same computational cost as that of the SpM method.