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arXiv:2109.08370 · Physical review. B./Physical review. B

Robust analytic continuation combining the advantages of the sparse modeling approach and the Padé approximation

Yuichi Motoyama, Kazuyoshi Yoshimi, Junya Otsuki

Published Jan 24, 2022• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.105.035139

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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.
Analytic continuation (AC) from imaginary-time Green's function to spectral function is essential in the numerical analysis of dynamical properties in quantum many-body systems. However, this process faces a fundamental challenge: it is an ill-posed problem, leading to unstable spectra against the noise in the Green's function. This instability is further complicated in multi-orbital systems with hybridization between spin-orbitals, where off-diagonal Green's functions yield a spectral matrix with off-diagonal elements, necessitating the matrix's semi-positive definiteness to satisfy the causality. We propose an advanced AC method using sparse modeling for multi-orbital systems, which reduces the effect of noise and ensures the matrix's semi-positive definiteness. We demonstrate the effectiveness of this approach by contrasting it with the conventional sparse modeling method, focusing on handling Green's functions with off-diagonal elements, thereby demonstrating our proposed method's enhanced stability and precision.
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.
The ill-posed analytic continuation problem for Green's functions or self-energies can be carried out using the Padé rational polynomial approximation. However, to extract accurate results from this approximation, high precision input data of the Matsubara Green function are needed. The calculation of the Matsubara Green function generally involves a Matsubara frequency summation, which cannot be evaluated analytically. Numerical summation is requisite but it converges slowly with the increase of the Matsubara frequency. Here we show that this slow convergence problem can be significantly improved by utilizing the Padé decomposition approach to replace the Matsubara frequency summation by a Padé frequency summation, and high precision input data can be obtained to successfully perform the Padé analytic continuation.
We critique a Pad\'e analytic continuation method whereby a rational polynomial function is fit to a set of input points by means of a single matrix inversion. This procedure is accomplished to an extremely high accuracy using a symbolic computation algorithm. As an example of this method in action, it is applied to the problem of determining the spectral function of a single-particle thermal Green's function known only at a finite number of Matsubara frequencies with two example self energies drawn from the T-matrix theory of the Hubbard model. We present a systematic analysis of the effects of error in the input points on the analytic continuation, and this leads us to propose a procedure to test quantitatively the reliability of the resulting continuation, thus eliminating the black-magic label frequently attached to this procedure.
Analytic continuation is a critical step in quantum many-body computations, connecting imaginary-time or Matsubara Green's functions with real-frequency spectral functions, which can be directly compared to experimental results. However, because of the ill-posed nature of the analytic continuation problems, they have not been completely solved so far. In this paper, we suggest a simple, yet highly efficient method for analytic continuations of Matsubara Green's functions. This method takes advantage of barycentric rational functions to directly interpolate Matsubara Green's functions. At first, the nodes and weights of the barycentric rational functions are determined by the adaptive Antoulas-Anderson algorithm, avoiding reliance on nonconvex optimization. Next, the retarded Green's functions and the related spectral functions are evaluated by the resulting interpolants. We systematically explore the performance of this method through a series of toy models and realistic examples, comparing its accuracy and efficiency with other popular methods, such as the maximum entropy method. The benchmark results demonstrate that the new method can accurately reproduce not only continuous but also discrete spectral functions, irrespective of their positive definiteness. It works well even in the presence of intermediate noise, and outperforms traditional analytic continuation methods in computational speed. We believe that this method should stand out for its robustness against noise, broad applicability, high precision, and ultra-efficiency, offering a promising alternative to the maximum entropy method.
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.