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arXiv:2305.03449 · Journal of the Physical Society of Japan

Bosonic Nevanlinna Analytic Continuation

Kosuke Nogaki, Hiroshi Shinaoka

Published Feb 17, 2023• 21 citations
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Abstract

Analytical continuation (AC) connects theoretical calculations and experimentally measurable quantities. The recently proposed Nevanlinna AC method is capable of accurately reproducing the sharp features of spectral functions at high frequencies while maintaining the causality of the response function. However, their use is currently limited to fermions. Here, we present an extension of this method to bosons using the hyperbolic tangent trick, allowing us to transform bosons into auxiliary fermions to which the Nevanlinna analytic continuation can be applied.

DOI
10.7566/jpsj.92.035001

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Analytical continuation (AC) connects theoretical calculations and experimentally measurable quantities. The recently proposed Nevanlinna AC method is capable of accurately reproducing the sharp features of spectral functions at high frequencies while maintaining the causality of the response function. However, their use is currently limited to fermions. Here, we present an extension of this method to bosons using the hyperbolic tangent trick, allowing us to transform bosons into auxiliary fermions to which the Nevanlinna analytic continuation can be applied.
Simulations of finite temperature quantum systems provide imaginary frequency Green's functions that correspond one to one to experimentally measurable real-frequency spectral functions. However, due to the bad conditioning of the continuation transform from imaginary to real frequencies, established methods tend to either wash out spectral features at high frequencies or produce spectral functions with unphysical negative parts. Here, we show that explicitly respecting the analytic "Nevanlinna" structure of the Green's function leads to intrinsically positive and normalized spectral functions, and we present a continued fraction expansion that yields all possible functions consistent with the analytic structure. Application to synthetic trial data shows that sharp, smooth, and multipeak data is resolved accurately. Application to the band structure of silicon demonstrates that high energy features are resolved precisely. Continuations in a realistic correlated setup reveal additional features that were previously unresolved. By substantially increasing the resolution of real frequency calculations our work overcomes one of the main limitations of finite-temperature quantum simulations.
We introduce a Julia implementation of the recently proposed Nevanlinna analytic continuation method. The method is based on Nevanlinna interpolants and inherently preserves the causality of a response function due to its construction. For theoretical calculations without statistical noise, this continuation method is a powerful tool to extract real-frequency information from numerical input data on the Matsubara axis. This method has been applied to first-principles calculations of correlated materials. This paper presents its efficient and full-featured open-source implementation of the method including the Hamburger moment problem and smoothing.
Green's functions of fermions are described by matrix-valued Herglotz-Nevanlinna functions. Since analytic continuation is fundamentally an ill-posed problem, the causal space described by the matrix-valued Herglotz-Nevanlinna structure can be instrumental in improving the accuracy and in enhancing the robustness with respect to noise. We demonstrate a three-pronged procedure for robust analytic continuation called PES: (1) projection of data to the causal space; (2) estimation of pole locations; and (3) semidefinite relaxation within the causal space. We compare the performance of PES with the recently developed Nevanlinna and Carath\'eodory continuation methods and find that PES is more robust in the presence of noise and does not require the usage of extended precision arithmetics. We also demonstrate that a causal projection improves the performance of the Nevanlinna and Carath\'eodory methods. The PES method is generalized to bosonic response functions, for which the Nevanlinna and Carath\'eodory continuation methods have not yet been developed. It is particularly useful for studying spectra with sharp features, as they occur in the study of molecules and band structures in solids.
We introduce a Julia implementation of the recently proposed Nevanlinna analytic continuation method. The method is based on Nevanlinna interpolants and inherently preserves the causality of a response function due to its construction. For theoretical calculations without statistical noise, this continuation method is a powerful tool to extract real-frequency information from numerical input data on the Matsubara axis. This method has been applied to first-principles calculations of correlated materials. This paper presents its efficient and full-featured open-source implementation of the method including the Hamburger moment problem and smoothing.
In this paper, we propose an analytic continuation method to extract real-frequency spectral functions from imaginary-frequency Green's functions of quantum many-body systems. This method is based on the pole representation of Matsubara Green's function and a stochastic sampling procedure is utilized to optimize the amplitudes and locations of poles. In order to capture narrow peaks and sharp band edges in the spectral functions, a constrained sampling algorithm and a self-adaptive sampling algorithm are developed. To demonstrate the usefulness and performance of this method, we at first apply it to study the spectral functions of representative fermionic and bosonic correlators. Then we employ this method to tackle the analytic continuation problems of matrix-valued Green's functions. The synthetic Green's functions, as well as realistic correlation functions from finite-temperature quantum many-body calculations, are used as input. The benchmark results demonstrate that this method is capable of reproducing most of the key characteristics in the spectral functions. The sharp, smooth, and multipeak features in both low- and high-frequency regions of spectral functions could be accurately resolved, which overcomes one of the main limitations of the traditional maximum entropy method. More importantly, it exhibits excellent robustness with respect to noisy and incomplete input data. The causality of spectral function is always satisfied even in the presence of sizable noises. As a by-product, this method could derive a fitting formula for the Matsubara data, which provides a compact approximation to the many-body Green's functions. Hence, we expect that this method could become a pivotal workhorse for numerically analytic continuation and be broadly useful in many applications.
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.