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arXiv:2010.04572 · Physical Review Letters

Nevanlinna Analytical Continuation

Jiani Fei, Chia-Nan Yeh, Emanuel Gull

Published Feb 4, 2021• 115 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.126.056402

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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 investigate the possibility to assist the numerically ill-posed calculation of spectral properties of interacting quantum systems in thermal equilibrium by extending the imaginary-time simulation to a finite Schwinger-Keldysh contour. The effect of this extension is tested within the standard maximum entropy approach to analytic continuation. We find that the inclusion of real-time data improves the resolution of structures at high energy, while the imaginary-time data are needed to correctly reproduce low-frequency features such as quasiparticle peaks. As a nonequilibrium application, we consider the calculation of time-dependent spectral functions from retarded Green function data on a finite time interval, and compare the maximum entropy approach to direct Fourier transformation and a method based on Padé approximants.
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.
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.
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.
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 report multipronged progress on the stochastic averaging approach to numerical analytic continuation of imaginary-time correlation functions computed by quantum Monte Carlo simulations. After reviewing the conventional maximum-entropy approach and established stochastic analytic continuation methods, we present several new developments in which the configurational entropy of the sampled spectrum plays a key role. Parametrizing the spectrum as a large number of δ-functions in continuous frequency space, an exact calculation of the entropy lends support to a simple goodness-of-fit criterion for the optimal sampling temperature. We also compare spectra sampled in continuous frequency with those from amplitudes sampled on a fixed frequency grid. Insights into the functional form of the entropy in different cases allow us to demonstrate equivalence in a generalized thermodynamic limit (large number of degrees of freedom) of the average spectrum and the maximum-entropy solution, with different parametrizations corresponding to different forms of the entropy in the prior probability. These results revise prevailing notions of the maximum-entropy method and its relationship to stochastic analytic continuation. In further developments of the sampling approach, we explore various adjustable (optimized) constraints that allow sharp low-temperature spectral features to be resolved, in particular at the lower frequency edge. The constraints, e.g., the location of the edge or the spectral weight of a quasi-particle peak, are optimized using a statistical criterion based on entropy minimization under the condition of optimal fit. We show with several examples that this method can correctly reproduce both narrow and broad quasi-particle peaks. We next introduce a parametrization for more intricate spectral functions with sharp edges, e.g., power-law singularities. We present tests with synthetic data as well as with real simulation data for the spin-1/2 Heisenberg chain, where a divergent edge of the dynamic structure factor is due to deconfined spinon excitations. Our results demonstrate that distortions of sharp edges or quasi-particle peaks, which arise with other analytic continuation methods, propagate and cause artificial spectral features also at higher energies. The constrained sampling methods overcome this problem and allow analytic continuation of spectra with sharp edge features at unprecedented fidelity. We present results for S=1/2 Heisenberg 2-leg and 3-leg ladders to illustrate the ability of the methods to resolve spectral features arising from both elementary and composite excitations. Finally, we also propose how the methods developed here could be used as “pre processors” for analytic continuation by machine learning. Edge singularities and narrow quasi-particle peaks being ubiquitous in quantum many-body systems, we expect the new methods to be broadly useful and take numerical analytic continuation to a new quantitative level in many applications.