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arXiv:2410.14000 · Physical review. B./Physical review. B

Minimal pole representation and analytic continuation of matrix-valued correlation functions

Lei Zhang, Yang Yu, Emanuel Gull

Published Dec 12, 2024• 15 citations
arXivHTMLPDF

Abstract

The authors present here a method for analytically continuing matrix-valued correlation functions to the real axis, accurately capturing off-diagonal elements to enhance the interpretation of self-energies and susceptibilities. Based on a minimal information criterion, a selection of poles in the complex plane results in a universal noise-resistant approximation that converges reliably with increasing data precision. Comparisons with state-of-the-art techniques highlight its resolution and efficiency, establishing it as a significant advancement in computational quantum physics.

DOI
10.1103/physrevb.110.235131

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The authors present here a method for analytically continuing matrix-valued correlation functions to the real axis, accurately capturing off-diagonal elements to enhance the interpretation of self-energies and susceptibilities. Based on a minimal information criterion, a selection of poles in the complex plane results in a universal noise-resistant approximation that converges reliably with increasing data precision. Comparisons with state-of-the-art techniques highlight its resolution and efficiency, establishing it as a significant advancement in computational quantum physics.
Analytic continuation is a central step in the simulation of finite-temperature field theories in which numerically obtained Matsubara data are continued to the real frequency axis for a physical interpretation. Numerical analytic continuation is considered to be an ill-posed problem where uncertainties on the Matsubara axis are amplified exponentially. Here, we present a systematic and controlled procedure that approximates any Matsubara function by a minimal pole representation to within a predefined precision. We then show a systematic convergence to the exact spectral function on the real axis as a function of our control parameter for a range of physically relevant setups. Our methodology is robust to noise and paves the way towards reliable analytic continuation in many-body theory and, by providing access to the analytic structure of the functions, a direct theoretical interpretation of physical properties.
Finite-temperature quantum field theories are formulated in terms of Green's functions and self-energies on the Matsubara axis. In multiorbital systems, these quantities are related to positive semidefinite matrix-valued functions of the Carath\'eodory and Schur class. Analysis, interpretation, and evaluation of derived quantities such as real-frequency response functions requires analytic continuation of the off-diagonal elements to the real axis. We derive the criteria under which such functions exist for given Matsubara data and present an interpolation algorithm that intrinsically respects their mathematical properties. For small systems with precise Matsubara data, we find that the continuation exactly recovers all off-diagonal and diagonal elements. In real-materials systems, we show that the precision of the continuation is sufficient for the analytic continuation to commute with the Dyson equation, and we show that the truncation of the off-diagonal self-energy elements leads to considerable approximation artifacts.
Representing spectral densities, real-frequency, and real-time Green's functions of continuous systems by a small discrete set of complex poles is a ubiquitous problem in condensed matter physics, with applications ranging from quantum transport simulations to the simulation of strongly correlated electron systems. This paper introduces a method for obtaining a compact, approximate representation of these functions, based on their parameterization on the real axis and a given approximate precision. We show applications to typical spectral functions and results for structured and unstructured correlation functions of model systems.
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.
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.
This dissertation develops the complex pole representation as a unified framework for compactly representing Green's functions in quantum many-body simulations, with applications to numerical analytic continuation, hybridization fitting, compact representation and quantum impurity solvers, accompanied with software implementation. The central idea is to replace continuous spectral and correlation functions by a small number of complex poles and complex weights, while preserving the analytic structure needed for physical interpretation and numerical simulation. By connecting analytic continuation, hybridization fitting, compact representation, impurity solvers, and software implementation within a single pole-based perspective, this dissertation provides a systematically improvable approach to extracting, representing, and directly manipulating real-frequency information in quantum many-body simulations.