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arXiv:2605.26586 · APS Open Science

Analytical Singular-Value Structure of Analytic-Continuation Kernels from Slepian Information Theory

Masayuki Ohzeki

quant-phcond-mat.str-elSubmitted May 26, 2026• Announced May 27, 2026• 0 citations
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Abstract

Analytic continuation from imaginary-time Green's functions to real-frequency spectra is a central ill-posed inverse problem in quantum many-body physics. We show that the thermal kernel admits an analytical generalized singular-value structure once its purely dynamical part is separated from the statistical weight imposed by the heat bath. The dynamical kernel is the imaginary-bandwidth continuation of Slepian's finite Fourier transform and is governed by the same Sturm-Liouville algebra that yields prolate spheroidal wave functions. Fermionic and bosonic statistics then enter as gauge transformations of the frequency-space inner product, producing self-adjoint effective potentials but no numerical kernel diagonalization. The Shannon number, $N_c=β\wmax/π$, fixes the upper information capacity of this pure Laplace channel. Finally, the optimal sampling points are obtained as eigenvalues of a Legendre colleague matrix, giving a deterministic compressed-sensing grid without iterative root searches.

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4 pages, 2 figures

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Analytic continuation from imaginary-time Green's functions to real-frequency spectra is a central ill-posed inverse problem in quantum many-body physics. We show that the thermal kernel admits an analytical generalized singular-value structure once its purely dynamical part is separated from the statistical weight imposed by the heat bath. The dynamical kernel is the imaginary-bandwidth continuation of Slepian's finite Fourier transform and is governed by the same Sturm-Liouville algebra that yields prolate spheroidal wave functions. Fermionic and bosonic statistics then enter as gauge transformations of the frequency-space inner product, producing self-adjoint effective potentials but no numerical kernel diagonalization. The Shannon number, Nc=β\wmax/πN_c=β\wmax/πNc​=β\wmax/π, fixes the upper information capacity of this pure Laplace channel. Finally, the optimal sampling points are obtained as eigenvalues of a Legendre colleague matrix, giving a deterministic compressed-sensing grid without iterative root searches.
We develop a reliable parameter-free analytic continuation method for quantum many-body calculations. Our method is based on a kernel grid, a causal spline, a regularization using the second-derivative roughness penalty, and the L-curve criterion. We also develop the L-curve averaged deviation to estimate the precision of our analytic continuation. To deal with statistically obtained data more efficiently, we further develop a bootstrap-averaged analytic continuation method. In the test using the exact imaginary-frequency Green's function with added statistical error, our method produces the spectral function that converges systematically to the exact one as the statistical error decreases. As an application, we simulate the two-orbital Hubbard model for various electron numbers with the dynamical-mean field theory in the imaginary time and obtain the real-frequency self-energy with our analytic continuation method, clearly identifying a non-Fermi-liquid behavior as the electron number approaches the half filling from the quarter filling. Our analytic continuation can be used widely, and it will facilitate drawing clear conclusions from imaginary-time quantum many-body calculations.
Information compression plays a central role in diverse fields of modern science and technology, from communication theory to machine learning. In condensed-matter physics, the intermediate representation (IR) basis has recently been developed as an efficient method for compressing imaginary-time Green functions, which are fundamental quantities for describing quantum many-body systems. This compression relies on the rapid decay of the singular values with the basis index and the unusually weak growth of the effective rank with inverse temperature. Because of these useful features, the IR basis is now widely used as a standard method in quantum many-body calculations. However, the analytic origin of its compression capability has remained unclear. Here we uncover a finite-Laplace-transform structure underlying the IR kernel, which reveals that the eigenfunctions of the IR kernel admit a natural expansion in terms of classical special functions, the oblate spheroidal wave functions. This finite-Laplace-transform structure also enables us to analytically clarify the compression mechanism of the IR basis. Our results provide a mathematical foundation for the compression of imaginary-time Green functions, connecting quantum many-body physics with theories of information compression and finite integral transforms.
An outstanding problem in the simulation of condensed-matter phenomena is how to obtain dynamical information. We consider the numerical analytic continuation of imaginary-time quantum Monte Carlo data to obtain real-frequency spectral functions. This is an extremely ill-posed problem similar to the inversion of a Laplace transform. We suggest an image-reconstruction approach, which has been widely applied to data analysis in experimental research. Specifically, we apply the maximum-entropy method (ME) to the analytic continuation of quantum Monte Carlo data. We report encouraging preliminary results for the Fano-Anderson model of an impurity state in a continuum. The incorporation of additional prior information, such as sum rules and asymptotic behavior, can be expected to significantly improve results. We compare (ME) to alternative methods. We also discuss statistical error propagation for the analytic continuation problem via the likelihood function, which is independent of the choice of image-reconstruction method. This includes the sensitivity of the data to structure in the spectral function, the optimization of Monte Carlo simulations, and how to incorporate covariance in the statistical errors of the Monte Carlo method.
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.
A method for analytic continuation of imaginary-time correlation functions (here obtained in quantum Monte Carlo simulations) to real-frequency spectral functions is proposed. Stochastically sampling a spectrum parametrized by a large number of δ functions, treated as a statistical-mechanics problem, it avoids distortions caused by (as demonstrated here) configurational entropy in previous sampling methods. The key development is the suppression of entropy by constraining the spectral weight to within identifiable optimal bounds and imposing a set number of peaks. As a test case, the dynamic structure factor of the S=1/2 Heisenberg chain is computed. Very good agreement is found with Bethe ansatz results in the ground state (including a sharp edge) and with exact diagonalization of small systems at elevated temperatures.