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arXiv:2605.24814

Analytic Origin of Green-Function Compression in the Intermediate Representation

Takahiro Misawa

Published May 24, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2605.24814

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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.
The imaginary-time Green's function is a building block of various numerical methods for correlated electron systems. Recently, it was shown that a model-independent compact orthogonal representation of the Green's function can be constructed by decomposing its spectral representation. We investigate the performance of this so-called intermediate representation (IR) from several points of view. First, we develop an efficient algorithm for computing the IR basis functions of arbitrary high degree. Second, for two simple models, we study the number of coefficients required to represent the Green's function within a given tolerance. We show that the number of coefficients grows only as $O(ln\ensuremath{\beta})$ for fermions and converges to a constant for bosons as temperature $T=1/\ensuremath{\beta}$ decreases. Third, we show that this remarkable feature is ascribed to the properties of the physically constructed basis functions. The fermionic basis functions on the real-frequency axis have features whose width is scaled as O(T)O(T)O(T), which is consistent with the low-TTT properties of quasiparticles in a Fermi liquid state. On the other hand, the properties of the bosonic basis functions are consistent with those of spin/orbital susceptibilities at low TTT. These results demonstrate the potential wide applications of the IR to calculations of correlated systems.
Model-independent compact representations of imaginary-time data are presented in terms of the intermediate representation (IR) of analytical continuation. We demonstrate the efficiency of the IR through continuous-time quantum Monte Carlo calculations of an Anderson impurity model. We find that the IR yields a significantly compact form of various types of correlation functions. This allows the direct quantum Monte Carlo measurement of Green's functions in a compressed form, which considerably reduces the computational cost and memory usage. Furthermore, the present framework will provide general ways to boost the power of cutting-edge diagrammatic/quantum Monte Carlo treatments of many-body systems.
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.
This lecture note reviews recently proposed sparse-modeling approaches for efficient ab initio many-body calculations based on the data compression of Green's functions. The sparse-modeling techniques are based on a compact orthogonal basis, an intermediate representation (IR) basis, for imaginary-time and Matsubara Green's functions. A sparse sampling method based on the IR basis enables solving diagrammatic equations efficiently. We describe the basic properties of the IR basis, the sparse sampling method and its applications to ab initio calculations based on the GW approximation and the Migdal--Eliashberg theory. We also describe a numerical library for the IR basis and the sparse sampling method, sparse-ir, and provide its sample codes. This lecture note follows the Japanese review article [H. Shinaoka et al., Solid State Physics 56(6), 301 (2021)].
We present an efficient basis for imaginary time Green's functions based on a low-rank decomposition of the spectral Lehmann representation. The basis functions are simply a set of well-chosen exponentials, so the corresponding expansion may be thought of as a discrete form of the Lehmann representation using an effective spectral density which is a sum of $\ensuremath{\delta}$ functions. The basis is determined only by an upper bound on the product $\ensuremath{\beta}{\ensuremath{\omega}}_{max}$, with $\ensuremath{\beta}$ the inverse temperature and ${\ensuremath{\omega}}_{max}$ an energy cutoff, and a user-defined error tolerance $\ensuremath{\epsilon}$. The number rrr of basis functions scales as $O(log(\ensuremath{\beta}{\ensuremath{\omega}}_{max})log(1/\ensuremath{\epsilon}))$. The discrete Lehmann representation of a particular imaginary time Green's function can be recovered by interpolation at a set of rrr imaginary time nodes. Both the basis functions and the interpolation nodes can be obtained rapidly using standard numerical linear algebra routines. Due to the simple form of the basis, the discrete Lehmann representation of a Green's function can be explicitly transformed to the Matsubara frequency domain, or obtained directly by interpolation on a Matsubara frequency grid. We benchmark the efficiency of the representation on simple cases, and with a high-precision solution of the Sachdev-Ye-Kitaev equation at low temperature. We compare our approach with the related intermediate representation method, and introduce an improved algorithm to build the intermediate representation basis and a corresponding sampling grid.
Two-particle Green's functions and the vertex functions play a critical role in theoretical frameworks for describing strongly correlated electron systems. However, numerical calculations at the two-particle level often suffer from large computation time and massive memory consumption. We derive a general expansion formula for the two-particle Green's functions in terms of an overcomplete representation based on the recently proposed ``intermediate representation'' basis. The expansion formula is obtained by decomposing the spectral representation of the two-particle Green's function. We demonstrate that the expansion coefficients decay exponentially, while all high-frequency and long-tail structures in the Matsubara-frequency domain are retained. This representation therefore enables efficient treatment of two-particle quantities and opens a route to the application of modern many-body theories to realistic strongly correlated electron systems.