Efficient ab initio calculations of correlated materials at finite temperature require compact representations of the Green's functions both in imaginary time and Matsubara frequency. In this paper, we introduce a general procedure which generates sparse sampling points in time and frequency from compact orthogonal basis representations, such as Chebyshev polynomials and intermediate representation (IR) basis functions. These sampling points accurately resolve the information contained in the Green's function, and efficient transforms between different representations are formulated with minimal loss of information. As a demonstration, we apply the sparse sampling scheme to diagrammatic GW and GF2 calculations of a hydrogen chain, of noble gas atoms and of a silicon crystal.arXiv:1908.07575v1 [cond-mat.str-el]

The open-source library, irbasis, provides easy-to-use tools for two sets of orthogonal functions named intermediate representation (IR). The IR basis enables a compact representation of the Matsubara Green's function and efficient calculations of quantum models. The IR basis functions are defined as the solution of an integral equation whose analytical solution is not available for this moment. The library consists of a database of pre-computed high-precision numerical solutions and computational code for evaluating the functions from the database. This paper describes technical details and demonstrates how to use the library.

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(log β) for fermions, and converges to a constant for bosons as temperature T = 1/β 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 ), which are consistent with the low-T 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 T . These results demonstrate the potential wide applications of the IR to calculations of correlated systems.

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 sparsemodeling techniques are based on a compact orthogonal basis representation, intermediate representation (IR) basis functions, 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, irbasis, and provide its sample codes. This lecture note follows the Japanese review article [H. Shinaoka et al., Solid State Physics 56(6), 301 (2021)].

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)].