Efficient ab initio many-body calculations based on sparse modeling of Matsubara Green's function

Abstract

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)].

Challenges with relativistic GW calculations in solids and molecules

“…All the dynamical quantities, i.e., imaginary-time and frequency dependent objects, in scGW are stored using sparse grids. [88][89][90][91][92][93] Sufficiently large grids are used to avoid loss of information in Fourier transforms performed over successive iterations of the GW cycle. To obtain spectral functions, IPs and band gaps, the Matsubara Green's function in our calculations is analytically continued to the real-frequency axis.…”

No superconductivity in Pb$_9$Cu$_1$(PO$_4$)$_6$O found in orbital and spin fluctuation exchange calculations

“…In all calculations, we used a k-mesh resolution of 30 × 30 × 30. For the imaginary-time and Matsubara frequency grids we applied the sparse-sampling approach [33,69,70] in combination with the intermediate representation (IR) basis [71][72][73], where we used an IR parameter of Λ = 10 4 and a basis cutoff of δ IR = 10 −15 .…”

A fast time domain solver for the equilibrium Dyson equation

“…The last decade has seen a revival of time domain Green's function methods [4], mainly driven by an interest in non-equilibrium phenomena. This has in turn spurred advances in the development of numerical algorithms both for imaginary time [5][6][7][8][9][10][11][12][13][14][15][16] and non-equilibrum real-time [17][18][19][20][21][22] Green's functions. However, the real-time Dyson equation can also be used to determine the dynamical properties of quantum many-body systems in equilibrium, by evolving the imaginary time Green's function along the real-time axis.…”