Orthogonal polynomial representation of imaginary-time Green’s functions

Abstract

We study the expansion of single-particle and two-particle imaginary-time Matsubara Green's functions of quantum impurity models in the basis of Legendre orthogonal polynomials. We discuss various applications within the dynamical mean-field theory (DMFT) framework. The method provides a more compact representation of the Green's functions than standard Matsubara frequencies and therefore significantly reduces the memory-storage size of these quantities. Moreover, it can be used as an efficient noise filter for various physical quantities within the continuous-time quantum Monte Carlo impurity solvers recently developed for DMFT and its extensions. In particular, we show how to use it for the computation of energies in the context of realistic DMFT calculations in combination with the local density approximation to the density functional theory (LDA+DMFT) and for the calculation of lattice susceptibilities from the local irreducible vertex function.PACS numbers: 71.27.+a, 71.10.Fd In recent years, significant progress has been made in the study of strongly-correlated fermionic quantum systems with the development of methods combining systematic analytical approximations and modern numerical algorithms. The Dynamical Mean-Field Theory (DMFT) (for a review see Ref. 1) and its various extensions. 2-6serve as successful examples for this theoretical advance. On the technical side, important progress was made in the solution of quantum impurity problems, i.e. local quantum systems coupled to a bath (self-consistently determined in the DMFT formalism). In particular, a new generation of continuous-time quantum Monte Carlo (CTQMC) impurity solvers 7-10 has emerged that provide unprecedented efficiency and accuracy (for a recent review, see Ref. 11).In practice, several important technical issues still remain. Firstly, while the original DMFT formalism is expressed in terms of single-particle quantities (Green's function and self-energy), two-particle quantities play a central role in the formulation of some DMFT extensions (e.g. dual-fermions 4,12-14 , DΓA 3 ) and in susceptibility and transport computations in DMFT itself. They typically depend on three independent times or frequencies, and spatial indices. Therefore, they are quite large objects that are hard to store, manipulate and analyze, even with modern computing capabilities. Developing more compact representations of these objects and using them to solve, e.g., the Bethe-Salpeter equations is therefore an important challenge.A natural route is to use an orthogonal polynomial representation of the imaginary-time dependence of these objects. While the application of orthogonal polynomials has had productive use in other approaches to correlated electrons, 15,16 in this paper we show how to use Legendre polynomials to represent various imaginary-time Green's functions in a more compact way and show their usefulness in some concrete calculations.A second aspect is that modern CTQMC impurity solvers still have limitations. One well-known problem is the high-fre...

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