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Progress of Theoretical Physics Supplement

A New Method to Calculate the Green Function by Polynomial Expansion

Wataru Kunishima, Masaki Itoh, Hiroshi L. TANAKA

Published Jan 1, 2000• 6 citations
Publisher

Abstract

An efficient scheme is introduced to calculate the Green function on the basis of polynomials expansion. This scheme has the following advantages: (i) it is possible to calculate both the diagonal and off-diagonal elements of the Green function, (ii) applicable to the system with discrete spectrum, (iii) integrated density of states can also be obtained by the same algorithm, and (iv) it is suitable for vector and parallel processing, and required CPU time and memory size are proportional only to the system size (i.e. ordered N). Applications are made to simple lattice systems for demonstration.

DOI
10.1143/ptps.138.149

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An efficient scheme is introduced to calculate the Green function on the basis of polynomials expansion. This scheme has the following advantages: (i) it is possible to calculate both the diagonal and off-diagonal elements of the Green function, (ii) applicable to the system with discrete spectrum, (iii) integrated density of states can also be obtained by the same algorithm, and (iv) it is suitable for vector and parallel processing, and required CPU time and memory size are proportional only to the system size (i.e. ordered N). Applications are made to simple lattice systems for demonstration.
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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.
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The Matsubara Green's function that is used to describe temperature-dependent behavior is expressed on a numerical grid. While such a grid usually has a couple of hundred points for low-energy model systems, for realistic systems with large basis sets the size of an accurate grid can be tens of thousands of points, constituting a severe computational and memory bottleneck. In this paper, we determine efficient imaginary time grids for the temperature-dependent Matsubara Green's function formalism that can be used for calculations on realistic systems. We show that, because of the use of an orthogonal polynomial transform, we can restrict the imaginary time grid to a few hundred points and reach micro-Hartree accuracy in the electronic energy evaluation. Moreover, we show that only a limited number of orthogonal polynomial expansion coefficients are necessary to preserve accuracy when working with a dual representation of the Green's function or self-energy and transforming between the imaginary time and frequency domain.