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arXiv:2405.06716 · Physical review. B./Physical review. B

Discrete Lehmann representation of three-point functions

Dominik Kiese, Hugo U. R. Strand, Kun Chen, Nils Wentzell, Olivier Parcollet, Jason Kaye

Published Jan 16, 2025• 8 citations
arXivHTMLPDF

Abstract

We present a generalization of the discrete Lehmann representation (DLR) to three-point correlation and vertex functions in imaginary time and Matsubara frequency. The representation takes the form of a linear combination of judiciously chosen exponentials in imaginary time, and products of simple poles in Matsubara frequency, which are universal for a given temperature and energy cutoff. We present a systematic algorithm to generate compact sampling grids, from which the coefficients of such an expansion can be obtained by solving a linear system. We show that the explicit form of the representation can be used to evaluate diagrammatic expressions involving infinite Matsubara sums, such as polarization functions or self-energies, with controllable, high-order accuracy. This collection of techniques establishes a framework through which methods involving three-point objects can be implemented robustly, with a substantially reduced computational cost and memory footprint.

DOI
10.1103/physrevb.111.035135

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We present a generalization of the discrete Lehmann representation (DLR) to three-point correlation and vertex functions in imaginary time and Matsubara frequency. The representation takes the form of a linear combination of judiciously chosen exponentials in imaginary time, and products of simple poles in Matsubara frequency, which are universal for a given temperature and energy cutoff. We present a systematic algorithm to generate compact sampling grids, from which the coefficients of such an expansion can be obtained by solving a linear system. We show that the explicit form of the representation can be used to evaluate diagrammatic expressions involving infinite Matsubara sums, such as polarization functions or self-energies, with controllable, high-order accuracy. This collection of techniques establishes a framework through which methods involving three-point objects can be implemented robustly, with a substantially reduced computational cost and memory footprint.
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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.
In this article, the numerical level representation of imaginary time Green's functions based on a low rank decomposition of the discrete Lehmann spectral representation (DLR) using an effective spectral density has been presented with illustrative efficiency. The DLR basis consists of a collection of exponentials chosen by interpolation decomposition to ensure stable and efficient recovery of imaginary time Green functions or Matsubara frequency samples (Kaye et al., 2022). This implementation to fit behavior of a low-temperature spinless free fermionics-electron gas quantum system requires much fewer degrees of freedom for the standard discretizations. The basic functions of DLR are explicit; carefully chosen to ensure a stable and accurate approach, simplifying standard operations. Importantly, the function can be explicitly transformed to the Matsubara frequency domain or obtained directly by interpolation on a Matsubara frequency grid.
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Several recent works have introduced highly compact representations of single-particle Green's functions in the imaginary time and Matsubara frequency domains, as well as efficient interpolation grids used to recover the representations. In particular, the intermediate representation with sparse sampling and the discrete Lehmann representation (DLR) make use of low rank compression techniques to obtain optimal approximations with controllable accuracy. We consider the use of the DLR in dynamical mean-field theory (DMFT) calculations, and in particular show that the standard full Matsubara frequency grid can be replaced by the compact grid of DLR Matsubara frequency nodes. We test the performance of the method for a DMFT calculation of Sr2RuO4{\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}Sr2​RuO4​ at temperature 50K50\phantom{\rule{0.16em}{0ex}}\mathrm{K}50K using a continuous-time quantum Monte Carlo impurity solver, and demonstrate that Matsubara frequency quantities can be represented on a grid of only 36 nodes with no reduction in accuracy, or increase in the number of self-consistent iterations, despite the presence of significant Monte Carlo noise.
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.