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arXiv:2110.06765 · Computer Physics Communications

libdlr: Efficient imaginary time calculations using the discrete Lehmann representation

Jason Kaye, Kun Chen, Hugo U. R. Strand

Published Jul 8, 2022• 32 citations
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Abstract

We introduce libdlr, a library implementing the recently introduced discrete Lehmann representation (DLR) of imaginary time Green's functions. The DLR basis consists of a collection of exponentials chosen by the interpolative decomposition to ensure stable and efficient recovery of Green's functions from imaginary time or Matsubara frequency samples. The library provides subroutines to build the DLR basis and grids, and to carry out various standard operations. The simplicity of the DLR makes it straightforward to incorporate into existing codes as a replacement for less efficient representations of imaginary time Green's functions, and libdlr is intended to facilitate this process. libdlr is written in Fortran, provides a C header interface, and contains a Python module pydlr. We also introduce a stand-alone Julia implementation, Lehmann.jl. Program Title: libdlr CPC Library link to program files: https://doi.org/10.17632/56z594pzsj.1 Developer's repository link: https://github.com/jasonkaye/libdlr Licensing provisions: Apache-2.0 Programming language: Fortran, C, Python, Julia Nature of problem: Discretization and compression of functions (Green's functions and self-energies) with an imaginary time variable. Solution method: Explicit basis functions and discretization points obtained by low rank compression of the analytical continuation kernel.

DOI
10.1016/j.cpc.2022.108458

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We introduce libdlr, a library implementing the recently introduced discrete Lehmann representation (DLR) of imaginary time Green's functions. The DLR basis consists of a collection of exponentials chosen by the interpolative decomposition to ensure stable and efficient recovery of Green's functions from imaginary time or Matsubara frequency samples. The library provides subroutines to build the DLR basis and grids, and to carry out various standard operations. The simplicity of the DLR makes it straightforward to incorporate into existing codes as a replacement for less efficient representations of imaginary time Green's functions, and libdlr is intended to facilitate this process. libdlr is written in Fortran, provides a C header interface, and contains a Python module pydlr. We also introduce a stand-alone Julia implementation, Lehmann.jl. Program Title: libdlr CPC Library link to program files: https://doi.org/10.17632/56z594pzsj.1 Developer's repository link: https://github.com/jasonkaye/libdlr Licensing provisions: Apache-2.0 Programming language: Fortran, C, Python, Julia Nature of problem: Discretization and compression of functions (Green's functions and self-energies) with an imaginary time variable. Solution method: Explicit basis functions and discretization points obtained by low rank compression of the analytical continuation kernel.
We introduce cppdlr, a C++ library implementing the discrete Lehmann representation (DLR) of functions in imaginary time and Matsubara frequency, such as Green's functions and self-energies. The DLR is based on a low-rank approximation of the analytic continuation kernel, and yields a compact and explicit basis consisting of exponentials in imaginary time and simple poles in Matsubara frequency. cppdlr constructs the DLR basis and associated interpolation grids, and implements standard operations. It provides a flexible yet high-level interface, facilitating the incorporation of the DLR into both small-scale applications and existing large-scale software projects.
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.
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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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 formalism stands as a powerful technique for computing the thermodynamic characteristics of interacting quantum many-particle systems at finite temperatures. In this manuscript, our focus centers on introducing MatsubaraFunctions.jl, a Julia library that implements data structures for generalized n-point Green’s functions on Matsubara frequency grids. The package’s architecture prioritizes user-friendliness without compromising the development of efficient solvers for quantum field theories in equilibrium. Following a comprehensive introduction of the fundamental types, we delve into a thorough examination of key facets of the interface. This encompasses avenues for accessing Green’s functions, techniques for extrapolation and interpolation, as well as the incorporation of symmetries and a variety of parallelization strategies. Examples of increasing complexity serve to demonstrate the practical utility of the library, supplemented by discussions on strategies for sidestepping impediments to optimal performance.