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Efficient ab initio many-body calculations based on sparse modeling of Matsubara Green's function
by Hiroshi Shinaoka, Naoya Chikano, Emanuel Gull, Jia Li, Takuya Nomoto, Junya Otsuki, Markus Wallerberger, Tianchun Wang, Kazuyoshi Yoshimi
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|As Contributors:||Hiroshi Shinaoka|
|Arxiv Link:||https://arxiv.org/abs/2106.12685v1 (pdf)|
|Date submitted:||2021-06-25 05:20|
|Submitted by:||Shinaoka, Hiroshi|
|Submitted to:||SciPost Physics Lecture Notes|
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 representation, intermediate representation (IR) basis functions, 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, irbasis, and provide its sample codes. This lecture note follows the Japanese review article [H. Shinaoka et al., Solid State Physics 56(6), 301 (2021)].
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 2021-10-3 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2106.12685v1, delivered 2021-10-03, doi: 10.21468/SciPost.Report.3598
- review of recent advances in a pedagogical manner
- open-source implementation
- several examples for a newcomer
Not much - some typos and connections with other works as described below.
These lecture notes review recent advances how to define and use an efficient representation for
Matsubara green functions and provide an associated numerical library. I found the presentation pedagogical and
could easily follow it without being an expert on the topic. The numerical library [at least the Python version] is intuitive,
easy to use, and easily embedded into already developed workflows. This will be a valuable material for researchers who want to use these numerical tools.
I will recommend the paper to be published but ask the authors to consider suggestions. I'd strongly recommend authors to reread the whole text carefully, as I found many typos which should not be there is a paper with 9 authors.
1. Authors made a nice overview of the recent intermediate representation progress, but a further search shows that they missed a recent alternative approach in
https://arxiv.org/abs/2107.13094 . Reading through the paper, my impression is that it gives an alternative but mostly comparable approach. I believe it would be useful to include a short discussion on the comparison between the two approaches, when to use one or another, etc.
2. Do you understand how strict is the condition that a number of IR basis functions grow logarithmically. You showed two examples, but are there known counterexamples.
Is there some physical reason for it?
3. It would be useful to gather weak points of the method in a paragraph of the main text and at which steps should an inexperienced be careful. For instance, the sparse sampling and its numerical stability, can user create precomputed coefficients if they are needed for some other values of $\Lambda \beta$, etc. Are these the only problems?
- Eq.2: missing \alpha on G
- Eq.3: you mark Matsubara propagators with a hat. Why G(\omega) has a hat? I thought its real frequency.
- below Eq.5: "while right singular functions [V_0(\omega)]" should probably be functions of omega, right?
- what is \epsilon_L in Eq.6
- We call these “appropriate frequencies” “sampling frequencies.” Missing or?
- Eq. 23: what is the .I at the end?
- it is difficult to calculate Eq. Using the IR basis,
- . interested readers are
Anonymous Report 2 on 2021-10-2 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2106.12685v1, delivered 2021-10-02, doi: 10.21468/SciPost.Report.3611
The authors propose a sparse representation method for Matsubara Green's functions that can be used for a reduction in terms of compute time and memory usage when used for DMFT, FLEX, GW and other many-body perturbation theory calculations.
The corresponding time and frequency grids are derived from sampling the extrema of the singular vector function of the Lehman kernel corresponding to the smallest singular value. The weights of the resulting representation are
obtained from a least-square fit (Eqs. 13-14). From a mathematical point of
view, using the infinity norm has typically better compression properties than
the L2 one (see for instance Takatsuka et. al. in JCP 129, 044112 (2008)). The
authors should mention this fact in the text.
Apart from this fact, I find the manuscript concise and ready for publication.
An analytic form of U and V might be obtained as follows. Write Eq. (33) as
an integral and find the corresponding minimax quadrature. This might reveal
the analytic form of the singular functions.
Authors should mention that L-infinity norm outperforms the used L2 in the text.