# Efficient ab initio many-body calculations based on sparse modeling of Matsubara Green's function

### Submission summary

 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 Academic field: Physics Specialties: Condensed Matter Physics - Computational Approaches: Theoretical, Computational

### Abstract

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)].

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 2106.12685v2 on 10 June 2022

Submission 2106.12685v1 on 25 June 2021

## 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

### Strengths

- review of recent advances in a pedagogical manner
- open-source implementation
- several examples for a newcomer

### Weaknesses

Not much - some typos and connections with other works as described below.

### Report

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.

### Requested changes

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?

4. Typos:
- 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,

• validity: top
• significance: good
• originality: high
• clarity: top
• formatting: good
• grammar: reasonable

### Author:  Hiroshi Shinaoka  on 2022-06-10  [id 2572]

(in reply to Report 1 on 2021-10-03)
Category:

We thank the Referee for the very positive assessment of the manuscript. We are sorry for the slow response, which was due to the release of a new Python library, sparse-ir. We have revised the manuscript according to the comments and fixed typos throughout the manuscript. We have updated the example source codes accordingly.

## Comment #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. We thank the Referee for the valuable suggestion. That preprint appeared after we uploaded our manuscript. They proposed a related approach based on a similar decomposition of the same kernel though its accuracy, numerical stability, and efficiency have been established in ab initio calculations. Following the suggestion, we compare the two approaches in Sections 3.4 and 3.5 of the revised manuscript.

## Comment #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? We thank the Referee for asking an important question. As long as $\rho(\omega)$ is bounded, the number of basis functions grows only logarithmically. This is based on the fact that the number of singular values above a certain cut-off value glows only logarithmically [see Fig. 4 of N. Chikano, J. Otsuki, H. Shinaoka, PRB 98, 035104 (2018)]. Although this property/scaling has been verified only numerically, we have not found any counterexamples.

## Comment #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? We thank the Referee for the useful suggestion. The choice of $\omega_\mathrm{max}$ is only a major problem in practical calculations. The user should always check if expansion coefficients $G_l$ of each object decay as fast as the singular values to some noise level. If not, the user should increase $\omega_\mathrm{max}$. We have recently developed an updated version of the python library (renamed as sparse-ir from irbasis). The new library allows us to compute IR basis functions for an arbitrary value of $\omega_\mathrm{max}$ on the fly.

We have extended Sec. 3.3 to explain how to check if $\omega_\mathrm{max}$ is large enough and how to tune the parameter otherwise. The sample codes in Section 5 have been adapted for the new python library.

## Comment #4

We thank the Referee for pointing out many typos. All the referees have made a critical reading of the manuscript and have fixed typos throughout the manuscript. Section 2 has been revised for better consistency with the new Python library.

### 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

### Report

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.

### Requested changes

Authors should mention that L-infinity norm outperforms the used L2 in the text.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Hiroshi Shinaoka  on 2022-06-10  [id 2573]

(in reply to Report 2 on 2021-10-02)

We thank the Referee for the very positive assessment of the manuscript. We are sorry for the slow response, which was due to the release of a new Python library, sparse-ir. We have updated the manuscript according to Comment #1 as described below.

## Comment #1

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.

We thank the Referee for letting us know the relevant paper. We have added the following comment as a footnote above Eq. (17):

Note that the $L_2$ norm is not only the choice in the fitting. It is claimed that using the infinite norm leads to smaller errors~\cite{doi:10.1063/1.2958921}.

## Comment #2

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.

We thank the Referee for the valuable suggestion for future studies. Our attempt to find the analytic form of the basis functions has not been successful.