## SciPost Submission Page

# Sparse sampling and tensor network representation of two-particle Green's functions

### by Hiroshi Shinaoka, Dominique Geffroy, Markus Wallerberger, Junya Otsuki, Kazuyoshi Yoshimi, Emanuel Gull, Jan Kuneš

### Submission summary

As Contributors: | Hiroshi Shinaoka |

Arxiv Link: | https://arxiv.org/abs/1909.07519v1 |

Date submitted: | 2019-09-18 |

Submitted by: | Shinaoka, Hiroshi |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Condensed Matter Physics - Computational |

Approaches: | Theoretical, Computational |

### Abstract

Many-body calculations at the two-particle level require a compact representation of two-particle Green's functions. In this paper, we introduce a sparse sampling scheme in the Matsubara frequency domain as well as a tensor network representation for two-particle Green's functions. The sparse sampling is based on the intermediate representation basis and allows an accurate extraction of the generalized susceptibility from a reduced set of Matsubara frequencies. The tensor network representation provides a system independent way to compress the information carried by two-particle Green's functions. We demonstrate efficiency of the present scheme for calculations of static and dynamic susceptibilities in single- and two-band Hubbard models in the framework of dynamical mean-field theory.

###### Current status:

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 1 on 2019-10-18 Invited Report

### Strengths

1. The manuscript is well written and concisely describes the problem, the algorithmic schemes, and the examples presented in the manuscript.

2. While the manuscript deals with a specific issue with less general appeal, a solution to the problems of storage and manipulation of numerical information, using a sampling algorithm and compact representation, should be broadly interesting to a wide range of computational physicists.

3. The authors provide a bridge between two different numerical worlds, bringing tensor network representations, and associated numerical schemes developed to manipulate them, into the realm of DMFT and related methods.

### Weaknesses

Minor weaknesses:

1. While the examples are sufficient to demonstrate the scheme, one really would want something more systematic, benchmarking in the single-band Hubbard model across a number of different parameters, to show the power of the technique, including changes in U and beta.

2. Computationally, one also would want a systematic analysis of the savings, particularly in storage, in applying the scheme across variations in the sampling thresholds, cutoffs, or tensor representations vs. the accuracy. Can one say anything general about limitations, at least for the simplest model?

### Report

The authors present a scheme for sampling and storing information for two-particle Green's functions which alleviates two problems: (1) computational cost of sampling and (2) storage of sampled information, with sufficient accuracy. The authors extend an approach developed for the sampling of the single-particle Green's function: a sparse grid in Matsubara space. The authors also present a tensor representation for storage, which itself borrows from tensor networks used in density-matrix methods, and can borrow from the algorithms already used in that community for tasks such as optimization.

The manuscript is well written and describes the general notions of the authors' schemes for sampling and storing the information from the two-particle Green's function, which themselves are necessary for understanding the behavior of numerous response functions, from simple charge correlations to superconductivity. The authors nicely describe the extension of the SVD decomposition used in sampling the single-particle Green's function; and the authors naturally connect the representation of the Green's function to low rank tensors, providing a graphical scheme that itself should be familiar to readers from tensor network theory. The authors also provide a framework for fitting and optimizing tensors -- a cost function -- although one likely could use a number of different methods, and the manuscript itself doesn't rest on this particular point.

Using some examples, the authors demonstrate the utility of this method. While one may want a more systematic approach to the examples, really benchmarking the method and establishing some limitations/trade-offs for accuracy, computational time, and storage requirements, the examples sufficiently demonstrate what one can expect to achieve using the technique.

While the manuscript may seem specialized, the authors pin-point at least a few groups of computational scientists who may benefit by employing this method, and quite generally the ideas presented in the manuscript should be of interest to a broad slice of computational condensed matter scientists engaged in related activities. Overall, I find the manuscript to be acceptable for publication.