Sparse Modeling in Quantum Many-Body Problems

TL;DR

This review explores how sparse modeling techniques can identify essential parameters in quantum many-body problems, leading to more efficient calculations and new methods for analytical continuation.

Contribution

It introduces sparse modeling concepts tailored for physicists and reviews novel applications in quantum many-body systems, especially involving Matsubara Green's functions.

Findings

Matsubara Green's function is sparse, containing minimal information.

Sparse modeling enables a new approach to analytical continuation.

Efficient, compact representations of Green's functions improve computational methods.

Abstract

This review paper describes the basic concept and technical details of sparse modeling and its applications to quantum many-body problems. Sparse modeling refers to methodologies for finding a small number of relevant parameters that well explain a given dataset. This concept reminds us physics, where the goal is to find a small number of physical laws that are hidden behind complicated phenomena. Sparse modeling extends the target of physics from natural phenomena to data, and may be interpreted as "physics for data". The first half of this review introduces sparse modeling for physicists. It is assumed that readers have physics background but no expertise in data science. The second half reviews applications. Matsubara Green's function, which plays a central role in descriptions of correlated systems, has been found to be sparse, meaning that it contains little information. This leads to (i) a new method for solving the ill-conditioned inverse problem for analytical continuation, and (ii) a highly compact representation of Matsubara Green's function, which enables efficient calculations for quantum many-body systems.