Efficient and Accurate Algorithms for Solving the Bethe-Salpeter Eigenvalue Problem for Crystalline Systems

TL;DR

This paper introduces new theoretical insights and two novel algorithms for efficiently solving the Bethe-Salpeter eigenvalue problem in crystalline systems, enhancing accuracy and reducing computational effort.

Contribution

It provides a new theoretical framework for the matrix structure and develops two algorithms that improve efficiency and accuracy for solving the Bethe-Salpeter eigenvalue problem.

Findings

One algorithm reduces computational effort without losing accuracy.

The other algorithm improves accuracy with comparable performance.

Both algorithms are suitable for high-performance computing environments.

Abstract

Optical properties of materials related to light absorption and scattering are explained by the excitation of electrons. The Bethe-Salpeter equation is the state-of-the-art approach to describe these processes from first principles (ab initio), i.e. without the need for empirical data in the model. To harness the predictive power of the equation, it is mapped to an eigenvalue problem via an appropriate discretization scheme. The eigenpairs of the resulting large, dense, structured matrix can be used to compute dielectric properties of the considered crystalline or molecular system. The matrix always shows a $2\times 2$ block structure. Additionally, certain definiteness properties typically hold. One form can be acquired for crystalline systems, another one is more general and can for example be used to study molecules. In this work, we present new theoretical results characterizing the structure of the two forms in the language of non-standard scalar products. These results enable us to develop a new perspective on the state-of-the-art solution approach for crystalline systems. This new viewpoint is used to develop two new methods for solving the eigenvalue problem. One requires less computational effort while providing the same degree of accuracy. The other one improves the expected accuracy, compared to methods currently in use, with a comparable performance. Both methods are well suited for high performance environments and only rely on basic numerical linear algebra building blocks.