Structure Preserving Parallel Algorithms for Solving the Bethe-Salpeter Eigenvalue Problem

TL;DR

This paper introduces structure-preserving parallel algorithms for solving the Bethe-Salpeter eigenvalue problem, which is crucial for computing exciton energies, by leveraging its equivalence to real Hamiltonian problems and addressing large-scale computational challenges.

Contribution

The paper establishes the equivalence between Bethe-Salpeter and real Hamiltonian eigenvalue problems and proposes structure-preserving algorithms with parallel implementations for large-scale computations.

Findings

Algorithms are efficient and accurate for large problems.

Tamm-Dancoff approximation overestimates eigenvalues.

Parallel algorithms outperform existing methods.

Abstract

The Bethe-Salpeter eigenvalue problem is a dense structured eigenvalue problem arising from discretized Bethe-Salpeter equation in the context of computing exciton energies and states. A computational challenge is that at least half of the eigenvalues and the associated eigenvectors are desired in practice. We establish the equivalence between Bethe-Salpeter eigenvalue problems and real Hamiltonian eigenvalue problems. Based on theoretical analysis, structure preserving algorithms for a class of Bethe-Salpeter eigenvalue problems are proposed. We also show that for this class of problems all eigenvalues obtained from the Tamm-Dancoff approximation are overestimated. In order to solve large scale problems of practical interest, we discuss parallel implementations of our algorithms targeting distributed memory systems. Several numerical examples are presented to demonstrate the efficiency and accuracy of our algorithms.