An efficient Fourier spectral algorithm for the Bogoliubov-de Gennes excitation eigenvalue problem

TL;DR

This paper introduces a fast, stable Fourier spectral algorithm for solving the Bogoliubov-de Gennes eigenvalue problem in spin-1 Bose-Einstein condensates, enabling efficient analysis of elementary excitations in large-scale systems.

Contribution

It develops a matrix-free, FFT-accelerated spectral method with a stable bi-orthogonal iterative solver for large eigenvalue problems in BECs, with rigorous analysis and extensive numerical validation.

Findings

The algorithm achieves spectral accuracy and efficiency.

It effectively computes excitation spectra in 1-3 dimensions.

The solver is memory-efficient and suitable for large-scale problems.

Abstract

In this paper, we propose an efficient Fourier spectral algorithm for an eigenvalue problem, that is, the Bogoliubov-de Gennes (BdG) equation arsing from spin-1 Bose-Einstein condensates (BEC) to describe the elementary/collective excitations around the mean-field ground state. The BdG equation is essentially a constrained eigenvalue/eigenfunction system. Firstly, we investigate its analytical properties, including exact eigenpairs, generalized nullspace, and bi-orthogonality of eigenspaces. Secondly, by combining the standard Fourier spectral method for spatial discretization and a stable Gram-Schmidt bi-orthogonal algorithm, we develop a subspace iterative solver for such a large-scale dense eigenvalue problem, and it proves to be numerically stable, efficient, and accurate. Our solver is matrix-free and the operator-function evaluation is accelerated by discrete Fast Fourier Transform (FFT) with almost optimal efficiency. Therefore, it is memory-friendly and efficient for large-scale problems. Furthermore, we give a rigorous and detailed numerical analysis on the stability and spectral convergence. Finally, we present extensive numerical results to illustrate the spectral accuracy and efficiency, and investigate the excitation spectrum and Bogoliubov amplitudes around the ground state in 1-3 spatial dimensions.