Efficient $\mathcal{O}(N^2)$ approach to solve the Bethe-Salpeter equation for excitonic bound states

TL;DR

This paper introduces an efficient quadratic-scaling method for solving the Bethe-Salpeter equation to determine exciton states, reducing computational costs while maintaining accuracy, demonstrated on model and real semiconductor systems.

Contribution

A novel quadratic-scaling approach for calculating exciton states from the Bethe-Salpeter equation, improving efficiency over traditional methods.

Findings

Method achieves quadratic computational complexity.

Accurate exciton binding energies for model and real materials.

Identifies convergence trends for k-point sampling.

Abstract

Excitonic effects in optical spectra and electron-hole pair excitations are described by solutions of the Bethe-Salpeter equation (BSE) that accounts for the Coulomb interaction of excited electron-hole pairs. Although for the computation of excitonic optical spectra in an extended frequency range efficient methods are available, the determination and analysis of individual exciton states still requires the diagonalization of the electron-hole Hamiltonian $\hat{H}$. We present a numerically efficient approach for the calculation of exciton states with quadratically scaling complexity, which significantly diminishes the computational costs compared to the commonly used cubically scaling direct-diagonalization schemes. The accuracy and performance of this approach is demonstrated by solving the BSE numerically for the Wannier-Mott two-band model in {\bf k} space and the semiconductors MgO and InN. For the convergence with respect to the $\vk$-point sampling a general trend is identified, which can be used to extrapolate converged results for the binding energies of the lowest bound states.