Symmetries in zero and finite center-of-mass momenta excitons

TL;DR

This paper develops a symmetry-based framework for analyzing excitonic states in materials, utilizing group theory to classify and organize excitons at various momenta, improving computational efficiency and understanding of their properties.

Contribution

The authors introduce a general symmetry classification method for excitons at zero and finite momenta, enabling systematic analysis and computational simplifications across materials.

Findings

Successfully applied to monolayer MoS₂, matching group theory predictions.

Enables block-diagonalization of the Bethe-Salpeter Hamiltonian at multiple momenta.

Reduces computational cost for optical and excitonic calculations.

Abstract

We present a symmetry-based framework for the analysis of excitonic states, incorporating both time-reversal and space-group symmetries. We demonstrate the use of time-reversal and space-group symmetries to obtain exciton eigenstates at symmetry-related center-of-mass momenta in the entire Brillouin zone from eigenstates calculated for center-of-mass momenta in the irreducible Brillouin zone. Furthermore, by explicitly calculating the irreducible representations of the little groups, we classify excitons according to their symmetry properties across the Brillouin zone. Using projection operators, we construct symmetry-adapted linear combinations of electron-hole product states, which block diagonalize the Bethe-Salpeter equation (BSE) Hamiltonian at both zero and finite exciton center-of-mass momenta. This enables a transparent organization of excitonic states and provides direct access to their degeneracies, selection rules, and symmetry-protected features. As a demonstration, we apply this formalism to monolayer MoS$_2$, where the classification of excitonic irreducible representations and the block structure of the BSE Hamiltonian show excellent agreement with compatibility relations derived from group theory. Beyond this material-specific example, the framework offers a general and conceptually rigorous approach to the symmetry classification of excitons, enabling significant reductions in computational cost for optical spectra, exciton-phonon interactions, and excitonic band structure calculations across a wide range of materials.