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

TL;DR

This work develops a general symmetry-based framework for excitons that unifies time-reversal and space-group symmetries to classify and connect excitonic states across the Brillouin zone. By constructing symmetry-adapted electron–hole bases with projection operators and analyzing the little-group representations, the authors achieve a block-diagonal form of the Bethe–Salpeter equation (BSE) at both zero and finite center-of-mass momentum $\mathbf{Q}$, while preserving correct transformation properties under $\mathcal{G}_{\mathbf{Q}}$ and time reversal. The framework is validated ab initio on monolayer MoS$_2$, where exciton irreducible representations and BSE block structures agree with compatibility relations derived from group theory, and where the approach yields clear insights into degeneracies and optical selection rules. Overall, the method offers substantial computational savings for optical spectra and exciton-phonon coupling calculations and provides a robust platform for symmetry-resolved exciton physics across a wide range of materials.

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.

Figures (2)

• Figure 1: (a) GW electronic band structure of monolayer MoS$_{2}$ along the high-symmetry path $\Gamma$-M-K-$\Gamma$ in the Brillouin zone. The valence band maximum is set to $0$ eV. The double-group spinor irreducible representations associated with the bands are indicated at the high-symmetry points. (b) Exciton band structure of monolayer MoS$_{2}$ along the path $\Gamma$-M-K-$\Gamma$ in the Brillouin zone. The irreducible representations of the excitonic bands are labeled at the high-symmetry points $\Gamma$, M, and K. The irreducible representations at the labeled points along the high symmetry lines are tabulated in Table \ref{['tab:table1']}. The evolution of the excitonic states at the transition between symmetry lines and high symmetry points illustrates the compatibility relations.
• Figure 2: Panels (a) and (c) depict the full spinor BSE Hamiltonian constructed from two valence and four conduction bands on a $24 \times 24 \times 1$$\mathbf{k}$-point grid, for exciton center-of-mass momenta $\mathbf{Q}=\Gamma$ and $\mathbf{Q}=\mathrm{K}$, respectively. Panels (b) and (d) show the corresponding block-diagonalized BSE Hamiltonians, resolved into blocks associated with the irreducible representations of the $D_{3h}$ and $C_{3h}$ symmetry groups at $\mathbf{Q}=\Gamma$ and $\mathbf{Q}=\mathrm{K}$, respectively. The dimensions of the blocks corresponding to each irreducible representation are indicated. The color bars represent the absolute values of the BSE Hamiltonian matrix elements for both the full and symmetry-adapted cases. For clarity, the diagonal matrix elements have been removed, and the color scale has been capped at a fixed maximum value to emphasize the block structure.