Symmetries of excitons

TL;DR

This work develops a comprehensive group‑theoretical framework for exciton symmetries in crystals by classifying excitonic states via the little group of the exciton momentum and introducing total crystal angular momentum as a conservation quantity. It provides a practical, ab initio method to assign irreducible representations to excitons from BSE solutions, derives robust selection rules for optical and phonon scattering, and demonstrates how crystal symmetries can dramatically accelerate BSE calculations through block diagonalization and symmetry‑based reconstructions. The methodology is validated on LiF, MoSe$_2$, and hBN, illustrating diverse exciton characters (Wannier/Frenkel/mixed), symmetry‑driven Raman enhancements, and phonon‑assisted luminescence. Overall, the framework enables precise symmetry‑driven insights into exciton dispersion, light–matter interactions, and exciton–phonon coupling, with practical implications for designing materials with tailored optical properties.

Abstract

Excitons, bound electron-hole pairs, are responsible for strong optical resonances near the bandgap in low-dimensional materials and wide-bandgap insulators. Although current ab initio methods can accurately determine exciton energies and eigenstates, their symmetries have been much less explored. In this work, we employ standard group-theory methods to analyse the transformation properties of excitonic states, obtained by solving the BSE, under crystal symmetry operations. We develop an approach to assign irreducible-representation labels to excitonic states, providing a state-of-the-art framework for analysing their symmetries and selection rules (including, for example, the case of exciton-phonon coupling). Complementary to the symmetry classification, we introduce the concept of total crystal angular momentum for excitons in the presence of rotational symmetries, allowing the derivation of conservation laws. Furthermore, we demonstrate how these symmetry properties can be exploited to greatly enhance the computational efficiency of exciton calculations with the BSE. We apply our methodology to three prototypical systems to understand the role of symmetries in different contexts: (i) For LiF, we present the symmetry analysis of the entire excitonic dispersion and examine the selection rules for optical absorption. (ii) In the calculation of resonant Raman spectra of monolayer MoSe2, we demonstrate how the conservation of total crystal angular momentum governs exciton-phonon interactions, leading to the observed resonant enhancement. (iii) In bulk hBN, we analyze the role of symmetries for the coupling of finite-momentum excitons to finite-momentum phonons and their manifestation in the phonon-assisted luminescence spectra. This work establishes a general and robust framework for understanding the symmetry properties of excitons in crystals, providing a foundation for future studies.

Figures (7)

• Figure 1: Feynman diagram for an arbitrary four-point function $F(1,2;3,4)$.
• Figure 2: Exciton dispersion of LiF, computed by solving the BSE without the TDA at selected $\mathbf{Q}$ points, followed by Fourier interpolation. Excitonic states at high-symmetry points are labeled with their corresponding irreducible representation labels. The red and blue dashed curves represent the exciton dispersion with and without LO–TO splitting only at the $\Gamma$ point, respectively. The direction of the $\mathbf{Q}$ vector used for the LO–TO splitting is taken along $(1,1,1)$, which breaks the symmetries of the excitonic states at the $\Gamma$ point.
• Figure 3: Real-space hole density, multiplied by the sign of the wavefunction as defined in Eq. \ref{['eq:hole_den_ex']}, for the first triply degenerate excitons of LiF at the $\Gamma$ point, with the electron fixed at the center of the Li atom located at the origin. Panels (a), (b), and (c) correspond to the three degenerate excitons, each selectively coupling to light polarized along the $\frac{\hat{x}+i\hat{y}}{\sqrt{2}}$, $\frac{\hat{x}-i\hat{y}}{\sqrt{2}}$, and $\hat{z}$ directions, respectively. The density is shown from the top view along the $z$ axis. Pink and green spheres denote lithium and fluorine atoms, while red and blue indicate the negative and positive lobes of the wavefunction.
• Figure 4: Absorption spectrum of LiF computed by solving the Bethe--Salpeter equation (blue solid line), compared with experimental data at $300K$ (green dots) from Ref. Roessler1967Jun. In the computed spectrum, we do not include temperature effects. The red and blue vertical lines indicate the positions of optically bright and dark excitons, respectively. The labels correspond to the irreducible representation labels of excitons. The computed spectrum is red shifted by 0.45 eV to align the position of the first peak.
• Figure 5: Excitons in monolayer MoSe$_2$. (a) Exciton energy spectrum at the $\Gamma$ point with corresponding irreducible representations. The blue/red vertical lines indicate the positions of dark/bright excitons, respectively. (b) Reciprocal space plot of the $\mathscr{A}_{1s}$ exciton wavefunction in the simultaneous eigenstate of the total crystal angular momentum matrix and the exciton Hamiltonian. (c) Resonant Raman spectrum as a function of incoming photon energy. The $A_1'$ and $E'$ Raman modes are represented by blue and orange lines, respectively. The grey shading corresponds to the imaginary part of the 2D polarizability tensor, representing the absorption spectrum. The black dots denote experimental data for the $A_1'$ mode, taken from Ref. McDonnell_2020. The absorption spectrum, and Raman spectrum are rigidly shifted in energy such that the first peak of the experimental Raman peak in (c) is aligned.