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

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.

💡 Research Summary

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The paper presents a comprehensive symmetry‑based formalism for the treatment of excitons with both zero and finite center‑of‑mass (c.m.) momentum Q within an ab‑initio framework. Starting from the well‑established use of space‑group and time‑reversal symmetries in one‑electron band‑structure and phonon calculations, the authors extend these concepts to the two‑particle Bethe–Salpeter equation (BSE) that describes bound electron‑hole pairs.

First, the transformation properties of Bloch wavefunctions under a generic space‑group operation {R|t} are derived, including the effect of spin‑orbit coupling through the double‑group (SU(2)) representation. The authors show that for non‑degenerate bands the plane‑wave coefficients transform with a simple phase factor, while for degenerate manifolds the operation mixes states according to a unitary matrix D that belongs to an irreducible representation (irrep) of the little group Gₖ. Time‑reversal symmetry, being anti‑unitary, connects states at k and –k (or Q and –Q) via complex conjugation, providing a second set of relations that can be used to reconstruct exciton wavefunctions throughout the full Brillouin zone (BZ) from those computed only in the irreducible wedge (IBZ).

The central technical advance is the construction of symmetry‑adapted electron‑hole product bases using projection operators. For a given Q, the little group G_Q is identified, and for each irrep α a projector

 P^{(α)} = (dim α / |G_Q|) Σ_{g∈G_Q} χ^{(α)}(g)^{*} R(g)

is applied to the set of all possible electron‑hole pairs (v,c, k). The resulting basis states transform solely according to α, which block‑diagonalizes the BSE Hamiltonian into independent subspaces. The size of each block equals the dimension of the corresponding irrep, automatically revealing degeneracies, selection rules, and symmetry‑protected dark or bright excitons. Because only the IBZ needs to be sampled, the computational cost scales with the number of Q points in the IBZ rather than the full BZ, yielding order‑of‑magnitude savings in both memory and CPU time.

To validate the method, the authors apply it to monolayer MoS₂, a prototypical transition‑metal dichalcogenide with strong spin‑orbit coupling and a D₃h point group at Γ and C₃h at the K/K′ valleys. They compute the irreps of the little groups at high‑symmetry Q points, construct the symmetry‑adapted bases, and solve the BSE in the IBZ. The resulting excitonic band structure exhibits block patterns that exactly match the compatibility relations derived from group theory. Bright excitons (A, B) belong to E′‑type irreps, while dark spin‑forbidden states fall into A₁′ or A₂′, confirming the predictive power of the symmetry classification.

Beyond the case study, the paper discusses broader implications. Exciton‑phonon coupling, indirect optical transitions, and exciton thermalization all require dense Q‑grids; the presented framework makes such calculations tractable for a wide class of materials (e.g., hBN, WSe₂, bulk silicon). The authors also outline how existing electronic‑structure packages (VASP, Quantum Espresso) and phonon tools (Phonopy) can be extended with symmetry‑analysis modules (e.g., using Bilbao Crystallographic Server data) to automate the generation of projection operators and the mapping of exciton wavefunctions across the BZ.

In summary, the work delivers three major contributions: (1) a rigorous derivation of how exciton wavefunctions transform under space‑group and time‑reversal operations, (2) a practical algorithm to generate symmetry‑adapted exciton bases that block‑diagonalize the BSE Hamiltonian, and (3) a demonstration of substantial computational savings and accurate symmetry labeling in a realistic 2D semiconductor. This framework opens the door to high‑throughput excitonic calculations, systematic exploration of exciton topology, and more efficient modeling of light‑matter interactions in complex materials.