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In many paradigmatic materials, such as transition metal dichalcogenides, the role played by the spin degrees of freedom is as important as the one played by the electron-electron interaction. Thus an accurate treatment of the two effects and of their interaction is necessary for an accurate and predictive study of the optical and electronic properties of these materials. Despite the fact that the GW-BSE approach correctly accounts for electronic correlations, the spin-orbit coupling effect is often neglected or treated perturbatively. Recently, spinorial formulations of GW-BSE have become available in different flavors in material-science codes. However, an accurate validation and comparison of different approaches is still missing. In this work, we go through the derivation of the noncollinear GW-BSE approach. The scheme is applied to transition metal dichalcogenides comparing the perturbative and full spinorial approaches. Our calculations reveal that dark-bright exciton splittings are generally improved when the spin-orbit coupling is included nonperturbatively. The exchange-driven intravalley mixing between the A and B excitons is found to play a role for Mo-based systems, being especially strong in the case of MoSe2. We finally compute the excitonic spin and use it to sharply analyze the spinorial properties of transition metal dichalcogenide excitonic states.
M. Marsili, A. Molina-Sánchez, M. Palummo, D. Sangalli, and A. Marini. Spinorial formulation of the GW -BSE equations and spin properties of excitons in two-dimensional transition metal dichalcogenides. Phys. Rev. B 103, 155152
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