Paulo E. Faria, Daniel Hernangómez‐Pérez, Tomer Amit, Jaroslav Fabian, Sivan Refaely‐Abramson
The magneto-optical response of excitons in monolayer transition metal dichalcogenides is governed by a complex interplay of Bloch-state quantum geometry---reflected in the electronic magnetic moment---coupled with interband mixing and many-body interactions. Here, we develop a robust and general first-principles framework for many-body exciton $g$ factors (magnetic moments) by incorporating off-diagonal terms for the spin and orbital angular momenta of single-particle bands and many-body states for magnetic fields pointing in arbitrary spatial directions. We implement our framework using many-body perturbation theory via the GW-Bethe-Salpeter equation and supplement our analysis with robust symmetry-based models. Focusing on the archetypal monolayer ${\mathrm{WSe}}_{2}$, we accurately reproduce the known results of the low-energy excitons including the Zeeman splitting and the dark/gray exciton brightening. Furthermore, our theory naturally reveals the magnetic-field hybridization of higher-energy excitons ($s, p$, and $d$ like) and shows that the magnetic moments of nodal excitons ($p$ and $d$ like) do not acquire additional contributions of $\ifmmode\pm\else\textpm\fi{}{m}_{j}{\ensuremath{\mu}}_{B}$ (${m}_{j}=1,2$), characteristic of the hydrogenic picture. Our general approach also allows us to resolve the long-standing puzzle of the experimentally measured nonmonotonic Rydberg series ($1s\ensuremath{-}4s$) of exciton $g$ factors. Our framework offers a comprehensive approach to investigate, rationalize, and predict the nontrivial interplay between magnetic fields, angular momenta, and many-body exciton physics in van der Waals systems, offering different opportunities to probe signatures of quantum geometry within many-body states.