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Signatures of Bloch-band geometry on excitons: non-hydrogenic spectra in transition metal dichalcogenides

The geometry of electronic bands in a solid can drastically alter single-particle charge and spin transport. We show here that collective optical excitations arising from Coulomb interactions also exhibit unique signatures of Berry curvature and quantum geometric tensor. A non-zero Berry curvature mixes and lifts the degeneracy of $l \neq 0$ states, leading to a time-reversal-symmetric analog of the orbital Zeeman effect. The quantum geometric tensor, on the other hand, leads to $l$-dependent shifts of exciton states that is analogous to the Lamb shift. Our results provide an explanation of the non-hydrogenic exciton spectrum recently calculated for transition metal dichalcogenides. Numerically, we find a Berry curvature induced splitting of $\sim 10$ meV between the $2p_x \pm i2p_y$ states of WSe$_2$.