Signatures of Bloch-Band Geometry on Excitons: Nonhydrogenic 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 nonzero Berry curvature mixes and lifts the degeneracy of $l\ne 0$ 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 for the nonhydrogenic exciton spectrum recently calculated for transition-metal dichalcogenides. Numerically, we find a Berry curvature induced splitting of $\sim 10\mathrm{meV}$ between the $2p_x\pm i2p_y$ states of ${\mathrm{WSe}}_2$.

Figure 1

A sketch of energy levels of the gapped-graphene model under assumptions of small, constant Berry curvature (${\mathrm{\Omega }}_0$) and large exciton Bohr radius. In the absence of a Bloch term in the numerical calculations, $n=2$ states remain degenerate, as with the hydrogen atom. Upon the inclusion of a Bloch term, the $2p$ states mix and split due to ${\mathrm{\Omega }}_0$, much like the orbital Zeeman effect. The shift in energy levels arises from the quantum geometric tensor ($g_{ij}$) and is analogous to Lamb shift in the hydrogen atom.

Figure 2

Calculated squared amplitude of $2p$ wave functions in reciprocal space for ${\mathrm{MoS}}_2$ in (a) the absence and (b) the presence of Bloch overlap. The Brillouin zone is chosen such that $\mathrm{\Gamma }$ points lie on the four vertices. The exciton wave function extends near the $\pm K$ points. Without the Bloch perturbation, the wave functions are degenerate and resemble $2p_x$ and $2p_y$ wave functions. Bloch perturbation mixes the states which resemble $2p_{\pm }$ states and have an energy splitting of about $\sim 10\mathrm{meV}$. (c) A sketch of energy levels with a scheme for optically determining the predicted splitting of $2p$ states using two-photon resonance spectroscopy.