Nonanalyticity, Valley Quantum Phases, and Massless Excitons in Monolayer Transition Metal Dichalcogenides*

Exciton dispersion as a function of center-of-mass momentum \textbf{Q} is essential to the understanding of exciton dynamics, relaxation, and condensation. We use the ab initio GW-Bethe-Salpeter equation(GW-BSE) method to calculate the dispersion of excitons in monolayer MoS$_{2}$ and find a nonanalytic lightlike dispersion. This behavior arises from the interplay of an unusual $|Q|$-term in both the intra- and intervalley exchange of the electron-hole interaction, which concurrently gives rise to a valley quantum phase of winding number two. We have derived a simple, effective Hamiltonian and analytic solutions, which quantitatively describe this physics, and we predict that signatures of this unusual dispersion can be measured with a linearly polarized optical beam tilted away from normal incidence. The existence of a nonanalytic exciton dispersion can be generalized to other 2D semiconductors with excitons whose amplitudes are localized in a small region of the Brillioun zone.