Magic of nonlocal geometric force: lighting up optical transition and transporting angular momentum by chiral phonons

We investigate the impact of the nonlocal geometric force -- arising from the molecular Berry curvature -- on the lattice dynamics of magnetic materials with broken time-reversal symmetry. A first-principles computational framework is established to evaluate this force across the entire Brillouin zone. We apply it to monolayer CoCl$_2$, a ferromagnetic half-semiconductor with a narrow bandgap forbidding direct dipolar optical transition. At the phonon Brillouin zone center, the pronounced nonlocal geometric force leads to a splitting of the two upper optical phonon branches by $3 \times 10^{-2}$ THz, transforming the phonons into chiral modes. Optical chiral phonons can light up the intravalley dark exciton via absorpting circularly polarized photons. Furthermore, acoustic chiral phonons induced by the nonlocal geometric force can transport angular momentum and contribute to a non-dissipative phonon Hall viscosity.