Accurate magneto-optical determination of radius of topological nodal-ring semimetals

The shape of the Fermi surface of topological nodal-ring semimetals at low carrier concentrations is characterized by the ring radius $b/\ensuremath{\hbar}{v}_F$. This peculiar topological property may not have a clear signature in measurable physical quantities. We demonstrate an accurate and definitive method to determine the radius of topological nodal-ring semimetals. Under a magnetic field along the ring axis, the axial magneto-optical response (${\ensuremath{\sigma}}_zz$) has a giant peak. The position of this ultrastrong response is at the frequency of exactly $2b$ and is independent of the strength of the magnetic field. The amplitude of the peak response is many times stronger than that of any other inter-Landau level transitions if the magnetic energy is greater than $b$ and is similar strength if $b$ is greater than the magnetic energy. The origin of the ultrastrong response is that the axial magnetic transition is governed by selection rules completely different to those governing ${\ensuremath{\sigma}}_xx$ where the giant response is absent [R. Y. Chen et al., Phys. Rev. Lett. 115, 176404 (2015)]. The present work provides a method to accurately determine parameters of the topological properties of semimetals.