  Topological semimetals are characterized by their intriguing Fermi surfaces
(FSs) such as Weyl and Dirac points, or nodal FS, and their associated surface
states. Among them, topological crystalline semimetals, in the presence of
strong spin-orbit coupling, possess a nodal FS protected by non-symmorphic
lattice symmetries. In particular, it was theoretically proposed that
$\mathrm{SrIrO}_{3}$ exhibits a bulk nodal ring due to glide symmetries, as
well as flat two-dimensional surface states related to chiral and mirror
symmetries. However, due to the semimetallic nature of the bulk, direct
observation of these surface states is difficult. Here we study the effect of
flat-surface states on phonon modes for $\mathrm{SrIrO}_{3}$ side surfaces. We
show that particular phonon modes, based on mirror symmetry, have qualitatively
different damping mechanisms due to the surface states which could be used to
infer their existence. Experimental techniques for such measurements are also
discussed.
