Degenerate topological line surface phonons in quasi-1D double helix crystal SnIP

Degenerate points/lines in the band structures of crystals have become a staple of the growing number of topological materials.The bulk-boundary correspondence provides a relation between bulk topology and surface states.While line degeneracies of bulk excitations have been extensively characterised,line degeneracies of surface states are not well understood.We show that SnIP,a quasi-one-dimensional van der Waals material with a double helix crystal structure,exhibits topological nodal rings/lines in both the bulk phonon modes and their corresponding surface states.Using a combination of first-principles calculations,symmetrybased indicator theories and Zak phase analysis,we find that two neighbouring bulk nodal rings form doubly degenerate lines in their drumhead-like surface states,which are protected by the combination of time-reversal symmetry T and glide mirror symmetry My.Our results indicate that surface degeneracies can be generically protected by symmetries such as T My,and phonons provide an ideal platform to explore such degeneracies.