  Topological semimetal states which are constrained by symmetries and give
birth to innovative excitations are the frontiers of topological quantum
matter. Nodal chains in which two nodal rings connect at one point were first
discovered in non-symmorphic electronic systems and then generalized to
symmorphic phononic systems. In this work, we identify a new class of planar
nodal chains in non-symmorphic phononic systems, where the connecting rings lie
in the same plane. The constituting nodal rings are protected by mirror
symmetry, their intersection is guaranteed by the combination of time-reversal
and non-symmorphic two-fold screw symmetry. In addition, the connecting points
are four-fold degenerate while those in previous works are two-fold degenerate.
We searched all 230 space groups and found 8 space groups that can host the
proposed planar nodal chain phonons. Taking wurtzite GaN (space group No.186)
as an example, the planar nodal chain is confirmed by first-principles
calculations. The planar nodal chains result in two distinct classes of
drumhead surface. The first category lies on the [10(-1)0] surface Brillouin
zone and the second lies on the [0001] surface Brillouin zone. Our finding
reveals a class of planar nodal chains in non-symmorphic phononic systems,
expands the catalog of topological nodal chains, and enriches the family of
topological surface states.
