LAUSR

We report the realization of a semimetal state in a phononic crystal, which hosts nodal-chain degeneracy in momentum space. The nodal-chain dispersion consists of interconnecting degenerate lines, and is protected by the nonsymmorphic and time-reversal symmetries together. The nontrivial Zak phases, induced by the nodal-chain configuration, give rise to the nonleaky drumhead surface modes on the surfaces. When the nonsymmorphic symmetry is broken, the nodal-chain semimetal transits to valley insulating state. The valley interface states present on the interface of two distinct valley phases, by which a topological focusing, which is free from reflection, is demonstrated. These findings enrich the knowledge of nodal-line materials and may pave the way to innovative acoustic devices.