Phonon analog of topological nodal semimetals

Topological band structures in electronic systems like topological insulators and semimetals give rise to highly unusual physical properties. Analogous topological effects have also been discussed in bosonic systems, but the novel phenomena typically occur only when the system is excited by finite-frequency probes. A mapping recently proposed by C. L. Kane and T. C. Lubensky [Nat. Phys. 10, 39 (2014)], however, establishes a closer correspondence. It relates the zero-frequency excitations of mechanical systems to topological zero modes of fermions that appear at the edges of an otherwise gapped system. Here we generalize the mapping to systems with an intrinsically gapless bulk. In particular, we construct mechanical counterparts of topological semimetals. The resulting gapless bulk modes are physically distinct from the usual acoustic Goldstone phonons and appear even in the absence of continuous translation invariance. Moreover, the zero-frequency phonon modes feature adjustable momenta and are topologically protected as long as the lattice coordination is unchanged. Such protected soft modes with tunable wave vector may be useful in designing mechanical structures with fault-tolerant properties.