Topological phonons

The interaction between phonons and an external elastic deformation within an elastic material can produce various controllable effects. It is known that these external elastic deformations endow the phononic dynamics (their wave equations) with a gauge structure similar to the one of electromagnetism, even though phonons are electrically neutral. An appropriately chosen deformation can generate an acoustic Hall effect among other experimental signals. We set out to study the elastic wave-dislocation interaction and then extend it to an interaction with a distribution of dislocations. Our formalism is based on finding an equation of motion for a wave in a medium with a static deformation in the regime in which the wave does not alter the deformation but its propagation is affected by the presence of this dislocation. With the above, we find an analogy between the Aharonov-Bohm effect and a dislocation. Our study provides a powerful tool to export topological properties to the context of elasticity and, with this, to produce more and better materials for nanoscience. The periodic case of the problem is also studied, and the new topological mechanical material arising from such periodic structure is discussed.