  Mechanical metamaterials are artificial structures with unusual properties,
such as negative Poisson ratio, bistability or tunable vibrational properties,
that originate in the geometry of their unit cell. At the heart of such unusual
behaviour is often a soft mode: a motion that does not significantly stretch or
compress the links between constituent elements. When activated by motors or
external fields, soft modes become the building blocks of robots and smart
materials. Here, we demonstrate the existence of topological soft modes that
can be positioned at desired locations in a metamaterial while being robust
against a wide range of structural deformations or changes in material
parameters. These protected modes, localized at dislocations, are the
mechanical analogue of topological states bound to defects in electronic
systems. We create physical realizations of the topological modes in prototypes
of kagome lattices built out of rigid triangular plates. We show mathematically
that they originate from the interplay between two Berry phases: the Burgers
vector of the dislocation and the topological polarization of the lattice. Our
work paves the way towards engineering topologically protected nano-mechanical
structures for molecular robotics or information storage and read-out.
