Topological boundary modes in isostatic lattices

Frames, or lattices consisting of mass points connected by rigid bonds or central-force springs, are important model constructs that have applications in such diverse fields as structural engineering, architecture and materials science. The difference between the number of bonds and the number of degrees of freedom in these lattices determines the number of their zero-frequency ‘floppy modes’. When these are balanced, the system is on the verge of mechanical instability and is termed isostatic. It has recently been shown that certain extended isostatic lattices exhibit floppy modes localized at their boundary. These boundary modes are insensitive to local perturbations, and seem to have a topological origin, reminiscent of the protected electronic boundary modes that occur in the quantum Hall effect and in topological insulators. Here, we establish the connection between the topological mechanical modes and the topological band theory of electronic systems, and we predict the existence of new topological bulk mechanical phases with distinct boundary modes. We introduce one- and two- dimensional model systems that exemplify this phenomenon. The mathematical connection between isostatic lattices—which are relevant for granular matter, glasses and other ‘soft’ systems—and topological quantum matter is as deep as it is unexpected.