Topological insulators are materials characterized by a dichotomy between the interior and the edge of a finite sample: The bulk of the system is gapped and no electronic current can flow. On the surface, however, the intricate topological features of the system guarantee stable conducting channels. These ideas were originally developed after the discovery of the quantum Hall effect in 1980. Over the last ten years, a plethora of new systems such as Mercury Telluride quantum wells or Bismuth Antimony compounds were found to show similar behavior.

In our research, we are interested in bringing the phenomenology of these topological insulators described by quantum mechanics to the stage of classical systems. We could show that the existence of topologically protected edge states has no quantum-mechanical origin but is a simple consequence of the structure of the underlying matrix describing the dynamics of a given mechanical system. We provided an experimental proof of the existence and stability of topological surface states for sound waves in a mechanical quantum spin Hall analog.

We are trying to keep a list of publication in the field of topomechanics up-to-date!

### Mechanical quantum spin Hall effect

In a recent publication (Science **349**, 47 (2015)) we could show how the quantum spin Hall effect in the form of two copies of the Hofstadter lattice model with flux ±1/3 can be implemented in a mechanical system. Using an array of classical pendula we could measure all key properties of the topological band structure. The following video provides an illustration of our coupling mechanisms:

When injected at the correct frequency, a wave-packet travels uni-directionally along the edge:

In order to measure the edge dispersion relation (arXiv) we excited the system into a steady state:

Finally, an edge state passing along a corner: