Topology ultimately unveils the roots of the perfect quantization observed in complex systems. The quantum Hall effect is the celebrated archetype of a two-dimensional topological invariant.  Remarkably, topology can manifest itself even in systems defined by control parameters playing the role of synthetic dimensions. Despite the practical importance of quantized responses, non-trivial topology is a fascinating example of a hidden system property that cannot be accessed by local measurements alone.

 

In this colloquium, I introduce superconducting multi-terminal structures as a platform to investigate a host of topological phenomena that can be experimentally implemented in taylored nanostructured devices. Such systems can realize an Andreev bound state spectrum displaying a stable so-called Weyl singularity that not only shows a quantized response but can be detected spectroscopically. First experimental hints for such topological Andreev bands have recently been found. The large variability of this devices allows to go beyond more traditional topologies. On one hand, is the number of synthetic dimensions not limited to the three spatial directions and allow to simulate exotic higher-dimensional systems. On the other hand, adding non-superconducting parts can implement non-Hermitian physics that realizes yet another topological structure with unique physical properties.

 

(Szilard Leo Colloquium)