Topology in multi-terminal superconducting structures

Topology ultimately unveils the roots of the perfect quantization observed in complex systems. The 2D quantum Hall effect is the celebrated archetype. Remarkably, topology can manifest itself even in higher-dimensional spaces defined by control parameters playing the role of synthetic dimensions. However, so far, a very limited number of implementations of higher-dimensional topological systems have been proposed, a notable example being the so-called 4D quantum Hall effect. In this talk show how to engineer non-trivial topological signatures like Weyl-nodes in synthetic dimensions created by multi-terminal superconductors and how Berry spectroscopy can be used to extract information about the systems quantum geometry [1]. Furthermore, I will show that mesoscopic superconducting systems can implement higher-dimensional topology and represent a formidable platform to study a quantum system with a purely nontrivial second Chern number [2]. I discuss that these systems also admit a non-Abelian Berry phase. Hence, they also realize an enlightening paradigm of topological non-Abelian systems in higher dimensions. I comment on our first experimental progress to implement synthetic dimensions in semiconductor-superconductor heterostructures [3]. Adding normal conducting contacts allows to investigate non-hermitian topology [4]. Finally, I introduce an experiment with four superconducting terminals that is investigated spectroscopically and shows low-energy Andreev bands. Theory shows a topological phase transition that is unfortunately masked by the limited experimental resolution.