Topological phases of matter have been a subject of intense studies in recent years. In many instances, topological properties are encoded in the band structure and one has to find the right material or combination of materials in order to realize them. More recently, an alternative approach to finding and exploring topological states of matter has emerged: namely, one can “imitate” necessary physical ingredients by using other degrees of freedom.
We show that, using the superconducting phases of the terminals in n-terminal Josephson junctions as variables, one may realize topological band structures in d=n-1 dimensions. In particular, we predict that a 4-terminal junction may realize a 3-dimensional Weyl semimetal, possessing topologically protected crossings in its Andreev bound state spectrum. As the phases can be controlled externally, furthermore, one has access to lower-dimensional subspaces. Namely, a 2-dimensional subsystem may have a finite Chern number that manifests itself in a quantized transconductance, like in the quantum Hall effect.
The analogy between the spectrum of Andreev bound states in an n-terminal Josephson junction and the bandstructure of an n-1-dimensional material opens the possibility of realizing topological phases in higher dimensions, not accessible in real materials.