Multi-terminal Josephson junctions as topological matter

Topological materials have attracted much interest in recent years. Topological insulators and superconductors possess a bulk gap and topologically protected surface states with unusual properties. A prime example are the Majorana bound states at the ends of onedimensional topological superconductors, whose non-Abelian statistics could be exploited for topological quantum computation. Topological semimetals possess topologically protected band crossings in the bulk and so-called Fermi arcs on their boundaries, which also lead to interesting phenomena. Here we show that n-terminal Josephson junctions with conventional superconductors may provide a straightforward realization of tunable topological materials in n−1 dimensions, the independent superconducting phases playing the role of quasi-momenta. In particular, we find zero-energy Weyl points in the Andreev bound state spectrum of 4-terminal junctions. The topological properties of the junction may be probed experimentally by measuring the transconductance between two voltage-biased leads, which we predict to be quantized. Further, 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.