Multi-terminal Josephson junctions as topological matter

Topological phases of matter have attracted much interest in recent years. Starting with gapped phases such as topological insulators and superconductors, more recently gapless topological phases possessing topologically protected band crossings have been discovered. 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 singularities in the Andreev bound state spectrum of 4-terminal junctions. Furthermore, we show that the presence of such Weyl singularities has important consequences on transport. Namely, it enables topological transitions that manifest themselves experimentally as changes of the quantized transconductance in units of 4e²/h between two voltage-biased terminals.