Analytically determined topological phase diagram of the proximity-induced gap in diffusive n-terminal Josephson junctions

TL;DR

This paper analytically and numerically maps the topological phase diagram of the proximity-induced gap in diffusive n-terminal Josephson junctions, revealing conditions for gapped and non-gapped states as functions of phase differences.

Contribution

It introduces an analytical equation for the phase diagram of n-terminal Josephson junctions, validated against numerical results, advancing understanding of topological transitions in these systems.

Findings

Derived an analytical phase diagram valid for any number of terminals n.

Validated the analytical results against numerical solutions for 2-, 3-, and 4-terminal cases.

Mapped regimes of gapped and non-gapped electronic excitations in 4-terminal junctions.

Abstract

Multiterminal Josephson junctions have recently been proposed as a route to artificially mimic topological matter with the distinct advantage that its properties can be controlled via the superconducting phase difference, giving rise to Weyl points in 4-terminal geometries. A key goal is to accurately determine when the system makes a transition from a gapped to non-gapped state as a function of the phase differences in the system, the latter effectively playing the role of quasiparticle momenta in conventional topological matter. We here determine the proximity gap phase diagram of diffusive n-terminal Josephson junctions, both numerically and analytically, by identifying a class of solutions to the Usadel equation at zero energy in the full proximity effect regime. We present an analytical equation which provides the phase diagram for an arbitrary number of terminals n. After briefly demonstrating the validity of the analytical approach in the previously studied 2- and 3-terminal cases, we focus on the 4-terminal case and map out the regimes where the electronic excitations in the system are gapped and non-gapped, respectively, demonstrating also in this case full agreement between the analytical and numerical approach.