Analytically determined topological phase diagram of the proximity-induced gap in diffusive n-terminal Josephson 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 ( ∈ N n ), 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.The interest in topological quantum phases of matter has grown steadily in recent years, and the fundamental importance of this topic in physics was recently recognized by Thouless, Haldane, and Kosterlitz being awarded the 2016 Nobel prize in physics for their contribution to this field. So far, specific material classes such as telluride-based quantum wells (HgTe, CdTe), bismuth antimony (Bi 1−x Sb x ) and bismuth selenide (Bi 2 Se 3 ) have received the most attention in the pursuit of symmetry-protected topological phases and excitations 1-4 . However, it was recently proposed 5 that similar physics could be obtained using conventional superconducting materials. More specifically, by using multiterminal Josephson junctions, the authors of ref. 5 showed that it was possible to create an artificial topological material displaying Weyl singularities under appropriate conditions. In multiterminal Josephson junctions hosting well-defined Andreev bound states, the crossing of these states with the Fermi level has been shown to be analogous to Weyl points in 3D solids with the Andreev bound state taking on the role of energy bands and the superconducting phase differences corresponding to quasiparticle momenta. A considerable advantage in utilizing multiterminal Josephson junctions rather than 3D solids to study exotic phenomena such as Weyl singularities and topologically different phases is that the phase differences are much more easily controlled experimentally than the quasiparticle momenta.In order to probe electronic excitations with topological properties, a key goal is to map out the phase diagram of the system in terms of when it is gapped or not. A gapped system here means that there are no available excitations in a finite interval around the Fermi level. The reason for why this...

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