# Topological supercurrents interaction and fluctuations in the multiterminal Josephson effect

## TL;DR

This paper investigates the topological properties and supercurrent fluctuations in multiterminal topological superconductor Josephson junctions, revealing protected band crossings, Majorana-Weyl nodes, and quantized conductance linked to topological invariants.

## Contribution

It introduces a symmetry-constrained scattering matrix approach to analyze sub-gap states and topological features in multiterminal Josephson junctions, including novel Majorana-Weyl nodes and conductance quantization.

## Key findings

- Identification of topologically protected band crossings as monopoles of Berry curvature.
- Discovery of finite energy Majorana-Weyl nodes in four-terminal junctions.
- Quantized nonlocal conductance related to the Chern number of bands.

## Abstract

We study the Josephson effect in the multiterminal junction of topological superconductors. We use the symmetry-constrained scattering matrix approach to derive band dispersions of emergent sub-gap Andreev bound states in a multidimensional parameter space of superconducting phase differences. We find distinct topologically protected band crossings that serve as monopoles of finite Berry curvature. Particularly, in a four-terminal junction the admixture of $2\pi$ and $4\pi$ periodic levels leads to the appearance of finite energy Majorana-Weyl nodes. This topological regime in the junction can be characterized by a quantized nonlocal conductance that measures the Chern number of the corresponding bands. In addition, we calculate current-phase relations, variance, and cross-correlations of topological supercurrents in multiterminal contacts and discuss the universality of these transport characteristics. At the technical level these results are obtained by integrating over the group of a circular ensemble that describes the scattering matrix of the junction. We briefly discuss our results in the context of observed fluctuations of the gate dependence of the critical current in topological planar Josephson junctions and comment on the possibility of parity measurements from the switching current distributions in multiterminal Majorana junctions.