# Topological Andreev bands in three-terminal Josephson junctions

## TL;DR

This paper investigates the topological properties of subgap Andreev bound states in three-terminal Josephson junctions, revealing how symmetry and magnetic flux influence topological phase transitions and conductance quantization.

## Contribution

It introduces a detailed analysis of the topological band structure of Andreev states in three-terminal junctions, highlighting the role of scattering matrix symmetry and magnetic flux in topological phase transitions.

## Key findings

- Time-reversal symmetry leads to trivial gap closings with zero Berry flux.
- Breaking time-reversal symmetry creates topological monopoles with nonzero Berry curvature.
- Nonlocal conductance quantization signals the topological regime.

## Abstract

We study the emergent band topology of subgap Andreev bound states in the three-terminal Josephson junctions. We scrutinize the symmetry constraints of the scattering matrix in the normal region connecting superconducting leads that enable the topological nodal points in the spectrum of Andreev states. When the scattering matrix possesses time-reversal symmetry, the gap closing occurs at special stationary points that are topologically trivial as they carry vanishing Berry fluxes. In contrast, for the time-reversal broken case we find topological monopoles of the Berry curvature and corresponding phase transition between states with different Chern numbers. The latter is controlled by the structure of the scattering matrix that can be tuned by a magnetic flux piercing through the junction area in a three-terminal geometry. The topological regime of the system can be identified by nonlocal conductance quantization that we compute explicitly for a particular parametrization of the scattering matrix in the case where each reservoir is connected by a single channel.