Reflectionless modes as a source of Weyl nodes in multiterminal Josephson junctions

TL;DR

The paper investigates how nontrivial topology and Weyl nodes arise in multiterminal Josephson junctions (MTJJs). It identifies zero-energy reflectionless modes (zero-RSMs) of the normal-state scattering matrix as the mechanism generating topological phase boundaries. Through detailed analysis of a four-terminal two-dot MTJJ and a three-dot MTJJ, it derives explicit phase-boundary conditions linked to zero-RSMs and shows many boundaries reduce to effective two-terminal Josephson junction behavior with a $\pi$ phase difference, yielding unity transmission. The results establish a bulk-boundary correspondence for MTJJs in the synthetic dimension, connecting normal-state S-matrix properties to the Andreev spectrum and guiding experimental exploration.

Abstract

Multiterminal Josephson junctions are a promising platform to study non-trivial topology in engineered quantum systems. Yet, experimentally meaningful insight into what exactly makes these systems topologically non-trivial remains elusive. In this work, we show that zero energy reflectionless scattering modes (RSMs) of the normal scattering matrix result in topological phase boundaries. By analyzing two different setups, we explain the origin of each topological phase boundary and furthermore provide generalizable insight into these systems. The considerations here can be of help for experimentalists as it connects the properties of the normal scattering region to the Andreev bound state spectrum of superconducting junctions in a multiterminal setup.

Figures (1)

• Figure 1: (a) Four-terminal MTJJ with two dots with energies $\epsilon_{\rm L/R}$ coupled to four SC terminals with phases $\phi_{1,2,3,4}$ via couplings $\Gamma_{1,2,3,4}$. (b) Topological index $\Lambda(\epsilon,t)$ indicating which parts in parameter space may host non-trivial topology. Red solid lines (I) and (II) correspond to zero-RSMs of reflection matrix $r_{13}$ with $\epsilon \pm t= 0$ ($E_\pm = 0$). The dashed line (III) corresponds to another zero-RSM of the reflection matrix $r_{12}$ with $\epsilon^2-t^2+4 = 0$ ($E_+E_-+4 = 0$). (c) Self-energies $\Sigma_{1/2}$ in the complex plane for the topological phase boundaries (I) and (II) and for the non-trivial case (III). (d) Four-terminal MTJJ with three dots with energies $\epsilon_{\rm L/M/R}$ coupled to four SC terminals with phases $\phi_{1,2,3,4}$. (e) Topological index $\Lambda(\epsilon,t)$ as in panel (b). Red solid lines (I) and (II) correspond to zero-RSMs of the reflection matrix $r_{145}$ with $\epsilon - t= 0$ ($E_- = 0$) and $\epsilon +2t= 0$ ($E_+ = 0$). The dashed line (III) corresponds to the zero-RSM of $r_{123}$ with $\tilde{\epsilon}^2-\tilde{t}^2+4 = 0$. The dotted line (IV) corresponds to the topological phase boundary following $\hat{\epsilon}^2-\hat{t}^2+1 = 0$. (f) Self-energies $\Sigma_{1/2/3}$ in the complex plane for the zero-RSMs (I) and (II), for the non-trivial zero-RSM (III) and for the boundary not corresponding to a zero-RSM (IV). (g) The vectorfield $\bm{V}_{12}$ in Eq. \ref{['vec']} of the reflection matrix $r_{12}(z)$ with complex energy $z$ for the parameter configurations seen in the insets and in panel (b). (h) The vectorfield $\bm{V}_{123}$ of the reflection matrix $r_{123}(z)$ for the parameter configurations seen inthe insets in panel (e).