Classification of Weyl point trajectories in multi-terminal Josephson junctions

TL;DR

This paper investigates how Weyl points (WPs) emerge and move in multi-terminal Josephson junctions (MTJJs) as external controls—such as gate voltages and magnetic flux—tune the superconducting-phase space $\{ \varphi_j \}$. Using a scattering-matrix framework with a random-matrix description of the normal region, it computes Andreev bound-state spectra and derives Berry-curvature–based Chern numbers $Ch_3$ to classify WP trajectories, revealing rich dynamics including pair creation/annihilation, closed loops, and open lines. For four-terminal junctions, WP trajectories exhibit symmetry and distinct topological phases, including transitions from four-WP to trivial states; for five-terminal junctions, the added phase $\varphi_4$ and coupling $T_4$ yield even more intricate phase diagrams with multiple creation/annihilation events and pair exchanges. The work highlights the possibility of tuning topological states and charge-flow in MTJJs, suggesting avenues to explore symmetry-dependent WP physics and potential experimental transconductance signatures of Berry curvature.

Abstract

Topological protection is an attractive signature in both fundamental and applied researches because it provides an exotic and robust state. Multi-terminal Josephson junctions have recently been studied extensively owing to the emergence of topologically protected Weyl points without the need for topological materials. In this study, we examine the dynamic properties of Weyl points in multi-terminal Josephson junctions. The junctions are modulated by external parameters, such as electric gate voltage, magnetic flux, bias voltage. The Weyl points are manipulated and draw trajectories accompanied by pair creation and annihilation. The trajectories form both closed loops and open lines. We classify these trajectories using the Chern number and the phase diagram.

Figures (14)

• Figure 1: Schematics of multi-terminal Josephson junction. (a) An example of multi-terminal Josephson junction using semiconductor nanowires with the QPC structures. (b) A model of multi-terminal Josephson junction depicted the scattering matrix.
• Figure 1: Numerical results of a sample with pair exchange in closed loop trajectories. (a) Number of the WPs as the phase diagram in the $\varphi_4$-$T_4$ plane. Four WPs are found at $T_4 = 0$. Red broken lines are reference to indicate $T_4 = 0.10$ (b), $0.30$ (c), $0.50$ (d), $0.70$ (e), $0.90$ (f). (b) Cross-sectional plot at fixed $T_4 = 0.10$ for the Chern number $Ch_3$ (b1) and the projected WP trajectory on $\varphi_1$-$\varphi_3$ (b2) and $\varphi_2$-$\varphi_3$ plane (b3). (c)-(f) Results of $Ch_3$ and the WP trajectory at $T_4 = 0.30$, $0.50$, $0.70$, and $0.90$, respectively, with the same manner as those in (b).
• Figure 2: The trajectory of WPs (a) and the Chern number (b) in one sample with the WPs at $T_3 = 1$. With the decrease of $T_3$, only pair annihilation occurs at $(\varphi_1,\varphi_2,\varphi_3) \approx \pm (0.762,-0.287,-0.366)\pi$ when $T_3 \approx 0.69$.
• Figure 2: Numerical results of a sample with pair exchange in open line trajectories. Four WPs are found at $T_4 = 0$. The parameters are the same as those in Fig. A\ref{['fig:A1']} except the transmission probability: $T_4 = 0.20$ (b), $0.40$ (c), $0.50$ (d), $0.60$ (e), $0.80$ (f).
• Figure 3: Numerical results of a typical sample for four-terminal Josephson junctions. The pair creation and annihilation of the WPs occur in the decrease of $T_3$. The two WP pairs draw separated two closed trajectories without pair exchange, as shown in the 3D space (a), in the projected 2D $\varphi_1$-$\varphi_2$ (b), $\varphi_2$-$\varphi_3$ (c), and $\varphi_1$-$\varphi_3$ planes (d). Red and blue lines indicate the WP trajectories with positive and negative topological charge, respectively. Yellow star and green hexagon marks indicate the positions of the pair creation and annihilation, respectively. (e) Chern number is $Ch_3 = +1$ (light red) and $-1$ (light blue) between the projected positions of WPs in $\varphi_3$.