Nontrivial Chern Numbers in Three-Terminal Josephson Junctions

Recently, it has been predicted that the Andreev bound state spectrum of four-terminal Josephson junctions may possess zero-energy Weyl singularities. Using one superconducting phase as a control parameter, these singularities are associated with topological transitions between time-reversal symmetry broken phases with different Chern numbers. Here we show that such topological transitions may also be tuned with a magnetic flux through the junction area in a three-terminal geometry.

Figure 1

Schema of the setup. Three superconducting reservoirs are connected via a normal scattering region that may be threaded by a magnetic flux. In equilibrium, the superconducting phase difference between any two leads can be controlled with a small magnetic field by connecting them into a large loop. To perform a transconductance measurement, two of the terminals are voltage biased with respect to the third one.

Figure 2

Minimal energy $E_{\min }(s)=\underset{{\phi }_1,{\phi }_2}{\min }{E}_{\mathrm{ABS}}({\phi }_1,{\phi }_2,s)$ for the symmetric case with $U=0.1$ and $t=1$. We find four Weyl points at $s=0.676$, 2.254, 4.029, and 5.608. The energy gap is largest at $s=1.479$ and 4.804, where $E_{\min }/\mathrm{\Delta }\approx 0.49$.

Figure 3

Conductances for the symmetric case with $U=0.1$ and $t=1$. Here the Dynes parameter is $\mathrm{\Gamma }/\mathrm{\Delta }=0.01$. (a) Conductances $G_{11}(V_1)$ and $G_{21}(V_1)$ as well as $G_{22}(V_2)$ and $G_{12}(V_2)$ as a function of the voltage at $s=\pi /2$. (b) Conductances $G_{11}$ and $G_{21}$ as a function of the magnetic flux at $e{V}_{\pm }/\mathrm{\Delta }=\pm 0.02$. While the plateau values are the same for both signs of the voltage, the dissipative regime close to the Weyl points strongly depends on how the bias is applied.

Figure 4

(a) Chern number $C_{21}^{\mathrm{ABS}}$ of the ABS with negative energy for the symmetric case with $U=0.1$ and $t=1$. (b) Flux dependence of the ABS at ${\phi }_1={\phi }_2=0$ close to $\mathrm{\Delta }$ when a weak energy dependence of the normal-state scattering matrix is taken into account. The charge-2 Weyl point at $s=0$ leads to a nontrivial Chern number of the lowest-energy state at $s\ne 0$.

Figure 5

Chern number $C_{21}^{\mathrm{ABS}}$ of the ABS with negative energy (solid line), Andreev gap $E_{\min }$ (dotted line), and maximal ABS energy $E_{\max }$ (dashed line) as a function of $x\sim s\sqrt{E_{\text{Th}}/\delta }$ for a three-terminal junction through a chaotic dot described by a scattering matrix generated randomly according to Eq. (14).