Weyl points in systems of multiple semiconductor-superconductor quantum dots

As an analogy to the Weyl point in $k$-space, we search for energy levels which close at a single point as a function of a three-dimensional parameter space. Such points are topologically protected in the sense that any perturbation which acts on the two-level subsystem can be corrected by tuning the control parameters. We find that parameter-controlled Weyl points are ubiquitous in semiconductor-superconductor quantum dots and that they are deeply related to Majorana zero modes. In this paper, we present several semiconductor-superconductor quantum-dot devices which host parameter-controlled Weyl points. Furthermore, we show how these points can be observed experimentally via conductance measurements.

Figure 1

Schematic of a three-quantum-dot system. (a) Three dots (blue) are proximity coupled to a parent superconductor with strength $\mathrm{\Delta }$, the potential on each dot (labeled ${\epsilon }_i$) is controlled by a back gate, the dots couple to their neighbor with a hopping strength $t$ and spin-orbit coupling $\alpha$, and the leftmost dot couples to a lead with strength $\tau$. There is a magnetic field gradient in the system. The magnetic field on the end dots points in the opposite direction and must have both an $\widehat{x}$ and a $\widehat{y}$ component. (b) Orientation of the ground state in the two-level system which forms the Weyl point. The Weyl point is controlled by the parameter space (${\epsilon }_1,{\epsilon }_3,B_y$).

Figure 2

A four-dot device that hosts Weyl points which can be fully controlled by the potentials of the outer dots. Each outer dot is coupled to the inner dot by a hopping term of strength $t$ and a $p$-wave superconducting term with a particular phase and a magnitude of $\mathrm{\Delta }$.

Figure 3

Emergence of the Weyl point. Each panel shows the energy difference, $E$, between the first two even-parity states as a function of the potential under dot 1, ${\epsilon }_1$, and dot 3, ${\epsilon }_3$, for different values of the magnetic field, $B_x$. All energy parameters are in units of the induced gap, $2\mathrm{\Delta }$. (a) $B_x=0.6\mathrm{\Delta }$, (b) $B_x=1.4\mathrm{\Delta }$, and (c) $B_x=2\mathrm{\Delta }$. For magnetic fields below the Andreev coupling $B_x<\mathrm{\Delta }$, as in (a), the energy levels are gapped in the ${\epsilon }_1\text{−}{\epsilon }_3$ plane and there is no Weyl point. The Weyl point emerges above $B_X>\mathrm{\Delta }$; however, near $B_x=\mathrm{\Delta }$, as in (b), the point is smeared. On the other hand, the Weyl point is sharp in the top right corner of (c).

Figure 4

Mapping out the Weyl point. Each panel shows the energy difference, $E$, between the first two even-parity states as a function of various control parameters which are defined in the main text. All parameters are in units of $2\mathrm{\Delta }$. (a) ${\epsilon }_1=1.86\mathrm{\Delta }$, (b) $B_y=0$, (c) $B_y=0.6\mathrm{\Delta }$, and (d) $B_y=0$. Panels (a) and (b) show two-dimensional slices of the Weyl cone. These show that all three parameters $({\epsilon }_1,{\epsilon }_2,B_y)$ control the energy-level separation. However, to prove that there is a Weyl point we need (c), which shows that $B_y$ opens the gap everywhere in the two-dimensional $({\epsilon }_1,{\epsilon }_2)$ space. Panel (d) shows that ${\epsilon }_2$ does not open the gap and therefore cannot be used as a control parameter.

Figure 5

Protection of the Weyl point. Each panel shows the energy difference, $E$, between the first two even-parity states under various small perturbations overlaid with the unperturbed case. All parameters are in units of $2\mathrm{\Delta }$. The yellow curve in each panel is the unperturbed case. The blue curve is for (a) $U=\mathrm{\Delta }$, (b) $t_1=0.5t_2=0.5t_3$ and ${\alpha }_1=0.5{\alpha }_2=0.5{\alpha }_3$, (c) $B_3=0.8B_2=0.8B_3$, and (d) ${\mathrm{\Delta }}_2=1.2{\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_3=1.4{\mathrm{\Delta }}_1$.

Figure 6

Measuring the Weyl point via tunnel conductance. Each scan is done in a different direction in the three-dimensional parameter space. The top row shows the different paths ${\delta }_i({\epsilon }_1,{\epsilon }_2,B_y)$ in the three-dimensional parameter space that are used. The bottom row shows the differential conductance as a function of bias voltage $V$ and the corresponding parameter ${\delta }_i$. All parameters are in units of $2\mathrm{\Delta }$.

Figure 7

Surfaces in the three-dimensional parameter space where the first and second even-parity states are degenerate. Going from right to left $B_x=0.6\mathrm{\Delta },1.4\mathrm{\Delta },2.0\mathrm{\Delta }$, matching the values in Fig. 3. The global charge, calculated by integrating over the blue box, is always ${\mathcal{C}}_0=+2$. The charges enclosed by the green boxes are ${\mathcal{C}}_0=+1$ for both (a) and (c).

Figure 8

Energy difference between the first two even-parity states (a) as a function of the spin-orbit angle ${\xi }_2$ and the potentials ${\epsilon }_3={\epsilon }_1$, (b) as a function of the dot potentials ${\epsilon }_1$ and ${\epsilon }_2$ for $\xi =\pi /4$, (c) as a function of the phase of the third superconducting contact ${\varphi }_{sc}=-iln({\mathrm{\Delta }}_3/|{\mathrm{\Delta }}_3\left|\right)$ while the other phases are zero, $\mathrm{Im}\left({\mathrm{\Delta }}_1\right)=\mathrm{Im}\left({\mathrm{\Delta }}_2\right)=0$, and (d) as a function of the dot potentials for ${\varphi }_{\mathrm{sc}}=\pi /4$.

Figure 9

Topological superconducting nanowires (blue) with Majorana end modes (red and green) in configurations that host Weyl points. The green Majoranas are active in the formation of the Weyl point while the red Majoranas are auxiliary.

Figure 10

Probability distribution of the two Majorana decompositions. The top row shows the probability distributions $P_{x,i\sigma }$ (yellow) and $P_{y,i\sigma }$ (blue) for a system with a magnetic field gradient as in the main text. Here the Majoranas cannot separate into different dots yet they still separate into different spin states at $\epsilon ={\epsilon }_1={\epsilon }_2={\epsilon }_3=0.85$. For comparison, in the bottom row we show an identical system with the exception that the magnetic field is uniform. Here the two Majorana decompositions separate into the dots at the opposite ends of the chain. $\epsilon$ is given in units of $2\mathrm{\Delta }$.