Electron transport in multiterminal networks of Majorana bound states

We investigate electron transport through multiterminal networks hosting Majorana bound states (MBS) in the framework of full counting statistics. In particular, we apply our general results to T-shaped junctions of two Majorana nanowires. When the wires are in the topologically nontrivial regime, three MBS are localized near the outer ends of the wires, while one MBS is localized near the crossing point, and when the lengths of the wires are finite adjacent MBS can overlap. We propose a combination of current and cross-correlation measurements to reveal the predicted coupling of four Majoranas in a topological T junction. Interestingly, we show that the elementary transport processes at the central lead are different compared to the outer leads, giving rise to characteristic nonlocal signatures in electronic transport. We find quantitative agreement between our analytical model and numerical simulations of a tight-binding model. Using the numerical simulations, we discuss the effect of weak disorder on the current and the cross-correlation functions.

Figure 1

Four Majorana bound states ${\gamma }_{1,2,3,4}$ (MBS; blue) in a T junction are located at the ends of the superconducting wires (green), as well as at the crossing point. Each of the MBS is tunnel coupled to a normal-metal lead (red) with chemical potential $\mu$. The superconductor (gray) is grounded.

Figure 2

Differential conductance $d{I}_{\alpha }/d\mu$ and current correlation functions $d{P}_{\alpha \beta }/d\mu$ for a T junction with four leads. The solid (dashed) lines indicate calculations based on FCS determinant formula Eq. (4) applied to the low-energy Hamiltonian Eq. (11) with parameters ${\varepsilon }_{14}={\varepsilon }_{24}=\xi /2,{\varepsilon }_{34}=\xi /\sqrt{2}$, ${\Gamma }_{1,2,3}=\Gamma ,{\Gamma }_4\approx \Gamma /4,\Gamma \approx \xi /5$ at zero temperature (temperature $kT=\xi /8$). For comparison, numerical results for a realistic T-junction setup with weak on-site disorder are also shown (dots with error bars); see Appendix pp3-s2 for details.

Figure 3

Electronic probability density in a Rashba 2DEG-based effective topological superconductor patterned into a T junction. The size of the blue point around the lattice sites shows the mean probability density on a given lattice site. Top graph: Majorana states at energy $E\approx 0$. Bottom graph: Nonlocal Dirac fermion at nonzero energy $E\approx \xi$.