Non-Hermiticity-stabilized Majorana zero modes in semiconductor-superconductor nanowires

Coupled Majorana zero modes with nonzero energies are generally detrimental to the non-Abelian statistics due to the additional dynamic phase. Nevertheless, we show that a well-connected lead can introduce a local non-Hermitian dissipation term to shift the energies of both coupled Majorana modes to zero, and surprisingly turn the coupled Majorana mode far from the lead into a dark Majorana mode with exponentially small dissipation. This dark Majorana mode can conquer the drawback of the partially overlapped Majorana zero modes and possess the properties of true Majorana zero mode such as the perfect fractional Josephson effect and the non-Abelian statistics.

Figure 1

(a) Schematic diagrams for an isolated nanowire (top panel) and one strongly coupled with a left lead (bottom panel). A right lead is present only when we calculate the differential conductance $d{I}_R/dV$. The coupling between two MZMs is $E_M$ and the $i\mathrm{th}$ lead introduces dissipation ${\mathrm{\Gamma }}_i$. The wavy curves denote the probability density of the lowest two eigenstates $|{\psi }_1\left(x\right)|,|{\psi }_2\left(x\right)|$. In the top panel, the two states $|{\psi }_1\left(x\right)|=|{\psi }_2\left(x\right)|$ have a common nonlocal distribution (dark gray). In the bottom panel, the two states are split into spatially separated $|{\psi }_L\left(x\right)|,|{\psi }_R\left(x\right)|$ by the self-energy ${\mathrm{\Gamma }}_1$ of the left lead. (b) The real parts of the lowest two eigenenergies for an isolated nanowire (red) and those for one strongly coupled with left lead (blue). Here ${\mathrm{\Gamma }}_R=0$. (c) $d{I}_R/dV(2e^2/h)$ versus $V$ and $\mu$, for ${\mathrm{\Gamma }}_L=0, {\mathrm{\Gamma }}_R=0.1\mathrm{\Delta }$. (d) $d{I}_R/dV(2e^2/h)$ versus $V$ and $\mu$, for ${\mathrm{\Gamma }}_L=1.0\mathrm{\Delta }, {\mathrm{\Gamma }}_R=0.1\mathrm{\Delta }$. (e) $d{I}_R/dV(2e^2/h)$ versus $V$ and ${\mathrm{\Gamma }}_L$, for ${\mathrm{\Gamma }}_R=0.1\mathrm{\Delta }, \mu =7.5\mathrm{\Delta }$.

Figure 2

The stability of the DMM. (a) The decay rate of lowest two levels versus the self-energy ${\mathrm{\Gamma }}_L$, with ${\mathrm{\Gamma }}_R=0, \mu =6\mathrm{\Delta }$. (b) The moduli of the two levels with ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=0$ (black, $|E_{+}|=|E_{-}|=E_M$), ${\mathrm{\Gamma }}_L=0.5\mathrm{\Delta },{\mathrm{\Gamma }}_R=0$ (blue, dashed for $|E_{+}|$ and solid for $|E_{-}|$), and ${\mathrm{\Gamma }}_L=\mathrm{\Delta },{\mathrm{\Gamma }}_R=0$ (red, dashed for $|E_{+}|$ and solid for $|E_{-}|$).

Figure 3

Topological Josephson junction. (a) A junction made by two nanowires (blue, with $N_{x}a=750\mathrm{nm}$) and a quantum dot (green) in the center. The finite-size effect causes the coupling of four MZMs ${\gamma }_1,{\gamma }_2,\widetilde{{\gamma }_1},\widetilde{{\gamma }_2}$. (b) The Andreev bound states versus the phase difference for (a), showing anticrossing and $2\pi$ period. (c) Same as (a) except the two outer ends of the system are connected to leads. The leads introduce ${\mathrm{\Gamma }}_L=\mathrm{\Delta }$ for the MZMs ${\tilde{\gamma }}_1,{\tilde{\gamma }}_2$ far from the quantum dot and ${\mathrm{\Gamma }}_R=0$ for the MZMs ${\gamma }_1,{\gamma }_2$ near the quantum dot. (d) The Andreev bound states versus the phase difference for (c), showing crossing and $4\pi$ period.

Figure 4

Non-Abelian braiding of the DMMs. (a) A device for braiding two MZMs ${\gamma }_1,{\gamma }_2$. W1, W2 are two nanowires, G1, G2 are the electrical gates between the nanowires and the quantum dot D. (b) The braiding fidelity versus $T$ for (a), where ${\psi }_{1\left(2\right)}\left(0\right)=({\gamma }_1\pm i{\tilde{\gamma }}_1)/\sqrt{2}$. (c) Same as (a) but W1, W2 are connected with leads L1, L2. (d) The time evolution of the wave functions ${\psi }_{1\left(2\right)}$ for (c), where ${\psi }_{1\left(2\right)}\left(0\right)={\gamma }_{1\left(2\right)}$ and time scale $T=100/\mathrm{\Delta }$. (e) The braiding fidelity versus $T$ for (c), where ${\psi }_{1\left(2\right)}\left(0\right)={\gamma }_{1\left(2\right)}$. $\mathrm{\Gamma }$ is the dissipation from the leads L1, L2.