Formation and detection of Majorana modes in quantum spin Hall trenches

We propose a novel realization for a topologically superconducting phase hosting Majorana zero modes on the basis of quantum spin Hall systems. Remarkably, our proposal is completely free of ferromagnets. Instead, we confine helical edge states around a narrow defect line of finite length in a two-dimensional topological insulator. We demonstrate the formation of a new topological regime, hosting protected Majorana modes in the presence of $s$-wave superconductivity and Zeeman coupling. Interestingly, when the system is weakly tunnel coupled to helical edge state reservoirs, a particular transport signature is associated with the presence of a non-Abelian Majorana zero mode.

Figure 1

Quantum spin Hall antiwire. (a) Schematic illustration of the system: A QSH antiwire, covered by a $s$-wave superconductor under the influence of a magnetic field weakly coupled to helical edge states at the boundary of the QSH stripe. (b) Sketch of the QSH constriction with the appearing scattering terms.

Figure 2

Topological phase diagram of the proximitized antiwire. (a) Eigenenergy spectrum of $H_0$. The different colors represent states with orthogonal spin with $t_c=t_0$. (b) Phase diagram as function of $\mu$ and $B_z$ (under the choice $t_0=t_c=1, \mathrm{\Delta }/t_0=0.3, v_F=1$). (c) Dependence of the topological phase on $t_c$. The different curves correspond to gap closures for $t_c/t_0=0.2,0.4,0.6,0.8,1.0$ (red to blue), $\mathrm{\Delta }=0.3t_0,t_0=1, v_F=1$. (d) Dependence of the topological phase on $t_0$. The curves correspond to to gap closures for $t_0/t_c=0.2,0.4,0.6,0.8,1$ (red to blue), $\mathrm{\Delta }/t_c=0.3,t_c=1, v_F=1$.

Figure 3

Majorana wave functions at the antiwire ends. (a) ${\lambda }_M$ (yellow) and $\delta {\mathrm{\Gamma }}_{{\lambda }_M}$ (blue) as a function of $L$. (b) $|U_0{\left(x\right)|}^2$ according to Eq. (9) with $U_0\left(0\right)={\nu }_{\lambda }$. (c) Schematic illustration of the probability distribution in the (folded) antiwire. The parameters of the calculation are: $B/t_0=0.6, \mu /t_0=\sqrt{2}, \mathrm{\Delta }/t_0=0.3, t_c=t_0=1, v_F=1$.

Figure 4

Transport measurements. (a) Two-terminal conductance as function of energy $\epsilon$ and Zeeman field $B_z$. (b)–(c) Multiterminal conductance between contacts 1 and 2 with respect to Fig. 1, as a function of $\mu$ and $B_z$ (b), $\epsilon$ and $B_z$ (c), respectively. In (b), all values $G_{1\to 2}>0$ are colored in blue. In (c), all values $G_{1\to 2}<0$ are colored in red. Other parameters of the plots are: $L=20\hslash v_F/t_0, \mathrm{\Delta }/t_0=0.3, \mu /t_0=\sqrt{2}$ [(a) and (c)], $\epsilon =0$ (b), $t_0=t_c=1v_F=1$. For computational reasons, the delta distribution separating the anti-wire from the leads is replaced with its step function approximation ${\delta }_a\left(x\right)=\mathrm{rect}(x/a)/a$ with $a=0.1$. Moreover, $T=1.5$ for (a)–(b) and $T=2$ for (c).

Figure 5

Majorana scatterer on the helical edge. (a) Schematic illustration of a Majorana mode ${\gamma }_1$ side coupled to a helical edge. (b) Multiterminal conductance as a function of energy $\epsilon$ with $v_F=1, t_{\uparrow }=1.2t_{\downarrow }$ and $t_{\downarrow }=0.2$ and ${\epsilon }_d/t_{\downarrow }=0,1$ (blue, orange).

Figure 6

Generic scatterer on the helical edge. (a) Schematic illustration of the coupling between the helical edge and the particle-hole-symmetric system $S$. (b) Conductance $G_{1\to 2}$ for ${\epsilon }_{\alpha }=0$, as a function of $\xi$ and $\epsilon$. Only negative values of $G_{1\to 2}$ are shown. The parameters are $t_{\uparrow }=1.2t_{\downarrow }$ with $t_{\downarrow }=0.2$. (c) Multiterminal conductance $G_{1\to 2}$ for the coupling to a generic eigenstate of a PHS system on resonance $\epsilon ={\epsilon }_{\alpha }$, as a function of $\xi$. The different lines correspond to ${\epsilon }_{\alpha }/t_{\uparrow }=2,0.2,0.02,$ and 0.002 (blue to red). Further parameters are $t_{\downarrow }=3/5t_{\uparrow }$ with $t_{\uparrow }=0.5$.

Figure 7

Conductance $G_{1\to 2}$ for the discussed toy models in the presence of TR breaking backscattering: (a) Schematic of the discussed system. (b) Comparison between TR invariant (purple) and TR breaking (blue to green) transport in a helical edge, side coupled to a Majorana. The TR breaking parameter is given by $B_x/t_{\uparrow }=1.25, 2.5$, 3.75, 6 (blue to green) with $\mu /t_{\uparrow }=5$ and ${\epsilon }_d=0$. Note that for $B_x/t_{\uparrow }=6$, we have $B_x>\mu$, which implies that no propagating modes are present for small $\epsilon$. (c) $G_{1\to 2}$ for side coupling a generic BdG state at energy ${\epsilon }_{\alpha }=0$ in dependence of the parametrization parameter $\xi$ and energy $\epsilon$. (d) $G_{1\to 2}$ on resonance ($\epsilon ={\epsilon }_{\alpha }$), where ${\epsilon }_{\alpha }/t_{\uparrow }=5\times {10}^{-2},5\times {10}^{-3},5\times {10}^{-4}$ (green to blue). Further parameters of the plot are: $x_i=-2.5v_F/t_{\uparrow }, x_f=5v_F/t_{\uparrow }, \mu /t_{\uparrow }=5, B_x/t_{\uparrow }=2.5$ [(c) and (d)], $t_{\downarrow }=3/5t_{\uparrow }, t_{\uparrow }=0.2$.

Figure 8

Coupling of six antiwires using gate potentials applied to the embedding quantum spin Hall insulator (orange regions).

Figure 9

Multiterminal conductances $G_{1\to 2}$ (a) and $G_{2\to 1}$ (b) as a function of energy $\epsilon$ and $B_z$. The parameters are the same as given in Fig. 4 of the main text. All negative values are colored in red.

Figure 10

(a) Multiterminal conductance $G_{1\to 2}$ for a Kitaev chain, side coupled to a helical edge as a function of the chains chemical potential $\mu$ and energy $\epsilon$. Negative values are colored red. (b) Eigenstates of the Kitaev chain as a function of the system parameter $\mu$ and $\epsilon$. The color code represents the absolute difference of electronic (${\zeta }_{\alpha ,1}^{\left(e\right)}$) and holelike wave function (${\zeta }_{\alpha ,1}^{\left(h\right)}$) at the first site of the chain normalized to the maximum value reached for all eigenstates indexed by $\alpha$. Further parameters of the plots are: $t=\mathrm{\Delta }=0.5$, the number of sites is $N=15$.