Effect of topological length on Bound states signatures in a Topological nanowire

We choose the parameter space of InSb-Al nanowire that are often used in theoretical and experimental studies[44, 46]. The effective mass considered is $m^{*}=0.015m_{e}$ with lattice constant $a=10$ nm and Rashba spin orbit coupling $\alpha_{R}=0.4$ eV$\AA$ which yields $t=\hbar^{2}/2m^{*}a^{2}=25.4$ meV and $\alpha=\alpha_{R}/2a=2$ meV. The effective superconducting coupling term considered is $\Delta=0.5$ meV [49, 46]. The ideal system enters the topological non-trivial region when the Zeeman strength is greater than the critical value given by $\Gamma_{c}\approx\sqrt{\mu^{2}+\Delta^{2}}$. Using the scattering matrix approach we have calculated the TI for a range of $\mu$ and $\Gamma$, and as shown in Fig. 4(a) the numerically obtained boundary between the topological and the non-topological region coincides with the theoretical prediction of the critical field shown by the red curve. For concretness, we consider $\mu=1$ meV as the onsite potential, the corresponding critical Zeeman strength for which the band closing takes place is $\Gamma_{c}\approx 1.12$ meV. From the plot of energy spectrum Vs $\Gamma$ shown in Fig. 4(b), we observe the expected appearance of zero energy modes after the band closing which takes place at the critical Zeeman field.

The numerically calculated tunneling conductance for a range of $\Gamma$ is plotted in Fig. 5. It is clear from the figure that ZBP in the local conductance appears after $\Gamma_{c}$. The ZBP is quantized at $2e^{2}/h$ which is also the predicted conductance for the MBS. The left($G_{LL}$) and right($G_{RR}$) local conductances exhibit coherence denoting the presence of MBS on either ends of the nanowire and at the same time the non-local conductance $G_{LR}$ and $G_{RL}$ exhibit the signature of band gap closing. These tunneling conductance signatures arising from a ideal nanowire with MBS at the edges of the pristine nanowire have been well studied in the past literature [49, 46].

Before we discuss the effect of the ZMP on the ZBP, we will first explore its effect on the system’s energy levels. Results of spectrum under ZMP for two different values of $n_{0}$ are shown in Fig. 4(c). From the plot of the energy spectrum it is clear that the MBS remain fixed at the zero energy since the protocol only changes the length of the topological region by putting some part of the wire in a Zeeman term below the critical term($\Gamma_{c}$). As long as the length of the topological region remains much larger than the Majorana localization length, the energies of the MBS will be fixed at zero energy due to the topological protection. Further insight into the effect of the protocol on the state is obtained by plotting the Majorana components of the low energy states. For uniform Zeeman term (which is the case for $n_{0}=-20$) the MBS are localized at the edges of the nanowire with the two Majorana components spatially separated (see Fig. 4(d)). However, as the Zeeman term profile is moved so that $n_{0}=50$, the length of the topological region as well as the bound state position changes but at the same time they remain localized at the edges of the topological region as shown in Fig. 4(e). The Majorana components are still separated so they stay at zero energy.

We next focus the effect of ZMP on the ZBP. For the ZMP, the calculated local conductance for a range of $n_{0}$ can be seen in Figs. 6(a-b). As $n_{0}$ is increased (i.e., the topological region is reduced while the position of the lead remains unchanged), a subsequent decrease in the width of the ZBP becomes apparent, while at the same time the height of the peak remains quantized to the value $2e^{2}/\hbar$. We also plot value of the conductance for two values of $n_{0}$ in Figs. 6(c-d), where the quantization of peak is clearly observed. This feature has a simple explanation due to the property of the MBS. Application of the moving protocol changes the topological length of the wire and therefore the MBS moves further away from the lead while always staying at the edge of the topological region. Since the edge states move further away from the lead the coupling between the lead and the MBS decreases, resulting in the reduced width of the ZBP. For sufficiently large separation between the lead and the edge mode, the ZBP disappears altogether. Through out this protocol the Majorana components remain spatially separated, as a result the MBS remains at the zero energy causing the peak to be at zero bias through out this protocol.

In quantum wires it is difficult to achieve a scenario wherein the parameters are homogeneous through out the length of the wire. Instead, the inhomogeneous profile in the heterostructure better captures the realistic scenario. We begin by considering an inhomogeneous NS heterostructure where the chemical potential and the superconducting profile used to model the heterostructure is given by Eq. (3) and shown in Fig. 1(a). The numerically calculated topological boundary invariant is shown in Fig. 7(a), which shows the presence of topological phase for Zeeman terms greater than $\Gamma_{c}$ for a given value of the chemical potential (in the $S$ region). This figure is similar to the corresponding Fig. 4(a) for the homogeneous case. However, the difference between the two scenarios can be seen from the presence of low lying energy states close to zero energy even in the non-topological regime , see Fig. 7(b). It turns out that the zero energy modes in the trivial region persist for large range of Zeeman coupling and chemical potential. The energy spectrum for the system for $\mu=1$ meV is shown in Fig. 7(c), which shows the appearance of ABS before the topological phase transition at $\Gamma_{c}\approx 1.12$ meV. For $\Gamma>\Gamma_{c}$ the Majorana components obtained from the low-lying states are localized in non-overlapping regions (at the edges) and are the MBS (see Fig. 7(e)), while for $\Gamma<\Gamma_{c}$ the components have significant overlap with each other and are the partially separated ABS, as shown in Fig. 7(d).

Consider next the Fig. 8 which depicts the tunneling conductance calculated for the NS heterostructure. For $\Gamma>\Gamma_{c}$ (denoted by the red vertical line) both the local conductances, $G_{LL}$ and $G_{RR}$, exhibit ZBP with a peak height of $2e^{2}/h$ which is consistent with the presence of MBS. It turns out that for the left lead this signature of MBS persists even for extended values of $\Gamma<\Gamma_{c}$. The trivial ZBP has a splitting at zero energy, however, the splitting is negligible making it difficult to resolve even at low non-zero temperatures [55, 56]. Another aspect which is different from the signatures in the topological regime is the absence of a ZBP on the right tunneling conductance (see Fig. 8(b)). The reason for this is the presence of bound states only on the left side of the wire, which couples to the corresponding left lead only. The absence of correlation in the local conductance is associated with the signature of ABS [44]. As for the non-local conductance, the plots in Figs. 8(c-d) exhibit weak signature to unambiguously distinguish bulk gap closing at $\Gamma_{c}$. Similar results for NS heterostructure have been demonstrated previously in the literature[47, 46].

In this scenario where the tunneling setup involves two leads it is more likely that the inhomogeneity will be present on both sides of the nanowire. The profile used to reproduce the NSN heterostructure is given in Eq. (4) and shown in Fig. 1(b). We find that for a range of parameters the NSN structure also hosts ABS. The same is verified from the energy spectrum in Fig. 9(a), with ABS appearing much before the critical Zeeman term similar to that for the NS system. Even if the energy spectrum for both the systems are nearly the same, the difference between the two can be seen from the Majorana component plots for the zero energy modes as shown in Fig. 9(b). For the NSN heterostructure the bound states for $\Gamma<\Gamma_{c}$ are also localized at both ends just like the MBS. These bound states are the ps-ABS, with the partially separated Majorana components at both ends of the wire.

The tunneling conductance of the NSN heterostructure is shown in Fig. 10. Now both the left ($G_{LL}$) and the right ($G_{RR}$) local conductance exhibit quantized ZBP and are also correlated. As before, these signatures are present even for $\Gamma<\Gamma_{c}$. However, even for this system the non-local conductance is unable to capture the band gap closing. This can again be attributed to the bulk states being localized in the superconducting region. Thus the correlated quantized ZBP produced by the ps-ABSs and the absence of band gap signature implies no distinguishing feature between the ABS and the MBS. These conclusions were reported previously by Hess et. al. [47].

We will now present the effect of ZMP on the trivial and topological ZBP for the inhomogeneous system. As we saw before, the NS system has both the topological and trivial quantized ZBP. The quantized ZBP for $\Gamma>\Gamma_{c}$ is present in both the left and the right local conductance, which on the application of ZMP persists for a range of $n_{0}$ values. This result is similar to the case of homogeneous wire with topological ZBP as discussed in Sec. III.1 . Interestingly, effect of ZMP on the trivial ZBP present on the left conductance for the NS system is significantly different (see Fig. 12). The ZBP splits into two separate peaks as $n_{0}$ increases. As discussed before, the topological and trivial peak for the NSN system is the most challenging to distinguish. So we apply ZMP to the tunneling conductance signature to the NSN system. The topological ZBP persists for an extended range of $n_{0}$, and the peak width decreases with an increase in $n_{0}$ as is the case for the NS system (see Fig. 13). In contrast, under ZMP, the trivial ZBP present on both local conductances splits into two separate peaks as shown in Fig. 14.

To get an insight into the behavior of topological and trivial peaks under the moving protocol, we focus our attention on the Majorana components of the state close to zero energy. Figure 15 shows the effect of ZMP on the Majorana components of the trivial states close to zero energy for the NS system. The trivial ZBP originates due to the ps-ABS which involves partially separated Majorana components. Under the application of ZMP, the overlap between the Majorana components increases, causing them to acquire a finite energy split, resulting in splitting of the ZBP into two peaks away from zero bias. For the case of NSN heterostructure we have two states close to zero energy, both of which are present at either ends of the wire. These states are ps-ABSs with small overlapping Majorana components. On the application of ZMP, the overlap increases for both the states (see Fig. 17), resulting in a finite energy split and the trivial ZBP on both local conductances splits into two separate peaks at finite bias.

Performing similar analysis for the topological states, we show the effect of ZMP on the Majorana components of MBS in NS and NSN systems in Fig. 16 and Fig. 18, respectively. Fully separated Majorana components represent the MBS, and the components remain separate throughout the protocol for both the NS and NSN systems. As a result, the MBS stays at zero energy resulting in the ZBP staying at zero bias throughout the protocol. Under the protocol, the components move away from the edge (see Fig. 18), resulting in weak coupling with the leads, leading to decreased peak width. For a large $n_{0}$ value($\approx 70$), the states will be far from the edge, making the peak disappear. Another thing to notice is that the decrease in peak height on both the local conductances of NS is different under the protocol as shown in Fig. 11(a-b). This is mainly due to the presence of the N region on the left side and MBS leaking to this region, see Fig. 18.

To showcase that the robustness of MBS energy to ZMP and the splitting is present for all parameter range of ABS, we plot the phase potrait corresponding to the lowest energy eigenvalues for the parameter range with the ZMP for different $n_{0}$ values. As shown in Fig. 19, with $n_{0}=-20$ we have normal NSN system with low energy ABS present before the topological region (separated by red line) with the application of the ZMP these low energy states moves away from zero energy and for $n_{0}=40$ there are no zero energy states outside the topological region. These plots show that the splitting is not specific to certain parameter values and the ABS move away from zero energy for all the parameter ranges. On the other hand the eigenvalues of the MBS in the topological region remains unaffected.

As discussed in Sec. II, external potential can also be used to modify the topological length of a nanowire and produce a trivial-topological-trivial structure. In earlier works, this strategy has been suggested in the context of braiding protocol [50, 51, 52, 53]. However, a moving protocol with tunneling measurements at the edges has not been considered before for the purpose of distinguishing topological states from trivial ones. In this subsection we will be presenting the effect of the PMP on trivial and topological ZBP.

In this last subsection we will study the tunneling conductance and moving protocol results for the S^′SS^′ system. We’ll utilize this system to highlight some of the shortcomings of the “moving protocol”. Nevertheless it turns out that compare to Zeeman moving protocol the Potential moving protocol is partially immune to some of the shortcomings and can clearly differentiate the topological ZBP signature from the trivial ZBP signature.

On the either ends of the wire, this system has zero energy states that appear even before the critical Zeeman term ($\Gamma_{c}\approx\sqrt{\mu_{S}^{2}+\Delta_{0}^{2}}$), however, as expected MBS emerges after the critical Zeeman term (see Fig. 26). The zero energy states before the critical zeeman field ($\Gamma_{c}$) are topological in their own right but have partially separable Majorana components, these states prior to the band gap closing are referred to as the quasi-MBS [46, 49]. The tunneling conductance of the system is shown in Fig. 27, where the ZBP corresponding to both the local conductances and the non-local conductance fail to detect the band closing signature. We note that this system’s tunneling conductance signatures are similar to that of the NSN system. Consequently one can not rely only on the tunneling signature to distinguish the trivial and the topological ZBP. Figures 28 and 29 demonstrate how ZMP affects trivial and topological ZBP, respectively. Here, the trivial ZBP peak splitting is only moderately noticeable. The peak splits, but before the split in the peak is pronounced, the peak height drops and disappears. On the other hand for the topological ZBP, the conductance peak is fixed to zero bias for a range of $n_{0}$ before it too vanishes. We plot the Majorana component for low energy states belonging to the trivial (Fig. 30) and topological (Fig. 31) regimes to determine why this is the case for various $n_{0}$ values. For trivial zero energy states, the overlap increases with the increase in $n_{0}$, leading to a rise in the energy split. Additionally, the states drift away from the lead, weakening the coupling between the lead and the states. As a result, the peak vanishes long before the peak split can be seen clearly. Figure 31 for the topological regime with the MBS demonstrates the presence of the exact shifting of Majorana component away from the lead, which accounts for the decreased peak width and the peak’s disappearance for higher $n_{0}$ values as depicted in Fig. 29.

This shortcoming of ZMP is mainly due to the reliance on the local probes to pick the energy split signature. The protocol moves the states away from the lead, which causes the signature to disappear. As we demonstrate in subsection III.3, the PMP is immune to this effect because of the leakage of states towards the ends of the wire, retaining the coupling to the lead. For trivial and topological ZBP, we show the results corresponding to the PMP on the S^′SS^′ system in Figs. 32 and 33, respectively. Figure 32 clearly illustrates the trivial ZBP splitting during the moving protocol. As shown in Fig. 34, the states leak towards the ends,thereby improving the split visibility. From tunneling conductance plot in Fig. 33 and the analogous Majorana component plot in Fig. 35, it is clear that the resilience of the topological ZBP is also present for this scenario. We want to stress the fact that both moving protocols can induce energy splitting in trivial zero energy states, but as we use local tunneling conductance probe to detect this splitting the PMP is better at capturing the split as well as the robustness in the ZBP.