Probing zero modes of a defect in a Kitaev quantum wire

The Kitaev quantum wire (KQW) model with open boundary possesses two Majorana edge modes. When the local chemical potential on a defect site is much higher than that on other sites and than the hopping energy, the electron hopping is blocked at this site. We show that the existence of such a defect on a closed KQW also gives rise to two low-energy modes, which can simulate the edge modes. The energies of the defect modes vanish to zero as the local chemical potential of the defect increase to infinity. We develop a quantum Langevin equation to study the transport of KQW for both open and closed cases. We find that when the lead is contacted with the site beside the defect, we can observe two splitted peaks around the zero-bias voltage in the differential conductance spectrum, whereas if the lead is contacted with the bulk of the quantum wire far from the the defect or the open edges, we cannot observe any zero-bias peak.

Figure 1

Demonstration of the energy spectrum of electrons and holes for (a) homogenous open quantum wire (b) closed quantum wire with a defect (${\mu }_p=10$). The existence of a defect gives rise to zero modes similar to open quantum wire. The chain has 30 sites, and other parameters are $\Delta =0.5,\mu =0.1$.

Figure 2

Mode energy of the defect mode in a closed wire ($N=30$). We set $J=1,\Delta =0.6,\mu =0.1$. The energy of defect mode decreases when ${\mu }_p$ increases.

Figure 3

Spatial profile [${\phi }^{\mu }{}_i$ and ${\varphi }^{\mu }{}_i$ in Eq. (4)] of [(a) and (b)] edge mode in homogenous open wire and [(c) and (d)] defect mode in closed wire. Panels (e) and (f) show the spatial profile of the Majorana operator ${\widehat{\gamma }}_{\mu ,\pm }$ [$g^{\mu }{}_i$ and $h^{\mu }{}_i$ in Eq. (6)]. The chain has 30 sites, and we set $J=1,\Delta =0.5,\mu =0.1$. There is a defect with ${\mu }_p=10$ at the 10-th site (represented by the green vertical line).

Figure 4

Two contacts transport measurement setup for (a) a homogenous open wire (b) a closed wire with a defect. The leads can be contacted with different sites.

Figure 5

Differential conductance for a homogenous tight-binding chain ($N=60$). We set $J=1,\mu =0.3,\Delta =0,\lambda =0.2,{\Omega }_c=20$. (a) The energy spectrum. (b) The differential conductance $dI/dV$. The leads contact with the two ends.

Figure 6

Differential conductance in topological phase regime for [(a) and (b)] a homogenous open wire and [(c) and (d)] a closed wire with a defect (${\mu }_p=15$ at site 20). We set $J=1,\mu =0.1,\Delta =0.4,\lambda =0.3,{\Omega }_c=20$. The chain contains 60 sites. The position of the right lead is fixed, while the left lead is contacted with different sites.