Even-odd effect on robustness of Majorana edge states in short Kitaev chains

We construct two-band effective models describing Majorana edge states in a short Kitaev chain. We derive an analytical formula for the effective Hamiltonian as a function of model parameters. Then, we discuss the robustness of Majorana edge states as a function of model parameters. We have found an even-odd effect on the robustness of Majorana states, which is characteristic of short Kitaev chains. It is experimentally observable as a differential conductance in quantum dot systems. We also study effects of coupling to an environment based on non-Hermitian Hamiltonians derived from the Lindblad equation. It is found that the Majorana zero-energy edge states acquire nonzero energy such as $E\propto \pm {\left(i\mathrm{\Gamma }\right)}^L$ for the local dissipation, where $\mathrm{\Gamma }$ is the magnitude of the dissipation and $L$ is the length of the chain. The even-odd effect is manifest for small $L$ in this formula. On the other hand, the Majorana zero-energy edge states acquire nonzero energy such as $E\propto \pm i\mathrm{\Gamma }$ for small $\mathrm{\Gamma }$ irrespective of the length $L$ for the global dissipation.

Figure 1

Illustration of the Kitaev model with length $L=4$ coupled with two leads, where $\mu$ is the chemical potential, $t>0$ is the nearest-neighbor hopping strength, and $\mathrm{\Delta }$ is the $p$-wave pairing amplitude of the superconductor. Magenta disks indicate the Majorana edge states. ${\mathrm{\Gamma }}_{\text{L}}$ (${\mathrm{\Gamma }}_{\text{R}}$) is a coupling between the left (right) quantum dot and the left (right) lead.

Figure 2

(a1)–(a5) Energy $E/t$ as a function of $\mathrm{\Delta }/t$. (b1)–(b5) Differential conductance in the $\mathrm{\Delta }/t\text{−}E/t$ plane. We have set $\mu =0$. (c1)–(c5) The energy $E/t$ as a function of $\mu /t$. (d1)–(d5) Differential conductance in the $\mu /t\text{−}E/t$ plane. We have set $\mathrm{\Delta }=t$. (a1)–(d1) $L=2$, (a2)–(d2) $L=3$, (a3)–(d3) $L=4$, (a4)–(d4) $L=5$, and (a5)–(d5) $L=6$. Blue dotted curves are the energy derived from the effective two-band model. (e) Color palette for (a1)–(a5) and (c1)–(c5) indicating the amplitude at the edge sites, where red color indicates the edge states and the green color indicates the bulk states. (f) Color palette for (b1)–(b5) and (d1)–(d5) indicating the magnitude of the differential conductance. We have set ${\mathrm{\Gamma }}_{\text{L}}={\mathrm{\Gamma }}_{\text{R}}=0.1t$.

Figure 3

(a1)–(a4) Real part and (b1)–(b4) imaginary part of the energy as a function of the local dissipation $\hslash \mathrm{\Gamma }/2t$. (a1) and (b1) $L=2$, (a2) and (b2) $L=3$, (a3) and (b3) $L=4$, (a4) and (b4) $L=5$, (a5) and (b5) $L=6$. We have set $\mathrm{\Delta }=t$ and $\mu =0$. Blue dotted curves are the energy derived from the effective two-band model. See the color palette in Fig. 2.