Majorana zero modes on a one-dimensional chain for quantum computation

Numerical calculations for Majorana zero modes on finite chains are performed using the technique of block diagonalization. It is found that Majorana zero modes occur near the ends of the chain and decay exponentially away from the ends. The phase diagrams show that Majorana zero modes of a sufficiently long chain indeed have a parameter domain of $2t>\left|\mu \right|$ as predicted from the bulk property of an infinite chain, but a short chain has a much smaller parameter domain than the prediction. Through a numerical simulation Majorana zero modes are found to be robust under the disturbance of noise. Finally the reversion of the parity of the ground states is studied by applying a bias voltage on the quantum dot at an end of the chain. It is found that for a weak coupling between a chain and a quantum dot the parity of the ground states can be reversed through adiabatically tuning the bias voltage.

Figure 1

Components $W_{1i}$ and $W_{2i}$ on ${\gamma }_i$ of zero modes of a 50-site chain with different parameters (a) $t=1$, $\Delta =0.8$, $\mu =0.4$, (b) $t=1$, $\Delta =0.2$, $\mu =0.4$, and (c) $t=1$, $\Delta =0.2$, $\mu =1.97$. The solid (blue online) and dashed (red online) lines correspond to the two Majorana zero modes ${\tilde{\gamma }}_1$ and ${\tilde{\gamma }}_2$, respectively.

Figure 2

Phase diagrams of Majorana zero modes on chains with (a) 8 sites, (b) 31 sites, and (c) 60 sites. Shadow regions stand for the parameter domains of Majorana zero modes.

Figure 3

(a) Electron densities in the ground states for different parameter settings as shown in the labeling and (b) single-particle energies ${\epsilon }_m$ on a 50-site chain with parameters as shown.

Figure 4

Majorana zero modes (blue [gray] crosses) of a 31-site chain with noise (a) $V_0=1$ and (b) $V_0=2$.

Figure 5

(a) Energies of a quantum dot coupled to a 50-site chain with parameters $\Delta =0.8,\mu =0.4$; (b) amplitudes $C_i$ in Eq. (17) with the original coupling constant $S$; (c) amplitudes $C_i$ with a 10-time reduced coupling constant $0.1S$.