Influence of long-range interaction on Majorana zero modes

Majorana bound states, with their non-Abelian properties, are candidates for the realization of fault-tolerant quantum computation. Here we study the influence of long-range many-body interactions on Majorana zero modes present in Kitaev chains. We show that these interactions can suppress the lifetime of the Majorana zero mode. We also discuss the role of long-range interactions on the Majorana state's spatial structure, and the overlap of the Majorana states localized at opposite ends of the chain. We have determined that increasing the interaction strength leads to decreasing of the stability of the Majorana modes. Moreover, we found out that interaction between particles located at more distant sites plays a more destructive role than the interaction between nearest neighbors.

Figure 1

MZM correlation function $\lambda$ as a function of chemical potential $\mu$ and $r$-nearest-neighbor interaction strength $V_r$ for $\mathrm{\Delta }/t=0.8, \tau =50$, and $L=10$. Panels correspond to different long-range interaction $V_r$, as labeled. Black and white contour marks $\lambda =0.9$ and 0.1, respectively.

Figure 2

The same as in Fig. 1, but as a function of $\mathrm{\Delta }$ and $V_r$. Black and red contour marks $\lambda =0.9$ and 0.5, respectively ($\mu =0$).

Figure 3

Finite time $\tau$ scaling. The same as in Fig. 1, but for different times $\tau =1,10,100,1000$. Black and white contour marks $\lambda =0.9$ and 0.1, respectively.

Figure 4

Finite-size $L$ scaling. The same as in Fig. 2, but for different system sizes $L=6,8,10,12$ and time $\tau =100$. Black contour marks $\lambda =0.9$.

Figure 5

Ground-state degeneracy $\delta E$ as a function of interaction $V_r$ and chemical potential $\mu$. Panels correspond to the different long-range interaction $V_r$, as labeled. System parameters the same as in Fig. 1.

Figure 6

Energy gap $\mathrm{\Delta }E$ as a function of interaction $V_r$ and chemical potential $\mu$. Panels correspond to different long-range interaction $V_r$, as labeled. System parameters same as in Fig. 1.

Figure 7

Spatial structure of the Majorana states ${\mathrm{\Gamma }}^{+}$ and ${\mathrm{\Gamma }}^{-}$. Results for $L=10, \mathrm{\Delta }/t=0.4, V_r/t=1$, and $\mu /t=0.7$.

Figure 8

Overlap $\mathrm{\Omega }$ between left, and right Majorana states, as a function of interaction $V_r$ and chemical potential $\mu$. Panels correspond to different long-range interaction $V_r$, as labeled. System parameters are the same as in Fig. 1.

Figure 9

The site number $i$ for which $|a_i^{+}{a}_i^{-}|$ is maximal as a function of interaction $V_r$ and chemical potential $\mu$. Panels correspond to different long-range interaction $V_r$, as labeled. System parameters are the same as in Fig. 1.

Figure 10

Local overlap $|{\alpha }_i^{+}{\alpha }_i^{-}|$ between MZM ${\mathrm{\Gamma }}^{+}$ and ${\mathrm{\Gamma }}^{-}$. System parameters are the same as in Fig. 7.