Half-integer quantized charge pumping induced by a Majorana fermion

We investigate the adiabatic topological charge pumping in a topological superconductor utilizing a quantum anomalous Hall insulator proximity coupled to an $s$-wave superconductor. We show that topological pumping is characterized by the appearance of protected Majorana edge states during the course of a pumping cycle. In a topological superconductor with a single Majorana edge state, the Majorana state will be pushed into the electrode in a cycle. This leads to a half-integer quantized pumped charge, because a Majorana fermion can be viewed as half of a fermion. The half-integer quantized charge pumping would serve as a fingerprint of a Majorana fermion.

Figure 1

Illustration of a pump/normal-metal junction and its energy band. The transport is along the $y$ direction, and the transverse width of the pump is $L_x$.

Figure 2

Plot of charge pumped per cycle as a function of the pairing potential $\mathrm{\Delta }$. We have taken the units $m_1=1$ and $A=1$, and set $m_0=0.5, e{A}_x=0.1, k_x=0$. The red solid line represents the total charge pumped; the black dashed line and blue dot-dashed line are the contributions of the normal reflection and Andreev reflection, respectively. Panel (a) is the result in the absence of quadratic term, and (b) contains the correction of the quadratic term.

Figure 3

Instantaneous spectrum of $H_{\mathrm{BdG}}$ versus time $t$ in a ribbon for (a) $\mathrm{\Delta }=0.3$ and (c) $\mathrm{\Delta }=1$. The other parameters are the as same as in Fig. 2. Panels (b) and (d) show the real-space probability distribution of the edge states corresponding to (a) and (c), respectively. The green lines in (d) represent the probability distribution of $A$ and $B$ in (c), which was magnified 10 times.

Figure 4

Charge pumped versus $\mathrm{\Delta }$ and $k_x$. The other parameters are the same as in Fig. 2.