Majorana zero modes choose Euler numbers as revealed by full counting statistics

We study transport properties of a quantum dot coupled to a Majorana zero mode and two normal leads. We investigate the full counting statistics of charge tunneling events and obtain complete information on current fluctuations through the dot. Using the Keldysh path-integral approach, we compute the cumulant generating function of the current. We first consider a spinless case and find that for the symmetric dot-lead couplings, the zero-frequency cumulants are independent of the microscopic parameters and exhibit a universal pattern described by Euler numbers. We then consider the spinful system and investigate the effect of both weak and strong Coulomb interactions. We show that cases with and without Majorana coupling exhibit qualitatively different full counting statistics of charge tunneling events despite the fact that differential linear conductance might have zero-bias features in both cases.

Figure 1

Proposed experimental setup to measure the distribution function of the transmitted charge, namely the full counting statistics. The QD is created near the center of the semi-conductor nanowire T junction, two normal metal leads are attached to the upper and lower legs, and an $s$-wave superconductor is in proximity to the third lead. The latter realizes the topological superconductor hosting two MZMs ${\gamma }_1$ and ${\gamma }_2$.

Figure 2

The cumulant spectrum $C_n\left(\omega \right)$ for $n=2,3,4$ for different $\kappa , \delta =0$ (left panel) and $\delta /\mathrm{\Gamma }=0.02$ (right panel). Here we set ${\epsilon }_d/\mathrm{\Gamma }=-2.0, {\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R$.

Figure 3

The ${\epsilon }_d=0.0$ result of the second cumulant spectrum $C_2\left(\omega \right)$ for different $\kappa , \delta =0$ (left panel) and $\delta /\mathrm{\Gamma }=0.02$ (right panel). Here we take ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R$.

Figure 4

(a) Diagrammatic representation of the normal and anomalous impurity Green functions and the Coulomb interaction; (b) and (c) diagrammatic representation of the self-energy due to leading order corrections of the Coulomb interaction.

Figure 5

Diagrammatic representation of the cumulant generating function up to the leading order corrections of the Coulomb interaction.

Figure 6

Diagrammatic representation of the cumulant generating function up to the leading order corrections of the $J_3$. The solid lines describe the Majorana propagators, and the dashed lines describe the propagators of the free electron in the leads.

Figure 7

Dependence of the cumulants $C_n\left(0\right)=\langle \langle {\delta }^{n}q\rangle \rangle /M$ ($\delta =0$ and $eV\to 0$ limit) on the Majorana coupling $\kappa$ in the Kondo regime from SBMF calculation. We choose ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R, {\epsilon }_d/\mathrm{\Gamma }=-10.0$, and bandwidth $\mathrm{\Lambda }/\mathrm{\Gamma }=30.0$.

Figure 8

The cumulants $C_n\left(0\right)$ for $n=1,2,3,4$ as a function $\kappa$ and $\delta$. Here ${\epsilon }_d/\mathrm{\Gamma }=-10.0, {\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R, eV/\mathrm{\Gamma }=0.001$, and $\mathrm{\Lambda }/\mathrm{\Gamma }=30.0$.