Tunneling between helical Majorana modes and helical Luttinger liquids

We propose and study the charge transport through single and double quantum point contacts setup between helical Majorana modes and an interacting helical Luttinger liquid. We show that the differential conductance decreases for stronger repulsive interactions and that the point contacts become insulating above a critical interaction strength. For a single-point contact, the differential conductance as a function of bias voltage shows a series of peaks due to Andreev reflection of electrons in the Majorana modes. In the case of two point contacts, interference phenomena make the structure of the individual resonance peaks less universal and show modulations with different separation distance between the contacts. For small separation distance, the overall features remain similar to the case of a single-point contact.

Figure 1

Proposed setup for realizing a heterojunction of a helical Luttinger lead and helical Majorana modes using a thin film of a three-dimensional topological insulator. The top and bottom superconductors' order parameters carry different signs as proposed in Ref. [22].

Figure 2

Schematic figure of single tunneling junction.

Figure 3

Schematic figure for two tunneling junctions separated by spatial distance $\mathrm{\Delta }x$ and $\mathrm{\Delta }y$.

Figure 4

Differential conductance as a function of voltage for different Luttinger parameters in the helical Luttinger lead connected by a single quantum point contact with grounded helical Majorana modes. The Luttinger parameters are $K=1$ (blue solid), $0.5$ (purple dot dashed), and $0.3$ (brown dashed). Other parameters are $\overline{t}=0.05\mathrm{\Lambda }$, the length of the edge of the topological superconductor $L={10}^3{a}_0$, and the linear spectrum cutoff $\mathrm{\Lambda }={10}^{-2}{\epsilon }_F=10\hslash v_M\frac{2\pi }{L}$.

Figure 5

Differential conductance as a function of voltage for different Luttinger parameters with a single Majorana level: $K=1$ (blue solid), $0.5$ (purple dot dashed), $0.3$ (brown dashed), and $0.2$ (green dotted). Other parameters are the same as in Fig. 4.

Figure 6

Differential conductance vs voltage for different Luttinger parameters $K=1, 0.5$, and $0.3$ with a separation distance $x_{12}=y_{12}=10a_0$. We choose the tunneling term ${\overline{t}}_1/\mathrm{\Lambda }={\overline{t}}_2/\mathrm{\Lambda }=0.01$, and the length of the helical Majorana modes $L={10}^3{a}_0$ with $a_0$ denoting lattice spacing. $\mathrm{\Lambda }={10}^{-2}{\epsilon }_F=10\hslash v_M\frac{2\pi }{L}$ is the linear spectrum cutoff in the helical Luttinger lead.

Figure 7

Differential conductance vs voltage for a Luttinger parameter $K=0.5$ with different separation lengths. We choose $x_{12}=y_{12}$ with $x_{12}=3.18a_0$ (blue solid), $x_{12}=7.96a_0$ (purple dot dashed), and $x_{12}=10a_0$ (brown dashed). The other parameters are ${\overline{t}}_1/\mathrm{\Lambda }={\overline{t}}_2/\mathrm{\Lambda }=0.01, L={10}^3{a}_0$, and $\mathrm{\Lambda }={10}^{-2}{\epsilon }_F=10\hslash v_M\frac{2\pi }{L}$. Top right inset shows the enlarged figure for peaks around $eV/\mathrm{\Lambda }=0.3$.

Figure 8

Differential conductance vs voltage for different Luttinger parameters $K=1$ (blue solid), $0.5$ (purple dot dashed), and $0.3$ (brown dashed) with a separation distance $x_{12}=y_{12}={10}^2{a}_0$. We choose the tunneling term $t_1/\mathrm{\Lambda }=t_2/\mathrm{\Lambda }=0.01$, and the length of the helical Majorana modes $L={10}^3{a}_0$ with $a_0$ denoting lattice spacing. $\mathrm{\Lambda }={10}^{-2}{\epsilon }_F=10\hslash v_M\frac{2\pi }{L}$ is the linear spectrum cutoff in the helical Luttinger lead.

Figure 9

Differential conductance vs tunneling amplitude phase difference $\varphi$ for different Luttinger parameters $K=1$ (blue solid), $0.5$ (purple dot dashed), and $0.3$ (brown dashed). $t_1=t_2{e}^{i\varphi }$ with $|t_1|/\mathrm{\Lambda }=0.01$. Other parameters are $x_{12}=y_{12}=10a_0, \mathrm{\Lambda }={10}^{-2}{\epsilon }_F=10\hslash v_M\frac{2\pi }{L}$, and $L={10}^3{a}_0$.

Figure 10

Zero bias differential conductance v.s. linear momentum cutoff $\mathrm{\Lambda }$ for fixed tunneling amplitude $\overline{t}$.

Figure 11

(Left) Contour chosen to evaluate $I^{+-}(\omega ,x)$. (Right) Contour chosen to evaluate $I^{++}(\omega ,x)$.