Universal transport signatures of Majorana fermions in superconductor-Luttinger liquid junctions

One of the most promising proposals for engineering topological superconductivity and Majorana fermions employs a spin-orbit coupled nanowire subjected to a magnetic field and proximate to an $s$-wave superconductor. When only part of the wire's length contacts to the superconductor, the remaining conducting portion serves as a natural lead that can be used to probe these Majorana modes via tunneling. The enhanced role of interactions in one dimension dictates that this configuration should be viewed as a superconductor-Luttinger liquid junction. We investigate such junctions between both helical and spinful Luttinger liquids, and topological as well as nontopological superconductors. We determine the phase diagram for each case and show that universal low-energy transport in these systems is governed by fixed points describing either perfect normal reflection or perfect Andreev reflection. In addition to capturing (in some instances) the familiar Majorana-mediated “zero-bias anomaly” in a new framework, we show that interactions yield dramatic consequences in certain regimes. Indeed, we establish that strong repulsion removes this conductance anomaly altogether while strong attraction produces dynamically generated effective Majorana modes even in a junction with a trivial superconductor. Interactions further lead to striking signatures in the local density of states and the line shape of the conductance peak at finite voltage, and also are essential for establishing smoking-gun transport signatures of Majorana fermions in spinful Luttinger liquid junctions.

Figure 1

(a) Topological superconductor forming a junction with a helical Luttinger liquid. We assume that the helical Luttinger liquid couples only to the Majorana ${\gamma }_1$ at the interface, while its partner ${\gamma }_2$ remains a robust zero-energy mode. (b) Flow diagram for the junction as a function of the interaction parameter $g$ for the Luttinger liquid. Provided $g>1/2$, coupling to the Majorana ${\gamma }_1$ causes the Luttinger liquid to flow onto a fixed point where perfect Andreev reflection occurs at the junction. Here, the zero-bias conductance for the junction is quantized at $2e^2/h$. For Luttinger liquids with strong repulsive interactions such that $g<1/2$, however, coupling to ${\gamma }_1$ is irrelevant. Here, the system flows onto a fixed point where the Luttinger liquid undergoes perfect normal reflection at the junction, leading to a vanishing zero-bias conductance.

Figure 2

(a) Ordinary superconductor forming a junction with a helical Luttinger liquid. If the superconductor is fully gapped throughout, then in the noninteracting limit the helical wire necessarily flows to a fixed point at which perfect normal reflection occurs at the junction. Accessing the perfect Andreev reflection fixed point in the noninteracting limit requires the presence of a zero-energy Andreev bound state $f=({\gamma }_1+i{\gamma }_2)/2$ at the junction as shown schematically in (b), with fine tuning such that the wire couples only to (say) ${\gamma }_1$. (c) Flow diagram for the junction as a function of the interaction parameter $g$ for the Luttinger liquid. For helical Luttinger liquids with $g<2$, the perfect normal reflection fixed point is stable. When $g>2$, however, this fixed point is unstable toward the Andreev fixed point. Remarkably, here, the Andreev bound state required to achieve perfect Andreev reflection will be generated dynamically, and with no fine tuning required.

Figure 3

(a) Trivial superconductor forming a junction with a spinful Luttinger liquid. (b) Flow diagram for the junction as a function of the charge-sector interaction parameter $g_{\rho }$ for the Luttinger liquid, in the limit where the spin-sector interaction parameter is $g_{\sigma }=1$. Note that the free-fermion limit is very special; here, there are marginal boundary couplings which lead to a nonuniversal zero-bias conductance ranging anywhere from 0 to $4e^2/h$ depending on parameters. For any repulsive interaction strength ($g_{\rho }<1$), however, the junction flows to the perfect normal reflection fixed point where the zero-bias conductance vanishes.

Figure 4

(a) Topological superconductor forming a junction with a spinful Luttinger liquid. (b) Flow diagram for the junction as a function of the charge-sector interaction parameter $g_{\rho }$ for the Luttinger liquid when the spin-sector interaction parameter is $g_{\sigma }=1$. For $g_{\rho }<1/3$, the system flows to a perfect normal reflection fixed point characterized by a vanishing zero-bias conductance. When $g_{\rho }>1/3$, however, the junction flows to a novel fixed point corresponding to perfect Andreev reflection for one species and perfect normal reflection for the other, yielding a quantized $2e^2/h$ conductance.

Figure 5

Sample $dI/dV$ curve as a function of voltage $V$ (in arbitrary units) for $g=0.95$ in a helical nanowire/topological superconductor junction. The blue and red portions are the perturbative results around the normal and Andreev fixed points respectively, calculated according to Eq. (91).

Figure 6

Sample $dI/dV$ curve as a function of voltage $V$ (in arbitrary units) for $g=0.7$ in a helical nanowire/nontopological superconductor junction. The red portion is calculated according to Eq. (95), while at sufficiently low voltages (blue curve), the Fermi liquid leads dominate and we have a crossover to $g=1$ scaling.