Exact Nonequilibrium Transport in the Topological Kondo Effect

A leading candidate for the experimental confirmation of the nonlocal quantum dynamics of Majorana fermions is the topological Kondo effect, predicted for mesoscopic superconducting islands connected to metallic leads. We identify an anisotropic, Toulouse-like, limit of the topological Kondo problem where the full nonequilibrium conductance and shot noise can be calculated exactly. Near the Kondo fixed point, we find novel asymptotic features including a universal conductance scaling function and fractional charge quantization observable via the Fano factor. In the universal regime, our results apply for generic anisotropy and even away from the Kondo limit as long as the system supports an emergent topological Kondo fixed point. Our approach thus provides key new qualitative insights and exact expressions for quantitative comparisons to future experimental data.

Figure 1

Sketch of an $M=5$ topological Kondo setup: a mesoscopic superconducting island hosting Majorana fermions (red dots), coupled to $M$ leads of conduction electrons. We focus on the conductance $G=\partial I/\partial V$ and zero-frequency shot noise $P$ associated with the current $I$ in one of the leads when it is biased with voltage $V$ with respect to the rest.

Figure 2

Conductance of an $M=3$ device calculated from the Toulouse limit. Main figure: Linear conductance (solid line) compared to numerical renormalization group results [10] for an isotropic device (dots) and to power law asymptotics (dashed). Inset: The conductance correction $f(x,T/T_K)=\delta G(x,T/T_K)/\delta G(0,T/T_K)$ for temperatures $T={10}^{-p}{T}_K$ with integer $p=2,\dots ,5$ (solid lines) compared to the scaling function in Eq. (8) (dashed). With increasing $p$, the scaling function is followed for an increasing range of $x$. The arrows indicate $eV=0.1T_K$ (for $p=2$, 3), beyond which the BSG description breaks down for isotropic systems and is expected to be replaced by more pronounced departures from scaling.