Nonlocal conductance of a Majorana wire near the topological transition

TL;DR

This paper develops a theory for the nonlocal conductance in disordered Majorana wires near the topological transition, highlighting the role of critical modes and localization length singularities.

Contribution

It introduces a model describing how critical modes influence conductance near the topological transition in Majorana wires, emphasizing the impact of localization length divergence.

Findings

Antisymmetric conductance dominates near the transition

Localization length diverges logarithmically at the critical point

Conductance magnitude scales with the ratio of wire length to localization length

Abstract

We develop a theory of the nonlocal conductance $G_{RL}(V)$ for a disordered Majorana wire tuned near the topological transition critical point. Under these conditions, the antisymmetric part of the differential conductance, $[G_{RL}(V) - G_{RL}(-V)] /2$, is the dominant one for a sufficiently long wire. This reflects the charge-neutral nature of the critical modes in the wire. We factorize the conductance into a term describing propagation of the critical modes along the wire, and terms describing the contacts between the wire and the normal leads. Topological transition affects only the former term. At the critical point, the localization length has a logarithmic singularity at the Fermi level, $l(E) \propto \ln(1 / E)$. This singularity directly manifests in the conductance magnitude, as $\ln |G_{RL}(V) / G_Q| \sim L / l(eV)$ for the wire of length $L \gg l(eV)$. Tuning the wire away from the immediate vicinity of the critical point changes the monotonicity of $l(E)$. This change in monotonicity allows us to define the width of the critical region around the transition point.