Nonlocal conductance of a Majorana wire near the topological transition

TL;DR

The paper develops a universal low-energy theory for the nonlocal differential conductance $G_{RL}(V)$ of a disordered Majorana wire near the topological transition, separating propagation along the wire from contact effects. It predicts a logarithmic divergence of the localization length at the critical point, $l(E) = v\tau \ln\bigl(1/(E\tau)\bigr)$, and provides a factorized conductance form $G_{RL}(V)/G_Q = (T^{R}_{em}-T^{R}_{hm})\, T^{L}_{me}\, Q(E)|_{E=eV}$ with a universal propagation factor $Q(E)$; the theory applies to both clean and disordered wires and is supported by numerical simulations for realistic material parameters. Majorana zero modes yield subgap resonances in $G_{RL}(V)$ that are exponentially small in wire length but produce quantized signatures in nonlocal shot noise, while disorder induces Dyson-type singularities in the DOS and a log-normal distribution of transmission. Collectively, the work links nonlocal transport to the infinite-randomness critical point and provides robust, transport-based signatures to identify and characterize the topological transition in Majorana wires.

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.

Figures (10)

• Figure 1: a. Sketch of the considered setup. A semiconducting wire (green) is proximitized by a superconductor (blue) and contacted at its ends by normal leads (yellow). Magnetic field $B$ tunes the wire to the vicinity of the topological phase transition. We find the nonlocal differential conductance $G_{RL}(V) = dI(V) / dV$; $V$ is the bias applied to the left lead and $I$ is the current collected at the right lead. b. At the critical point, $B = B_{\rm c}$, the low-energy degrees of freedom in the wire are a pair of counter-propagating Majorana fermions with linear dispersion relation $E(k)$. c. Neutral character of Majorana fermions renders $G_{RL}(V)$ an odd function at low biases. d. Phase diagram of the Majorana wire. $\Delta_0 \propto B - B_{\rm c}$ characterizes the detuning from the critical point. Positive and negative values of $\Delta_0$ correspond to the topological and trivial phase, respectively. The critical region is the range of fields for which $|\Delta_0| \leq 1 / 2 \tau$; its width is determined by the Drude mean free time $\tau$. The properties of the wire differ qualitatively inside and outside of the critical region, as illustrated in panels e-- g. e. Evolution of the energy dependence of the localization length $l(E)$ with $B$. At the critical point, $l(E)$ diverges logarithmically at $E \rightarrow 0$ [see Eq. \ref{['eq:log_div_intro']}]. The monotonicity of $l(E)$ changes as the wire is tuned across the boundary of the critical region. Inset: field-dependence of the localization length at the Fermi level [see Eq. \ref{['eq:loc_at_FL_intro']}]. f. In the critical region, the density of states is singular at $E = 0$. g. Outside of the critical region, the singularity is replaced by a "soft" gap, $\nu (0) = 0$.
• Figure 2: a. Energy dependence of the localization length $l(E)$ for a wire tuned away from the critical point by $\Delta_0 \gg 1 / \tau$ (solid line). Dashed lines are the asymptotes for the subgap and above-the-gap behavior of $l(E)$. In the crossover region (shaded green), the dependence of $l(E)$ on $E - \Delta_0$, $\Delta_0$, and $1/\tau$ has a scaling form [see Eqs. \ref{['eq:scaling']} and \ref{['eq:scaling_2']}]. Insets shows the scaling function $f(\varepsilon)$. b. Density of states $\nu (E)$ plotted with the help of Eq. \ref{['eq:dos_full']} (solid line). Dashed line is the respective result for a clean wire [Eq. \ref{['eq:dos_0']}]. Disorder smears the "coherence" peak but a relatively well-pronounced spectral "gap" remains for $\Delta_0 \gg 1 / \tau$.
• Figure 3: Setup of the scattering problem. Electron and hole waves in the leads are depicted with black and white circles, respectively. Gray circles represent Majorana modes in the wire. Scattering at the junctions between the wire (green) and the normal leads (yellow) is described by matrices $S^{L}(E)$ and $S^{R}(E)$ [see Eqs. \ref{['eq:Sl_def']} and \ref{['eq:Sr_def']}]. $S^M(E)$ is the scattering matrix of Majorana modes across the wire [see Eq. \ref{['eq:Sm_def']}].
• Figure 4: Nonlocal differential conductance $G_{RL}(V)$ of a clean wire at (panels a) and away from (panel b) the critical point at zero temperature. The curves are produced with the help of Eqs. \ref{['eq:G_RL_low_bias']}, \ref{['eq:clean_crit_PE']}, \ref{['eq:FM']}, and \ref{['eq:FM_above-the-gap']}, in which we take $T_{\rm em}^L(0) = T^R_{\rm me}(0) = 0.4$ and $\Sigma_R = 50 \pi v / L$. In panel b, we choose $\Delta_0 = 2.5 \pi v / L$; $\Delta_0 > 0$ corresponds to the topological phase. The nonlocal conductance is an odd function of bias. At the critical point, it has a sequence of Fabry-Pérot peaks with a spacing $e\delta V = \pi v / L$. Tuning the wire away from the critical point leads to opening of the "transport" in gap of magnitude $|\Delta_0| \propto |B - B_{\rm c}|$. Majorana zero modes are reflected in $G_{RL}(V)$ as subgap resonances (inset of panel b). To make the resonances in $G_{RL}(V)$ well-resolved, in the inset we take $T_{\rm em}^L(0) = T^R_{\rm me}(0) = 10^{-3}$. In contrast to the resonance in the local conductance [see Fig. \ref{['fig:local']} in Appenidx \ref{['sec:local']}], the resonances in $G_{RL}(V)$ are exponentially small in the system size, cf. Eqs. \ref{['eq:splitting']} and \ref{['eq:resonant']}.
• Figure 5: Schematic plot of the bias dependence of the nonlocal conductance at and away from the critical point. At $B = B_{\rm c}$, $G_{RL}(V)$ has a low-bias peak at $V = E_L / e$ [see Eq. \ref{['eq:EL_scale']}]. Both the position and the height of the peak scale exponentially with the length of the wire, $\propto \exp(-c_L L / v\tau)$. Tuning magnetic field away from the the critical point reduces the conductance magnitude. On the trivial side of the transition, the peak in $G_{RL}(V)$ disappears entirely. On the topological side, the peak remains but its height is reduced compared to $B = B_{\rm c}$ value. Outside the critical region, the peak height is $\propto \exp(-L \Delta_0 / v)$, see Eq. \ref{['eq:splitting']}. The conductance at negative biases satisfies $G_{RL}(-V) = - G_{RL}(V)$.