Spin responses of a disordered helical superconducting edge under Zeeman field

TL;DR

This work analyzes spin transport and correlation functions on a partially mixed helical edge of a 2D topological insulator under Zeeman fields and proximity-induced superconductivity, incorporating both single-impurity and disorder effects. Using bosonization, RG, and memory-function formalisms, it shows that the Zeeman field tunes the competition between superconductivity and impurity scattering via the renormalized Luttinger parameter $K$, with attractive interactions enhancing superconducting gaps and repulsive interactions amplifying impurity effects. The authors derive explicit scaling relations for spin conductance corrections, provide analytic forms for density and pairing correlations in the relevant regime, and reveal logarithmic corrections to these correlations due to marginal disorder, highlighting distinct signatures in spin transport and pairing stability. Overall, the results elucidate how disorder and Zeeman-induced symmetry breaking modify the low-energy physics of topological edge channels, offering experimentally accessible fingerprints of the interplay between disorder and superconductivity in helical edge systems.

Abstract

We investigate analytically and numerically the effects of disorder on the helical edge of the 2D topological insulator in the presence of the Zeeman field and superconductivity. Employing bosonization and a renormalization-group analysis, we study how impurity potentials modify charge- and spin-density wave correlations as well as superconducting pair correlations. Our results reveal that the Zeeman field controls the competition: in the attractive regime, it amplifies the superconducting gap, while in the repulsive regime, it stabilizes impurity effects by keeping the system longer in the relevant regime for disorder. We also find that disorder induces logarithmic suppression of transverse density-wave correlations, while at the same time introducing positive logarithmic corrections that enhance superconducting pair correlations and contribute to their stability. These effects directly modify the scaling of spin conductance, providing experimentally accessible signatures of the interplay between disorder and superconductivity in topological edge channels.

Figures (7)

• Figure 1: (Color online) Schematic of a superconducting PMH edge in the presence of insulator impurities. We look for the effect of a single impurity placed at $x=0$ represented in yellow color.
• Figure 2: (Color online) Luttinger parameter as function the Zeeman gap and interactions in the regimes (a) repulsive and (b) attractive. Here $g=\frac{g_{fb}}{8\pi}$ indicates interactions effects.
• Figure 3: (Color online) Flow trajectories and relevant regimes diagram in the disordered super-PMH edge.
• Figure 4: (Color online) Schematic of the superconducting partial mixed helical edge in the presence of many impurities.
• Figure 5: (Color online) Temperature dependence of the conductance correction in a super-PMH edge for clean (dash–dotted) and disordered (solid) cases in the interaction regimes (a) of attraction and (b) of repulsion, for Zeeman strengths of 4, 5, and 6, a superconductivity gap of 2, and disorder coefficient of 0.002.