Cascade of topological phase transitions and revival of topological zero modes in imperfect double helical liquids

TL;DR

This work analyzes two proximitized, interacting helical edge channels subject to local and nonlocal pairing and random spin-flip backscattering. Using bosonization and renormalization-group methods, it derives flow equations for pairing and backscattering, and connects these flows to transport and topological properties by mapping to an effective single-particle model and a Z-invariant Majorana-counting criterion. The results reveal that realistic imperfections can induce new topological phases, including a single Majorana mode per corner (N_mzm = 1) via detuning of Majorana conditions, and can trigger cascades of topological transitions with a disorder-tunable landscape; there is also a disorder-enabled revival of Majorana zero modes and imperfection-induced topological windows. These findings show that imperfections can serve as practical tuning knobs to realize and control Majorana physics in proximitized double helical liquids, with implications for experiments on quantum spin Hall edges and fractional variants.

Abstract

Two parallel helical edge channels hosting interacting electrons, when proximitized by local and nonlocal pairings, can host time-reversal-invariant pairs of topological zero modes at the system corners. Here we show that realistic imperfections substantially enrich the physics of such proximitized double helical liquids. Specifically, we analyze this platform and its fractional counterparts in the presence of pairing and interaction asymmetries between the two channels, as well as random spin-flip terms arising from either magnetic disorder or coexisting charge disorder and external magnetic fields. Using renormalization-group analysis, we determine how Coulomb interactions, pairings, and magnetic disorder collectively influence the transport behavior and topological properties of the double helical liquid. As the system transitions from class DIII to class BDI, an additional topological phase supporting a single Majorana zero mode per corner emerges. We further show how additional pairing or Coulomb asymmetry influences the stability of various topological phases and uncovers a revival of Majorana zero modes and cascades of transitions through topological phases characterized by a $\mathbb {Z}$ invariant, which are accessible through controlling the electrical screening effect. In contrast to conventional understanding, disorder is not merely detrimental, as it in general allows for a tuning knob that qualitatively reshapes the topological superconductivity in imperfect helical liquids.

Figures (19)

• Figure 1: Schematic of a proximitized double helical liquid consisting of two parallel edge channels of time-reversal-invariant topological insulators (blue), separated by a distance $d$. The channels are in contact with an $s$-wave superconductor (orange). (a) Possible realization based on twisted bilayer structures. (b) Side view of the setup along the channel coordinate $r$. Both local and nonlocal pairings are induced in the edge region $r\in[0,L]$, and only local pairing occurs for $r<0$ and $r>L$. The edges are subject to spin-flip backscatterings with strength $V_{{\rm rs}, n}$.
• Figure 2: RG flow and renormalized coupling strengths with $\tilde{\Delta}_+(0) =0.03$, $\tilde{\Delta}_-(0) = 0.01$, $\tilde{\Delta}_c(0)=0.01$ and $\tilde{D}_-(0) =K_-(0) =0$. (a) RG Flow for the initial parameter set $P$ with $\tilde{D}_+(0) = 10^{-6}$ and $K_{+} (0) = 0.57$. The label $l^*$ marks the cutoff scale where the flow stops. (b--f) Color maps of the renormalized couplings in the $[K_+(0),\tilde{D}_+(0)]$ plane for (b--d) the pairing strengths $\tilde{\Delta}_{n}(l^*)$ and $\tilde{\Delta}_{c}(l^*)$ and (e--f) the backscattering strengths $D_{n}(l^*)$. The marked dot $P$ corresponds to the parameter set used in Panel (a). See Table \ref{['Table:Parameters']} for the complete set of the adopted parameter values.
• Figure 3: Similar plots to Fig. \ref{['fig:pointPflow']} but with color maps in the $[ \Delta_-(0)/ \Delta_+(0),\,\tilde{D}_+(0) ]$ plane, $\tilde{\Delta}_+(0) =0.03$ and $\tilde{\Delta}_-(0) = 0.015$. For the parameter set $P'$, we additionally set $\tilde{D}_+(0) = 10^{-5}$ and $K_{+} (0) = 0.7$. See Table \ref{['Table:Parameters']} for the complete set of the adopted parameter values.
• Figure 4: Phase diagrams based on the transport properties in the $[K_+(0),\,\tilde{D}_+(0)]$ plane. We consider (a) the symmetric-channel case with $\tilde{\Delta}_-(0) = K_-(0) =0$, (b) the pairing asymmetry case with $\tilde{\Delta}_-(0) = 0.01$, and (c) the Coulomb asymmetry case with $K_-(0)=0.05$. We identify the dominant (and subdominant) phases among local superconductivity in channel $n \in \{1,2\}$ (labeled as "SC $n$"), nonlocal superconductivity ($\times$ SC), insulating channel $n$ (Ins. $n$), helical liquid in channel $n$ (HL $n$), double helical liquid (DH), fully insulating channels (Ins.), and superconductivity with comparable local and nonlocal pairings (all SC). See Table \ref{['Table:Parameters']} for the complete set of the adopted parameter values.
• Figure 5: Phase diagrams similar to Fig. \ref{['fig:PD_transport1']} but in the $[ \Delta_-(0)/ \Delta_+(0),\,\tilde{D}_+(0) ]$ plane. The adopted parameter values include (a) $\tilde{\Delta}_c(0)=0.01$ and $K_+(0)=0.6$, (b) $\tilde{\Delta}_c(0) = 0.005$ and $K_+(0)=0.6$, and (c) $\tilde{\Delta}_c(0)=0.01$ and $K_+(0)=0.7$. See Table \ref{['Table:Parameters']} for the complete set of the adopted parameter values.