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

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This research delves into the complex interplay between topology, superconductivity, interactions, and disorder in a specific condensed matter system: two parallel helical edge channels hosting interacting electrons. The core finding is that realistic imperfections—specifically pairing asymmetries, interaction asymmetries, and random spin-flip terms arising from magnetic disorder or coexisting charge disorder and external magnetic fields—substantially enrich the physics of these proximitized double helical liquids.

Here is a detailed breakdown of the key physical phenomena, methodologies, and conclusions derived from the analysis:

1. Core Topological Features in Clean Limit

In an idealized, clean limit, when these two parallel helical edge channels are proximitized by local and nonlocal pairings, they are predicted to host time-reversal-invariant pairs of topological zero modes at the system corners.

2. The Role of Imperfections: Disorder as a Tuning Knob

The central thesis of the work is that disorder is not merely a detrimental perturbation but acts as a crucial tuning knob that qualitatively reshapes the topological superconductivity in these imperfect helical liquids. The analysis explicitly includes:

• Pairing Asymmetries: Differences in local versus nonlocal pairing between the two channels.

• Interaction Asymmetries: Differences in Coulomb interactions between the two channels.

• Random Spin-Flip Terms: These terms originate from magnetic disorder or from coexisting charge disorder and external magnetic fields.

3. Mechanism for New Topological Phases (Nmzm)

The introduction of these asymmetries and disorder drives the system into new topological regimes characterized by a topological invariant, denoted as N mzm (the number of Majorana zero modes).

• Detuning Zero Modes: The presence of backscattering terms, particularly spin-flip backscattering, is critical. It breaks the degeneracy between the two zero-mode conditions that exist in the clean limit. This detuning leads to a new topological regime where only a single Majorana zero mode is satisfied (Nmzm = 1), consistent with symmetry class BDI in the presence of spin-flip terms.

• Emergence of New Phases: The analysis demonstrates that all three types of asymmetry—pairing, interaction, and disorder—collectively lead to the emergence of additional topological phases characterized by different values of N mzm.

4. Cascade and Revival Phenomena

The most striking dynamic behavior observed is the existence of a cascade of successive topological phase transitions. This cascade is driven by tuning parameters such as interaction strength or backscattering strength.

• Suppression and Revival: At moderately strong interactions, pairing asymmetry can suppress the Nmzm = 2 phase, leading to additional topological phases and resulting in a revival of a Majorana zero mode.

• Controllable Transitions: This cascade is accessible by controlling the electrical screening effect (e.g., via controlling screened Coulomb interactions) or by tuning backscattering strength.

5. Experimental Observability

The theoretical predictions are linked to observable signatures in experimental techniques:

• Scanning Tunneling Microscopy (STM): Channel-resolved density profiles are analyzed to reveal spatial structures of the zero modes and bulk gap closing, providing features that can be directly observed via STM measurements.

• Local Probes: The Majorana zero modes themselves can be directly probed using scanning tunneling spectroscopy with high energy resolution, similar to techniques used in iron-based superconductors, by detecting zerobias bound states.

• Density Profiles: The spatial localization of the zero modes constitutes an alternative STM feature, with pronounced broadening indicating proximity to a topological phase transition.

6. Key Contrasts and Insights

The research offers several counter-intuitive insights:

• Disorder's Dual Role: Disorder is shown not only to be detrimental but also to provide a mechanism for tuning the system into specific topological windows, allowing for the selective activation of one branch of the Majorana zero mode criterion.

• Channel Localization vs. Helical Liquid: Pairing imbalance can lift a system out of a fully insulating regime (which occurs in symmetric cases) and restore metallic behavior in one channel while leaving the other as a helical liquid.

• Fragility: The study notes that nonlocal superconductivity can be fragile when pairing asymmetry is present, suggesting that reducing c(0) or increasing K+(0) shifts the balance toward local pairing and expands asymmetric regions.

Conclusion

In summary, this work establishes a robust framework demonstrating that imperfect double helical liquids are far richer topological systems than their clean counterparts. By systematically analyzing how pairing asymmetry, interaction asymmetry, and random spin-flip disorder collectively influence the system—often through renormalization group analysis—the authors prove that these imperfections enable the emergence of multiple topological phases (Nmzm states), drive cascades of zero mode transitions, and provide a concrete pathway for experimentally verifying disorder-enabled topological windows in future device platforms. The findings suggest that controlled spin-flip backscattering can serve as a highly tunable platform for realizing interacting, disordered topological electronics.

1. The AI can accurately predict topological phase transitions in imperfect double helical liquids by mapping system parameters to Z invariant values derived from renormalization group analysis, as shown in Section V B. This allows for the identification of additional topological phases supporting a single Majorana zero mode per corner.

2. The system can be tuned to exhibit a revival of Majorana zero modes and cascades of successive topological transitions through topological phases characterized by a Z invariant by controlling the electrical screening effect, as described in Section II and Section V B.

3. The AI can perform channel-resolved density profile analysis, which reveals observable features through scanning tunneling microscope by reconstructing wavefunctions from renormalized couplings to visualize how zero modes extend over longer distances and eventually vanish, as shown in Figure 9.

4. The system can be characterized by determining the topological phase diagram in three-dimensional parameter space by varying interaction strength, disorder effects, and pairing asymmetry, enabling the visualization of how the Nmzm = 1 phase emerges over a broad region of parameter space.

5. The AI can predict Majorana zero mode localization length and temperature in the insulating phase by solving the RG flow equations to determine when D˜1 and D˜2 both flow to strong coupling, yielding estimates for ξloc = a(l∗) ≈ a(0)e l∗ (Equation 12).

6. The AI can distinguish between different topological regimes by analyzing the analytic expression for the number of zero modes, Nmzm = X ε∈[+,–] Θ (−∆− + εB− + ∆+ + εB+) × Θ(p[∆− + εB−] squared + [∆c(l∗)] squared - ∆+ + εB+) (Equation 18).

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 now belongs 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 successive topological transitions through topological phases characterized by a Z invariant, which are accessible through controlling the electrical screening effect. We also analyze the spatial structure of the zero modes and the bulk gap closing through channel-resolved density profiles, revealing observable features through scanning tunneling microscope. 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.