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

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. We also analyze the spatial structure of the zero modes and the bulk gap closing through channel-resolved density profiles. 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.

💡 Research Summary

The authors investigate a realistic platform consisting of two parallel helical edge channels (a double helical liquid) that host interacting electrons and are proximitized by both local (intra‑channel) and non‑local (inter‑channel) superconducting pairings. While idealized models assume perfect symmetry between the two channels, real devices inevitably exhibit several imperfections: (i) different Luttinger parameters K₁≠K₂ due to unequal Coulomb screening, (ii) asymmetric local pairing amplitudes Δ₁≠Δ₂ because of non‑uniform proximity effect, and (iii) random spin‑flip back‑scattering arising from magnetic impurities, charge disorder combined with in‑plane magnetic fields, or other sources of magnetic disorder.

Using bosonization, the authors write the low‑energy Hamiltonian in terms of dual bosonic fields φₙ and θₙ (n=1,2). The Hamiltonian comprises four parts: (1) Tomonaga‑Luttinger liquid kinetic terms, (2) intra‑channel s‑wave pairing Δₙ, (3) inter‑channel crossed‑Andreev pairing Δ_c that splits a Cooper pair between the two edges, and (4) a random spin‑flip back‑scattering term V_{rs,n}(r). After disorder averaging via the replica method, they obtain an effective Euclidean action containing cosine operators for the pairings and a cosine‑cosine term for the disorder.

They introduce dimensionless couplings ˜Δₙ = Δₙ a/(ℏuₙ), ˜Δ_c = Δ_c a/(ℏ√(u₁u₂)), and ˜Dₙ = 2a²V_n²π/(ℏ²uₙ²). The one‑loop renormalization‑group (RG) flow equations (Eqs. 11a‑11e) show that the scaling dimensions depend on the Luttinger parameters and the fractional index m (m=1 for ordinary helical edges, m>1 for fractional counterparts). In particular:

• The non‑local pairing has the smallest scaling dimension and is therefore the most relevant perturbation in a wide parameter range; it tends to flow to strong coupling first, establishing a “×SC” phase where the crossed‑Andreev process dominates.
• Local pairings flow with exponent (2‑mKₙ); if Kₙ is small (strong repulsion) they become less relevant, but any initial asymmetry Δ₁≠Δ₂ is amplified during the flow.
• Spin‑flip back‑scattering scales as (3‑2mKₙ); for sufficiently strong repulsion (Kₙ<3/(2m)) the disorder becomes relevant and can grow to dominate over the pairings.

Numerical integration of the RG equations (Figs. 2‑4) reveals several distinct regimes:

1. Symmetric case (Δ₁=Δ₂, K₁=K₂, D₁=D₂): The non‑local pairing wins, the system enters a DIII‑class topological superconducting phase with a two‑fold Kramers‑pair of Majorana zero modes (MZMs) localized at each corner.

2. Pairing asymmetry (Δ₁≠Δ₂) with weak disorder: The stronger intra‑channel pairing (say Δ₁) reaches strong coupling before Δ₂, producing a mixed phase where channel 1 is locally superconducting while channel 2 is still governed by the crossed‑Andreev term. The competition splits the degeneracy of the two MZMs, effectively leaving a single MZM per corner—a transition from class DIII to class BDI.

3. Coulomb asymmetry (K₁≠K₂): Different interaction strengths modify the RG exponents of the disorder and pairing terms. When one channel is more strongly interacting (smaller K), disorder in that channel becomes more relevant, further favoring the BDI phase.

4. Strong disorder (large ˜D₊): Even if the clean system would be topologically trivial, a sufficiently large spin‑flip back‑scattering can suppress one of the two MZMs and drive the system into the BDI regime. Thus disorder acts as a tuning knob rather than a purely detrimental factor.

5. Cascade of Z‑indexed phases: By simultaneously tuning the screening (hence Kₙ) and the ratio Δ₋/Δ₊, the RG flow can pass through a sequence of fixed points characterized by an integer topological invariant ℤ. Each step corresponds to the emergence or annihilation of an additional MZM (or parafermionic zero mode for m>1) at the corners. The authors term this a “cascade of topological phase transitions.”

The paper also presents channel‑resolved density profiles obtained from solving the Bogoliubov‑de Gennes equations with the renormalized couplings. These profiles show how the bulk gap closes locally in one channel while remaining open in the other during a transition, providing experimentally accessible signatures for scanning‑probe techniques (STM, spin‑polarized ARPES).

In the discussion, the authors emphasize that realistic imperfections—pairing asymmetry, interaction asymmetry, and magnetic disorder—can be harnessed to engineer richer topological phase diagrams than previously recognized. The ability to switch between DIII (Kramers‑pair MZMs), BDI (single MZM per corner), and higher‑ℤ phases without applying external magnetic fields opens new routes for Majorana‑based quantum computation and for realizing parafermionic excitations in fractional helical liquids. They outline possible material platforms (twisted bilayer graphene, moiré transition‑metal dichalcogenides, quantum spin Hall insulators) where the required gate‑tunable screening and proximity‑induced pairing asymmetries are experimentally feasible.

Overall, the work demonstrates that disorder and asymmetry are not merely obstacles but powerful control parameters that reshape the topological superconductivity of double helical liquids, enabling electrically tunable cascades of topological phases and the revival of Majorana zero modes in imperfect, realistic devices.