Proximitized Double Helical Liquid

• Proximitized double helical liquids are systems of two helical Luttinger channels coupled to a superconducting or magnetic environment, exhibiting spin-momentum locking and exotic pairing effects.
• The analysis employs bosonization, coupled-channel theories, and lattice discretization techniques to capture helical kinematics, local/nonlocal pairing, and interaction-driven renormalization.
• Key implications include the emergence of Majorana zero modes, robust spin-to-charge conversion, and tunable transitions among distinct topological superconducting phases.

A proximitized double helical liquid is a system of two helical one-dimensional channels—typically two helical Luttinger liquids—coupled to a gapped environment such as an $s$-wave superconductor or a magnetic insulator. In the concrete device geometries discussed in the literature, it appears as two spatially separated helical edges of a quantum spin Hall bar, with one edge proximitized into a topological Josephson junction and the other coupled through a quantum point contact as a spin-sensitive probe (Calzona et al., 2021), or as two parallel interacting helical channels with both local and nonlocal pairing terms and random spin-flip perturbations (Ohorodnyk et al., 11 Dec 2025). Single-channel helical Luttinger liquid constructions, including bosonized field theories, experimentally tunable interaction parameters, and symmetry-preserving space-time lattice formulations, supply the building blocks for the double-channel proximitized problem (Zakharov et al., 2024, Jia et al., 2022).

1. Channel structure and helical kinematics

In the quantum spin Hall realization, the double helical structure is furnished by the two edges of a two-dimensional topological insulator bar. The upper edge has negative helicity, with spin-up left movers and spin-down right movers, while the lower edge has positive helicity, with spin-up right movers and spin-down left movers (Calzona et al., 2021). Both edges are described by a linearized helical Hamiltonian

$$H_0 = v \sum_{\sigma=\uparrow,\downarrow}\sum_{r=R,L} \int dx\; \psi^\dagger_{r\sigma}(x)(-ir\partial_x)\psi_{r\sigma}(x),$$

with spin-momentum locking encoded by the edge-dependent assignment of $(r,\sigma)$ (Calzona et al., 2021).

This helical structure is not restricted to quantum spin Hall bars. In topological-insulator nanowires pierced by a half-integer magnetic flux, a single gapless helical channel emerges when $|\mu|<\Delta_s=v_2/R$, while all other subbands remain gapped (Egger et al., 2010). In a different synthetic route, a quantum wire with uniform Dresselhaus coupling and spatially periodic Rashba coupling can support two gapless helical branches when the commensurability condition that gaps one branch is not satisfied; in that regime the low-energy sector is effectively a double helical liquid with two Kramers pairs (Japaridze et al., 2013).

The basic interacting building block is the helical Luttinger liquid. In bosonized form, a single channel is described by

$$H_{\rm hLL}=\frac{v}{2}\int dz\left[K(\partial_z\varphi)^2+K^{-1}(\partial_z\theta)^2\right],$$

or, for two channels $n=1,2$,

$$H_{\rm el}=\sum_{n=1,2}\int \frac{\hbar\,dr}{2\pi}\left[u_nK_n(\partial_r\theta_n)^2+\frac{u_n}{K_n}(\partial_r\phi_n)^2\right],$$

with $K_n$ the Luttinger parameters and $u_n=v_F/(mK_n)$ in the fractional generalization (Egger et al., 2010, Ohorodnyk et al., 11 Dec 2025). Repulsive interactions correspond to $K_n<1$.

2. Proximity terms and effective superconducting structure

The minimal proximitized double-helical geometry combines helical kinematics with superconducting pairing. In the spectrometer device of a quantum spin Hall bar, only the upper edge is directly proximitized: two $$H_0 = v \sum_{\sigma=\uparrow,\downarrow}\sum_{r=R,L} \int dx\; \psi^\dagger_{r\sigma}(x)(-ir\partial_x)\psi_{r\sigma}(x),$$0-wave superconductors cover the upper edge, leaving a normal region of length $$H_0 = v \sum_{\sigma=\uparrow,\downarrow}\sum_{r=R,L} \int dx\; \psi^\dagger_{r\sigma}(x)(-ir\partial_x)\psi_{r\sigma}(x),$$1 that forms a topological Josephson junction, while the lower edge remains normal and couples to the junction through a quantum point contact at $$H_0 = v \sum_{\sigma=\uparrow,\downarrow}\sum_{r=R,L} \int dx\; \psi^\dagger_{r\sigma}(x)(-ir\partial_x)\psi_{r\sigma}(x),$$2 (Calzona et al., 2021). A common misconception is that both channels must be directly proximitized for the system to qualify as a proximitized double helical liquid; the device analyzed in (Calzona et al., 2021) shows that one proximitized helical channel plus a second helical probe channel already realizes the essential double-helical proximity architecture.

For a single helical edge with helicity $$H_0 = v \sum_{\sigma=\uparrow,\downarrow}\sum_{r=R,L} \int dx\; \psi^\dagger_{r\sigma}(x)(-ir\partial_x)\psi_{r\sigma}(x),$$3, the proximitized upper edge is described in Nambu space by the Bogoliubov–de Gennes Hamiltonian density

$$H_0 = v \sum_{\sigma=\uparrow,\downarrow}\sum_{r=R,L} \int dx\; \psi^\dagger_{r\sigma}(x)(-ir\partial_x)\psi_{r\sigma}(x),$$4

with $$H_0 = v \sum_{\sigma=\uparrow,\downarrow}\sum_{r=R,L} \int dx\; \psi^\dagger_{r\sigma}(x)(-ir\partial_x)\psi_{r\sigma}(x),$$5 in the normal weak link and $$H_0 = v \sum_{\sigma=\uparrow,\downarrow}\sum_{r=R,L} \int dx\; \psi^\dagger_{r\sigma}(x)(-ir\partial_x)\psi_{r\sigma}(x),$$6 in the superconducting regions (Calzona et al., 2021). Because of helicity, the induced $$H_0 = v \sum_{\sigma=\uparrow,\downarrow}\sum_{r=R,L} \int dx\; \psi^\dagger_{r\sigma}(x)(-ir\partial_x)\psi_{r\sigma}(x),$$7-wave pairing couples counterpropagating opposite spins and is effectively equivalent to $$H_0 = v \sum_{\sigma=\uparrow,\downarrow}\sum_{r=R,L} \int dx\; \psi^\dagger_{r\sigma}(x)(-ir\partial_x)\psi_{r\sigma}(x),$$8-wave pairing in the helical basis (Calzona et al., 2021).

For two interacting helical channels, proximity generates both local and nonlocal pairing terms,

$$H_0 = v \sum_{\sigma=\uparrow,\downarrow}\sum_{r=R,L} \int dx\; \psi^\dagger_{r\sigma}(x)(-ir\partial_x)\psi_{r\sigma}(x),$$9

and

$(r,\sigma)$0

where $(r,\sigma)$1 for integer quantum spin Hall edges and $(r,\sigma)$2 for fractional helical liquids (Ohorodnyk et al., 11 Dec 2025). The local term injects a Cooper pair into the same edge; the nonlocal term implements crossed Andreev pairing between the two channels (Ohorodnyk et al., 11 Dec 2025).

In the clean limit, the proximitized double-edge system is time-reversal invariant and can realize a class DIII topological superconductor when nonlocal pairing dominates (Ohorodnyk et al., 11 Dec 2025). This clean DIII regime supports Kramers pairs of Majorana zero modes at corners or ends (Ohorodnyk et al., 11 Dec 2025).

3. Interactions, local observables, and correlation structure

Interactions are not a perturbative detail but a defining control parameter. In monolayer $(r,\sigma)$3-WTe$(r,\sigma)$4, tunneling spectroscopy extracted $(r,\sigma)$5 from 69 locations along different edges and substrates, yielding $(r,\sigma)$6 for the Y-edge on bilayer graphene, $(r,\sigma)$7 for the X-edge on bilayer graphene, $(r,\sigma)$8 for the Y-edge on HOPG, and $(r,\sigma)$9 for the X-edge on HOPG (Jia et al., 2022). The same work identifies $|\mu|<\Delta_s=v_2/R$0 as a strongly interacting regime and $|\mu|<\Delta_s=v_2/R$1 as a very strongly interacting regime in which two-particle backscattering and more exotic correlated phases become important (Jia et al., 2022).

The helical liquid has distinctive local observables. In a weakly interacting helical Luttinger liquid, density-density correlation functions do not exhibit Friedel or Wigner oscillations, while spin-spin correlation functions are strongly anisotropic and reveal a planar spin wave (Ziani et al., 2015). Concretely, the absence of the usual $|\mu|<\Delta_s=v_2/R$2 density oscillation follows from spin-momentum locking: the bilinear $|\mu|<\Delta_s=v_2/R$3 contributes to transverse spin density rather than scalar charge density (Ziani et al., 2015). A localized impurity does not generate charge Friedel oscillations, and only magnetic impurities can pin the planar spin density wave (Ziani et al., 2015).

These single-channel facts remain important in the double-channel setting. A plausible implication is that proximitized double helical liquids inherit a competition between superconducting pinning of $|\mu|<\Delta_s=v_2/R$4-type fields and magnetic or backscattering pinning of $|\mu|<\Delta_s=v_2/R$5-type fields, but the detailed balance is channel- and coupling-dependent (Ziani et al., 2015). The strong-interaction thresholds measured in $|\mu|<\Delta_s=v_2/R$6-WTe$|\mu|<\Delta_s=v_2/R$7 therefore directly delimit whether a proximitized double-edge device lies in a pairing-dominated, backscattering-dominated, or fractional regime (Jia et al., 2022).

4. Lattice formulations and numerical building blocks

A central technical difficulty in helical-liquid numerics is the fermion-doubling obstruction: a local and symmetry-preserving discretization of a one-dimensional massless Dirac Hamiltonian produces a spurious second low-energy species, while a nonlocal discretization opens a gap at the Dirac point (Zakharov et al., 2024). The space-time lattice construction of the helical Luttinger liquid resolves this by discretizing both space and time. The tangent discretization,

$|\mu|<\Delta_s=v_2/R$8

produces a single Dirac point with dispersion

$|\mu|<\Delta_s=v_2/R$9

and, after a local field redefinition $$H_{\rm hLL}=\frac{v}{2}\int dz\left[K(\partial_z\varphi)^2+K^{-1}(\partial_z\theta)^2\right],$$0, yields a local Lagrangian with only nearest-neighbor hoppings in space-time (Zakharov et al., 2024).

The resulting Euclidean action admits a local Hubbard interaction and a sign-problem-free Hubbard–Stratonovich decoupling. For attractive $$H_{\rm hLL}=\frac{v}{2}\int dz\left[K(\partial_z\varphi)^2+K^{-1}(\partial_z\theta)^2\right],$$1, $$H_{\rm hLL}=\frac{v}{2}\int dz\left[K(\partial_z\varphi)^2+K^{-1}(\partial_z\theta)^2\right],$$2, and for repulsive $$H_{\rm hLL}=\frac{v}{2}\int dz\left[K(\partial_z\varphi)^2+K^{-1}(\partial_z\theta)^2\right],$$3, $$H_{\rm hLL}=\frac{v}{2}\int dz\left[K(\partial_z\varphi)^2+K^{-1}(\partial_z\theta)^2\right],$$4, so in both cases the Monte Carlo weight is nonnegative, $$H_{\rm hLL}=\frac{v}{2}\int dz\left[K(\partial_z\varphi)^2+K^{-1}(\partial_z\theta)^2\right],$$5 (Zakharov et al., 2024). The calculated propagator and transverse spin-spin correlator reproduce the expected helical Luttinger liquid scaling, with very good agreement with continuum bosonization up to $$H_{\rm hLL}=\frac{v}{2}\int dz\left[K(\partial_z\varphi)^2+K^{-1}(\partial_z\theta)^2\right],$$6 (Zakharov et al., 2024).

The same paper does not explicitly construct a proximitized double helical liquid, include superconducting pairing, or simulate multiple helical channels (Zakharov et al., 2024). A plausible implication is that its local, symmetry-preserving single-channel lattice action can be duplicated and augmented with inter-channel tunneling, density couplings, or Bogoliubov–de Gennes pairing kernels to build a numerically tractable double-channel proximitized theory, but that extension is not carried out in the work itself (Zakharov et al., 2024).

5. Spectroscopy, Andreev structure, and transport

In a helical Josephson junction, repeated Andreev reflection generates helical Andreev bound states (hABSs). Because of spin-momentum locking and the absence of elastic backscattering in a time-reversal-symmetric helical channel, a right-moving electron with spin $$H_{\rm hLL}=\frac{v}{2}\int dz\left[K(\partial_z\varphi)^2+K^{-1}(\partial_z\theta)^2\right],$$7 can only be Andreev-reflected as a left-moving hole with spin $$H_{\rm hLL}=\frac{v}{2}\int dz\left[K(\partial_z\varphi)^2+K^{-1}(\partial_z\theta)^2\right],$$8, and vice versa (Calzona et al., 2021). The bound-state condition is

$$H_{\rm hLL}=\frac{v}{2}\int dz\left[K(\partial_z\varphi)^2+K^{-1}(\partial_z\theta)^2\right],$$9

with $n=1,2$0 labeling the hABS class and $n=1,2$1 the superconducting phase difference (Calzona et al., 2021). For symmetric junctions, the two hABS branches display a protected zero-energy crossing at $n=1,2$2, where they form a Kramers pair of Majorana zero modes (Calzona et al., 2021).

The lower helical edge in the double-edge spectrometer converts spin information into charge transport. The quantum point contact couples upper and lower edges through

$n=1,2$3

with $n=1,2$4 the spin-preserving tunneling amplitude and $n=1,2$5 the spin-flipping amplitude (Calzona et al., 2021). In the ideal spin-preserving limit $n=1,2$6, the nonlocal conductances $n=1,2$7 and $n=1,2$8 probe opposite hABS classes, and $n=1,2$9 by time reversal (Calzona et al., 2021). When $$H_{\rm el}=\sum_{n=1,2}\int \frac{\hbar\,dr}{2\pi}\left[u_nK_n(\partial_r\theta_n)^2+\frac{u_n}{K_n}(\partial_r\phi_n)^2\right],$$0, the asymmetry

$$H_{\rm el}=\sum_{n=1,2}\int \frac{\hbar\,dr}{2\pi}\left[u_nK_n(\partial_r\theta_n)^2+\frac{u_n}{K_n}(\partial_r\phi_n)^2\right],$$1

remains finite over a broad range and quantifies robust spin-to-charge conversion (Calzona et al., 2021). A second diagnostic is

$$H_{\rm el}=\sum_{n=1,2}\int \frac{\hbar\,dr}{2\pi}\left[u_nK_n(\partial_r\theta_n)^2+\frac{u_n}{K_n}(\partial_r\phi_n)^2\right],$$2

which satisfies $$H_{\rm el}=\sum_{n=1,2}\int \frac{\hbar\,dr}{2\pi}\left[u_nK_n(\partial_r\theta_n)^2+\frac{u_n}{K_n}(\partial_r\phi_n)^2\right],$$3 for pure hABSs with only perfect Andreev reflection and generally becomes nonzero when electronic backscattering contaminates the subgap states (Calzona et al., 2021).

A related transport realization couples a helical Luttinger liquid to helical Majorana modes through single or double quantum point contacts (Chao et al., 2015). For a single contact, the differential conductance shows a series of peaks due to Andreev reflection of electrons in the Majorana modes, while stronger repulsive interactions suppress the peaks and eventually render the point contact insulating above a critical interaction strength (Chao et al., 2015). For a double contact, interference between the two tunneling paths produces nonuniversal peak reshaping and distance-dependent modulations; for small separation the overall structure remains similar to the single-contact case (Chao et al., 2015).

6. Imperfections, symmetry class changes, and topological phase structure

Realistic proximitized double helical liquids generically contain channel asymmetries and spin-flip disorder. The relevant perturbations include pairing asymmetry $$H_{\rm el}=\sum_{n=1,2}\int \frac{\hbar\,dr}{2\pi}\left[u_nK_n(\partial_r\theta_n)^2+\frac{u_n}{K_n}(\partial_r\phi_n)^2\right],$$4, interaction asymmetry $$H_{\rm el}=\sum_{n=1,2}\int \frac{\hbar\,dr}{2\pi}\left[u_nK_n(\partial_r\theta_n)^2+\frac{u_n}{K_n}(\partial_r\phi_n)^2\right],$$5, disorder asymmetry $$H_{\rm el}=\sum_{n=1,2}\int \frac{\hbar\,dr}{2\pi}\left[u_nK_n(\partial_r\theta_n)^2+\frac{u_n}{K_n}(\partial_r\phi_n)^2\right],$$6, and random spin-flip backscattering

$$H_{\rm el}=\sum_{n=1,2}\int \frac{\hbar\,dr}{2\pi}\left[u_nK_n(\partial_r\theta_n)^2+\frac{u_n}{K_n}(\partial_r\phi_n)^2\right],$$7

which can arise either from magnetic impurities or from the combination of charge disorder and an external transverse magnetic field (Ohorodnyk et al., 11 Dec 2025).

The one-loop RG equations for the dimensionless local pairing $$H_{\rm el}=\sum_{n=1,2}\int \frac{\hbar\,dr}{2\pi}\left[u_nK_n(\partial_r\theta_n)^2+\frac{u_n}{K_n}(\partial_r\phi_n)^2\right],$$8, nonlocal pairing $$H_{\rm el}=\sum_{n=1,2}\int \frac{\hbar\,dr}{2\pi}\left[u_nK_n(\partial_r\theta_n)^2+\frac{u_n}{K_n}(\partial_r\phi_n)^2\right],$$9, disorder strength $K_n$0, and Luttinger parameters $K_n$1 are

$K_n$2

$K_n$3

with $K_n$4 flowing as $K_n$5 (Ohorodnyk et al., 11 Dec 2025). These flows determine whether the low-energy phase is nonlocal superconducting, locally superconducting on one channel, mixed, insulating, or gapless double-helical (Ohorodnyk et al., 11 Dec 2025).

The topological classification changes qualitatively in the presence of spin-flip terms. The clean system is DIII-like and supports only the $K_n$6 distinction between trivial and Kramers-paired Majorana phases (Ohorodnyk et al., 11 Dec 2025). With effective spin-flip fields $K_n$7, the renormalized Bogoliubov–de Gennes theory belongs to class BDI with a $K_n$8 invariant, allowing 0, 1, or 2 Majorana zero modes per corner (Ohorodnyk et al., 11 Dec 2025). The analytic criterion is that, for each $K_n$9, one zero mode appears when

$u_n=v_F/(mK_n)$0

with all couplings understood as renormalized at the RG stopping scale (Ohorodnyk et al., 11 Dec 2025). In the clean symmetric limit this reduces to $u_n=v_F/(mK_n)$1, yielding either 0 or 2 Majoranas; once spin-flip disorder detunes the two $u_n=v_F/(mK_n)$2 branches, an intermediate phase with a single Majorana per corner becomes possible (Ohorodnyk et al., 11 Dec 2025).

This leads to a cascade of topological phase transitions and to a revival of zero modes as screening, disorder, or asymmetry is tuned (Ohorodnyk et al., 11 Dec 2025). A second common misconception is that disorder is merely destructive. In these imperfect proximitized double helical liquids, disorder can also be a control parameter: it can drive transitions from a trivial phase to a phase with one Majorana per corner, and it can reopen topological regimes after an apparently trivial interval (Ohorodnyk et al., 11 Dec 2025). Candidate material platforms include HgTe and InAs/GaSb quantum wells, monolayer quantum spin Hall materials such as WTe$u_n=v_F/(mK_n)$3, bismuthene, and TaIrTe$u_n=v_F/(mK_n)$4, and moiré quantum spin Hall systems including twisted WSe$u_n=v_F/(mK_n)$5 and MoTe$u_n=v_F/(mK_n)$6 (Ohorodnyk et al., 11 Dec 2025).