---
title: Proximitized Double Helical Liquid
url: https://www.emergentmind.com/topics/proximitized-double-helical-liquid
type: topic
---

# Proximitized Double Helical Liquid

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 [2111.07696], or as two parallel interacting helical channels with both local and nonlocal pairing terms and random spin-flip perturbations [2512.10335]. 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 [2401.10828], [2209.10422].

## 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 [2111.07696]. 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 \((r,\sigma)\) [2111.07696].

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 [1006.4496]. 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 [1311.4716].

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 [1006.4496], [2512.10335]. 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 \(s\)-wave superconductors cover the upper edge, leaving a normal region of length \(D\) that forms a topological Josephson junction, while the lower edge remains normal and couples to the junction through a quantum point contact at \(x=\bar x\) [2111.07696]. 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 [2111.07696] 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 \(\zeta=\pm1\), the proximitized upper edge is described in Nambu space by the Bogoliubov–de Gennes Hamiltonian density
\[
\mathcal H_{\rm BdG}^{\zeta}(x)=
\begin{pmatrix}
\zeta p & 0 & 0 & \Delta(x)e^{-i\chi}\\
0 & -\zeta p & -\Delta(x)e^{-i\chi} & 0\\
0 & -\Delta(x)e^{i\chi} & \zeta p & 0\\
\Delta(x)e^{i\chi} & 0 & 0 & -\zeta p
\end{pmatrix},
\qquad p=-i\partial_x,
\]
with \(\Delta(x)=0\) in the normal weak link and \(\Delta(x)=\Delta_{L,R}\) in the superconducting regions [2111.07696]. Because of helicity, the induced \(s\)-wave pairing couples counterpropagating opposite spins and is effectively equivalent to \(p\)-wave pairing in the helical basis [2111.07696].

For two interacting helical channels, proximity generates both local and nonlocal pairing terms,
\[
H_{\rm s}=\sum_{n=1,2}\frac{\Delta_n}{\pi a}\int dr\,\cos[2m\theta_n(r)],
\]
and
\[
H_{\times}=\frac{2\Delta_c}{\pi a}\int dr\,
\cos[m(\theta_1+\theta_2)]\,\cos[m(\phi_1-\phi_2)],
\]
where \(m=1\) for integer quantum spin Hall edges and \(m>1\) for fractional helical liquids [2512.10335]. The local term injects a Cooper pair into the same edge; the nonlocal term implements crossed Andreev pairing between the two channels [2512.10335].

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 [2512.10335]. This clean DIII regime supports Kramers pairs of Majorana zero modes at corners or ends [2512.10335].

## 3. Interactions, local observables, and correlation structure

Interactions are not a perturbative detail but a defining control parameter. In monolayer \(1T'\)-WTe\(_2\), tunneling spectroscopy extracted \(K\) from 69 locations along different edges and substrates, yielding \(K = 0.27 \pm 0.01\) for the Y-edge on bilayer graphene, \(K = 0.21 \pm 0.01\) for the X-edge on bilayer graphene, \(K = 0.33 \pm 0.01\) for the Y-edge on HOPG, and \(K = 0.24 \pm 0.01\) for the X-edge on HOPG [2209.10422]. The same work identifies \(K<1/2\) as a strongly interacting regime and \(K<1/4\) as a very strongly interacting regime in which two-particle backscattering and more exotic correlated phases become important [2209.10422].

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 [1511.03157]. Concretely, the absence of the usual \(2k_F\) density oscillation follows from spin-momentum locking: the bilinear \(\psi_+^\dagger\psi_-\) contributes to transverse spin density rather than scalar charge density [1511.03157]. A localized impurity does not generate charge Friedel oscillations, and only magnetic impurities can pin the planar spin density wave [1511.03157].

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 \(\theta\)-type fields and magnetic or backscattering pinning of \(\phi\)-type fields, but the detailed balance is channel- and coupling-dependent [1511.03157]. The strong-interaction thresholds measured in \(1T'\)-WTe\(_2\) therefore directly delimit whether a proximitized double-edge device lies in a pairing-dominated, backscattering-dominated, or fractional regime [2209.10422].

## 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 [2401.10828]. The space-time lattice construction of the helical Luttinger liquid resolves this by discretizing both space and time. The tangent discretization,
\[
i\partial_t \mapsto \frac{2}{\tau}\tan(\hat\omega\tau/2),\qquad
i\partial_x \mapsto -\frac{2}{a}\tan(\hat k a/2),
\]
produces a single Dirac point with dispersion
\[
\tan(\omega\tau/2)=\sigma\gamma\,\tan(ka/2),
\]
and, after a local field redefinition \(\psi_\sigma=\hat D\phi_\sigma\), yields a local Lagrangian with only nearest-neighbor hoppings in space-time [2401.10828].

The resulting Euclidean action admits a local Hubbard interaction and a sign-problem-free Hubbard–Stratonovich decoupling. For attractive \(U<0\), \(M_-^*=M_+\), and for repulsive \(U>0\), \(M_-^\dagger=-M_+\), so in both cases the Monte Carlo weight is nonnegative, \(P[s]=|\det M_+|^2\ge 0\) [2401.10828]. The calculated propagator and transverse spin-spin correlator reproduce the expected helical Luttinger liquid scaling, with very good agreement with continuum bosonization up to \(|\kappa|\lesssim 0.4\) [2401.10828].

The same paper does not explicitly construct a proximitized double helical liquid, include superconducting pairing, or simulate multiple helical channels [2401.10828]. 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 [2401.10828].

## 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 \(\sigma\) can only be Andreev-reflected as a left-moving hole with spin \(-\sigma\), and vice versa [2111.07696]. The bound-state condition is
\[
-\exp\!\big[i(\nu\chi+\Phi_R(E)+\Phi_L(E))\big]=1,
\]
with \(\nu=rc=\pm1\) labeling the hABS class and \(\chi=\chi_R-\chi_L\) the superconducting phase difference [2111.07696]. For symmetric junctions, the two hABS branches display a protected zero-energy crossing at \(\chi=\pi\), where they form a Kramers pair of Majorana zero modes [2111.07696].

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
\[
H_{\rm QPC}
=2v\lambda_p\sum_\sigma \psi_{R\sigma}^\dagger\psi_{L\sigma}
+2v\lambda_f\sum_r r\,\psi_{r\uparrow}^\dagger\psi_{r\downarrow}
+\text{h.c.},
\]
with \(\lambda_p\) the spin-preserving tunneling amplitude and \(\lambda_f\) the spin-flipping amplitude [2111.07696]. In the ideal spin-preserving limit \(\lambda_f=0\), the nonlocal conductances \(G_{12}\) and \(G_{21}\) probe opposite hABS classes, and \(G_{21}(E,\chi)=G_{12}(E,-\chi)\) by time reversal [2111.07696]. When \(\lambda_f\neq 0\), the asymmetry
\[
\delta G^\nu(\chi)=G_{12}(E_\nu,\chi)-G_{21}(E_\nu,\chi)
\]
remains finite over a broad range and quantifies robust spin-to-charge conversion [2111.07696]. A second diagnostic is
\[
\Sigma \equiv G_{12}+G_{11}-G_{21}-G_{22},
\]
which satisfies \(\Sigma(E,\chi)=0\) for pure hABSs with only perfect Andreev reflection and generally becomes nonzero when electronic backscattering contaminates the subgap states [2111.07696].

A related transport realization couples a helical Luttinger liquid to helical Majorana modes through single or double quantum point contacts [1503.08254]. 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 [1503.08254]. 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 [1503.08254].

## 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 \(\Delta_1\neq\Delta_2\), interaction asymmetry \(K_1\neq K_2\), disorder asymmetry \(D_1\neq D_2\), and random spin-flip backscattering
\[
H_{\rm rs}=\sum_n \int \frac{dr}{2\pi a}
\Big[V_{{\rm rs},n}(r)e^{2mi\phi_n(r)}+\text{H.c.}\Big],
\]
which can arise either from magnetic impurities or from the combination of charge disorder and an external transverse magnetic field [2512.10335].

The one-loop RG equations for the dimensionless local pairing \(\tilde\Delta_n\), nonlocal pairing \(\tilde\Delta_c\), disorder strength \(\tilde D_n\), and Luttinger parameters \(K_n\) are
\[
\frac{d\tilde{\Delta}_n}{dl}
=
\Big(2-\frac{m}{K_n}\Big)\tilde{\Delta}_n,\qquad
\frac{d\tilde{\Delta}_c}{dl}
=
\Big[2-\frac{m}{4}\Big(K_1+K_2+\frac{1}{K_1}+\frac{1}{K_2}\Big)\Big]\tilde{\Delta}_c,
\]
\[
\frac{d\tilde D_n}{dl}=(3-2mK_n)\tilde D_n,\qquad
\frac{dK_n}{dl}
=
m\left[\tilde{\Delta}_n^2+\frac{1}{2}(1-K_n^2)\tilde{\Delta}_c^2-\frac{K_n^2}{2}\tilde D_n\right],
\]
with \(u_n\) flowing as \(du_n/dl=-m(u_nK_n/2)\tilde D_n\) [2512.10335]. These flows determine whether the low-energy phase is nonlocal superconducting, locally superconducting on one channel, mixed, insulating, or gapless double-helical [2512.10335].

The topological classification changes qualitatively in the presence of spin-flip terms. The clean system is DIII-like and supports only the \(\mathbb{Z}_2\) distinction between trivial and Kramers-paired Majorana phases [2512.10335]. With effective spin-flip fields \(V_\pm\), the renormalized Bogoliubov–de Gennes theory belongs to class BDI with a \(\mathbb{Z}\) invariant, allowing 0, 1, or 2 Majorana zero modes per corner [2512.10335]. The analytic criterion is that, for each \(\varepsilon=\pm\), one zero mode appears when
\[
|\Delta_-+\varepsilon V_-|
<
|\Delta_+ + \varepsilon V_+|
<
\sqrt{(\Delta_-+\varepsilon V_-)^2+\Delta_c^2},
\]
with all couplings understood as renormalized at the RG stopping scale [2512.10335]. In the clean symmetric limit this reduces to \(0<|\Delta_+|<|\Delta_c|\), yielding either 0 or 2 Majoranas; once spin-flip disorder detunes the two \(\varepsilon\) branches, an intermediate phase with a single Majorana per corner becomes possible [2512.10335].

This leads to a cascade of topological phase transitions and to a revival of zero modes as screening, disorder, or asymmetry is tuned [2512.10335]. 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 [2512.10335]. Candidate material platforms include HgTe and InAs/GaSb quantum wells, monolayer quantum spin Hall materials such as WTe\(_2\), bismuthene, and TaIrTe\(_4\), and moiré quantum spin Hall systems including twisted WSe\(_2\) and MoTe\(_2\) [2512.10335].

Source: https://www.emergentmind.com/topics/proximitized-double-helical-liquid