Transport in helical Luttinger liquids in the fractional quantum Hall regime

Domain walls in fractional quantum Hall ferromagnets are gapless helical one-dimensional channels formed at the boundaries of topologically distinct quantum Hall (QH) liquids. Naïvely, these helical domain walls (hDWs) constitute two counter-propagating chiral states with opposite spins. Coupled to an s-wave superconductor, helical channels are expected to lead to topological superconductivity with high order non-Abelian excitations1–3. Here we investigate transport properties of hDWs in the ν = 2/3 fractional QH regime. Experimentally we found that current carried by hDWs is substantially smaller than the prediction of the naïve model. Luttinger liquid theory of the system reveals redistribution of currents between quasiparticle charge, spin and neutral modes, and predicts the reduction of the hDW current. Inclusion of spin-non-conserving tunneling processes reconciles theory with experiment. The theory confirms emergence of spin modes required for the formation of fractional topological superconductivity.


II. SUPPLEMENTARY NOTE 1: DEVICES CHARACTERIZATION
After cooldown from room temperature to 4 Kelvin devices are illuminated with a red LED at 100 µA for 1 min and left to relax for 12 hours before cooling to the base temperature. This sequence is found to result in the best quality and uniformity of a 2D gas in our experiments, with the sharpest spin transition and widest fractional quantum Hall states.
With zero gate voltages densities of 2D gases under gates G1 and G2 are approximately 0.9 · 10 11 cm −2 , see Fig. 2. Small differences in the quality of 2D gases under two gates is attributed to different thickness of gate oxides ( 50nm under G1 and 100nm under G2), as well as to a small difference in the thickness of semi-transparent titanium layers which form gate electrodes. Application of negative gate voltage and reduction of the density to ≈ 0.7 · 10 11 cm −2 does not degrade quality of 2D gases as shown in Fig. 2b. \caption{Magnetoresistance $R_{xx}$ of a 2D gas as a function of magnetic field at (a) zero gate voltages and (b) gate voltages where $\nu=2/3$ is close to the spin transition. Inset shows $R_{xx1}$ and $R_{xy1}$ for $V_{g1}=0$ (different cooldown from the main trace).} where ν = 2/3 is close to the spin transition measured at the base temperature. Inset shows R xx1 and R xy1 for V g1 = 0 from the same cooldown but different LED sequence.
Evolution of the 2D gas resistance in the vicinity of ν = 2/3 as a function of magnetic field and gate voltage is shown in Supplementary Fig. 3. Electron density is extracted from the position of ν = 3/5 state (within this field range ν = 3/5 is far from the spin transition).
Electron density is a linear function of gate voltages n(V g ) = n 0 + βV g , where coefficients β 1 = 4.76 · 10 8 cm −2 /mV and β 2 = 3.21 · 10 8 cm −2 /mV are almost the same for different cooldowns while the zero voltage density n 0 varies within 5%. -100 -90 -80 -70 -60 -50 -40 -30 -20   Finally, Supplementary Fig. 6 shows resistance in the vicinity of ν = 2/3 for two opposite field directions. As expected for a helical channel hDW resistance does not depend on the field direction, at least close to the center of the ν = 2/3 state (at the edges of the ν = 2/3 state there is an onset of chiral channels formation, resistance of chiral channels depends strongly on field direction as discussed in more details in the next section).
for chiral channels is plotted as a function of a filling factor, error bars are standard deviations obtained by averaging R 34 within ∆ν = ±0.003 around the center of the IQHE or the FQHE state.

IV. SUPPLEMENTARY NOTE 3: MODELING OF TRANSPORT THROUGH A HELICAL DOMAIN WALL AT ν = 2/3
A. Description of edges of ν = 2/3 state in terms of bosonic fields and quasiparticle bosonic fields Luttinger liquid action for ν = 2/3 edge states in terms of bosonic fields of electron and the charge density is where matrix K and vector q are given by For interaction matrixV e , we assume that diagonal matrix elements for intra-mode Coulomb interaction is defined by the same matrix element V 1 and off-diagonal matrix elements for inter-mode Coulomb interaction is defined by matrix element V 2 , As was discussed by Wen 3 , equal couplings V 1 of both composite fermion modes are the consequences of the long-range Coulomb interaction.
The commutation relation is For composite fermion creation operators, we have and Luttinger liquid action in terms of quasiparticle bosonic fields χ , defined by Φ =Kχ is given by whereV The two sets of fields are orthogonal: The charge mode ϕ c , and the neutral mode ϕ n are expressed as follows: which in vector form reads (χ 1 , χ 2 ) T =Ŵ (ϕ c , ϕ n ) T , where the operation T transposes a row vector into a column vector. The transformation matrixŴ is given by and the following relations hold The neutral mode ϕ n can be interpreted as a difference in the occupation of edge modes corresponding to the first and second Λ-levels of composite fermions. In the unpolarized phase it coincides with the spin density and we will use spin index s instead of n.
For the transformed matrixV qp , we obtain where in terms of couplings V 1 and V 2 in Supplementary Eq. (4), and off-diagonal terms v cn = 0 as a result of our choice of equal diagonal couplings V 1 of both modes. This allows separation of charge and neutral modes in polarized phase and charge and spin modes for non-polarized phase. As was discussed by Wen 3 , in the absence of the mechanism of electron scattering on impurities, this separation in both phases originate from equal couplings V 1 of both composite fermion modes Supplementary Eq. (4) are the consequences of the long-range Coulomb interaction.
In order to distinguish Luttinger liquid modes in p and u phases, we characterize all modes, including bosonic electron modes, quasiparticle modes, and separated charge and neutral (spin) modes by indices p and u. The transformed action for the p phase is This action coincides with the one expressed in terms of the charge and neutral fields (Supplementary Eqs. (7)- (8)) in the seminal paper by Kane, Fisher and Polchinsky 4 , in the case if no electron scattering that leads to the composite fermion tunneling between the different modes and no coupling between modes takes place. The Luttinger liquid action in u phase is similar and is presented in the main text, Eq. (1), with spin modes entering instead of neural modes. In both phases, charge and neutral, or charge and spin modes separate.
The commutation relations for separated charge and neutral modes in the p phase are given by In the u phase, the commutation relatons for separated charge and spin modes are given by these equations with substitution pc → uc, pn → us. To acquire non-zero average charge density and current, density of the charge mode ϕ pc is shifted, ϕ pc → ϕ pc (x, t) +φ pc . A non-zero average appears due to a charge current injection, where V is the applied voltage. Then the average current carried by the edge is The shift of the charge mode density is described by an addition to the Luttinger liquid so that S(ϕ pc ) + ∆S(ϕ pc ) = S(ϕ pc −φ pc ). The case of injection from the unpolarized phase into polarized phase, when ϕ uc is shifted due to applied voltage instead of ϕ pc , is described The Luttinger liquid action for the edge states at ν = 2/3 consisting of the two phases, polarized p and unpolarized u, is given by where 4x4 matricesK with matrixK defined by Supplementary Eq. (3) and with matrixV e is defined by Supplementary Eq. (4). For convenience, in order to keep the form of relations for the current as described above for both p and u phases, we reverse the sign of the quasiparticle field χ u → −χ u , so that Φ u = Kχ u .

D. Tunneling and charge currents
The point contact (junction) x = 0 electron tunneling between polarized and unpolarized phases is described by the tunnel Hamiltonian where the neutral mode in the unpolarized phase is a spin mode describing the difference in spin density between modes.
The tunneling charge current is given by a shift in ϕ pc due to the applied voltage V described by Supplementary Eqs. (21, 23): where ρ pc = 1 √ 2π ∂ x ϕ pc is the charge density.

E. Tunneling in the model of zero length hDW
In the strong coupling limit the tunneling current can be found by imposing boundary conditions. A zero-length hDW is the limit x 1 = x 2 = 0 of the model of the hDW shown in Imposing H T = −t cos (Φ p1 (0, t) − Φ u1 (0, t)) att → ∞ as a boundary condition that leads to a jump in the tunneling mode and a continuity in the orthogonal, non-tunneling mode, we obtain: Using the expressions for quasiparticle modes, Supplementary Eq. (13), we have We obtain two more equations defining boundary conditions by imposing them on the two modes orthogonal to the modes described by Supplementary Eq. (13). For these modes we have and the boundary conditions for these modes are Using explicit expressions for the quasiparticle modes, we obtain that Supplementary Eqs.(30,36) result in the following four equations defining the outgoing fields via the incoming fields: The current injected into the polarized phase due to the applied voltage V shifts only the incoming field Using Supplementary Eq. (37), we obtain that this change leads to the following changes in outgoing fields: Describing the currents, we will use indices p and u for the currents on the polarized and unpolarized side, correspondingly. Upper indices in, out correspond to the incoming and outgoing currents, lower indices c, n and s correspond to charge, neutral and spin currents.
On the polarized side, charge and spin currents coincide; on the unpolarized side, neutral and spin currents coinside The average incoming or outgoing current due to quasiparticle where for our choice of signs in action Supplementary Eq. (24) charges q p = 1 = −q u , q p is the quasiparticle charge in the polarized phase and q u is the quasiparticle charge in the unpolarized phase, and coefficients a pc = a uc = 1, a pn = a us = −1 reflect counting direction of currents in the direction of chirality of modes rather than from the left to the right. We will assume first that the externally induced incoming charge current comes from the contact with potential V to the polarized liquid, and the current flows into the grounded (V = 0) contact to the unpolarized liquid. Then the incomig current due to Supplementary Eq. (38) is and the outgoing currents are given by where σ 0 = e 2 /h is the conductance quantum.
Using the voltage-induced shift of the incoming and outgoing bosonic fields, ϕ pc , ϕ ps , ϕ uc and ϕ us , given by Supplementary Eqs (38), (39), we can analyze the shift of quasiparticle edge state fields χ p(1,2) = 1 √ 2 (ϕ pc ± ϕ pn ) and χ u(1,2) = 1 √ 2 (ϕ uc ± ϕ us ), that describes the distribution of current over the two quasiparticle modes. The calculation shows that the resulting currents associated with these modes are given by where θ-function θ(x) = 1 at x > 0 and θ(x) = 0 at x < 0 is used to describe incoming and outgoing modes in a single equation. We observe that it follows from these equations that tunneling occurs only between χ p1 and χ u1 edges, and tunneling charge current is given by In contrast, states χ p2 and χ u2 flow, correspondingly, in the polarized and unpolarized region.
Modes χ p1 and χ u1 include modes with the same spin up, while modes χ p2 and χ u2 carry the opposite spin. This is a consequence of our tunnel Hamiltonian allowing only transmission of like spins. Generalization of the model permitting tunneling with a spin flip, e.g,. induced by interaction with nuclear spins, makes possible some tunneling processes between χ p2 and χ u2 modes. We will consider modification of the current flow in the presence of spin flips in the subsection J below.

F. Ballistic domain wall of finite length with scattering at the ends
We now analyse a system of two point scatterers at points x 1 and x 2 separated by a ballistic domain wall of a finite length L = x 2 − x 1 . In the experimental setting, these scatterers are the tri-junctions between edge modes in the p phase, the u phase and modes in the domain wall.
Each of the scatterers is described by Supplementary Eq. (37). To calculate the transmission through two junctions, it is convenient to present the connection Supplementary Eq. (37) between 4-vector of incoming modes ϕ in = (ϕ → pc (x − 0), ϕ → us (x − 0), ϕ ← uc (x + 0), ϕ ← pn (x + 0)) T and 4-vector of outgoing modes ϕ out = (ϕ → pc (x + 0), ϕ → us (x + 0), ϕ ← uc (x − 0), ϕ ← pn (x − 0)) T in a matrix form: where 4x4 matrix P is given by and 2x2 matrices P ij ,i, j = ± are defined by The matrix P ++ describes propagation of chiral modes through a single zero-length scatterer, and the matrix P +− describes the reflection of chiral modes. The total 4x4 transfer matrix T connects incoming and outgoing modes of two scatterers, via  where x 1 and x 2 are positions of the scatterers and ∆ V denotes a voltage-dependent shift of the corresponding field.
Matrix T is defined by propagation of modes through the scatterers, and potentially by multiple reflections of modes between them. However, as a result of identity the total transfer matrix for the two-junction system is defined only by a single act of reflection, T −+ = P −+ or by a single propagation through both scatterers, T ++ = P ++ P ++ = P ++ , while any contribution from processes involving subsequent propagations and reflections from scatterers vanishes due to disentanglement property of chiral channels described by Eq. (56).
G. Account for tunneling between the same spin modes in the polarized region We now consider whether tunneling between the modes with the same spin in the polarized region between two zero length scatterers will change transmission through the domain wall. The boundary condition introduced by the tunneling Hamiltonian corresponding to this process where x 1 < x < x 2 , in the strong coupling limitt p → ∞ is Taking into account this process amounts to changing one of the P matrices in the model with two scatterers We then observe that P 1++ = P ++ , P 1−+ = P −+ , as described by Supplementary Eqs. (53,54) and Therefore, disentangling relations P ++ P 1+− = 0 (62) hold, and no contributions from processes with consequent propagation and reflection occur in the total transfer matrix in the presence of P 1 → P 2 scattering. The total transfer matrix T c for this case is also defined by It means that T c = P despite scattering in the same spin channel in the polarized region.
That leads us to the conclusion that localization and backscattering by the domain wall that results in length-dependent resistance requires spin flips or inelastic processes. We also note that in the Φ 1p , Φ 1u , χ 2p , χ 2u representation, only χ 2 modes of opposite spin propagate along the domain wall, while Φ modes having the same spin pass along the edges.This directly correlates with the T c matrix properties discussed in the present section.

H. General case of the domain wall of finite length
Here we demonstrate that the coincidence of the currents flowing outside the domain wall in the cases of single-junction, two junctions, as well as two junctions with scatterers in between follows from the imposed strong coupling boundary conditions in a general case of the domain wall of finite length. At the same time, inside the domain wall, the chiral evolution of modes is controlled by the average voltage shifts at their corresponding boundaries.
In the representation of modes Φ u1 , χ u2 , Φ p1 , χ p2 , the kinetic matrix K x has the form We write the tunneling Hamiltonian as Here the tunneling constant U T has dimensions of energy/length, in contrast to tunneling constantt describing point contact/junction tunneling introduced above that has dimensions of energy. At U T → ∞, the Φ 1 fields become fixed, and there remains the free chiral evolution of the quasiparticle fields χ u2 , χ p2 described by the action for u and p modes with the charges q x u2 = 1 = −q x p2 and relations These equations, as in the above sections, reflect the fact that the neutral mode in the unpolarized region coincides with the spin current mode. In the polarized region, spin and charge currents coincide. Boundary conditions for the voltage-dependent average shifts of the modes describe the continuity of the polarized and unpolarized components of the free and boundary conditions due to tunneling of like spins given by Supplementary Eq. (72) in the strong coupling limit and coincide with those at +0 → L.
The two other point contact (junction) boundary conditions with +0 changed into L read: This is an identity due to conservation of the spin current in unpolarized phase and the neutral current in the polarized phase. Therefore, the relation between the averages ∆φ in and ∆φ out follows from the identities for a single junction, in which +0 is substituted by L.
Indeed, for voltage-dependent shifts defining outgoing charge currents we have For voltage-dependent shifts defining outgoing neutral current in the polarized region and spin current in the unpolarized region we have These relations reflect in and out tunneling current conservation.
Analyzing the relations Supplementary Eqs. (91, 93), we find that currents outside the domain wall of finite length are the same as in the previous settings: for single junction (point contact), two junctions, and two junctions with allowed scattering. This property is a result of boundary conditions in the strong coupling limit.
Inside the domain wall, the chiral evolution of χ u2 , χ p2 is controlled by the average voltage shifts at their corresponding boundaries ∆ vχp2 (L) and ∆ vχu2 (0): If the only nonzero incoming voltage shift is ∆ vφ in pc (0) = − e √ 2 3 V t, and the only nonzero incoming current is j pc = 2 3 σ 0 V , we find and The currents carried by various modes are defined by equations: Therefore, the average currents along the domain wall are where the upper index 1 corresponds to the incoming (outgoing) currents at the junction x 1 and the upper index 2 corresponds to the incoming (outgoing) currents at the junction x 2 .
For the currents inside the domain wall, we have It is easy to see that at the each junction the charge current is conserved and so are the spin x 2 junctions on the edge of the sample. The domain wall in both cases, when the current injected from the p phase into the u phase, and when the current injected from the u phase into the p phase, can be described by the ratio i DW = 1/3.

J. Effect of spin flip processes
So far we considered tunneling and/or scattering between modes assuming that no spin flips are allowed. Experiment, however, clearly demonstrates substantial role of spin-flip processes associated with nuclear spins. Such spin-flip processes due to hyperfine interactions can occur in the vicinity of the domain wall, where it is possible to match small nuclear spin splitting and electron energy splitting near the crossing of composite fermion levels.
We will model partial spin flip by assuming that nuclear spin flips may result in an admixture of polarized and unpolarized propagating density modes inside the domain wall.
In the model discussed above, tunneling of like spins is defined by tunneling Hamiltonian Supplementary Eq. (72) that completely determines the electronic mode Φ 1 . In the absence of spin flip processes, the χ 2 quasiparticle modes are free chiral propagating modes, which correspond to uncoupled propagation of χ 2 modes with opposite spins, as given by Supplementary Eqs. (81,82). In the presence of spin flips in the domain wall, the χ 2u mode, which is a superposition of a charge and a spin mode, and the χ 2p mode, which is a superposition of a charge and a neutral mode, become coupled. In particular, the incoming χ 2p mode at the junction x = x 2 = L that propagates as charge and neutral mode towards x = x 1 = 0, is partially reflected into u phase through junction x 2 = L as χ 2u mode with probability r, and partially continues propagation towards x 1 = 0 with probability 1 − r. The χ 2u charge and spin modes incoming at junction x = x 1 = 0, are reflected with probability r as χ 2p modes and partially propagate with probability 1 − r towards x = x 2 = L as χ 2u modes.
These processes are summarized as boundary conditions At r = 0, these boundary conditions coincide with boundary conditions Supplementary Eqs.
where we used the same notation as in Supplementary Eq.(108), with index 1 corresponding x = 0 and index 2 corresponding x = L. Therefore, we see that allowing all spin-flip processes in electron tunneling and scattering inside the domain wall, r = 1, leads to the absence of current through the domain wall, so that the current is flowing only along the sample edges, with the Hall resistance quantized to 3/2 h/e 2 and zero longitudinal resistance.
This result is naturally expected, as r = 1 essentially describes the absence of the domain wall. The domain wall current and currents flowing along the edge are the functions of the spin-flip probability r. The ratio i DW = I DW /I changes continuously between 1/3 for r = 0 to zero for r = 1, where I DW is the current diverted from the edge through the domain wall, and I is the injected current.