Topological $π$-junctions from crossed Andreev reflection in the Quantum Hall regime
Francesca Finocchiaro Francisco Guinea Pablo San-Jose




F. Finocchiaro
F. Guinea
P. San-Jose


^1Materials Science Factory, ICMM-CSIC, Sor Juana Ines de La Cruz 3, 28049 Madrid, Spain
^2IMDEA Nanociencia, Calle de Faraday 9, 28049 Madrid, Spain
^3Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom






We consider a two-dimensional electron gas (2DEG) in the Quantum Hall regime in the presence of a Zeeman field, with the Fermi level tuned to filling factor ν=1. We show that, in the presence of spin-orbit coupling, contacting the 2DEG to a narrow strip of an s-wave superconductor produces a topological superconducting gap along the contact as a result of crossed Andreev reflection (CAR) processes across the strip. The sign of the topological gap, controlled by the CAR amplitude, depends periodically on the Fermi wavelength and strip width and can be externally tuned. An interface between two halves of a long strip with topological gaps of opposite sign implements a robust π-junction, hosting a pair of Majorana zero modes that do not split despite their overlap. We show that such a configuration can be exploited to perform protected non-Abelian tunnel-braid operations without any fine tuning.


Topological π-junctions from crossed Andreev reflection in the Quantum Hall regime
    F. Finocchiaro^1,2, F. Guinea^2,3 and P. San-Jose^1
    December 30, 2023
==================================================================================








During the last decade we have witnessed a surge in both theoretical and experimental progress towards the realisation of Majorana-based quantum computation.<cit.> Majorana zero modes (MZMs) are zero-energy bound quasiparticles of topological origin that are their own self-adjoint and obey non-Abelian anyon statistics. As a result, the adiabatic exchange (or `braiding') of a pair of MZMs rotates the wavefunction of the degenerate ground state in a non-commutative fashion.<cit.> Such a process or its generalisations<cit.> can be viewed as a coherent manipulation of qubit states realised by pairs of MZMs. The interest in Majorana-based topological quantum computation stems from the fact that, as a result from the non-locality of the MZMs, local sources of noise do not

affect the fidelity of the braiding operation, nor do they induce decoherence of the ground state manifold. This property has inspired implementations of fault-tolerant computation schemes able in principle to beat decoherence at the hardware level.<cit.>

The fundamental ingredient needed to create MZMs is topological superconductivity, either intrinsic, like in p-wave superconductors,<cit.> or artificially designed, like in proximitised superconducting wires with strong spin-orbit coupling (SOC) in an external magnetic field.<cit.>

More recently, two dimensional electron gases (2DEGs) with induced superconductivity are being actively investigated as platforms for topological superconductivity.<cit.>
In addition to the increased freedom afforded by the planar geometry, these systems allow for the formation of a new type of topological quasi-one dimensional (1D) system, confined on both sides by two different superconductors with a phase difference π.  For transparent enough contacts, such π junctions can greatly reduce the magnetic fields required for MZMs to emerge.<cit.>





In this work we show that planar junctions allow for yet another implementation of 1D topological superconductivity, with a geometry dual to the above. It is achieved by contacting a long and narrow strip of a conventional superconductor to a 2DEG in the Quantum Hall (QH) regime at filling factor ν=1. The proximitised region acquires a superconducting gap Δ, and as a results develops gapless QH edge states along each side. Due to local Andreev reflection (LAR) processes, these edge states are a mixture of electrons and holes,<cit.> see Fig. <ref>. Assuming that spin-orbit coupling (SOC) is present in the system, and that the strip width is comparable with or smaller than the superconducting coherence length, the QH edge states may become Cooper-paired through additional crossed Andreev reflection (CAR) processes<cit.> across the strip. We show that a topologically non-trivial superconducting gap Δ^* then opens in the edge state dispersion, and MZMs emerge at either end of the strip. This possibility was suggested by Lee et al. in Ref. Lee:NatPhys17, where the requisite CAR processes were experimentally demonstrated in the case of graphene, although they concentrated on the ν=2 regime and not on the ν=1 condition required for the formation of MZMs.


Here we theoretically investigate the conditions for CAR-induced topological superconductivity at ν=1. [We note that for ν=2 <cit.> (or even fillings in general), pairs of Majoranas will be generated which will not be protected against hybridisation into conventional fermions. Odd fillings, however, will always generate one protected unpaired Majorana zero mode.]
(Related approaches have been explored in fractionalized QH systems supporting parafermions<cit.>.)
We find that both the magnitude and, more importantly, the sign of the topological gap depends on the amplitude of the CAR processes. As a result, the sign of Δ^* can be controlled by adjusting the width of the strip and/or the electronic density of the proximitised region, which in turn determine the CAR amplitude. Reeg et al. anticipated such a possibility while studying a related system of two parallel nanowires coupled through a superconductor.<cit.>
We show that this effect may be used to induce a sign change Δ^*→ -Δ^* along the strip by e.g. electrostatic gating. This situation corresponds to a one-dimensional topological π-junction along the strip which is host to two degenerate MZMs that do not hybridise despite their spatial overlap.<cit.>
Since the original induced Δ does not change sign (only the edge state gap Δ^* does), no external fine-tuning is required to mantain the π phase difference, and the MZMs remain protected at zero energy. As we will show, this allows for a powerful generalisation of tunnel-braiding strategies (originally proposed by Flensberg <cit.>) on the two MZMs in the junction, without the need to carefully control external parameters in the process.

Consider a normal (N) 2DEG  with a proximitised superconducting strip (S) of width W_S along the x direction, see Fig. <ref>a. The N region is in the QH regime and is subject to a Zeeman field along x allowing the electron density to be tuned to an odd filling factor ν=1. (Other mechanisms such as interaction-induced spin instabilities may play the role of the Zeeman field in some systems<cit.>). The S region has uniform superconducting pairing Δ induced by proximity to the parent superconductor. We also assume that SOC is present in the system, either in the N region and/or in the S region (e.g. inherited from a superconductor made of heavy elements, such as NbN or NbTiN). The electronic structure of this system, obtained using a tight-binding approximation on a square lattice (see Supplementary Information<cit.> for details), is studied in the following.

 


 

Since the N region is in the ν=1 QH regime and the S strip is trivially gapped, each of the two NS interfaces hosts a single spin-polarized edge state. These states travel in opposite directions at opposite interfaces (see Fig. <ref>a).
Local Andreev reflections at each interface transform the edge states into coherent superpositions of electrons and holes,<cit.> but do not open a gap because of the chiral nature of the carriers.


However, in our geometry with two parallel NS interfaces at either side of the strip, another type of Andreev reflection process can take place, wherein an electron on one interface is scattered as a hole into the other interface.
This crossed Andreev reflection process has a significant amplitude only for strips narrower than the coherence length ξ≈ħ v_F/Δ. Unlike local Andreev reflection, CAR processes may open a superconducting gap Δ^* in the presence of SOC, since electron and hole edge states at opposite interfaces propagate in opposite directions at the same wave vector. 
The role of the SOC is to cant the spin away from the Zeeman field in opposite directions in the two edge states, so that they can pair to form a spin singlet.
At ν=1 the gap resulting from CAR is topological, as can be seen by a direct mapping of the two spin-canted edge states plus pairing into an Oreg-Lutchyn model<cit.> [Eq. (B4) in Supplementary Information].
Fig. <ref>a shows the gapped bandstructure of an infinite strip with significant CAR processes (left, W_S ≃ξ) and the gapless case without CAR (right, W_S ≫ξ). 
The topological nature of Δ^* manifests in the appearance of MZMs when the strip is terminated inside the 2DEG (Fig. <ref>d). 




The value of the topological gap Δ^* is entirely determined by the CAR amplitude, that in turn depends on the strip width W_S, the Fermi wavelength λ_F and the singlet amplitude governed by the proximity gap Δ and the SOC strength α. 

We have performed tight-binding simulations which show, specifically, that Δ^* is a real periodic function of the W_S/λ_F with alternating sign, see Fig. <ref>b. This behavior is confirmed by an analytical calculation in terms of Green's functions, which yields







    Δ^*≈4π^2t'^2 a^3/W_Sλ̃_F^2 μ̃× Im(z z)sinθ

where θ is the spin canting angle due to spin-orbit coupling, λ̃_F=2π/√(2m μ̃), μ̃=μ-k_F^2/2m, μ is the strip Fermi energy, k_F is the edge-state Fermi wavevector, and z=2π√(1+iΔ/μ̃)× W_S/λ̃_F (See Supplementary Information<cit.> for details). This formalises the central finding of our work. The sign of Δ^* follows the change in the number of normal modes in the strip, given by ⌊ 2W_S/λ̃_F⌋. It is therefore likely to be realistically tuneable with electrostatic gating of the strip region that may modify both its effective width W_S and electronic density, or by adjusting the width lithographically.<cit.>


The possibility of changing the sign of the topological gap

along the strip opens a new opportunity for the generation of MZMs. A long strip with uniform induced gap Δ but edge-state gaps of opposite sign in its two halves (Δ_1^*Δ_2^*<0) forms a topological π-junction, similar to a topological Josephson junction tuned to phase difference ϕ=π. Such a system then develops two MZMs localised at the junction [see Fig. <ref>(c,d)], that stay at zero energy despite their spatial overlap as long as the phase difference across the junction remains π. The π phase difference between the two halves of the strip is robust. Since Δ^* on both sides is finite and real, its sign does not depend on perturbations. The CAR π-junction is furthermore stabilised by the phase rigidity of the strip order parameter Δ.<cit.> Unlike in ϕ=π Josephson junctions, it does not require fine tuning of any external parameter such as the flux across the superconducting circuit or the strip parameters.  As a result, CAR-induced topological superconductivity enables the creation of MZMs that remain decoupled regardless of their overlap. This offers great advantages in the context of coherent Majorana qubit manipulation and braiding, as outlined in the following.



We now present a possible application of the CAR π-junction to the challenge of non-Abelian Majorana braiding.
Plenty of proposals for the demonstration of the non-Abelian statistics of Majorana excitations have been presented which hinge on the physical exchange (or braiding) in real space of pairs of Majoranas.<cit.> Some other approaches, however, rest upon schemes that involve rotation of the wavefunction without the need for actual MZMs to move spatially.<cit.> Among these, it has been suggested<cit.> that adiabatic tunnel processes of single electrons from a quantum dot into pairs of Majorana zero modes can result in arbitrary non-Abelian rotations of the ground-state manifold.


These so-called tunnel-braid operations are extremely versatile as they allow a universal set of single-qubit gates, in contrast to braiding that only allows a limited set of operations.

Unfortunately, tunnel-braiding has the drawback of requiring a precise, typically fine-tuned, phase difference of π between the MZMs involved throughout the operation.
If the phase deviates from this value, the result of the operation becomes time-dependent and is no longer protected against decoherence. 


The robustness and lack of fine-tuning of CAR π-junctions promises to overcome this problem. 
In Fig. <ref>a we present a possible geometry to implement a CAR-protected tunnel-braiding scheme. We deposit two narrow superconducting strips on a ν=1 2DEG such that two independent CAR-induced topological gaps Δ_1^* and Δ_2^* open on each. One end of each strip terminates inside the 2DEG, so that the corresponding MZMs γ_1,2 lie within a finite distance of each other. The MZMs γ̃_1,2 on far end of the strips are assumed sufficiently far from the junction so as to become decoupled from γ_1,2. We control the Fermi level of the two strips, μ_1 and μ_2, by means of two independent gates, in order to tune the magnitude and sign of the topological gaps Δ^*_1,2. 

The two `inner' MZMs γ_1 and γ_2 are then coupled to a quantum dot through two tunnel barriers that may be tuned externally. The tunnelling couplings t_1,2 control the specific non-Abelian opearation to perform. The dot is in the Coulomb-blockade regime, with occupation N. We adiabatically tune the dot level ε_D across an N→ N-1 transitions between two adjacent Coulomb valleys. This transfers a single electron to the composite state of the two Majorana modes. Figure <ref>b shows the evolution of the low-energy single-particle Bogoliubov spectrum of the full dot-2DEG-strip system across this process, with dashed lines corresponding to mostly-dot states, and solid lines to MZMs states in the strip. The two cases with equal (blue, ϕ=0) and opposite (red, ϕ=π) signs for Δ^*_1,2 show markedly different structure. The conventional ϕ=0 case splits the MZMs away from zero close to the N→ N-1 transition, as they become resonantly coupled via the dot state.<cit.> Such an operation is not protected against noise and its result depends on timing. In contrast, the ϕ=π case shows MZMs that remain exactly at zero energy throughout the operation, as their hybridisation across the dot is forbidden by the opposite sign of Δ^*_1,2. The state after emptying the dot is then independent of timing and insensitive to noise in ε_D. As shown by Flensberg,<cit.> the transformation P within the degenerate ground state manifold associated to this process is a rotation by an angle π around an axis in the xy plane, controlled by the tunnel couplings t_1,2.
If the couplings are then changed to t'_1,2, and the reverse adiabatic transition N - 1 → N on the dot is performed, the composite operation P'P rotates the quantum state of the Majoranas by an arbitrary angle around the z axis.
In comparison, braiding two MZMs can only rotate the wavefunction about the z axis by an angle of π / 2.


As no fine-tuning is required to maintain the ϕ=π condition in the CAR π-junction, the tunnel-braiding process should enjoy similar topological protection as a standard spatial-braiding.
In Fig. <ref>c we show the MZM splitting across a resonant dot as we vary the Fermi energy under one of the strips, while the other is kept fixed. As expected, we find alternating ϕ=0 (red) and ϕ=π (blue) regions, in which the MZM splitting is finite and zero, respectively. The width in parameter space of the ϕ=π regions with MZMs pinned to zero is finite, unlike in topological Josephson junctions.


In essence, we have presented here a scheme towards one-dimensional topological superconductivity that extends  previous approaches that are based on the proximity effect, i.e. local Andreev reflections, of spinless helical electronic phases coupled to superconductors. While such approaches indeed produce a topological order parameter, its phase is fixed by the parent superconductor. In contrast, crossed Andreev reflections, relevant in geometries as those discussed here, also produces a topological order parameter, but its sign may be either the same as or opposite to that of the parent, depending on the CAR amplitude itself. Controlling the sign of the topological gap in a stable way has many ramifications. We have shown how it may be exploited to produce stable, self-tuned π-junctions, wherein sizeable Majorana overlaps, which are problematic in more conventional Majorana devices, are no longer a concern, at least for pairs of MZMs at the junction. As a result, parametric braiding of Majoranas through e.g tunnel-braiding schemes becomes significantly more realistic.
The specific implementation of the CAR-induced topological gap described here is just one conceptually simple possibility, but it is not unique. Other phases, such as quantum anomalous Hall states, could also exhibit the requisite ν=1 spin-singlet states.

The temperature requirements for using our protocol are limited by both the Zeeman splitting and Δ^*, which gives a conservative estimate between 0.1 K and 1 K, well within reach of current experiments on this type of systems. CAR-induced topological superconductivity is thus proposed as a promising road forward towards the next landmark in the field, the realisation of protected non-Abelian operations in the lab. 





§ 
 
For the numerical calculations, we consider a two-dimensional square lattice that extends from -L/2 to L/2 along the x axis, and from -W/2 to W/2 along the y axis. The central superconducting strip, oriented along the x axis, occupies the area that goes from y = -W_S/2 to y = W_S/2. We use either periodic boundary conditions (PBC) or open boundary conditions (OBC) along both directions, as specified in the main text. The tight binding Hamiltonian that we use for all the calculations in the paper is given by


    H = H_0 + H_Z + H_S + H_SOC

where

    H_0 = -∑_mnμ_n c_mn^†c_mn - t ∑_⟨ m n,m'n'⟩ c_mn^†c_m'n' e^-iϕ_mn,m'n'


    H_Z = ∑_mn V_n^Z c_mn^†σ_x c_mn


    H_S = ∑_mnΔ_n[ c_mn,↓ c_mn,↑ + c_mn,↑^† c_mn,↓^†]


    H_SOC = i∑_⟨ m n,m'n'⟩ (α_n / a^2)    c_mn^†(σ×r_mn)_z c_m'n'


Where


  * r_mn=(ma, na), with a the lattice parameter of the square lattice.


  * ⟨ m n,m' n'⟩ indicates restriction to nearest neighboring sites.


  * μ_n=μ_N for n ∈ [-W/2,-W_S/2] and n ∈ [W_S/2,W/2], and μ_n=μ≠μ_N for n ∈ [-W_S/2,W_S/2].


  * ϕ_mn,m'n' is the Peierls phase acquired by the electrons under an external magnetic field, defined as ϕ_mn,m'n'=∫_r_m'n'^r_mnA· dr if n ∈ [-W/2,-W_S/2] and n ∈ [W_S/2,W/2] and that is 0 if n ∈ [-W_S/2,W_S/2] due to the Meissner effect. Under the choice of the Gauge A=(A_x(na),0,0), with

    A_x(na) =
    {[      B(na + W_S/2)  n ∈ [-W/2,-W_S/2];                  0 n ∈ [-W_S/2,W_S/2];      B(na - W_S/2)    n ∈ [W_S/2,W/2] ].

and performing the integral, ϕ_mn,m'n' becomes


    ϕ_mn,m'n' =
    {[ Ba(m-m')[a(n+n')/2 + W_S/2]           n ∈ [-W/2,-W_S/2];                           0          n ∈ [-W_S/2,W_S/2]; Ba(m-m')[a(n+n')/2 - W_S/2]             n ∈ [W_S/2,W/2] ].



  * V_n^Z=V_Z≠ 0 for n ∈ [-W/2,-W_S/2] and n ∈ [W_S/2,W/2] and V_n^Z=0 for n ∈ [-W_S/2,W_S/2].


  * Δ_n=0 for n ∈ [-W/2,-W_S/2] and n ∈ [W_S/2,W/2] and Δ_n=Δ≠ 0 for n ∈ [-W_S/2,W_S/2].


  * α_n=α≠ 0 for n ∈ [-W/2,W/2].


  * The creation and annihilation operators are two-component vectors in spin space

    c_mn^†=(c^†_mn,↑,c^†_mn,↓)



The parameters used for the simulations that are common to all the results presented in the main text are m^*=0.015m_e, B=0.34 T, Δ=0.38 meV, α = 3 · 10^-11 eV m, V_Z = 0.3 meV.
In addition, in Fig. <ref>a we have employed a chemical potential of the proximitized region of μ=4 meV for both panels, while changing the width from W_S=300 nm (left panel) to W_S=2 μm (right panel).
The chemical potential employed in Fig. <ref>b is fixed to μ=0.74 meV, while the width W_S varies from 0 to 950 nm. The ratio μ/Δ is therefore equal to 1.95. In Fig. <ref>c we have used a strip 3 μm long with PBC. The blue points represent the lowest eigenvalues corresponding to a uniform chemical potential of μ_1=10.5 throughout the strip, whereas the red ones represent the case of a strip that is cut in two halves, one with μ_1=10.5 meV and the other with μ_2=13.1 meV, characterized by gaps Δ^* of opposite sign. In Fig. <ref>d we have used W_S=220 nm, μ_1=13.3 meV and μ_2=16.5 meV. The total length of the strip is of 3.4 μm, and the length of the strip is of 2.4 μm. The same width and chemical potentials have been used in Figures <ref>b and <ref>c, except for the fact that the strips are now spatially separated by 1 μm and long 2 μm each. We have considered a system with PBC and excluded the eigenvalues associated to the external MZMs (identically zero) that are present in the case of Δ^*_1Δ^*_2<0 (cfr Fig. <ref>a). The hopping amplitudes from the MZMs to the dot are t_1=0.32 meV and t_2=0.51 meV.





§ 
 
To derive the analytical dependence of the gap Δ^* on the parameters of the system, we consider an infinite system with a superconducting strip coupled to its surrounding 2DEG by a real hopping t' that could in principle be different from t in Eq. (<ref>). For t'=0 gapless edge states circulate along the 2DEG surface. A finite t' couples edge states at either side of the strip, opening a gap Δ^* in their spectrum. This can be understood by considering the effective Hamiltonian of the 2DEG H_eff = H_0 + H_Z + H_SOC + Σ(ω) once the strip is integrated out, which introduces a self-energy Σ(ω) that pairs opposite edge states. The induced superconducting pairing Δ̃ is given by the off-diagonal (pairing) elements of the self-energy at ω=0 between opposite edges (the actual gap Δ^* depends on this pairing Δ̃, but also on the singlet amplitude of the 2DEG edge states, determined by H_SOC and to be discussed later).[We may neglect the frequency dependence of Σ(ω) for the purpose of computing Δ^* as long as Δ^*≪Δ, which is the physically relevant situation.] The self-energy from the strip reads

    Σ(ω) = .t'^*G^(0)_tb(ω;y,y')t'|_y=0, y'=W_S.

Here G^(0)_tb is the tight-binding Nambu-Green function of the decoupled strip, evaluated above at y, y' in opposite edges. A given k_x wavevector is implicit here, as we assume x translation symmetry. For simplicity we have shifted the strip to y ∈ [0,W_S] here. In the continuum limit a→ 0 the Green's function is G^(0) = lim_a→ 0 G^(0)_tb/a. We may decompose G^(0) in terms of the continuum eigenvalues ϵ_λ and eigenvectors φ_λ as 

    G^(0)(ω;y,y')=∑_λφ_λ(y)⊗φ_λ^†(y')/ω-ϵ_λ

This is a 4× 4 matrix, as φ contains both spin and electron/hole amplitudes. The continuum Green's function, evaluated at the boundaries of the decoupled strip, vanishes by definition. One cannot, therefore, simply replace G^(0)_tb with G^(0) in Eq. (<ref>).

As shown in [Prada:EPJB04], the Green's function at the outermost sites of a system described by a simple tight binding model can be written, in the limit where the lattice constant is the smallest length scale in the problem,  as:

    G^(0)_tb(ω;y,y') = - a^3∂_y∂_y'G^(0)(ω,y,y')

Hence, the pairing induced on the 2DEG edge state through crossed-Andreev reflection (CAR) processes reads

    Δ̃≈ -a^3 t'^2[∂_y∂_y'F^(0)(ω=0;y,y')]_y=0,y'=W_S

where F^(0)=1/4Tr(τ_yσ_yG^(0)) is the off-diagonal (pairing, or anomalous) component of the continuum Green's function and τ,σ are Pauli matrices in the particle-hole and spin sectors, respectively.

To compute G^(0) analytically we assume spin-orbit to be negligible inside the strip (it is assumed finite in the 2DEG only). Hence G^(0) is spin degenerate, and can be obtained by diagonalising the 2× 2 continuum Hamiltonian of the strip

    H_S=(k_y^2/2m-μ̃)τ_z
    +Δτ_x,
        μ̃=μ-k_x^2/2m

where m is the effective mass, μ is the chemical potential and Δ the pairing potential. The τ matrices are now Pauli matrices acting in a Cooper-pairing sector of a given spin, defined by the basis ψ=(ψ_↑,ψ_↓^†)^T. The eigenvalues of this Hamiltonian are given by

    ϵ_η=η√((k_y^2/2m-μ̃)^2 + Δ^2)=η√(ξ^2 + Δ^2),   η=± 1

where we have defined ξ=k_y^2/2m-μ̃. The associated normalized spinors are

    φ_η=
     (
    [ u_η; v_η;     ])
    =
    1/√(2ϵ_η)(
    [ η√(ϵ_η+ξ);  √(ϵ_η-ξ);           ])







For a given eigenvalue of the problem, the most general eigenstate solution is given by

    φ_η(y)=
     (
    [ u_η; v_η;     ])
    [
    A_ηe^i k_y y + B_ηe^-i k_y y]

The coefficients A_η and B_η are found by imposing the boundary conditions that the wavefunction of the isolated strip needs to vanish at the boundaries:

    φ_η(y=0)=φ_η(y=W_S) = 0

that yields A_η=-B_η and the quantization of the wavevector along the y direction

    k_y^n=nπ/W_S

The eigenvalues ϵ_λ and eigenvectors φ_λ of the isolated strip, indexed by λ=(n,η) quantum numbers, therefore read

    ϵ_η^n=ημ̃√((n^2λ̃_F^2/4W_S^2-1)^2 + (Δ/μ̃)^2)

and

    φ_η^n(y)=
    
    
    
    
    
    
    1/√(2ϵ^n_η)(
    [ η√(ϵ^n_η+ξ_n);  √(ϵ^n_η-ξ_n);               ])
    √(%s/%s)2W_Ssin(nπ y/W_S),

where λ̃_F=2π/√(2mμ̃) and ξ_n=μ̃[(nλ̃_F/2W_S)^2 -1]. The Green's function of the isolated system is

    G^(0)(ω;y,y')=∑_n=1^∞ G_n^(0)(ω;y,y')

where

    G_n^(0)(ω;y,y')=
    ∑_η=± 1φ_η^n(y) ⊗[φ_η^n(y')]^†/ω-ϵ_η^n






The off-diagonal component of this matrix, F_n^(0)=1/2Tr(τ_x G_n^(0)), evaluated at ω=0, reads




    F_n^(0)(ω=0; y,y')=2sin(k_y^n y) sin(k_y^n y')/W_S∑_η=± 1√((ϵ_η^n)^2-ξ_n^2)/2(ϵ_η^n)^2

The double derivative evaluated at the boundaries is

    [∂_y∂_y'F^(0)_n(ω=0;y,y')]_y=0,y'=W_S      
    
    = 1/W_S^3Δ(nπ)^2 cos(nπ)/1+(μ̃/Δ)^2[1-(nλ̃_F/2W_S)^2]^2

Performing the sum over n we get that the effective pairing induced by the strip in the external edge states is





    Δ̃   =   -a^3 t'^2∑_n=1^∞[∂_y∂_y'F^(0)_n(ω=0;y,y')]_y=0,y'=W_S
       =   
    a^3 t'^21/Δ W_S^34π^2/(λ̃_F/W_S)^2μ̃/Δ
          × Im[2π√(i+μ̃/Δ)/√(μ̃/Δ) λ̃_F/W_S(2π√(i+μ̃/Δ)/√(μ̃/Δ) λ̃_F/W_S)].


Related expressions were derived in Ref. 24 corresponding to various limiting cases of the general result above.
Recall that all the dependence on the k_x wavevector is inside μ̃=μ-k_x^2/2m and λ̃_F=2π/√(2mμ̃). At k_x=0 these quantities become the actual Fermi energy μ and Fermi wavelength λ_F=2π/√(2mμ) of the superconducting strip, respectively. The ratio ⌊ 2W_S/λ̃_F⌋ represents the total number of open modes in the quasi-1D strip with a given k_x  in the absence of superconductivity. Equation <ref> then shows that the sign of the CAR-induced pairing is given by the parity of the number of open modes, see Fig. <ref>.








We now consider how the presence of spin-orbit coupling in the 2DEG allows for the pairing Δ̃ to open a gap Δ^* in the edge state spectrum. 

We employ a simplified low-energy description of the edge states. Given that the 2DEG bulk is insulating, we consider just the 1D chiral channels generated at the two sides of the strip in the ν=1 QH regime. These can be modelled by the 4x4 continuum Hamiltonian

    H=(k^2/2m-μ_N+V_Zσ_x +α kσ_y)τ_z -Δ̃τ_yσ_y

where k=k_x, μ_N is the chemical potential, α is the spin-orbit coupling and Δ̃ is the CAR-induced pairing derived above (evaluated at k_x=k_F, i.e. at the wavevector for which the edge states cross zero energy). We recall that the σ matrices act in spin space and the τ matrices in particle/hole space. This Hamiltonian is akin to the Oreg-Lutchyn model<cit.>, and is a valid description of the edge states at either side of the strip at low energies. For Zeeman fields V_Z<μ_N the model has two carrier species propagating along each direction, which corresponds to filling ν=2 of the QH state. The model is then in a topologically trivial phase. For strong enough Zeeman fields V_Z>√(Δ̃^2+μ_N^2), however, it can be driven into a topologically non-trivial phase. One spin sector is thus depleted, so that the corresponding filling is ν=1 in the absence of the superconducting strip (one mode propagating along each direction on each side of the strip). The pairing Δ̃ then creates a topologically non-trivial gap Δ^* that leads to Majorana bound states.


The wavefunction satisfying the Schroedinger equation is now a 4-component spinor ψ=(ψ_↑,ψ_↓,ψ_↑^†,ψ_↓^†)^T. At μ_N=0, the system opens a gap of

    Δ^*(k)= √(k^4 + 4 m (m γ_k - Γ_k))/m

where

    γ_k = V_Z^2 +  α^2 k^2+ Δ̃^2

and

    Γ_k=√(α^2 k^6 + V_Z^2 k^4 + 4 m^2 V_Z^2 Δ̃^2)

If we work within the limit in which Δ̃ is the smallest scale of the problem, then the wavevector at which the gap opens is well approximated by the Fermi wavevector at zeroth order in Δ̃. Thus

    k=k_F ≈√(2 m^2 α^2 + √(m^2 (V_Z^2 + m^2 α^4)))

and, therefore,

    Δ^*=Δ^*(k_F)= 2 √(β[1 + Δ̃^2/β - √(1 + 
     Δ̃4 V_Z^2/β^2)])

where

    β = 2 V_Z^2 + 4 m^2 α^4 + 
      4 m α^2 √(V_Z^2 + m^2 α^4)=
      2 (V_Z^2 + α^2 k_F^2)

Now one can expand Δ^* in series as a function of Δ̃ up to first order, obtaining

    Δ^*≈√(1 - 2 V_Z^2/β)Δ̃=
     α k_F/√(V_Z^2 + α^2 k_F^2)Δ̃

By writing the Zeeman and Rashba part of the Hamiltonian as

    H_Z+SOC=h·σ

where h=(V_Z, α k_F,0). This expression allows to define a canting angle θ such that

    θ=arcsin(α k_F/√(V_Z^2 + α^2 k_F^2))

and

    Δ^*=sinθΔ̃

This angle represents the spin-orbit-induced deviation of the edge state spins away from the Zeeman axis σ_x. Now, plugging in the values that we used in the main text for the numerical calculations, we obtain the behaviour of Δ^* as a function of W_S/λ̃_F shown in Fig. <ref>a, in excellent agreement with the full numerics shown in the main text. (Note that in the main text W_S is normalized to λ_F in spite of the λ̃_F used in Fig. <ref>).



§ 
 
In this section, we study the stability of the π-junction to perturbations in the phase difference between Δ^*_1 and Δ^*_2. To confirm that a π-junction in Δ^* is indeed a stable solution for the system, one must demonstrate that a phase difference ϕ^*=π between Δ^*_1 and Δ^*_2 corresponds to a minimum in the Josephson free energy under variations of ϕ^*. Bardeen et al.<cit.> and Beenakker and van Houten<cit.> demonstrated, using complementary approaches, that the free energy of a generic Josephson junction may be written, at finite temperature T and up to a phase-independent constant, as

    E_J(ϕ)= -k_B T ∑_ϵ_n<0ln( 2 coshϵ_n(ϕ)/2k_B T)

where the sum is performed over both spin flavours and over both particle- and hole-like levels.  In the low temperature limit E_J(ϕ) reduces to

    lim_T → 0E_J(ϕ)= - 1/2∑_ϵ_n<0ϵ_n(ϕ)

Variations in the phase of the induced gap ϕ^* can be generated through variations in the phase ϕ of the left and right portions of the parent superconductor on top of the two half-strips, which is a controllable parameter in the model. Hence, computing the free energy E_J(ϕ) and knowing the relation between ϕ^* and ϕ, one may establish whether the π-junction is stable. Intuitively the total free energy, which includes the energy of the parent superconductor, will be a competition between the phase rigidity of ϕ and the phase rigidity of ϕ^*, which in the π-junction configuration are out of phase.

We have performed numerical calculations of the Josephson free energy of the junction as a function of parent phase difference ϕ ∈ [- 2 π: 2 π] in two cases. In case (a) the ratio W_S/λ_F on the two sides of the junction is such that for ϕ=0 the induced gap has also ϕ^*=0 (panel a of Fig. <ref>), and the π-junction case (b) which has ϕ^*=π at ϕ=0 (panel b of Fig. <ref>). The results for the free energy (panels c,d, respectively) show that the minimum is obtained in both configurations when ϕ=0. In other words, the phase rigidity of the parent superconductor is stronger than that of the induced gap, and hence determines the equilibrium configuration. It is more expensive to generate a phase difference of π in ϕ than in ϕ^*. Thus, in case (b), the ϕ^*=π configuration is thermodynamically stable.





We are grateful to L. Chirolli, E. Prada, C. Reeg, J. Klinovaja and D. Loss for fruitful discussions. F. F. and F. G. acknowledge the financial support by Marie-Curie-ITN Grant No. 607904-SPINOGRAPH. F. F. and P.S-J. acknowledge financial support from the Spanish Ministry of Economy and Competitiveness through Grant No. FIS2015-65706-P (MINECO/FEDER).



