Signatures of Rashba spin-orbit interaction in the superconducting proximity effect in helical Luttinger liquids

Signatures of Rashba spin-orbit interaction in the superconducting proximity effect in helical Luttinger liquids

Pauli Virtanen Institute for Theoretical Physics and Astrophysics, University of Würzburg, D-97074 Würzburg, Germany    Patrik Recher Institute for Theoretical Physics and Astrophysics, University of Würzburg, D-97074 Würzburg, Germany Institute for Mathematical Physics, TU Braunschweig, 38106 Braunschweig, Germany
July 14, 2019
Abstract

We consider the superconducting proximity effect in a helical Luttinger liquid at the edge of a 2D topological insulator, and derive the low-energy Hamiltonian for an edge state tunnel-coupled to a -wave superconductor. In addition to correlations between the left and right moving modes, the coupling can induce them inside a single mode, as the spin axis of the edge modes is not necessarily constant. This can be induced controllably in HgTe/CdTe quantum wells via the Rashba spin-orbit coupling, and is a consequence of the 2D nature of the edge state wave function. The distinction of these two features in the proximity effect is also vital for the use of such helical modes in order to split Cooper-pairs. We discuss the consequent transport signatures, and point out a long-ranged feature in a dc conductance measurement that can be used to distinguish the two types of correlations present and to determine the magnitude of the Rashba interaction.

pacs:
74.45.+c, 71.10.Pm, 73.23.-b

I Introduction

The helical edge states of a 2D topological insulator (TI) consist of a Kramers pair of right- and left-moving electron modes of opposite spin situated inside the bulk gap Kane and Mele (2005); Bernevig et al. (2006); Wu et al. (2006); Xu and Moore (2006), and they have so far been observed in HgTe/CdTe quantum wells (HgTe-QW) Bernevig et al. (2006); König et al. (2007); Roth et al. (2009). In 3D topological insulators, the edge states cover the surface of the material and consist of a single-valley Dirac cone with spin-momentum locking, which leads to unique electromagnetic properties and quantum interference effects Qi and Zhang (2011); *hasan2010-cti. In both 2D-TI and 3D-TI the coupling of spin and orbital motion can lead to interesting effects when combined with superconductivity. Superconducting correlations induced by the proximity of a singlet -wave superconductor can inside the TI obtain a -wave character, which can be used to engineer Majorana bound states. Fu and Kane (2008); Linder et al. (2010); Stanescu et al. (2010) A somewhat similar induction of non-conventional correlations has also been proposed to occur in other semiconductor systems in the combined presence of the spin-orbit interaction and superconductivity. Sau et al. (2010a); *sau2010-rom; *alicea2010-mfi; *linder2010-mfm

When the edge state of a 2D-TI is coupled to a singlet superconductor, the transfer of electrons between the systems can, first of all, induce singlet-type proximity correlations between electrons in the right and left moving modes (the channel). Fu and Kane (2008) This already leads to several effects of interest. For instance, the helicity of the electron liquid lifts the spin degeneracy and enables Majorana states, Fu and Kane (2009) causes Cooper pairs to split, Sato et al. (2010) and affects transport properties. Adroguer et al. (2010) Tight-binding calculations studying the pair amplitude have also been made Black-Schaffer (2011). There is, however, also a possibility of inducing correlations only within the right-moving (or the left-moving) channel at a nonzero total momentum (the and channels). Such a channel is not forbidden by symmetries in the problem: due to the spin-orbit coupling, the spin axis of the edge state is not necessarily constant, so that the electrons forming a Cooper pair singlet can both enter the same mode on the TI edge, even when spin is conserved in the tunneling process and time-reversal symmetry is present. In HgTe-QW, a non-constant spin axis can be induced externally by the Rashba spin-orbit coupling that breaks inversion symmetry. Roth et al. (2009) Momentum conservation is required to be broken, but this can occur e.g. due to inhomogeneity or a finite size of a tunneling contact. Moreover, unlike in metals, in 2D-TI the momentum non-conservation can in principle be made arbitrarily small by tuning the Fermi level near the Dirac point ().

A straightforward way to probe the existence of superconducting correlations is to observe the Josephson effect or other interference effects that can be modulated with superconducting phase differences. The Josephson effect has been studied previously in various one-dimensional Luttinger liquid systems. Fisher (1994); Fazio et al. (1996, 1995) The finite-momentum channel has, however, received limited attention, Pugnetti et al. (2007) and is usually negligible. As shown below, certain experiments with superconducting contacts attached to the helical edge states can nevertheless probe such microscopic aspects of the tunneling, including the role of the Rashba interaction.

Here, we first derive a low-energy Hamiltonian describing the superconducting proximity effect in the edge states of a 2D TI coupled to a conventional superconductor by tunnel contacts. We use it to find the signatures of both types of tunneling events in a transport experiment. Because of the reduced number of propagating modes in the helical liquid, correlations within the same channel occur at a finite momentum and, as in chiral liquids, Fisher (1994) are affected by the exclusion principle. It turns out that although this component of the proximity effect gives a negligible correction to the dc Josephson effect, in the NS tunneling conductance [see Fig. 1(c)] it manifests as a long-ranged interference effect, oscillating as a function of the superconducting phase difference, and unlike the part, is not exponentially suppressed at length scales longer than the thermal wavelength. The ratio of the contributions of the two possible channels scales as (in the noninteracting case), where characterizes the strength of Rashba interaction, is the Fermi velocity of the edge channels, is the energy gap of the TI, and the distance between two superconducting contacts forming the interferometry setup. The amplitude of the effect is proportional to the amount of spin rotation achieved by Rashba interaction, and the quadratic temperature dependence is due to the exclusion principle. We also discuss how - interactions modify this result.

This paper is organized as follows. In Section II, we introduce the model for the helical Luttinger liquid (HLL), the coupling to the superconductors, and the electronic structure of HgTe-QWs. Section III discusses the effective low-energy Hamiltonian, and Section IV transport signatures in the dc and ac Josephson effects and the NS conductance. Section V concludes the manuscript with a discussion on the results and remarks on experimental realizability.

Ii Model

We consider the setup depicted in Fig. 1. The edge states of a 2D-TI are coupled to two superconducting terminals via two tunnel junctions. Below, we in general assume that the distance between the contacts is longer than the superconducting coherence length .

The left- and right moving edge states and have a linear dispersion, and are described by the bosonized Hamiltonian Wu et al. (2006)

 H0=12∫∞−∞dxu[g−1(∂xϑ)2+g(∂xϕ)2] (1)

where the Fermi field operator is , the standard boson fields , satisfy , and are the Klein factors. is the renormalized Fermi velocity. Here and below, we let , unless otherwise mentioned. The parameter is the short-distance cutoff. In the noninteracting case, the Luttinger interaction parameter , and with repulsive electron-electron interactions one has .

The coupling to the superconductors is modeled with a tunneling Hamiltonian

 HT=∑α=±,σ′=↑,↓∫dxd3r′tασ′(x,→r′)ψ†α(x)ψSσ′(→r′)+h.c., (2)

where the tunneling amplitude describes the tunneling from the state in the superconductor to state in the edge mode. For what follows, it is useful to introduce also the corresponding one-particle operator , in terms of which,  . The momentum along the edge is a good quantum number for straight TI edges, and we define the state in the momentum representation: , where is the edge eigenstate with momentum and propagation direction .

We assume that the Hamiltonian is time-reversal symmetric, which implies that the tunneling operator in general satisfies . Here, we choose the phases of the wave functions so that the time reversal operations read and . We also assume that the tunneling is spin-conserving, that is, written in terms of real electron spin states in the TI and the superconductor, we have .

To describe tunneling to HgTe-QWs, we need some knowledge of the structure of the edge states. This can be obtained from the four-band model used in Ref. Bernevig et al., 2006. In this approach, the low-energy properties of the TI are described using a 2D envelope function in the basis of four states localized in the quantum well. Bernevig et al. (2006); Novik et al. (2005) The edge states at the boundaries of the TI can be solved within this four-band model; Zhou et al. (2008) for which we give a full analytical solution in Appendix A.

We assume the terminals are conventional spin-singlet superconductors. As usual, Abrikosov et al. (1975) they are characterized by the correlation function that has a singlet symmetry . In the bulk, the correlation function obtains its equilibrium BCS form, which in imaginary time can be written as

 F(→r1,→r2;ω)=∫d3k(2π)3e−i(→r1−→r2)⋅→kΔω2+ξ2k+|Δ|2, (3)

with the dispersion relation and the gap of the superconductor.

Iii Effective Hamiltonian

Integrating out the superconductors using perturbative renormalization group theory (RG) and considering only energies reduces the Hamiltonian of the total system to one concerning only the one-dimensional edge states:

 H =H0+∫dx[Γ+−(x)ψ+(x)ψ−(x) (4) +Γ++(x)ψ+(x)ψ+(x+a) +Γ−−(x)ψ−(x)ψ−(x+a)+h.c.].

Here, describe the coupling to the superconductor, and is the new short-distance cutoff in the theory. Details of the derivation are discussed in Appendix B.

The coupling factors in the noninteracting case () are given by the expressions (see Appendix B for general discussion):

 Γ++(x)=π2∫d3r′1d3r′2∑KeiKxF(r′1,r′2;0) (5) ×Δv−1F∂kP++(K2+k,r′1;K2−k,r′2)∗|k=0,

where is given in Eq. (3), and

 Γ+−(x)=π∫d3r′1d3r′2∑KeiKxF(r′1,r′2;0) (6) ×P+−(K2−kF,r′1;K2+kF,r′2)∗,

with in the presence of time reversal symmetry. The main contributions should arise around for (due to the oscillations in the Fermi operators), and around for .

The coupling is proportional to the factor

 Pα1α2(k1,→r′1;k2,→r′2) ≡[tα1↓(k1,→r′1)tα2↑(k2,→r′2) (7) −tα1↑(k1,→r′1)tα2↓(k2,→r′2)]+[→r′1↔→r′2],

which describes two-particle tunneling of a singlet, from two points and in the superconductor, to momentum states , in the TI edge modes (cf. Fig. 2). Here, is a Fourier transform of the tunneling matrix element.

One can also verify that in the absence of interactions, the expression for the amplitude coincides with the leading term in the zero-bias conductance in the normal state, up to a replacement . Within a quasiclassical approximation in the superconductor, Rammer and Smith (1986) one then finds a relation to the normal-state conductance per unit length, , of the tunnel interface:

 Γ+−(x)≃14ℏvFRKg(x)=ℏvFlTRK4RN. (8)

Such a relation is typical for NS systems, and connects the amplitude to observable quantities. The latter expression assumes the total resistance is uniformly distributed in a junction of length . When the interface resistance decreases, the effective pairing amplitude grows — and although not included in our perturbative calculation, one expects that this increase is cut off when the effective gap reaches the bulk gap of the superconductor, .

Unlike , the amplitudes do not have a direct relation to the normal-state conductance, and they depend on the factors , which are proportional to the spin rotation between the states involved in the pair tunneling (see Fig. 2). Estimating this factor is necessary for determining how large the same-mode tunneling is in a given system.

iii.1 Two-particle tunneling

Making use of the time-reversal symmetry, it is possible to rewrite the factors in a more transparent form:

 Pα1α2(k1,→r′1;k2,→r′2)=⟨α1,−k1|^Z(→r′1,→r′2)T|α2,−k2⟩, (9) ^Z(→r′1,→r′2)≡^hT[1σ⊗(|→r′1⟩⟨→r′2|+|→r′2⟩⟨→r′1|)]^hT, (10)

where is the identity matrix in the spin space of the superconductor. Unlike the starting point, this expression is explicitly independent of the choice of the spin quantization axis. We also note the symmetry:

 Pα1α2(k1,r′1;k2,r′2) =−Pα2α1(k2,r′1;k1,r′2), (11)

following from the definition Eq. (7).

We can now make some remarks on the possibility of tunneling. First, suppose that the state describes an electron wave function with a fixed -independent and spatially constant spin part, and that the tunneling is spin conserving. In this case it is easy to see that , as the inner product of a spinor and its time reversed counterpart vanishes. Such a situation is realized, for instance, within the plain Kane-Mele model. Kane and Mele (2005) Breaking such conditions can, however, lead to . We demonstrate in the next section that this can occur in HgTe-QW.

iii.2 Effect of Rashba interaction in HgTe/CdTe quantum wells

We now discuss a simple model for tunneling into the helical edge states of a HgTe-QW, taking spin axis rotation from the Rashba interaction into account. We make the following assumptions: the tunneling is spin-conserving and local [] on the length scales of the four-band model. This results to all contributions to coming solely from the Rashba mixing. While we cannot estimate the actual values of or within this simplified a model, we can study their relative magnitudes, which is now determined by the low-energy four-band physics only.

Under the locality and spin-conservation assumptions, the tunnel matrix element introduced above obtains the following form in terms of envelope spinor wave functions in the four-band basis :

 Pα1α2(k1,→r′1;k2,→r′2)=^Ψα1,−k1(x′1,y′1)†^Z(→r′1,→r′2)×T^Ψα2,−k2(x′2,y′2)+[→r′1↔→r′2], (12) ^Z(→r′1,→r′2)jj′ =⟨j|hT[1σ⊗|→r′1⟩⟨→r′2|]hT|j′⟩. (13)

Time reversal for the four-band spinor reads with the complex conjugation, and the matrix acts on the Kramers blocks (, ). For simplicity, we use now a length-scale separation between the scales appearing in the four-band model () and the atomic ones (tunneling , in the superconductor, unit cell). We consider only the long-wavelength part of , and replace with a constant describing the tunnel coupling to the quantum well basis states, obtained by averaging it together with [cf. Eqs. (5), (6)] over and :

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯[Z(→r′1,→r′2)+Z(→r′2,→r′1)]F(→r′1,→r′2) ∼⎛⎜ ⎜ ⎜ ⎜ ⎜⎝A(→r′1)C(→r′1)0D(→r′1)C(→r′1)∗B(→r′1)−D(→r′1)00−D(→r′1)∗A(→r′1)C(→r′1)∗D(→r′1)∗0C(→r′1)B(→r′1)⎞⎟ ⎟ ⎟ ⎟ ⎟⎠F(0)δ(→r′1−→r′2), (14)

with and real-valued. This form follows from the time reversal symmetry and hermiticity of the matrix elements of the operator in Eq. (10). We have also assumed here that the decay length for the function () is short on the scales of the 4-band model. Finally, we for simplicity neglect the coupling to the band, and set . For a lateral contact (SC on top of HgTe-QW), the main tunnel coupling is expected to involve the E1 band, which extends deeper Bernevig et al. (2006) into the CdTe barrier than H1. Including additional couplings would however cause no essential qualitative differences in the estimated ratio between the and terms.

Without additional spin axis rotation from the Rashba interaction, the edge states are in separate Kramers blocks (see Appendix A), and , and we can see that whereas . Note that a contribution proportional does not arise: the unperturbed edge state wave functions are both proportional to the same constant real-valued spinor, , so that the -dependent contribution would be proportional to . This structure also implies that contributions proportional to do not arise in the leading order of the Rashba coupling.

Rashba and other related spin-orbit interactions in the four-band model can be represented as Rothe et al. (2010)

 HR=(0hRh†R0),hR=i(−R0k−δ+iS0k2−−δ−iS0k2−T0k3−), (15)

where . For the QW parameters used in Ref. Rothe et al., 2010, , , and , where is the electric field perpendicular to the QW plane. The model could also include the bulk inversion asymmetry terms . König et al. (2008)

To obtain the effect of the Rashba interaction on the wave functions, we find the low-energy eigenstates of numerically. For given , this is a 1-D eigenvalue problem in the -direction, which can be discretized and solved by standard approaches. Analytical results can be obtained by perturbation theory in restricted to the low-energy subspace spanned by the unperturbed edge states. For typical experimental parameters, the whole wave functions however turn out to have a significant component also in the continuum of bulk modes above the gap, which is not adequately captured by such an approach. Our estimates for the matrix elements below are therefore based on the numerical solutions for the eigenstates.

However, qualitative understanding can be obtained on the basis of the model restricted to the low-energy subspace. Projecting to this basis (see Appendix A), we find the effective low-energy Hamiltonian of the system 111 Because the analytical edge state wave functions have a discontinuous derivative due to the boundary condition, the matrix element is better rewritten as , to remove the need to evaluate boundary terms.

 H′R =(0−i[R0w1(k)+T0w3(k)]c.c.0), (16) w1(k) =χ21∫∞−∞dyf+,k(y)[k+∂y]f−,k(y) (17) w3(k) =χ22∫∞−∞dyf+,k(y)[k+∂y]3f−,k(y), (18)

where . This result is valid to the leading order in . The constant and quadratic in terms (proportional to and ) give no contribution, as . Using typical HgTe-QW parameters, Rothe et al. (2010) the integrals evaluate to and near the Dirac point, as illustrated in Fig. 3. The prefactor is essentially independent of the mass parameter , and . Note here that the matrix element of the Rashba interaction with the edge states is significantly smaller than the appearing in the bulk Hamiltonian. The effective Hamiltonian yields the wave functions:

 ^Ψ+,k≃⎛⎝^Φ+,kiw02vF^Φ−,k⎞⎠,^Ψ−,k≃⎛⎝iw02vF^Φ+,k^Φ−,k⎞⎠, (19)

where . The Rashba interaction mixes the two Kramers blocks, but in the leading order does not modify the energy dispersion. Although the mixing angle of the spinors is independent of , the total four-band spinor is not: the decay lengths of in the -direction depend on and are different for the and states: time-reversal symmetry only guarantees . This makes the electron spin axis to rotate both spatially and with energy , which ultimately is required for a finite .

For comparison, we show in Fig. 4 the component of the numerically computed total edge state wave function , and its projection to the low-energy subspace, which can be seen to match Eqs. (19) to a very good accuracy. The component is proportional to the Rashba coupling and contributes to . As is clearly visible in the figure, neglecting the bulk states underestimates the total amount of spin rotation, for experimentally relevant parameters. For a larger (but unphysical) value for the gap , the low-energy theory works slightly better, as visible in the inset of Fig. 4.

We can now estimate the relative order of magnitude between and within this model. From the results above, one can see that the representative quantities to be compared are

 c++(K,→r′1,→r′2) =ΔiℏvF∂kP++(K2+k,→r′1;K2−k,→r′2)|k=0, (20)

and

 c+−(K,→r′1,→r′2) =P+−(K2−kF,→r′1;K2+kF,→r′2). (21)

In Fig. 5 we show the ratio of these amplitudes for (i.e., the value at a distance from the edge). The amplitude increases when the energies of the edge states involved approach the TI energy gap edge. The general order of magnitude of the factors can be estimated to be of the order

 c++∼Δ|M|z0ℏvFc+−. (22)

A similar relation is then expected also between the and factors for surface contacts to area near . Here and below, we characterize the strength of the Rasbha interaction with the quantity .

Using the above results for the order of magnitude of and we find (see Appendix B) the estimates for the general case with - interactions:

 Γ+− ≃(a0Δ)g+g−12−1Γg=1+− (23a) Γ++ ≃12(a0Δ)g+g−12−1z0ΔvF|M|Γg=1+−, (23b)

corresponding to cutoff . The relation (8) for the noninteracting value fixes the magnitudes relative to experimental parameters.

With finite electron-electron interactions () in the helical liquid, all the effective tunnel rates obtain identical scaling in the original short-distance cutoff . This reflects the renormalization of the single-particle tunneling elements by the electron-electron interactions.

We can also estimate the Rashba coupling factor appearing in . With a typical TI gap , we see that the factor of can be made of the order of with conventional superconductors, and can be even larger for smaller TI gaps. The second factor is , and as visible in Eq. (19), measures the rotation of the spin axis caused by the Rashba mixing. An upper limit for the field that can be applied in practice is likely of the order , as for fields larger than that, the potential difference across the QW becomes comparable to the energy gap of the barrier material (CdTe). Based on this we find an estimate for the achievable ratio, .

Finally, let us remark that tunneling that is local in real space, , does not lead to tunneling that is local in the edge state Hamiltonian, . This follows in a straightforward way from the extended 2-D nature of the edge states and the mixing due to the spin-orbit interactions: . If the spatial profile of the wave function has -dependence on the scale , the sum resembles a rounded function of width . For HgTe QW edge states, is a low-energy length scale. Because of this, a pointlike contact to a superconductor can produce a finite , even though assuming in Eq. (7) leads to the opposite conclusion.

Iv Transport signatures

To study the experimental signatures implied by the above model, we consider the transport problem in the setups depicted in Fig. 1. There, two superconducting contacts are coupled to a helical liquid, whose potential is tuned by additional terminals at the ends. There are three related transport effects one can study here: the equilibrium dc Josephson effect, the ac Josephson effect, and the NS conductance.

We consider a general nonequilibrium case of a time-dependent pair potential in the left contact and in the right one, with and . In Eq. (4), the factors inherit this time dependence. We also assume that only sub-gap energies are involved in the transport, so that the quasiparticle current to the superconductors remains exponentially suppressed by the superconducting gap.

The current is obtained as an expectation value of a current operator where is the particle number in the HLL. From the effective Hamiltonian, we identify

 ^I =^IS1+^IS2 (24) ^IS1 =∑αβ∫S1dx2iΓαβ(x)ψα(x)ψβ(x)+h.c., (25) ^IS2 =∑αβ∫S2dx2iΓαβ(x)ψα(x)ψβ(x)+h.c., (26)

where and must be interpreted as the parts corresponding to currents injected through the interfaces at and . The sums over run over , , and .

Considering only the Cooperon terms [cf. Fig. 7(a)], using perturbation theory up to second order in we find

 IJ,S1(t) =−8Im∑αβ∫S1dx1∫S2dx2Xαβ(x1,x2) (27) ×eiφ1(t)∫∞0dt′e−iφ2(t−t′)Im[χαβ(x1−x2,t′)], IJ,S2(t) =IJ,S1(t)|φ1↔φ2, (28)

where

 Xαβ(x1,x2) ≡e2ikF(α+β)(x1−x2)Γαβ(x1)Γαβ(x2)∗ (29) χαβ(x;t) =⟨eiϕα(x,t)eiϕβ(x,t)e−iϕα(0,0)e−iϕβ(0,0)⟩0(2πa)2. (30)

The component of the current coincides with the result obtained in Ref. Fazio et al., 1995. Note that the terms included here contain the leading order of the dependence in the phase difference .

The above correlation functions can be evaluated via standard bosonization techniques: Giamarchi (2004)

 χαα(x,t) =(2πa)−2Bα(x,t)g+g−1+2B−α(x,t)g+g−1−2, (31a) χ+−(x,t) =(2πa)−2B+(x,t)1/gB−(x,t)1/g, (31b) B±(x,t) =−iazsinh[z(ut−ia∓x)], (31c)

where .

In the noninteracting case (), we can evaluate the time integrals analytically, to order :

 IJ,S1(t) =∫S1dx1∫S2dx2[j++J,S1+j−−J,S1+j+−J,S1] (32) j++J,S1 =|X++|3πvFV2[(V2/Δ)2+4π2(T/Δ)2] (33) ×cos(φ0+2V1t−2V2(t−|x1−x2|/vF)+ϕ0), j−−J,S1 =0, (34) j+−J,S1 =−|X+−|πvF2zsinh(2|x1−x2|z) (35) ×sin(φ0+2V1t−2V2(t−|x1−x2|/vF)),

where is a dynamical phase shift.

Below, we discuss the implications of these results first at equilibrium and then at finite biases.

iv.1 Equilibrium

At equilibrium, the leading contribution to the supercurrent comes from the channel. As shown in Fig. 6, the supercurrent is finite at zero temperature, and decays exponentially as the temperature is increased above , in a way that depends on the strength of electron-electron interactions. The qualitative features are the same as those found in Ref. Fazio et al., 1995.

The contribution from the and channels to the equilibrium current is not more relevant than even in the interacting case, unlike in Ref. Pugnetti et al., 2007. Based on scaling dimensions in the effective Hamiltonian (, ), one finds the scaling and for the low-energy scale , which implies that will be more relevant than whatever the interaction parameter. This difference arises from the exclusion principle, which makes the channel less favorable for the supercurrent, although note that with decreasing (larger repulsive e-e interaction), the () contribution grows relative to the one. However, as noted in Section B, the scaling with the bare short-distance cutoff as opposed to is identical for and .

iv.2 Nonequilibrium

When the superconductors are biased with a finite voltage, currents generically start to flow between all the terminals, and they may also be time dependent due to the ac Josephson effect. To fully understand these effects, it is illuminating to compute the spatial distribution of the currents in the system.

The spatial dependence of the currents in the helical liquid can be obtained by making use of the following expression for the current operator in the Heisenberg picture (see App. C): Virtanen and Recher (2011)

 ^I(x,t) =^I0(x,t)+vF∫∞−∞dx′dt′∑α=±α( (36) 1+g2gDα(x,t;x′,t′)−1−g2gD−α(x,t;x′,t′) )^j