Parity dependent Josephson current through a helical Luttinger liquid

We consider a superconductor-two dimensional topological insulator- superconductor junction (S-2DTI-S) and study how the 2{\pi}- and 4{\pi}-periodic Josephson currents are affected by the electron-electron interaction. In the long-junction limit the supercurrent can by evaluated by modeling the system as a helical Luttinger liquid coupled to superconducting reservoirs. After having introduced bosonization in the presence of the parity constraint we turn to consider the limit of perfect and poor interfaces. For transparent interfaces, where perfect Andreev reflections occur at the boundaries, the Josephson current is marginally affected by the interaction. On the contrary, if strong magnetic scatterers are present in the weak link, the situation changes dramatically. Here Coulomb interaction plays a crucial role both in low and high temperature regimes. Furthermore, a phase-shift of Josephson current can be induced by changing the direction of the magnetization of the impurity.


Introduction
Two-dimensional topological insulators (2DTI) are characterized by a gapped bulk spectrum and gapless edge states which are robust against time-reversal invariant perturbations [1,2]. Originally predicted for HgTe/CdTe quantum wells in Refs. [3,4], and observed in Ref. [5], these systems have attracted great attention in the last few years because of their peculiar electronic transport properties. Edge states possess a helical nature, namely electrons have spin direction and momentum locked to each other and constitute Kramer partners. Central for the present paper is the study of hybrid 2DTI-Superconductor (S) systems, a topic which has lately gained an increasing interest both theoretically and experimentally. A comprehensive description of the activity in this field can be found in Ref. [6]. The proximity effect into a 2DTI has been largely investigated (see for example [7,8,9,10]). On the experimental side Andreev reflection at S-2DTI interfaces has been recently observed by Du in Ref. [11]. The, not yet observed, Josephson effect through a topological insulator is expected to show spectacular features. Indeed in 2009 Fu and Kane [12] considered a Josephson junction with two s-wave superconductors connected by a weak link of length L, obtained by a single edge of a 2DTI. In the short-junction regime (i.e. when L ξ, with ξ the BCS coherence length), they showed that the S-2DTI-S junction exhibits a fractional Josephson effect [13,14,15,16] in which the current phase relation has a 4π-periodicity, rather than the standard 2πperiodicity, if the fermion parity (parity of the number of electrons in the system) is preserved. This phenomenon is related to the presence of Majorana fermions [14] at the S-2DTI interfaces. More recently Beenakker et al . [17] addressed the long-junction regime (L ξ) showing that the amplitude of the 4π-periodic critical current, at zero temperature, is doubled with respect to the 2π-periodic one. The AC Josephson effect in S-2DTI-S has also been considered [18].
In all papers mentioned above, the effect of the electron-electron interaction on the parity dependent Josephson current (JC) was neglected. Aim of this paper is to study the 2π-and 4π-periodic Josephson effect in the long-junction regime for a S-2DTI-S system taking into account the Coulomb interaction within the framework of the helical Luttinger liquids [19,20]. Firstly we consider the case of transparent S-2DTI interfaces and then we address the presence of magnetic impurities in the weak link [21,22,23,24]. Non-magnetic impurities cannot induce elastic back scattering in a helical liquid [25]. We present analytical results for both high and low temperature regimes. If no impurities are present, at low temperature the JC exhibits a saw-tooth behavior, the 4π-periodic critical current is doubled with respect to the 2π-periodic one as in the non-interacting case. At high temperatures both the 2π-and 4π-periodic currents are sinusoidal and the 2π-periodic current is suppressed with respect to the 4π-periodic one. Our results agree with a recent paper by Crépin et al . [26]. If point-like magnetic impurities are present within the weak link the situation changes significantly. A single impurity with magnetization along the z-direction, which is the spin quantization axis of the helical states, induces a phase shift of the JC with respect to the transparent regime. Both the 2π-and 4π-periodic critical currents are not affected by such barrier. Otherwise, when the impurity magnetization lies in an arbitrary direction of the xy-plane, the current phase relation is always sinusoidal and the critical current depends on fermion parity and on the Luttinger interaction parameter. The previous results can be generalized if two barriers are present at the 2DTI-S interfaces.
The paper is organized as follows. In Section 2 we give a brief introduction to the bosonization technique with Andreev boundary conditions and discuss how the fermion parity can be implemented within the bosonization formalism. In Section 3 we discuss the 2π-and 4π-periodic JC for different regimes of the system. In Section 4 we summarize our results together with the conclusions of our work.

Model of the S-2DTI-S system
The setup we consider is depicted in Figure 1. A single edge state of a 2DTI is sandwiched between two s-wave superconducting terminals. The pair potential profile is with χ 1 , χ 2 the macroscopic phases of the superconductors, ∆ the modulus of the superconductive gap, and L the distance between the two superconductors. We ignore self-consistency in determining the order parameter this is why we can safely model the space profile of ∆ using the step function H(x). Figure 1. Helical states of a single edge of a two-dimensional topological insulator sandwiched between two s-wave superconductors (shaded areas).
The free Hamiltonian of the edge of the topological insulator is ( = 1): The operator ψ +/− annihilates right/left moving spin up/down electrons and v F is the Fermi velocity. Short-range interactions between two electrons in the weak link can be analyzed in the so called g-ology [27,28] framework. In a helical Luttinger liquid, one has a forward scattering term We have here neglected umklapp terms which are not relevant in the renormalization-group sense [19] if interactions are not too strong.
Point-like magnetic barriers in a generic point 0 ≤ x ≤ L of the weak link are described by the hamiltonian where M = (M x , M y , M z ) is the magnetization vector and σ = (σ x , σ y , σ z ) are the Pauli matrices acting on the spinor space Ψ = e ik F x ψ + , e −ik F x ψ − , with k F the Fermi momentum. By expanding the scalar product in Eq. (3), one obtains three terms: Finally the coupling of the edge to the superconducting electrodes, in the limit in which the superconducting gap is the largest energy scale in the problem, can be introduced through simple boundary conditions for the edge fermion field. Electrons impinging at normal-superconductor (in this case the normal part is the edge of the topological insulator) interface are retro-reflected as holes, this is the well known Andreev reflection. By solving the Bogoliubov -de Gennes equations, one can find the boundary conditions for fermionic operators which are essentially determined by Andreev reflections. In the limit ∆ → +∞, fermionic boundary conditions take a simple form because they are independent on the energy of the excitations (electrons can only be Andreev reflected and normal reflections do not occur), actually one finds [29] (at the two interfaces): As shown in Ref. [29] such conditions, known as Andreev boundary conditions, are equivalent to twisted periodic boundary conditions for ψ − on an interval of length twice the length of the original system supplemented by the connection between ψ + and ψ − following from the chiral symmetry Boundary conditions (6) and (7) can be conveniently done in the bosonization language which we introduce in the following sub-section.

Bosonization
Fermionic operators can be put into the form ψ ± (x) = exp [±iΦ ± (x)]/ √ 2πa in terms of the bosonic fields Φ ± (x) [27]. The Andreev boundary conditions (6) and (7) are automatically satisfied if the boson fields are chosen to be [29]    with In the previous equations a → 0 + is a convergence factor for the theory, χ ≡ χ 2 − χ 1 ; ϕ and N are conjugated zero-mode operators and the bosonic operators satisfy [a q , a † q ] = δ q,q with q = πn/L, n ∈ Z. Note that Φ + and Φ − obey canonical commutation relations and [N, ϕ] = 2i, the eigenvalues of N are even, N = 2k, k ∈ Z as implied by the boundary condition (6).
It is convenient to define the bosonic fields Φ = (Φ − + Φ + ) /2 and Θ = (Φ − − Φ + ) /2, thus fermion operators take the form In terms of the bosonic operators (10) the Hamiltonian of the system H = H F +H S2 +H S4 (the superconducting electrodes enter only via the boundary conditions) is the sum of a zero mode term H 0 and of a bosonic term H B 1 2 is the renormalized Fermi velocity and g = ((2πv F + g 4 − g 2 )/(2πv F + g 4 + g 2 )) 1 2 is the Luttinger parameter which is related to the attractive (g > 1) or repulsive (g < 1) nature of the interaction.
In the bosonized form the Hamiltonian (3) associated to the scattering from the magnetic impurities takes the form: We have defined φ x ≡ φ(x).

Fermion parity
In order to see how fermion parity is implemented in the bosonization language let's start by noting that the superconductive phase difference χ is defined up to multiples of 2mπ with m ∈ Z. Consequently if we substitute χ → χ + 2nπ (n ∈ Z) in Φ + and Using Eq. (7) with χ 1 set to zero through a proper gauge transformation, we get n = −m in Eqs. (15). We obtain the periodicity requirements satisfied by the even eigenvalues of the operator N , N = 2k, k ∈ Z. The field Φ(x) corresponding to the periodicity requirements (16) which obeys the condition Φ(x + 2L) = Φ(x) + 2mπ, will be called parity independent in the rest of the paper. We now assume that the superconductive phase difference χ is defined up to multiples of 4mπ, instead of 2mπ, with m ∈ Z. By carrying out the same procedure used for the parity independent case we obtain the analogous of Eqs. (16): that lead to Φ (E) (x + 2L) = Φ (E) (x) + 4mπ, which will be called even parity dependent [30]. Periodicity requirements given in Eqs. (17) are satisfied if and only if the eigenvalues of the operator N take the form N = 4k, k ∈ Z. Furthermore, by inducing an additional shift of the superconductive phase χ of 2π in Eqs. (17) we obtain the conditions π, which will be called odd parity dependent [30]. Periodicity requirements (18) are now satisfied if and only if the eigenvalues of the operator N take the form N = 4k + 2, k ∈ Z.
We can reach the same conclusions by studying the spectrum of the Hamiltonian (11) which is unchanged by a proper shift of the superconductive phase χ. If fermion parity is not conserved, namely the hamiltonian is invariant with respect to a shift of the form χ → χ + 2πm, one obtains that N → N − 2m and concludes the eigenvalues of N must be of the form N = 2k. If fermion parity is conserved χ → χ + 4πm, one obtains the constraint N → N − 4m or N → N − (4m + 2) and concludes that the eigenvalues must be of the form N = 4k in the even case and N = 4k + 2 in the odd one. Incidentally note that the number of fermions in the weak link is N/2 [31]. It is important to stress that constraints imposed by fermion parity conservation involve only the eigenvalues of the zero mode operator N and leave the bosonic excitation modes unaffected. Summarizing, the possible eigenvalues of the operator N are N = 2k, k ∈ Z if parity is not conserved and N = 4k (even) or N = 4k + 2 (odd), k ∈ Z in the opposite case.

The Josephson current
Equipped with the definitions given above we calculate the 2π-and 4π-periodic JC for different configurations of the system. Firstly we focus on the transparent regime where perfect Andreev reflections occur in correspondence of the S-2DTI interfaces, then we introduce magnetic barriers which induce normal reflections with spin-flip. The Josephson current can be computed from the partition function Z(χ) as where e is the elementary charge, β = 1/T , T the temperature and the Boltzmann constant k B = 1. The partition function can be expressed in the form where ∆ϕ = ϕ(β) − ϕ(0) = 2πm and m ∈ Z; S M is the Euclidean action corresponding to the Lagrangian of magnetic barriers, S 0 and S B to the Lagrangian of the Luttinger Hamiltonian introduced in Eq. (11): where Here n ≡ N/α, n ∈ Z and α = 2 if the fermion parity is not conserved, i.e. 2π-periodic case, and α = 4 if the fermion parity is preserved, i.e. 4π-periodic even case. In the next section we will also show how to calculate the 4π-periodic odd current. Moreover, we stress again that constraints on the eigenvalues of the zero mode operator N due to the fermion parity symmetry affect the Lagrangian L 0 , as one can see from Eqs. (22,23), but not L B .

Transparent interfaces
Firstly we calculate the 2π-and 4π-periodic supercurrent in the transparent regime for both low and high temperatures. The evaluation of the partition function is straightforward because Eq. (20) can be put into the form Z B is independent on χ and does not contribute to the Josephson current, as one can easily verify from Eq. (19). In order to perform the path integral in Eq. (24), we parametrize ϕ(τ ) as ϕ(τ ) = 2πmτ /β +φ(τ ), withφ(0) =φ(β) obtaining The integral in Eq. (25) does not contribute to the supercurrent since it is independent on χ. Using Poisson's summation formula [32] and neglecting constants which do not contribute to the JC, one has θ 3 is the elliptic Jacobi's function and A ≡ βu/L. In the low temperature regime A 1, the supercurrent exhibits a saw-tooth behavior As shown by Beenakker et al . [17] in the non-interacting case, the 4π-periodic critical current is twice the 2π-periodic one even in the presence of interactions. In the high temperature regime A 1, the JC Aα 2 g sin 2χ α has a sinusoidal behavior and the ratio between the 4π-periodic critical current (α = 4) and the 2π-periodic critical one (α = 2) is much larger than 2, since A 1. Let's now focus on the role played by interactions. For a generic interaction with g 2 = g 4 , one has ug = v F + (g 4 − g 2 )/(2π), i.e. the forward scattering term and the dispersive one act differently on the critical valued of the JC. Conversely, for the Coulomb interaction, one has g 2 = g 4 , as shown in [33]. In this case, the JC is completely unaffected by the Coulomb interaction because ug = v F .
In the odd parity conserving case the partition function has the form from which one can easily see that the corresponding current is equal to the even one translated of 2π.

One impurity
We start by considering a single magnetic impurity described by the Hamiltonian (14) in a generic point of the weak link whose magnetization has the same direction of the spin quantization axis, namely z-axis. The calculation proceeds similarly to the transparent regime, where L 0 is now given by with From the partition function we get the Josephson current in the low temperature regime and in the high temperature regime The values of the critical currents remain unchanged with respect to the transparent regime found in Sub-section 3.1. The current-phase relation, however, exhibits a phase shift whose magnitude depends on M z and, for the high-temperature regime, on the periodicity of the JC. Note that the supercurrent remains finite even at χ = 0 because time reversal symmetry is broken by the magnetic barrier.
We now consider a magnetic impurity in x = 0, or equivalently in x = L, with magnetization lying in an arbitrary direction of the xy-plane. The corresponding Hamiltonian is the sum of the Hamiltonians given in Eqs. (12,13) that can be conveniently written as with |M | = M 2 x + M 2 y and tan(δ 0 −π/2) = M y /M x , −π ≤ δ 0 ≤ π; φ 0 = φ(x = 0). The impurity potential is a relevant term in the renormalization group sense [34] if g < 1, then we consider the strong barrier limit, where the argument of the cos function in Eq. (36) is strongly pinned in the minima. The partition function of the system is given by Eq. (20), where S M = − β 0 dτ H M and S 0 consists of a linear term S 0,l in ∂ τ ϕ and of a quadratic term S 0,q , as shown in Eq. (22). Consequently Z(χ) can be expressed in the form where Here is an effective action obtained by integrating the degrees of freedom of S B away from the impurity (more details are given in Appendix A where the effective action is calculated for a generic point x in the weak link); ω = 2πn/β, n ∈ Z are the Matsubara frequencies and is a high-frequency cut-off action with M 0 = 1/Λ ≈ 1/∆. We evaluate the partition function in the semiclassical limit simply searching the stationary path of the action S ≡ S 0,q +S Λ +S M which gives the most relevant contribution to the functional integral. Such procedure is justified by the strong magnetic barrier limit. Then, we include the contributions of S ef f B , which plays the role of a dissipative environment, by integrating out the low energy fluctuations of ϕ and φ 0 around the stationary path [35,36,37,38].
It is convenient to introduce the fields where m L = 4L/(α 2 πug). The action S takes then the form The saddle point solution for φ r (τ ) obeys a Sine-Gordon equation which admits the instanton solution with J j=m e j = m, J ∈ N is the number of instantons. By calculating the saddle point action [39], one can show that the quantity Z m /Z 0 is proportional to δ J 1, where δ = λ − 1 2 exp −2M r √ λ , with λ ≡ |M |/(πaM 0 ) = Λ|M |/(πa) 1. In this limit, we can take m = 1 with J = 1 and Eq. (45) can be approximated with l ≡ l 0 − l. Due to the strong impurity potential, ϕ and φ 0 can fluctuate under the condition that ϕ + φ 0 is strongly pinned. Such low energy fluctuations can be taken into account by introducing the field ψ The partition function becomes with and integration over ψ gives In the previous expression = α 2 /4 − 1: if α = 2, = 0; if α = 4, = 3; remarkably no contribution toS ins arises from the ω = 0 term. By using the definition (19) of the JC definition one finds (more details are given in Appendix B.1) for the high temperature regime which is sinusoidal both in the 2π-and 4π-periodic case. Moreover the ratio between the 4π-periodic critical current and the 2π-periodic critical one is much larger than 1.
Keeping fixed the parity, the critical current is reduced by the Coulomb interaction according to the power 2/g. In the low temperature regime one obtains for the 2π-periodic JC and for the 4π-periodic JC (even). As expected, the odd 4π-periodic current can be obtained by translating the even 4π-periodic current of 2π. In the non-interacting case the 4πperiodic current is larger than the 2π-periodic one, as in the transparent regime. The sinusoidal behavior is a direct consequence of the strong barrier limit and it is not related to helical nature of the weak link: Andreev reflections at S-2DTI interfaces are strongly suppressed with respect to normal reflections induced by magnetic barriers. Interestingly, if the repulsive interaction is strong, the 4π-periodic critical current is more robust with respect to the 2π-periodic because it lacks of the exponential term exp[−2g −1 (γ + 2 ln 2)], thus the ratio between the 4π-periodic JC and the 2π-periodic one is bigger than 2. We note that the JC is unaffected by changing the direction of the magnetization in the xy-plane because Eqs. (52, 53,54) do not depend on δ 0 . Finally we point out that by varying the position of the barrier keeping fixed the direction of the magnetization in the xy-plane, different power laws are obtained. For example, if the barrier is in L/2 of the weak link, one obtains 4/g instead of 2/g for both high and low temperature regimes, i.e. in the middle of the junction the barrier acts strongly than at the interfaces.

Two impurities
Finally we discuss the case of two magnetic impurities at the S-2DTI interfaces, namely in x = 0 and x = L. The two magnetizations, equal in magnitude, lying in the xy plane and not collinear, are taken into account by the Hamiltonian with φ 0 ≡ φ(0, τ ) and φ L ≡ φ(L, τ ), or equivalently The parameters δ 1 and δ 2 specify the angle of magnetizations with respect to the y-axis, δ ≡ (δ 1 + δ 2 )/2 andδ ≡ δ 2 − δ 1 , k F L has been chosen proportional to π. The partition function takes the form S ef f B is the effective action obtained by integrating out the bosonic modes away from x = 0 and x = L of the Hamiltonian (56) [37]. and provides the high frequency cut-off (M = 2/Λ ≈ 2/∆ andM = 1/(2Λ) = 1/(2∆)). We proceed as in the single impurity problem and we calculate the functional integral in the semiclassical approximation. By introducing the fields with m L = 4L/(α 2 πug). The saddle point for S 0,q + S Λ + S M leads to the equations with the boundary conditions φ R (β) = φ R (0) + 2mπm L /(m L + M ) and φ r (β) = φ r (0) + 2πm, where M r = m L M /(m L + M ) and M R = m L + M . In the strong barrier limit it is sufficient to consider only m = 1. The instanton solutions take the form withl = l 1 + l 2 − 2l and l = l 1 − l 2 . We substitute J(ω) andJ(ω) in the action S 0,q + S ef f B and we take into account the low energy fluctuations of ϕ + φ as in the single impurity problem by introducing the auxiliary fields ψ, whileφ is kept strongly pinned. Integration over ψ gives two contributions with τ ≡ τ 2 − τ 1 , ≡ α 2 /2 − 2, for α = 2 one has = 0; while = 6 for α = 4; and Integration over τ 1 and τ 2 or equivalently over τ = τ 2 − τ 1 and τ = (τ 1 + τ 2 )/2 gives From Eq. (19), in the high-temperature regime, the supercurrent (more details on the calculations are given in Appendix B.2) exhibits a power law in β whose exponent depends on the strength of the interactions and it is unaffected by the angleδ between the two magnetizations. In the opposite limit of low temperatures the 2π-periodic JC shows a dependance on the angleδ between the two magnetizations through the modulation function with Γ(x) the Euler-Gamma function. The modulating function has a minimum iñ δ = 0, i.e. the two magnetizations are parallel and maxima inδ = ±π, i.e. the two magnetizations are anti-parallel and exhibits a weak dependance on g.
The 4π-periodic JC is given by where the modulation function η 4 (δ, g) can be expressed in terms of the Gaussian Hypergeometric function as which has a minimum inδ = 0 and maxima inδ = ±π.

Conclusions
In this paper we studied the parity-dependent Josephson current in a S-2DTI-S junction if L ξ taking into account the Coulomb interaction. For transparent S-2DTI interfaces no significant corrections arise with respect to the non-interacting case. When a single magnetic impurity whose magnetization is collinear with the spin quantization z-axis, the Josephson current is only shifted with respect to the transparent regime. If the magnetization lies in the xy plane, the current is sinusoidal and is strongly renormalized by the interaction. In particular, for a single barrier at the S-2DTI interface, the current is proportional to β − 2 g in the high temperature regime and to L − 2 g in the low temperature regime. The 2π-periodic critical current is more suppressed by repulsive interactions with respect to the 4π-periodic. If two impurities are present at the S-2DTI interfaces new power laws are obtained. Remarkably in the low temperature regime both the 2π-and 4π-periodic currents exhibit a dependance on the angle between the two magnetizations.

Appendix A.
In this appendix we calculate the effective bosonic action S ef f B in the single impurity problem. Let's consider the Hamiltonian H B defined in Eq. (11) which can be written as where φ and θ are the bosonic modes of the fields Φ and Θ and the corresponding lagrangian If L M is the lagrangian of a magnetic impurity in x, the functional integral Dφ exp β 0 dτ (L(τ ) + L M (τ )) can be simplified by integrating out the degrees of freedom not involved by the lagrangian L M [34,38]: where we have introduced the auxiliary fields λ(τ ) and φ x (τ ) whose Fourier series are λ(τ ) = 1/β ω λ(ω)e −iωτ and φ x (τ ) = 1/β ω φ x (ω)e −iωτ . Substituting in (A.3), we get Dφ x Dλ exp − i β ω λ(ω)φ x (ω) Dη q exp 1 β ω q>0 1 4πg uq + ω 2 uq |η q (ω)| 2 − π Lq λ(ω)η q (−ω) cos qx (A. 4) with η q (τ ) = a † q (τ ) − a q (τ ), q = πn L , n ∈ Z, which can be integrated over η q and then over the auxiliary field λ, obtaining As the sum on q can be exactly evaluated for an arbitrary x π L q>0 cos 2 qx 1 + u 2 ω 2 q 2 = − π 2L + πω 4u 1 + cosh Lω u 1 − 2x  Let us now consider the contributions to the partition function arising fromS ins2,ω . In this case one has to evaluate the integral (67) which can be cast in the form where 2 F 1 is the Gaussian Hypergeometric function.