Josephson effects in the junction formed by DIII-class topological and s-wave superconductors with an embedded quantum dot

We investigate the Josephson effects in the junction formed by the indirect coupling between DIII-class topological and s-wave superconductors via an embedded quantum dot. Due to the presence of two kinds of superconductors, three dot-superconductor coupling manners are considered, respectively. As a result, the Josephson current is found to oscillate in period 2π. More importantly, the presence of Majorana doublet in the DIII-class superconductor renders the current finite at the case of zero phase difference, with its sign determined by the fermion parity of such a junction. In addition, the dot-superconductor coupling plays a nontrivial role in adjusting the Josephson current. When the s-wave superconductor couples to the dot in the weak limit, the current direction will have an opportunity to reverse. It is believed that these results will be helpful for understanding the transport properties of the DIII-class superconductor.

This formula indicates that the calculation about the Josephson current is dependent on the solution of the ground-state (GS) level of this system.
In the low-energy region, the DIII-class TS only contributes Majorana doublets to the Josephson effect, so H p describes the coupling between two Majorana doublets. For the extreme case of one infinitely-long TS, the coupling strength between the Majorana doublets decreases to zero 35 . Following this idea, we project H T onto its zero-energy subspace. As a result, H can be rewritten as  . γ 0σ , the Majorana operator, which obeys the anti-commutation relationship of {γ 0σ , γ 0σ′ } = 2δ σσ′ . Based on the renewed expression of H, we next try to diagonalize the Hamiltonian of such a Josephson junction.

Diagonalization of the Junction Hamiltonian
The continuum state in the s-wave superconductor hinders the diagonalization of the system's Hamiltonian. In order to present a comprehensive analysis, we would like to consider three cases, i.e., the cases of t T ≫ t S , t T ≈ t S , and t T ≪ t S , followed by the application of different approximation methods. The following are the detailed discussion processes. For convenience, they are named as Case I, Case II, and Case III, respectively.

Diagonalization of H in Case I
In Case I where t T ≫ t S , the subsystem formed by the QD and Majorana doublet can be considered to be one system, whereas the s-wave superconductor can be viewed as a perturbation factor. We next simplify the system Hamiltonian with the help of the perturbation theory. Ignoring the Coulomb interaction term in the QD, we can write out the action of the subsystem of QD and s-wave superconductor kS z with σ α being the pauli matrix (σ = x, y, z). As for the field operators, they are given by With the action S, we can express the partition function as a path integral, i.e., S , in which the measure Ψ k  denotes all the possible integral paths. After integrating out the fermion field Ψ Ψ † , k k with a Gaussian integral, the partition function will become a "generating functional" is defined in the s-wave superconductor. It obeys the Fourier expansion   Via a unitary transformation, the system Hamiltonian can be expressed as the following form . Such a result indicates that the weak Andreev reflection between the s-wave superconductor and QD induces a weak s-wave pairing potential on the QD, which is exactly the so-called the proximity effect.
For the sake of diagonalizing such a Hamiltonian, we need to introduce local Majorana operators η iσ through

. And then, by defining Dirac fermionic operators
Based on Eq. (12), the Bogoliubov-de Gennes equation H eff Ψ = EΨ can be built up, and then the eigenvalues of H eff can be worked out. On the basis of {|000〉 , |001〉 , |010〉 , |100〉 , |110〉 , |101〉 , |011〉 , |111〉 }, the matrix form of H can be obtained (| 〉 = | 〉| 〉| 〉  n n n n n n Note that in the TS-existed system, only the parity of the average particle occupation number is the good quantum number, thus the matrix form of H eff should be given according to FP. In the case of odd FP, . Thus, the Josephson effect can be clarified by paying attention to the current oscillation result in one FP.

Diagonalization of H in Case II
In Case II where t T ≈ t S , H is difficult to diagonalize due to the presence of continuum state in the s-wave superconductor. However, according to the previous works, the zero band-width approximation is feasible to solve the Josephson effect contributed by the s-wave superconductor 36 . Within such an approximation, the Hamiltonian can be simplified as 2 , we get the Hamiltonian in the spinless-fermion representation , and t N = ξ + iΔ s cosθ. On the basis of even FP, the matrix of H e eff ( ) can be given by

Diagonalization of H in Case III
In Case III, we turn to the discussion about the diagonalization of H when t T ≪ t S . In such a case, the QD will dip in the s-wave superconductor, leading to the formation of a composite s-wave superconductor. Consequently, the considered structure will be transformed into a junction in which the Majorana doublet couples to a s-wave superconductor directly. Its Hamiltonian can thus be written as In Eq. (19), F kσ originates from the unitary transformation that

Numerical Results and Discussions
Following the theory in the section above, we proceed to calculate the Josephson current in our considered junction. As a typical case, the system temperature is taken to be zero in the context. Besides, we take Δ s = 1.0 to be the energy unit of the structural parameters.
First of all, we would like to investigate the odd-FP Josephson current in Case I. In Fig. 2, we plot the spectra of the Josephson current as a function of phase difference θ. As for the QD level, we change ε 0 from − 3.0 to 2.0. Besides, the intradot Coulomb strength is assumed to be 0.0 and 2.0, respectively. In this figure, we can find that despite the change of ε 0 and U, the leading oscillation property of the Josephson current is relatively robust, since it reaches the maximum at the point θ = nπ with its profile as . Meanwhile, the roles of QD level and Coulomb strength can be clearly observed. For instance, with the departure of the QD level from energy zero point, the current amplitude will be suppressed gradually. Such a result is relatively apparent in Fig. 2(a,b) which correspond to the case of the zero Coulomb interaction. The reason can be explained as the weakness of quantum coherence when the QD level departs from energy zero point. For the effect of Coulomb interaction, it is more apparent in the region of ε 0 < 0, where the QD level is occupied. It can be seen that the Coulomb interaction suppresses the current amplitude as well. This should be attributed to the destruction of the quantum coherence induced by the QD-level splitting (ε 0 → ε 0 and ε 0 + U) in the presence of Coulomb interaction.
According to the discussion about Case II in the second part of Sec. II, the s-wave superconductor should not be viewed as perturbation when t S gets close to t T . It is easy to think that in such a case, the Josephson current will show new properties. Thus, we would like to increase the coupling strength between the QD and s-wave superconductor to discuss the change of Josephson effect. The numerical results are shown in Fig. 3 where the t T = t S = 0.5. In this figure, we see that in Case II, the current properties are completely different from those in Case I. To be concrete, the current amplitude is efficiently enhanced and the current direction is completely reversed by the increase of t S . The other result is that the current profile deviates from the relationship of θ I cos J o ( ) when ε 0 ≠ 0. Such a phenomenon can be understood as follows. In the case of t T = t S , practical Cooper-pair tunneling occurs between the TS and s-wave superconductor via the QD. When the QD level departs from the energy zero point, the phases of electron and hole are modified, and then the Cooper-pair tunneling process is changed.
In what follows, if the coupling between the QD and s-wave superconductor further increases, the QD will be submerged in the s-wave superconductor. Consider the extreme case of the weak-coupling limit (i.e., Case III), the perturbation method can also be employed to evaluate the Josephson current, as displayed in the third part in Sec. II. It clearly shows that in such a case, Majorana doublet couples weakly to the composite s-wave superconductor. Consequently, the s-wave superconductor contributes an effective coupling between the two MBSs of one . In such case, the current properties become very well-defined, as described by the results in Fig. 4.
In view of the current results in Case I, Case II, and Case III, one can observe that at the case of t T ≫ t S (i.e., Case I), the current oscillation manner is opposite to that in the other two cases. In order to clarify the change of Josephson current from Case I to Case III, we plot the geometries of these three cases in the Nambu representation, as shown in Fig. 5. In Fig. 5(a-c), we notice that the finite coupling between the two MBSs of Majorana doublet, despite the direct or indirect coupling, gives rise to the occurrence of anomalous Josephson effect. On the other hand, the coupling strength between the QD and s-wave superconductor can vary the inter-MBS coupling property, leading to the change of the current oscillation manner. In the case of t T ≪ t S , the MBSs couple directly to each other with a constant coupling parameter. In such a situation, the current direction is only dependent on the FP of the Majorana doublet with θ =       I Jcos J e P . In the other case where t T ≫ t S , the coupling between the QD and Majorana doublet induces the indirect inter-MBS coupling, as shown in Fig. 5(a). With respect to the inter-MBS coupling in these two cases, we find that in the former case, the MBSs couple to each other via a nonresonant Andreev reflection process, whereas in the latter case, one bound state is involved in the Andreev reflection process. It is well known that the quasi-particle phase will undergo a π-phase shift due to the presence of one bound state in the Andreev reflection process. Accordingly, for the case of identical FP, the current oscillations in Case I and Case III are opposite to each other. By the same token, we can easily see that in Case I and Case II, the oscillation manners of the Josephson current are opposite to each other, because an additional bound state is presented in the Andreev reflection process in Case II. Up to now, we have known the reason that in the considered junction, the current in the case of t T ≫ t S is different from that in the other cases. Note, additionally, that in such a structure, the role of the s-wave superconductor is to provide a channel for the coupling between the MBSs in the Kramers doublet and the QD is to change the channel property. Due to this reason, the change of ε 0 and U cannot induce any phase transition behaviors for the Josephson effect.

Summary
To summarize, we have discussed the Josephson effects in the junction formed by the indirecting coupling between a one-dimensional DIII-class TS and a s-wave superconductor via an embedded QD. Via considering three QD-superconductor coupling manners, i.e., t T ≫ t S , t T = t s , and t T ≪ t s , we have presented a comprehensive analysis about the Josephson effect in this system. As a consequence, it has been found that the Josephson current oscillates in period 2π. Moreover, the presence of Majorana doublet in the DIII-class TS renders the Josephson current finite in the case of zero phase difference between the superconductors. The other interesting result is that in addition to the FP of this system, the coupling strength between the QD and s-wave superconductor can affect the current direction. To be concrete, when the coupling between the QD and s-wave superconductor decreases to its weak limit, the direction of the Josephson current will have an opportunity to reverse. After analyzing the particle motion in this structure, we have demonstrated the reason for such a result. Namely, the QD-superconductor coupling manner can modulate the property of the Andreev reflection between the MBSs in the Majorana doublet. We believe that this work can be helpful for understanding the transport properties of the DIII-class TS.
At last, we notice that some previous work has also reported the nonzero-supercurrent phenomenon in the case of zero phase difference between two superconductors. For instance, ref. 37 describes a Josephson junction between two s-wave superconductors with an embedded QD. It shows that in the presence of spin-orbit interaction and a suitably oriented Zeeman field in the QD, the spontaneously-broken TRS leads to an anomalous