Impact of Majorana fermions on the Kondo state in the carbon nanotube quantum dot

The quantum dot can be realized in the carbon nanotubes due to the quantization effect of the confinement potential [1, 2]. One of the interesting features in the low-dimensional systems is the Kondo effect [3, 4, 5, 6, 7, 8], where the localized pseudospin or spin states on the quantum dot are screened by the conduction electrons in the leads. The system is in the new ground state, called the Kondo singlet, which opens the transport window in the Coulomb blockade region [4].

The screening effect plays one of the most important roles in the formation of the Kondo singlet [9]. One of the popular methods to solve the Kondo cloud problem is the numerical renormalization group (NRG) method [10], where the screening strongly depends on the number of sites in the Wilson chain and is based on deleting the states in the growing Hilbert space. In another method, the poor man’s scaling method, we solve the Kondo Hamiltonian (the effective Hamiltonian of the Anderson model reduced by the Schrieffer-Wolff canonical transformation using the projection operator technique). In the solution we observe an asymptotic strengthening of the Kondo exchange coupling, logarithmically proportional to the Kondo temperature. Other methods, which include the problem of spin-orbital fluctuations, the finite Coulomb interaction, the Fermi liquid behavior [11] and the scaling problem [12], are the symmetrized finite-U NCA (non-crossing approximation) [13], the equation of motion (EOM) method [14, 15], or the extended Kotliar-Ruckenstein slave-boson mean-field approach (KR-sbMFA)[16] based on path integral methods [17]. On the other hand, the Bethe ansatz approach (BAA) is reserved to the strong coupling limit (infinite-U) and is also limited to the number of states in the chain [18], which is a mapping between a set of quantum numbers and a set of momenta. This map is nonlinear and fully coupled. In this paper [19, 20], the authors showed that the single impurity Anderson model is completely integrable with finite U using the Bethe ansatz approach and solved the transcendent equations with the energies lying not far from the Fermi level. Formally, for non-trivial values of the interaction, Bethe’s equations form a transcendental system of equations that can’t be solved in closed form. However, as is often the case in statistical mechanics, it is hoped that moving to a thermodynamically large system will simplify matters somewhat. In any case, all these methods have some limitations, but in most general aspects they describe the N-orbital Anderson model with modifications.

The Kondo singlet has been realized in the heterostructure [21, 7] and molecular devices [22, 23, 24, 25], but was first discovered in the dilute alloys [9]. In contrast to heterostructure quantum dots [4, 7, 6], the intra- and inter-Coulomb interactions in carbon nanotube quantum dots (CNTQD) are comparable, which is an essential condition for the formation of the SU(4) Kondo effect [5, 2, 26, 27, 28]. In CNTQD, the Kondo temperature [26, 27, 3, 5] is three orders of magnitude higher than the milikelvin temperatures observed in GaAs quantum dots [4, 6]. These two aspects are the features that determine the experimental attractiveness of CNTQDs in the strongly correlated electron regime [29, 12, 17]. The Seebeck coefficient for the Kondo-correlated single quantum dot transistor is suppressed at the electron hole symmetry point and changes sign beyond this point [30, 31]. This is a result of the linear response theory [32] and is a consequence of Fermi liquid (FL) behavior. For temperatures below the Kondo temperature $T_{K}$ only the linear coefficient contributes to the Seebeck effect and the thermoelectric power (TEP) is proportional to the ratio of the derivative of the quasiparticle densities to itself (the Mott’s formula) [33]. The Seebeck effect has also been observed in the Coulomb blockade regime [34]. The authors observed the large enhancement of the thermoelectric power (TEP) in a double quantum dot system due to interference and Coulomb correlation effect.

The model of CNTQD is described by the linear approach of graphene bands [8, 35]. The quantum numbers that characterize the carbon quantum dots are the spin, isospin (valley) and pseudospin (lattice) numbers. The source of the energy gap in semiconducting CNTQDs is the chirality of the nanotube (the gap is proportional to one over the diameter of the nanotube) and the perturbation effects (the perturbation gap is smaller than the geometry gap and proportional to one over the square of the diameter of the nanotube) [8, 2]. Despite the fact that the spin-orbit interaction (SOI) is negligible in graphene, the SOI has been uncovered in the semiconducting CNTQD [36, 5, 2]. In flat graphene, the symmetry forbids direct hopping between orbitals with opposite parity under the inversion. In a nanotube, the symmetry is broken by the curvature, and the SOI arises from the direct hybridization between the non-orthogonal orbitals on the A and the B sites in graphene [37, 38, 39, 2]. The spin-orbit coupling breaks the fourfold degeneracy in the shell of the CNTQD and leads to the ground state with two Kramers doublets [8] (opposite pairs in the spin and isospin sectors). Increasing the SOI changes the symmetry of the Kondo effect from SU(4) to SU(2) [35, 40, 41, 42, 43]. The SOI in CNTQD has two contributions: Zeeman (diagonal part in the A(B) lattice basis) and orbital (off-diagonal part). The value and sign of these interactions determine the ground state in the energy spectrum of holes and electrons on the multishell quantum dot. In general, the SOI is of the order of tens of meV [44, 2], but for the ultraclean semiconducting CNTQD the coupling is comparable to the Coulomb interaction. Special attention has been paid to the problem of the square dependence in the magnetic field of the CNTQD states [44] as a consequence of the narrow band gap in semiconducting CNTQD, which contributed to the spinful SU(3) Kondo state [45]. The observation of the SU(3) Kondo effect can be identified by localizing the quantum conductance to the characteristic value $(9/4)(e^{2}/h)$ [46].

One of the most important differences between SU(2) and SU(4) Kondo states in CNTQD, besides the conductance measurements, is the nonlinear shot noise detection using the lock-in technique [47, 28, 48, 49, 50]. The electronic transport is described by the free non-interacting quasiparticles around the equilibrium state (low bias). In the non-equilibrium regime, two-particle scattering processes dominate due to the residual interaction. Using the Fermi liquid theory [11, 51, 52, 53, 54, 55, 56, 57], the authors showed that the interaction parameter [58], called the Wilson ratio, takes the quantized values $W=2$ for SU(2) and $W=4/3$ for SU(4) Kondo states and changes the corresponding effective charges e^⋆ [59]. For single Kondo QDs, the experimental measurements showed the three-body correlation function in the nonlinear conductance at finite magnetic field, validating the recent Fermi liquid theory in the nonequilibrium Kondo regime [60].

Majorana fermions are it own antiparticles originally proposed by Ettore Majorana [61]. They are called real fermion quasiparticles because of the real nature of the creation and annihilation operators [62]. These quasiparticles exhibit non-Abelian braiding properties [63, 62, 64] and the Majorana fermion states are associated with zero energy modes that occur in the Bogoliubov-de Gennes description of a paired condensate with non-Abelian exchange statistics [65]. The search for Majorana quasi-particle bound states in condensed matter systems is motivated in part by their potential use as topological qubits and possible applications in quantum computation. The Majorana qubits are predicted for quantum states in the fault-tolerant non-Abelian quantum processors [66, 67, 68, 63]. A pair of Majorana fermions can be combined into a complex Dirac fermionic state. The Majorana fermion in this composite complex fermion is half of a normal fermion, and is obtained as a superposition of two Majorana fermions. Each of the Majorana fermions is basically split into a real and an imaginary part of a fermion. The Majorana fermions exist at the edges of proximitized quantum wire by p-wave superconductor [69]. The states are spatially separated and protected from most types of decoherence. However, in [70] as a result, the authors showed that the Majorana qubit coherence and the fermion parity conservation cannot be immune to local perturbations during the braiding operations.

The statistical thermodynamic calculation based on the NRG method showed that the entropy of the quantum dot coupled to a single Majorana fermion leads to $S_{tot}/k_{B}=\ln[2]/2$ and corresponds to the NFL behavior, confirming the anyonic non-Abelian nature of the hybrid device [71]. The author presented the contribution of the impurity to the electron entropy as a function of the temperature for different values of the ratio of the Majorana tunneling rate.

Rasetti and Castagnoli argued that anyons could be used to perform quantum computations [72]. The idea of statistical mechanics in anyons was originated with Arovas et al. [73] and had previously been studied by Frank Wilczek [74]. The author mentioned that the interchange of two particles orbiting around the magnetic flux manifests itself as an arbitrary phase between bosons and fermions, and called the exotic state anyons. Other authors showed that the excitation spectrum of a half-quantum vortex in a p-wave superconductor contains a zero-energy Majorana fermion with non-Abelian statistics [75, 76, 77]. Haldane proposed that the reduction of the apparent Hilbert space dimension by non-orthogonality of states describes localized topological defects at different points in space is also seen in the fractional quantum Hall effect (FQHE), and seems to be the fundamental feature of the fractional statistics [78]. In [76] the authors suggest, that p-wave superfluids, and the Moore-Read state are predicted to support the simplest non-Abelian anyons - the Ising anyons. Their behavior can be understood in terms of Majorana fermion modes at the vortex cores, and it is argued that in 1d, non-Abelian and in particular SU(2) level-k statistics manifest themselves in fractional statistics. For $k=2$, the authors have observed for the Ising anyons that the state counting of the internal Hilbert space associated with the non-Abelian statistics is equivalent to that of the Majorana fermion states coupled to the spinons [76].

The simplest model of the Majorana fermion is predicted by the Kitaev toy model which assumes the spinless topological superconductor (TSC) [69]. In real TSC, we should consider the polarization of the Majorana fermions in the Rashba and Dresselhaus 1d wire. The authors [79] introduced the Majorana pseudospin and showed that the local Majorana polarization is correlated with the transverse spin polarization. Other authors [80, 81] studied the selective equal spin Andreev reflections (SESARs) spectroscopy to detect the polarized Majorana quasiparticles appearing at the edges of the proximitized Rashba chain. In this paper the authors show under which conditions a pseudo-spin degree of freedom can be attributed to Majorana bound states (MBS). MBS correspond to class D and are related to the Z-topological invariant. Class DIII with mirror symmetry supports multiple MBS and is described by the Z₂-invariant with an additional time-reversal symmetry [82].

Discussions remain divided regarding over the preparation and actual implementation of Majorana fermions in low-dimensional systems [83, 84, 85]. Regardless of the scientific dispute, the SU(2) Kondo effect in quantum dots can be used as a very precise detector of topologically protected Majorana states [86, 87]. There are currently several candidates for host boundary Majorana quasiparticles: vortices in two-dimensional (2d) $p_{x}+ip_{y}$ spinless superconductors [88], Moore-Reed type states in FQHE [89, 90], the surface of a 3d topological insulator in proximity to an s-wave superconductor [91], 2d semiconductors with strong spin-orbit coupling coupled to an s-wave superconductor with broken time-reversal symmetry (using a local ferromagnet [92, 93] or an external magnetic field [94]), domain walls in 1d p-wave topological superconductors [95], and helical Majorana modes appearing at the two ends of a 1d wire [96, 97]. In particular, the authors of the references [98] demonstrated in InSb nanowires with strong Rashba-type spin-orbit coupling, the artificial realization of a p-wave superconductor and the observation of a magnetic field-induced zero-bias conductance peak, as expected for a zero-energy Majorana fermion signature [99, 100]. In this setup, where the presence of a topological superconductor is controlled by the Zeeman gap, these systems require a delicate balance of the (spin-orbit coupling, magnetic field and chemical potential) to create the topological superconductor. The idea of 1d spinless p-wave superconductor based on the semiconducting nanowire, where spin-orbit coupling is used to shift the spin-up/down levels in the momentum space and Zeeman field, leading to spin splitting and spin texture at the Fermi level. A proximity induced SC gap within the spin-split levels would lead to an effective spinless p-wave SC [101].

Recently, a topological superconductor has been realized in 1d ferromagnetic atom chains [102], where the 1d system with a strong spin-orbit interaction is placed in proximity to a conventional s-type superconductor. Using high-resolution spectroscopic imaging techniques, the authors have demonstrated the spatially resolved signature of edge-bound Majorana fermions in Fe atom chains and the appearance of zero-energy states in the electronic density of states of the chains. Majorana fermion states are expected to be realized in class D topological superconductors (TSCs with broken time-reversal symmetry)[82]. However, Majorana zero modes can also appear in pairs in time-reversal invariant DIII class topological superconductors. These interesting types of Majorana fermions are called Majorana Kramers pairs [103, 104]. For example, chiral superconductors with $p_{x}+ip_{y}$ pairing state in 2d have a sharp topological distinction between the strong and weak pairing regime [103, 104, 87]. In the weak pairing regime, the gapless chiral Majorana states at the edge are topologically protected. In two dimensions, a time-reversal invariant topological defect of a Z₂ non-trivial superconductor carries a Kramers pairs of Majorana fermions [105].

One of the interesting papers focuses on the aspect of Majorana-Klein hybridization: in [106, 107] the authors demonstrate a topological Kondo effect that implements the SO(M) Kondo problem for M Majorana lead couplings. These topological Kondo states give rise to robust non-Fermi liquid behavior, even for Fermi liquid leads, and to a quantum phase transition between the insulating and Kondo regimes when the leads form Luttinger liquids.

In another paper, the authors studied the interacting Majorana fermions [108]. This is quite a challenge. The simplest interactions between the Majorana degrees of freedom show an unusual non-local structure involving four distinct Majorana sites [109]. The authors [108] solved the Sachdev-Ye-Kitaev model and showed that correlated phases of matter with Majorana building blocks can lead to emergent spacetime supersymmetry (SUSY), topological order or Fibonacci topological phase, which are more exotic generalizations of Majorana fermions known as parafermions.

Returning to the issues discussed in this article, the previous theoretical work investigated the problem of the Majorana zero mode coupled to the spin Kondo state using NRG methods: in single [71, 110, 111] and double quantum dots [112, 113]. Several papers have discussed the thermoelectric effects of quantum dots coupled to side-attached TSCs in the Coulomb blockade regime [114, 115, 116] and in the T-shaped DQD system in the Kondo state side-attached to the Majorana fermion [117]. In these thermoelectric quantum devices, the thermopower changes sign and is fully spin polarized. Measurements of the Seebeck coefficient beyond the e-h symmetry point show strong enhancement and a violation of the Wiedeman-Franz law.

The Kondo cloud plays the role of an interference detector for Majorana fermions. In this article, we discuss the influence of weakly and strongly coupled multi-Majorana fermions on the spin-orbital SU(4) Kondo effect in the carbon nanotube quantum dot. We have coupled three TSC devices proximitized by AB superconducting pairing coat. SOI and SC leads to the Majoranization of the wire states of the zigzag CNT in the absence of a magnetic field. The Majorana states are indexed by the spin-orbital numbers and coupled to the fourfold degenerate states of the CNTQD. The Majorana-Kondo effect is observed for the strong coupling strength limit and manifests itself as the coexistence of the strongly correlated electrons and the topological Majorana states in the system. The spin-orbital type of the Majorana quasiparticles is chosen by the sign of the SOI. An alternative realization is proposed in the armchair CNT, where the electric field can induce the Majorana fermions [118].

The shot noise for the QD-TSC device in the weak coupling limit and for the SU(2) Kondo quantum dot is discussed in [119], where the the shot noise power for the linear voltage is quantized to $(1/2)(e^{2}/h)$. Alternatively results, using the Keldysh field integral description are presented in [120, 121], where the author suggests that in the Majorana state, for positive atomic level of the QD, we should observe two different fractional effective charges at low and high energies, e^⋆/e$=1/2$ and e^⋆/e$=3/2$, accessible at low and high bias voltages. In another paper [122] the authors analyzed the shot noise in a 1d Majorana chain fermion coupled to a normal metal, and found that the Fano factor is quantized to $F=2$ for the single Majorana bound state (MBS) and to non-integer $F$ when both MBSs couple to the lead.

The paper is organized as follows. In Sec. II we discuss the Hamiltonian of the two-orbital Anderson model coupled with one, two and three topological superconducting wires (1TS, 2TS and 3TSC). In the subsections of Sec. III we demonstrate the detection of the symmetry reduction of the spin-orbital SU(4) Kondo effect to exotic SU^⋆(3) and SU^⋆(2) Kondo states in the quantum transport measurements (i.e. quantum conductance, thermoelectric power, linear and nonlinear shot noise). In the last part of the results we study the influence of the SOI on the Majorana-Kondo states. Finally, we summarize the results in the conclusions.

We address the calculation to the system with quantum dot tunneling coupled to multi-Majorna fermions (Fig. 1). We model the carbon nanotube quantum dot (CNTQD) by using the two-orbital Anderson Hamiltonian with side-attached topological superconductors (TSCs):

where the first term describes the energy of the spin-orbital quantum dot level ($E_{ls}=E_{d}(V_{g})+ls\Delta/2$). $\Delta$ is the spin-orbit interaction (SOI) observed in the CNTQD [8, 2], which arises from the curvature of the carbon nanotube [39]. $s=\uparrow(\downarrow)$ and $l=\pm 1$ are the spin and orbital numbers. The second and third terms are the intra- and inter-Coulomb interactions in the system. The next two parts of the Hamiltonian (1) are related to the energy of the left and right electrodes $E_{k\alpha ls}$ ($\alpha=L,R$) and the tunneling strength between the CNTQD and the normal electrodes ($t_{0}$). The last two terms are the tunneling terms of the Majorana fermions $\gamma_{ls}$ with the QD states. The tunneling strength is given by $t_{ls}$, and in the paper we considered three types of hybrid systems: CNTQD-TSC ($t_{+1\uparrow}=t$), CNTQD-2TSC ($t_{+1\uparrow}=t_{-1\uparrow}=t$) and CNTQD-3TSC device (where $t_{+1s}=t_{-1\uparrow}=t$). The Majorana fermions are indexed by their spin-orbital number. All energies are given in $\Gamma$ units, where $\Gamma=\frac{\pi t_{0}^{2}}{2W}$ is the tunnel coupling . $1/2W$ is the flat density of states in the electrode, inversely proportional to the half bandwidth ($W$).

Using the extended slave-boson Kotliar-Ruckenstein mean-field approach [16, 40, 45], the Hamiltonian (1) can be written in the effective form:

where $\widetilde{E}_{ls}=E_{ls}+\lambda_{ls}$ is the renormalized energy level of the quasiparticle Kondo resonance. $\lambda$ and $\lambda_{ls}$ are the Lagrange multipliers associated with the completeness relation (${\cal{I}}=e^{\dagger}e+\sum_{ls}p^{\dagger}_{ls}p_{ls}+\sum_{\nu={20,02,s\overline{s}}}d^{\dagger}_{\nu}d_{\nu}+\sum_{ls}\overline{t}^{\dagger}_{ls}\overline{t}_{ls}+f^{\dagger}f$) and charge conservation ($Q_{ls}=\widetilde{z}^{\dagger}_{ls}\cdot\widetilde{z}_{ls}=p^{\dagger}_{ls}p_{ls}+d^{\dagger}_{l}d_{l}+\sum_{s^{\prime}}d^{\dagger}_{ss^{\prime}}d_{ss^{\prime}}+\overline{t}^{\dagger}_{ls}\overline{t}_{ls}+\sum_{s^{\prime}}\overline{t}^{\dagger}_{\overline{l}s^{\prime}}\overline{t}_{\overline{l}s^{\prime}}+f^{\dagger}f$). The $\cdot$ denotes the non-commutative multiplication in the charge operator. In the effective Hamiltonian, the quantum dot operators $d_{ls}$ are replaced by the product of the bosonic operator $z_{ls}$ and the pseudofermionic operator $f_{ls}$ ($d_{ls}\equiv z_{ls}f_{ls}$). Thus, all physical states are obtained by creating electrons and auxiliary bosons on the vacuum state $\lvert\textrm{vac}\rangle$. In this formalism, the empty and the fully occupied states are generated by the operators as follows: $\lvert e\rangle=\lvert 00\rangle=e^{\dagger}\lvert\textrm{vac}\rangle$ and $\lvert f\rangle=\lvert 22\rangle=f^{\dagger}\prod_{ls}f^{\dagger}_{ls}\lvert\textrm{vac}\rangle$. The single particle electron state is represented by $\lvert p_{+s(-s)}\rangle=\lvert s0(0s)\rangle=p^{\dagger}_{ls}f^{\dagger}_{ls}\lvert\textrm{vac}\rangle$. The triple occupied states are given by $\lvert\overline{t}_{\pm s}\rangle=\lvert s2(2s)\rangle=\overline{t}^{\dagger}_{\pm s}f^{\dagger}_{\pm s}f^{\dagger}_{\mp s}f^{\dagger}_{\mp\overline{s}}\lvert\textrm{vac}\rangle$. The auxiliary canonical particles for the double occupied state are represented by six states: $\lvert d_{20(02)}\rangle=\lvert 20(02)\rangle=d^{\dagger}_{20(02)}f^{\dagger}_{\pm\uparrow}f^{\dagger}_{\pm\downarrow}\lvert\textrm{vac}\rangle$ and $\lvert d_{ss(s\overline{s})}\rangle=\lvert ss^{\prime}\rangle=d^{\dagger}_{ss(s\overline{s})}f^{\dagger}_{+s(+s)}f^{\dagger}_{-s(-\overline{s})}\lvert\textrm{vac}\rangle$.

The tunnel rates $\widetilde{t}_{0ls}=t_{0}z_{ls}$ and $\widetilde{t}_{ls}=t_{ls}z_{ls}$ are renormalized by $z_{ls}=\widetilde{z}_{ls}/\sqrt{\delta n^{2}_{ls}}=(e^{\dagger}p_{ls}+p^{\dagger}_{l\overline{s}}d_{l}+\sum_{ls^{\prime}}p^{\dagger}_{ls^{\prime}}d_{ss^{\prime}}+\sum_{l}d^{\dagger}_{l}\overline{t}_{ls}+\sum_{ls^{\prime}}d^{\dagger}_{\overline{s}s^{\prime}}\overline{t}_{ls^{\prime}}+\overline{t}^{\dagger}_{\overline{l}\overline{s}}f)/\sqrt{Q_{ls}(1-Q_{ls})}$. $z_{ls}$ is the renormalization of the width of the Kondo resonance (compare with the amplitude of the quasiparticle wave function [29, 17]), and determines the Kondo temperature $T_{K}=\min\{T_{K,ls}\}=\sqrt{\widetilde{E}^{2}_{ls}+\widetilde{\Gamma}^{2}_{ls}}$, where $\widetilde{\Gamma}_{ls}=\Gamma|z_{ls}|^{2}$ is renormalized tunnel coupling. For the non-interacting system $U=0$, $z^{2}_{ls}\approx 1$ and we can say that the spin-orbital fluctuations are comparable to the tunneling processes involved in the in Kondo effect $\delta n_{ls}\approx\widetilde{z}^{2}_{ls}$.

In the calculations, we consider three models: a quantum dot coupled to a single Majorana fermion $\gamma_{+\uparrow}$ (CNTQD-TSC), with two Majorana fermions $\gamma_{\pm\uparrow}$ (CNTQD-2TSC), and with side-attached three Majoranas $\gamma_{\pm\uparrow}$ and $\gamma_{+\downarrow}$ (CNTQD-3TSC). In all systems we take the same value for the coupling strength $t$ in the Hamiltonian (2). Using the saddle-point approximation, we solved the following self-consistent equations:

where $b^{\dagger}$ is represented by auxiliary boson operators: $b^{\dagger}_{n=1...16}=\{e^{\dagger},p^{\dagger}_{ls},d^{\dagger}_{\nu},\overline{t}^{\dagger}_{ls},f^{\dagger}\}$.

and $\Delta\widetilde{E}_{n}=\{\lambda e,(\lambda_{ls}+\lambda)p_{ls},(U+\sum_{s}\lambda_{ls}+\lambda)d_{l},(U+\lambda_{ls}+\lambda_{\overline{l}s^{\prime}}+\lambda)d_{ss^{\prime}},(3U+\lambda_{ls}+\sum_{s}\lambda_{\overline{l}s}+\lambda)\overline{t}_{ls},(6U+\sum_{ls}\lambda_{ls}+\lambda)f\}$. The correlators in Eqs. (3-4) can be written in the form:

where $G^{<}_{ls,ls}$, $G^{<}_{k\alpha ls,ls}=\widetilde{t}_{0ls}(G^{R}_{ls,ls}g^{<}_{k\alpha ls}+G^{<}_{ls,ls}g^{A}_{k\alpha ls})$ and $G^{<}_{\underline{ls},ls}$ are the non-equilibrium Green’s functions calculated using the EOM and Keldysh formalism for the Hamiltonian (2) [123, 124, 56]. The retarded and lesser Green’s functions in the $\nu=ls$ channel (decoupled from the TSC) are given by $G^{R}_{\nu,\nu}(E)=(E-\widetilde{E}_{\nu}+i\widetilde{\Gamma}_{\nu})^{-1}$ and $G^{<}_{\nu,\nu}(E)=G^{R}_{\nu,\nu}\widetilde{\Sigma}^{<}_{\nu}G^{A}_{\nu,\nu}$. $\widetilde{\Sigma}^{<}_{\nu}=\sum_{\alpha}\widetilde{t}^{2}_{0\nu}g^{<}_{k\alpha\nu}=i\widetilde{\Gamma}_{\nu}\sum_{\alpha}f_{\alpha}$ is the lesser self-energy and $f_{\alpha}=(e^{\frac{E\pm V_{\alpha}}{k_{B}T}}+1)^{-1}$ is the Fermi-Dirac function. The Green’s functions in the $\nu^{\prime}$ channel (the channel coupled to the TSC) can be written in the matrix form as follows:

where $\hat{E}_{\nu^{\prime}}$ is the matrix of the diagonal energies $\{\pm\widetilde{E}_{\nu^{\prime}},0\}$, the remaining elements are the matrix of retarded self-energy $\hat{\Sigma}^{R}$. $\delta$ is the lifetime of the Majorana fermion, and is the lowest energy in the system $\delta\ll T_{K}$ (in our calculations $\delta=10^{-8}$). In practice this is the consequence of the disappearance of the overlap term between the Majorana fermions at the ends of the proximitized wire ($i\Delta_{(0)}\gamma_{Als}\gamma_{Bls}$) in the self-energy of the TSC $\widetilde{\Sigma}^{R}_{t}=(\widetilde{t}^{2}z)/(z^{2}-\Delta^{2}_{(0)})$ [69, 119]. Therefore, in our model we used the self-energy with the finite lifetime of the Majorana fermion $\delta$, where in the asymptotic limit: $\lim_{\Delta_{(0)}\mapsto 0}\widetilde{\Sigma}^{R}_{t}=\widetilde{t}^{2}/z=\widetilde{t}^{2}/(E+i\delta)$. The lesser Green’s function matrix can be written as $\hat{G}^{<}_{\nu^{\prime}}=\hat{G}^{R}_{\nu^{\prime}}\hat{\Sigma}^{<}_{\nu^{\prime}}\hat{G}^{A}_{\nu^{\prime}}$, where $\hat{\Sigma}^{<}_{\nu^{\prime}}=\hat{\Sigma}^{R}_{\nu^{\prime}}\sum_{\alpha}f_{\alpha}$. The mean-field slave-boson approach has self-consistent solutions for finite temperature $T$ and bias voltage $V_{\alpha=L(R)}=\pm V/2$ below and around the the Kondo temperature $T_{K}$.