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In this paper, we study the topological properties of the non-Hermitian Su-Schrieffer-Heeger (SSH) lattice by periodically introducing onsite imaginary potentials in the manner of ($i\gamma_1$, $-i\gamma_2$, $-i\gamma_1$, $i\gamma_2$) where $\gamma_1$ and $\gamma_2$ are the imaginary potential strengths. Results show that by changing the lattice to a tetramerized non-Hermitian system, such imaginary potentials induce the nontrivial transition of the topological properties of the SSH system. First, the topologically-nontrivial region is extended, followed by the non-Hermitian spontaneous breaking of the anti-$\cal PT$ symmetry. In addition, new edge state appears, but its locality is different from the state induced by the Hermitian SSH lattice. If such potentials are strong enough, the bulk states of this system can become purely imaginary states. We believe that these imaginary potentials play special roles in modulating the topological properties of the non-Hermitian SSH lattice., Comment: 10 pages, 7 figures

We investigate the topological properties of a dimerized Kitaev chain with long-range interactions, including the intercell hopping and superconducting pairing terms. It is found that even only when the intercell hopping term appears, the size of the energy gap, the proportion of topological phases, and the topological phase transition can be modulated. The notable result is that they lead to a new Kitaev-like phase featured by the twofold-degenerated Majorana zero-energy edge states. Next in the presence of the intercell superconducting pairing term, this kind of Majorana phase can be magnified. This work provides new proposals to realize the twofold degenerated Majorana modes based on the intercell hopping and superconducting pairing terms of the dimerized Kitaev chain., Comment: 9 pages, 10 figures

We theoretically study the transport properties in the T-shaped double-quantum-dot structure, by introducing the Majorana bound state (MBS) to couple to the dot in the main channel. It is found that the side-coupled dot governs the effect of the MBS on the transport behavior. When its level is consistent with the energy zero point, the MBS contributes little to the conductance spectrum. Otherwise, the linear conductance exhibits notable changes according to the inter-MBS coupling manners. In the case of Majorana zero mode, the linear conductance value keeps equal to $e^2\over 2h$ when the level of the side-coupled dot departs from the energy zero point. However, the linear conductance is always analogous to the MBS-absent case once the inter-MBS coupling comes into play. These findings provide new information about the interplay between the MBSs and electron states in the quantum dots., Comment: 10 pages, 9 figures

We theoretically investigate the Kondo effect of a T-shaped triple-quantum-dot structure, by means of the numerical renormalization group method. It is found that at the point of electron-hole symmetry, the system's entropy has opportunities to exhibit three kinds of transition processes for different interdot couplings, with the decrease of temperature. This leads to the different pictures of the Kondo physics, including the three-stage Kondo effect. Next when the electron-hole symmetry is broken or the structural parameters are changed, the Kondo resonance can also be observed in the conductance spectrum. However, it shows alternative dependence on the relevant quantities, i.e., the Coulomb interaction and interdot couplings. All these phenomena exhibit the abundant and interesting Kondo physics in this system. We believe that this work can be helpful for further understanding the Kondo effect in the triple-quantum-dot structures., Comment: To be published in Physical Review B

By introducing the next-nearest-neighboring (NNN) intersite coupling, we investigate the eigenenergies of the $\cal PT$-symmetric non-Hermitian Su-Schrieffer-Heeger (SSH) model with two conjugated imaginary potentials at the end sites. It is found that with the strengthening of NNN coupling, the particle-hole symmetry is destroyed. As a result, the bonding band is first narrowed and then undergoes the top-bottom reversal followed by the its width's increase, whereas the antibonding band is widened monotonously. In this process, the topological state extends into the topologically-trivial region, and its energy departs from the energy zero point, accompanied by the emergence of one new topological state in this region. All these results give rise to the complication of the topological properties and the manner of $\cal PT$-symmetry breaking. It can be concluded that the NNN coupling takes important effects to the change of the topological properties of the non-Hermitian SSH system., Comment: 8 pages and 8 figures

We investigate the influences of the electron-phonon interaction on the transport properties of one quantum-dot system with a side-coupled Majorana zero mode (MZM). Our calculation results show that at the zero-temperature limit, the MZM-governed zero-bias conductance value can be magnified, dependent on the interplay between electron-phonon interaction and dot-MZM coupling. In the case of finite temperature, the electron-phonon interaction makes leading contributions to the suppression of the magnitude of zero-bias conductance. We believe that this work can be helpful for understanding the signature of the MZM in electron transport through mesoscopic circuits., Comment: 9 pages 7 figures

We investigate the Andreev reflection in the parallel double-quantum-dot structure, by considering one metallic lead and one $s$-wave superconductor to couple to the quantum dots simultaneously. It is found that the Fano lineshhape has opportunities to appear in the linear conductance spectrum of the Andreev reflection, which can be reversed by tuning the dot level or local magnetic flux. However, the property of the Fano effect is very complicated, in comparison with the normal electron tunnelling case. This is manifested as the special Fano form of the linear-conductance expression and the interference manner among the Feynman paths. We believe that this work can be helpful for understanding the Fano interference in the Andreev reflection process., Comment: 10 pages, 10 figures

We study the quantum transport behavior of a normal-superconductor-normal junction based on type-II Weyl semimetal, which is arranged in the tilting direction of the Weyl semimetal. We find that both the crossed Andreev reflection and normal reflection are forbidden, while there will be double Andreev reflections and double electron transmissions for the incident electron from the semimetal side. Andreev reflections and transmissions occur both in the retro and specular directions simultaneously, symmetric about the normal of the interface but with different amplitudes, depending on the angle and energy of incident electrons. These transport processes make the junction here quite different from that based on the normal metal or graphene. In addition, the differential conductance is studied for experimental signatures. We find that the conductance is almost unaffected by the chemical potential and it is enhanced with increasing junction length., Comment: 8 pages, 6 figures

By considering Majorana zero modes to laterally couple to the quantum dot, we evaluate the thermoelectric effect in one single-dot system. The calculation results show that if one Majorana zero mode couples to the dot, the thermoelectric effect will exhibit its change, but the thermoelectric efficiency cannot be enhanced apparently. However, the thermoelectric effect can be efficiently strengthened when two Majorana zero modes are introduced simultaneously. We believe that the findings in this work provide an alternative method for the detection of Majorana bound state., Comment: 8 pages, 7 figures, submitted to PRB

With the help of the numerical renormalization group method, we theoretically investigate the Josephson phase transition in a parallel junction with one quantum dot embedded in each arm. It is found that in the cases of uniform dot levels and dot-superconductor couplings, the Josephson phase transition will be suppressed. This is manifested as the fact that with the enhancement of the electron correlation, the supercurrent only arrives at its $\pi'$ phase but cannot enter its $\pi$ phase. Moreover, when the dot levels are detuned, one $\pi'$-phase island appears in the phase diagram. Such a result is attributed to the nonlocal motion of the Cooper pair in this structure. We believe that this work can be helpful in understanding the Josephson phase transition modified by the electron correlation and quantum interference.

In this work, we study the effect of $\mathcal{PT}$-symmetric complex potentials on the transport properties of one non-Hermitian system, which is formed by the coupling between a triple-quantum-dot molecule and two semi-infinite leads. As a result, it is found that the $\mathcal{PT}$-symmetric imaginary potentials take pronounced effects on transport properties of such a system, including changes from antiresonance to resonance, shift of antiresonance, and occurrence of new antiresonance, which are determined by the interdot and dot-lead coupling manners. This study can be helpful in understanding the quantum transport behaviors modified by the $\cal PT$ symmetry in non-Hermitian discrete systems., Comment: 8 pages, 9 figures

We consider one system in which the terminal dots of a one-dimensional quantum-dot chain couple equally to the left and right leads and study the influence of $\mathcal{PT}$-symmetric complex potentials on the quantum transport process. It is found that in the case of the Hermitian Hamiltonian, remarkable decoupling and antiresonance phenomena have an opportunity to co-occur in the transport process. For the chains with odd(even) dots, all their even(odd)-numbered molecular states decouple from the leads. Meanwhile, antiresonance occurs at the positions of the even(odd)-numbered eigenenergies of the sub-chains without terminal dots. When the $\mathcal{PT}$-symmetric complex potentials are introduced to the terminal dots, the decoupling phenomenon is found to transform into the Fano antiresonance. In addition, it shows that appropriate magnetic flux can interchange the roles of the odd and even molecular states. These results can assist to understand the quantum transport modified by the $\mathcal{PT}$ symmetry in non-Hermitian discrete systems., Comment: 7 pages, 5 figures

We investigate the Kondo effect in a structure which is constructed by embedding one quantum dot between a normal metal and a \emph{DIII}-class topological superconductor supporting Majorana doublets at its ends. It is observed that Kondo correlation occurs between the localized state in the dot and two continuum states simultaneously, i.e., the continuum state in the metal and the continuum Andreev reflection state between the metal and topological superconductor. As a result, the Kondo model Hamiltonian is topologically protected by the $SU(2)_s\rtimes Z_2^T$ symmetry. Besides, the Kondo temperature has an opportunity to present three forms in turn, following the enhancement of the coupling between the dot and Majorana doublet. This phenomenon exactly reflects the special role of Majorana doublet in tuning the Kondo effect., Comment: 5 pages, 2 figures

We investigate the Josephson effect in one triple-terminal junction with embedded parallel-coupled double quantum dots. It is found that the inter-superconductor supercurrent has opportunities to oscillate in $4\pi$ period, with the adjustment of the phase differences among the superconductors. What is notable is that such a result is robust and independent of fermion parities, intradot Coulomb strength, and the dot-superconductor coupling manner. By introducing the concept of spinful many-particle Majorana modes, we present the analytical definition of the Majorana operator via superposing electron and hole operators. It can be believed that this work provide a simple but feasible proposal for the realization of Majorana modes in a nonmagnetic system., Comment: 4 pages, 3 figures

We discuss the change of the Kondo effect in the Josephson junction formed by the indirect coupling between a one-dimensional \emph{DIII}-class topological and s-wave superconductors via a quantum dot. By performing the Schrieffer-Wolff transformation, we find that the single-electron occupation in the quantum dot induces various correlation modes, such as the Kondo and singlet-triplet correlations between the quantum dot and the $s$-wave superconductor and the spin exchange correlation between the dot and Majorana doublet. Moreover, it plays a nontrivial role in modifying the Josephson effect, leading to the occurrence of anisotropic and high-order Kondo correlation. In addition, due to the quantum dot in the Kondo regime, extra spin exchange correlations contribute to the Josephson effect as well. Nevertheless, if the \emph{DIII}-class topological superconductor degenerates into \emph{D}-class because of the destruction of time-reversal invariance, all such terms will disappear completely. We believe that this work shows the fundamental difference between the \emph{D}- and \emph{DIII}-class topological superconductors., Comment: 10 pages, 3 figures. Any comment is welcome

We investigate the Josephson effects in the junction formed by \emph{DIII}-class topological and $s$-wave superconductors, by embedding a quantum dot in the junction. Three dot-superconductor coupling manners are considered, respectively. As a result, the Josephson current is found to oscillate in $2\pi$ period. Moreover, the presence of Majorana doublet in the \emph{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 jucntion. 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. The results in this work will be helpful for understanding the transport properties of the \emph{DIII}-class superconductor., Comment: 9 pages, 5 figures

One Majorana doublet can be realized at each end of the time-reversal-invariant Majorana nanowires. We investigate the Josephson effect in the Majorana-doublet-presented junction modified by different inter-doublet coupling manners. It is found that when the Majorana doublets couple indirectly via a non-magnetic quantum dot, only the normal Josephson effects occur, and the fermion parity in the system just affects the current direction and amplitude. However, in the odd-parity case, applying finite magnetic field on the quantum dot can induce the appearance of the fractional Josephson effect. Next, when the direct and indirect couplings between the Majorana doublets coexist, no fractional Josephson effect takes place, regardless of finite magnetic field on the quantum dot. Instead, the $\pi$-period current has an opportunity to appear in some special cases. All the results are clarified by analyzing the influence of the fermion occupation in the quantum dot on the parity conservation in the whole system. We ascertain that this work will be helpful for describing the dot-assisted Josephson effect between the Majorana doublets., Comment: 8 pages, 6 figures