Majorana tunneling in a one-dimensional wire with non-Hermitian double quantum dots

Peng-Bin Niu(牛鵬斌) and Hong-Gang Luo(羅洪剛)

1Institute of Solid State Physics,Shanxi Datong University,and Shanxi Provincial Key Laboratory of Microstructural Electromagnetic Functional Materials,Datong 037009,China

2School of Physical Science and Technology,Lanzhou University,Lanzhou 730000,China

3Beijing Computational Science Research Center,Beijing 100084,China

Keywords: Majorana fermion,non-Hermitian quantum dot,parity–time symmetry,exceptional point

1.Introduction

In quantum mechanics, there is a fundamental postulation: a physical observable must be expressed by a Hermitian operator, which ensures the eigenvalues of the operator are real and measurable.A generalization to this postulation can be made by considering non-Hermitian Hamiltonians with parity–time(PT)symmetry.[1–4]These non-Hermitian Hamiltonians can exhibit a real spectrum and exceptional points,[5–7]where eigenvalues are degenerate at certain values of non-Hermitian parameters.Physically, non-Hermitian Hamiltonians are usually used to describe open quantum systems,[6,8–11]which experience gain and loss of energy due to coupling to the environment.Non-Hermitian Hamiltonians can also be realized in fermionic systems,[12,13]where theoretical works have demonstrated that non-Hermitian topological phases can be realized.

In this work,we consider the transport properties of a hybrid fermionic system consisting of Majorana fermions and non-Hermitian double quantum dots.In fact, many works discuss Hermitian quantum dots interacting with Majorana fermions.[14–27]However, here we focus on the interplay of PT-symmetric complex potentials,Majorana tunneling and interdot tunneling in a non-Hermitian quantum dot system.We expect that the combination of PT-symmetric complex potentials,Majorana tunneling and interdot tunneling would induce an exceptional point that has not been discussed before.Then we explore the transport properties of this system.Some interesting results are found.For instance, we find that their joint effects mainly are splitting, a 1/4 conductance peak, asymmetry, and an exceptional point.More specifically, Majorana tunneling would split the single peak into three and manifest itself as a reduced 1/4 central peak.There is an exceptional point induced by the union of Majorana tunneling and interdot tunneling.When Majorana tunneling appears,the antiresonance induced by PT-symmetric complex potentials would disappear.The interdot tunneling would induce asymmetry,instead of moving the conductance peak.These effects are revealed in the transmission function.The model considered in this work is exactly solvable.In order to find analytical results for the retarded Green’s functions contained in the transmission function,we use the equation-of-motion method.

This paper is organized as follows.In Section 2 we present the model of the studied system, define the transmission function and describe the method used in calculations.To gain an intuitive understanding of the transport properties,we present the numerical results in Section 3.Finally,the paper is summarized in Section 4.

2.Model and method

We study the quantum transport of two parallel quantum dots coupled to each other through hopping, with each of them side-attached to a topological superconductor, which supports Majorana fermions at the endpoint.The system is placed between two semi-infinite wires, as as illustrated in Fig.1.We consider a poor man’s Majorana fermions, i.e.,only one Majorana fermion at the end of the superconductor wire is considered.[28–30]The whole system can be modeled by the Hamiltonian[29,31–34]H=Hwires+HT+Hc.The first term stands for the left and right semi-infinite wires,which are modeled as hopping of noninteracting quasiparticles in a line

whereα=L,R denotes the left or right wire,withthe creation operator for an electron in wireαat sitej.Heret0is the hopping amplitude between the nearest sites.The second term describes the tunneling coupling between the dots and the wires,

wherei=1,2 is the index for the first or second quantum dot.vαiis the tunneling amplitude between theith dot and wireα.diis the annihilation operator for theith dot.It is worth mentioning that,in normal metallic leads,the free electrons in the leads are tunnel-coupled with a quantum dot.Thus the metallic leads are three-dimensional in real space.However, compared with metallic leads,[35]the one-dimensional wire combined with the model in this work yields different transport behaviors in the weak-and strong-coupling regimes.We will show this in the discussion in Section 3.The last term of the total Hamiltonian models the hybrid subsystem consisting of two quantum dots and two Majorana fermions, which can be described by the following Hamiltonian:

whereεiis theith dot’s energy level.n1=andn2=are the number operators for each dot.tcis the hopping amplitude between double quantum dots.Following Refs.[31,32],we introduce PT-symmetric complex potentials into two quantum dots by consideringε1=ε+iβandε1=ε ?iβ.εis a single energy level, and it can be tuned by the gate voltage.For simplicity,we will consider the case of on-resonance(ε=0)in the following discussion.Whenεiis complex, the Hamiltonian becomes non-Hermitian.Note that in practice±iβrepresents the physical gain and loss during the interactions between the environment and the dots.[36]The Majorana operators,γ1andγ2, describe the Majorana zero modes sidecoupled with the quantum dot.λstands for the parameter of tunnel coupling between the dot and Majorana zero mode.A Hamiltonian is said to be PT-symmetric if it is invariant under the combination of parity and time reversal transformations.With respect to our model considered here, the effect of parity transformation is P:cLj →cRj,d1→d2,γ1→γ2,and the effect of time reversal transformation is T: i→?i.It is easy to check that the total Hamiltonian in this work preserves PT symmetry, under the condition ofvαi=vα'i'=v.It is worth discussing the experimental realization of this system.In fact,the PT-symmetric complex potentials are easy to realize by coupling quantum point contacts with the quantum dots.[13]On the other hand,although theorists have predicted the existence of Majorana fermions in topological superconductors,it is still challenging to provide convincing experiments.Fortunately, there is accumulating positive evidence that Majorana zero modes are being realized.[37,38]

Fig.1.Schematic representation of the Majorana–quantum-dot hybrid system, which is placed between two semi-infinite wires.The system is formed by the coupling between two quantum dots, and each dot is side-coupled to a Majorana zero-energy mode.

The Majorana operators fulfill the algebraic relation{γi,γj}=2δi j,and they can be expressed in terms of fermion operators asγ1=f?1+f1,γ2=f?2+f2.With these operators,the Hamiltonian in Eq.(3)becomes

We expect the model considered in this work would introduce the appearance of exceptional points.Hence the eigenenergies of the central Hamiltonian decoupled from the wires (denoted asHc) should be investigated.In the basis states{|0〉,|1〉}Dot1?{|0〉,|1〉}Majorana1?{|0〉,|1〉}Dot2?{|0〉,|1〉}Majorana2,the Hamiltonian is not diagonal.However,by calculation it is easy to know that these states are divided into two groups,i.e.,two subspaces.In each subspace the matrix of the HamiltonianHcis the same.With the matrix, we obtain the eigenenergies as follows:

which possess an exceptional point when

In this work, we would like to use Green’s function method to carry out the calculation.Therefore, the transmission function can be expressed as[39]

whereGris the retarded Green’s function in the frequency domain.It is composed of a 2×2 matrix and its elements are given byGrij(ω)=〈〈di|d?j〉〉, where the indexi,j=1,2.Gais the advanced Green’s function, which can be obtained by taking the complex conjugate ofGr.Γα=Σα ?Σ?αdenotes the tunnel coupling between wireαand the dots.Σαis the self-energy matrix,which will be given below.

In order to obtainGr, we use the standard equation of motion (EOM) technique,[40,41]ω〈〈A|B〉〉r=〈{A,B}〉+〈〈[A,H]|B〉〉r, whereAandBare arbitrary operators.For example,the equation of motion of〈〈d1|d?1〉〉generates

whereδ1,iis the Kronecker delta andi=1,2.The matrix reads

By using Cramer’s rule, equation (8) is solved and the analytical results for elements ofGrare obtained as

whereA11,A12,A13,A33, andA34are algebraic cofactors of the matrix in Eq.(9).W1,2are defined asWn=1,2=ω ?εn ?2λ2/ω ?Σnnfor brevity.Equations(10)–(13)forGrrepresent one of the main results in this work.

3.Results and discussion

In this section,we investigate the quantum transport properties induced by the interplay of PT-symmetric complex potentials,Majorana tunneling and interdot tunneling in the system.In numerical calculations, we taket0= 1, i.e., it is taken as the unit of energy.We discuss transport in the weakcoupling regime (v ?t0)first, and take dot–wire coupling asv=0.2,without loss of generality.

As a beginning,in Fig.2 we show the transport properties caused by interdot tunneling (tc), Majorana tunneling (λ) or PT-symmetric complex potentials (β) separately.In the case oftc=0,λ=0,andβ=0,it is seen in Fig.2(a)that the transmission function shows a Breit–Wigner lineshape whose peak reachesT(ω)=1 and is centered atω=0.When interdot tunneling is nonzero(tc/=0),the peak is shifted toω=tc.In the case ofλ/=0, it is seen that the effect of Majorana tunneling is to split the single peak into three.These peaks are located atω=0,±2λ.Another characteristic of the Majorana fermion in this model is that the central peak atω=0 is reduced to 1/4, as seen in Fig.2(b).In Fig.2(c), whenβ/=0,an antiresonance appears atω= 0, which is a typical phenomenon induced by PT-symmetric complex potentials.Withβincreasing, the height of peaks in transmission function is usually suppressed.

In Fig.2 the separate effects induced by interdot tunneling(tc),Majorana tunneling(λ)or PT-symmetric complex potentials (β) have been considered.We now consider their pairwise joint effects.In Fig.3(a), the joint effect oftcandλis investigated.One can readily find that the effect of interdot tunneling is to make the three-peak structure asymmetric.But the positions of these three peaks are pinned with differenttc,which is a demonstration of the Majorana fermion’s robustness.In the meantime, one can see that the 1/4 central peak maintains itself,i.e.,it is immune to the interdot tunneling.In Fig.3(b), the joint effect ofλandβis examined.The PTsymmetric complex potentials tune the two side peaks, while the 1/4 central peak is still robust.This behavior is visualized by the curve ofβ=0.5.With increasingβ,the two side peaks move towardsω=0.They meet each other and merge when the exceptional point appears,i.e.,whenβ=2λ,which is seen in the curve ofβ=1.Whenβ ＞2λ,the two side peaks disappear and the central 1/4 peak is left alone,as seen in the curve ofβ=1.5.In Fig.3(c),the interplay oftcandβis discussed.Whenβis small,the PT-symmetric complex potentials induce an antiresonance and two resonance peaks atω=±tc.This is seen in the curve ofβ=0.1.With increasingβ,the position of the antiresonance will not change, but the resonance peaks atω=±tcwill move towards each other.These two resonance peaks also merge when the exceptional point is reached, i.e.,whenβ=tc.This behavior is seen in the curve ofβ=0.2.Whenβ ＞tc, these two resonance peaks also disappear and the resonance peak atω=tcis restored; its magnitude will be suppressed with increasingβ.This is seen in the curve ofβ=0.3.Comparing Figs.2(c),3(b)and 3(c),a conclusion is drawn: the antiresonance induced by PT-symmetric complex potentials appears only whenλ/=0.Thus one can predict that the joint effects induced by PT-symmetric complex potentials,Majorana tunneling and interdot tunneling will be similar to those in Fig.3(b).Meanwhile,asymmetry is to appear there.

Fig.2.Spectra of the transmission function of the double quantum dots influenced by interdot tunneling, Majorana tunneling or PT-symmetric complex potentials separately.The parameters used are(a)tc=0.5,(b)λ =0.2,0.5,0.7,and(c)β =0.1,0.2,0.3.All energies are measured in units of t0.

Fig.3.Transmission function curves of the double quantum dots influenced by pairwise combination of interdot tunneling,Majorana tunneling and PT-symmetric complex potentials.The parameters used are(a)λ =0.2 and tc =0, 0.1, 0.3, (b)λ =0.5 and β =0.5, 1, 1.5,tc =0.2 and(c)β =0.1,0.2,0.3.All energies are measured in units of t0.

peak structure.The interdot tunneling induces the asymmetry of the transmission function,which can be seen clearly in this figure.

Fig.4.Energy spectrum of the system.The real and imaginary parts are shown, as a function of PT-symmetric complex potentials in the upper row,and as a function of interdot tunneling in the lower row.The parameters used in(a)and(b)are tc=0.1 and λ =0.3,while in(c)and(d)the parameters are β =0.8 and λ =0.3.All energies are measured in units of t0.

Fig.5.Transmission function curves of the double quantum dots influenced by the interplay of interdot tunneling,Majorana tunneling and PT-symmetric complex potentials.The parameters used are tc =0.1 and λ =0.3 in (a)–(c).The parameter used for PT-symmetric complex potentials is(a)β =0.1,0.4, 0.6, (b) β =βEP, and (c) β =1.1βEP,1.5βEP,2βEP.All energies are measured in units of t0.

Now we discuss the transport properties in the strongcoupling regime.We take dot–wire coupling asv=1.The results are shown in Fig.7.When the dot–wire coupling is strong, the window of resonant tunneling is broadened, and the transmission function shows a flat lineshape, as seen in Fig.7.The effect of interdot tunneling remains,i.e.,it still induces asymmetry in the transmission function.Interestingly,the participation of Majorana zero modes induces a 1/4 conductance dip atω=0,instead of the three-peak structure in the weak-coupling regime.When PT-symmetric complex potentials are added, they induce two side dips in the transmission function, which are located atWe notice that these two dips are pinned, no matter whetherβ ＜βEP,β=βEPorβ ＞βEP, as seen in the curves ofβ=0.25, 0.5 and 0.75 in Fig.7(a).At the exceptional point,the central dip disappears,as seen in the curve ofβ=0.5 in Fig.7(a) or in the merging point atω=0 andβ=0.5 in Fig.7(b).

Fig.6.Transmission coefficient as a function of ω and tc.The parameter used are v=0.2,β =1.5,and λ =0.5.All energies are measured in units of t0.

Fig.7.Transmission function in the strong-coupling regime.In(a)and(b)the parameters used are v=1,tc=0.3 and λ =0.2.In(a)the values of PT-symmetric complex potentials are chosen as β =0,0.25,0.5,0.75.The curves for β =0, 0.25, 0.5 are shifted upward for clarity.In(b)the value of β varies continuously.All energies are measured in units of t0.

4.Conclusion

In summary, we have studied a model with a combination of non-Hermitian quantum dots and Majorana fermions.Some new effects were revealed in this quantum transport system.The interplay of PT-symmetric complex potentials, Majorana tunneling and interdot tunneling were investigated.By using the equation-of-motion technique, an analytical result for the retarded Green’s functions has been obtained.It was demonstrated that in the weak-coupling regime Majorana tunneling splits the single peak of the transmission function and maintains itself as a reduced 1/4 conductance peak.The interdot tunneling only induces asymmetry,instead of moving the conductance peak, due to the robustness of Majorana modes.There is an exceptional point induced by the union of Majorana tunneling and interdot tunneling.With large enough PT-symmetric complex potentials and when the exceptional point is passed through, the three-peak structure of the transmission function is reduced to a single 1/4 peak.In the strongcoupling regime, a Majorana fermion induces a 1/4 conductance dip instead of the three-peak structure.PT-symmetric complex potentials induce two conductance dips pinned at the exceptional point.It was shown that this model generates some interesting results,which are accessible to experiments.These results may also be useful for the study of quantum transport models with gain and loss.

Acknowledgement

Project supported by the National Natural Science Foundation of China(Grant No.11834005).