Non-Abelian statistics of Majorana zero modes in the presence of an Andreev bound state
NNon-Abelian statistics of Majorana zero modes in the presence of an Andreev boundstate
Wenqin Chen, Jiachen Wang, Yijia Wu, Jie Liu, ∗ and X. C. Xie
2, 3, 4 Department of Applied Physics, School of Science, Xian Jiaotong University, Xian 710049, China International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Beijing Academy of Quantum Information Sciences, Beijing 100193, China CAS Center for Excellence in Topological Quantum Computation,University of Chinese Academy of Sciences, Beijing 100190, China
The low-energy Andreev bound states (ABSs) mixing with the Majorana zero modes (MZMs)may destroy the non-Abelian braiding statistics of the MZMs. We numerically studied the braidingproperties of MZMs when an ABS is present. Numerical simulation results support the argumentthat the ABS can be regarded as a pair of weakly coupled MZMs. The non-Abelian braidingproperties of MZMs exhibit oscillation behaviour with respect to the braiding time if the ABS-related dynamic phase is present. Remarkably, such dynamic phase can be eliminated by tuningthe magnetic field or gate voltage. In this way, the non-Abelian braiding statistics independent ofthe braiding time retrieves so that the topological quantum computation could still be robust evenwhen the ABS is engaged.
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Introduction — Majorana zero mode (MZM) is deemedas the most promising candidate for topological quantumcomputation (TQC) [1, 2] for its non-Abelian statistics.The exploration for MZMs in topological superconduc-tors (TSCs) has been drawing extensive attention in thelast decade [3–12]. To date, TSC has been realized in var-ious experimental platforms and the signals for MZMshave also been reported [13–26]. The semiconductor-conventional superconductor heterostructure, first exper-imentally realized one among these systems, is regardedas one of the most promising platforms for the realiza-tion of TQC. Experimentally, however, a roadblock insuch system is that other low-energy modes, e.g. Andreevbound states (ABSs), may blend with the MZMs [27–36].Such ABSs are widely viewed as a pair of weakly coupledMZMs with finite separation [30, 32, 33], it would concealthe information of one MZM due to the weak hybridiza-tion and masquerade as a single MZM. For example, inthe tunneling experiments, the ABS will also induce a2 e /h zero-bias conductance peak, which is very similarto the case of MZM [30]. Thus, ABSs and MZMs arehard to be distinguished through normal measurementsuch as the local electric transport.Recently, various experimental schemes have been pro-posed to distinguish these two types of states. Theseproposals can be classified into two categories: measur-ing the response to the local perturbations [30–33] ordetecting the non-local conductance correlations [37–43].Nevertheless, all these methods strongly depend on thedetailed properties of the materials, therefore other possi-bilities besides MZMs and ABSs are still hard to be ruledout. As a result, there is still no smoking-gun evidencefor the identification of the MZMs yet. One convincingway to distinguish the MZM from the ABS is based onits non-Abelian statistics [44–46] which is a global prop- (a)(b) Δ )1-10 E ne r g y ( Δ ) (d)
10 2 3 4 5 60101 step t/T (c)(e)
FIG. 1: (a) MZMs in a semiconductor-superconductornanowire. The density of states (DOS) of the MZMs (greencurve) is non-locally distributed at both ends of the nanowire.(b) ABS in the same nanowire. The DOS of the ABS (greencurve) is confined by the chemical potential of the QD. (c)The energy spectrum of the semiconductor-superconductornanowire (with the QD confinement) versus the Zeeman en-ergy. The ABS is presented before the system becomes topo-logically non-trivial. (d) The cross-shaped structure adoptedfor the braiding of the MZMs. All four arms are topologi-cally non-trivial with N x = 100 a , µ = − t , and the Zeemanenergy V z = 2∆ [vertical black line in Fig. 1(c)]. (e) Evo-lution of the wavefunction ψ − j ( t ). After swapping MZMs γ and γ twice in succession, ψ − j evolves into ψ + j due to thenon-Abelian braiding statistics of the MZMs ( j = 1 , erty instead of a local one. However, only few studieshave paid attention to this topic yet [47]. Therefore, asthe first step toward distinguishing the MZM from theABS, it is essential to theoretically investigate the differ-ence between the braiding properties of MZM and ABS.What’s more, since MZM and ABS are usually mixed a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y with each other, it is necessary to study the braidingstatistics in the presence of both MZM and ABS. Suchinvestigation will also shed light on how to realize TQCwhen ABS is also engaged.Based on the cross-shaped junction shown in Fig. 1(d)[48, 49], we numerically studied the non-Abelian braid-ing of the MZMs in a semiconductor-superconductornanowire. The braiding properties of the MZMs are ro-bust against perturbations as expected. On the contrary,the braiding results exhibit a dynamic-phase-induced pe-riodic oscillation if one pair of MZMs is replaced by anABS. Further investigation implies that an ABS can bedecomposed into two weakly coupled MZMs. In this way,the braiding results in the presence of an ABS can be ex-plained by combining the non-Abelian geometric phasewith the ABS-related dynamic phase. Such dynamicphase can be eliminated by reversing the eigenenergy atthe middle of the symmetric braiding protocol, which iswidely adopted in the geometric quantum computation(GQC). In this way, the non-Abelian braiding propertieswill come back to its original form, which implies that theTQC could still be robust even when the ABS is engaged. MZM and ABS in the semiconductor-superconductornanowire — The tight-binding model for a one-dimensional s -wave superconductor with Rashba spin-orbit coupling can be described as: H D = (cid:88) R , d ,α − t ( ψ † R + d ,α ψ R,α + h.c. ) − µψ † R ,α ψ R ,α + (cid:88) R , d ,α,β − iU R ψ † R + d ,α ˆ z · ( (cid:126)σ × d ) αβ ψ R ,β + (cid:88) R ,α ∆ e iφ ψ † R ,α ψ † R , − α + h.c. + (cid:88) R ,α,β ψ † R ,α ( V · (cid:126)σ ) αβ ψ R ,β . (1)where R denotes the lattice site, d is the unit vector that d x and d y connect the nearest neighbor sites along the x - and y -direction, respectively. Besides, α and β arespin indices, t denotes the hopping amplitude, µ is thechemical potential, U R is the Rashba coupling strength,and V is the Zeeman energy induced by the magneticfield along the x - or the z -direction. The superconductingpairing amplitude is denoted as ∆, and φ is the pairingphase. The practical parameters can be obtained froma recent experiment [13]. Here, we adopt ∆ = 250 µ e V , t = 10∆, and U R = 2∆ in the following calculations.As the localized state bound with the impurity or spa-tial defect, the ABS is absent in a clean system. Exper-imentally, the ABS is usually presented with the quan-tum dot (QD) confinement at the end of the nanowire[18, 30]. Thus, for obtaining an ABS, a QD confine-ment potential should be included. As depicted in Fig.1(b), a sinusoidal local chemical potential in the form of V d ( R ) = − V D cos(2 π R − L D L D ) is presented at the right endof the nanowire, in which L D = 10 a is the half lengthof the QD and V D = 0 . t is the depth of the poten- tial well in the QD. When the external magnetic fieldis turned on, as shown in Fig. 1(c), a low-energy ABSwill be trapped in the QD before the system enteringinto the topologically non-trivial phase. In the conditionthat the magnetic field is larger than ∆, the topologi-cal phase transition happens and therefore two MZMswill distribute non-locally at both ends of the nanowire[Fig. 1(a)]. On the contrary, the ABS is a localized statedistributed at one end of the nanowire [Fig. 1(b)]. Re-markably, the ABS’s energy is close to zero at V = 0 . Non-Abelian braiding of MZMs — The non-Abelianbraiding of MZMs can be simulated based on the cross-shaped junction [Fig. 1(d)]. Each of the four arms inthe junction is a topologically non-trivial semiconductor-superconductor nanowire [ V z = 2∆ as indicated by thevertical black line in Fig. 1(c)]. Four gates (G1, G2, G3,and G4) are situated near the cross point, each arm canbe connected to (separated from) the others by turningoff (on) the gate voltage in the corresponding gate. Ini-tially, gate voltages in G1 and G3 are turned on whilein G2 and G4 are turned off, hence three pairs of MZMs( γ j − and γ j with j = 1 , ,
3) are localized at the endsof the three divided parts. The aim of our braiding opera-tion is swapping the positions of γ and γ . The nanowirealong the y -direction is an auxiliary one in such opera-tion. The braiding protocol takes three steps (the timecost for each step is T ) to swap γ and γ spatially. Inthe first step, G1 is turned off and then G2 is turned on,hence γ is transmitted to the top of G2. In the secondstep, G3 is turned off and then G1 is turned on, so that γ is moved to the original position of γ . In the thirdstep, G2 is turned off and then G3 is turned on, as aresult, the spatial positions of γ and γ are swapped.Initially, the effective low-energy Hamiltonian describ-ing each separated arm is in the form of H j, eff = i(cid:15) j γ j − γ j ( j = 1 , , (cid:15) j is the coupling energybetween MZMs induced by the finite-size effect, whichis exponentially small and can be neglected in the mostcases. Thus, the eigenstates are in the wavefunctions of ψ ± j (0) = ( γ j − ± iγ j ) / √
2. During the braiding pro-cess, the wavefunction evolves as | ψ ± j ( t ) (cid:105) = U ( t ) | ψ ± j (0) (cid:105) ,where U ( t ) = ˆ T exp[ i (cid:82) t dτ H ( τ )] is the time-evolutionoperator ( ˆ T is the time-ordering operator). Since (cid:15) j isexponentially small, the dynamic phase accumulated canbe neglected. However, a topological geometric phase π ispicked up during the braiding hence we have γ → γ and γ → − γ [44]. Thus, if γ and γ are swapped once, thenthe wavefunction will evolve into ψ ± (3 T ) = ( γ ± iγ ) / √ ψ ± (3 T ) = ( − γ ± iγ ) / √
2. After γ and γ areswapped twice in succession, the wavefunctions are inthe forms of ψ ± (6 T ) = ( γ ∓ iγ ) / √ ψ ∓ (0) and ψ ± (6 T ) = ( − γ ± iγ ) / √ − ψ ∓ (0). Our numericalsimulation results [Fig. 1(e)] confirm that ψ + j evolves FIG. 2: (a) The braiding process with an ABS engaged. Thelow-energy ABS can be regarded as a pair of coupled MZMs γ and γ which are spatially separated with a limited dis-tance. In this point of view, the braiding in the presence ofABS is equivalent to the exchange between one “free” MZM γ and another MZM γ bounded in the ABS. (b) An illustra-tion for the braiding operation in the presence of the ABS. Acomplete braiding operation, which swaps γ and γ twice insuccession, can be decomposed into five operation steps: step1, 3, and 5 are the coupling (fusion) process with nonvanish-ing dynamic phase that forms an ABS; step 2 and 4 are theexchange operation swapping one “free” MZM and anotherMZM bounded in the ABS. into ψ − j ( j = 1 ,
2) after adiabatically swapping γ and γ twice in succession, which is consistent with the non-Abelian braiding rules discussed above. It is worth not-ing that the non-Abelian statistics originates from thetopological geometric phase, thus the braiding results areindependent of the braiding time T provided that the adi-abatic condition is satisfied. Non-Abelian braiding in the presence of ABS —Naively, the presence of the ABSs are supposed to ruinthe non-Abelian statistics of the MZMs. However, basedon the simple assumption that an ABS can be decom-posed into two MZMs [30, 32, 33], we found that thebraiding operation between MZMs can still be performedin the cross-shaped junction if only one pair of MZMsis replaced by an ABS. This suggests that the possiblenon-Abelian braiding properties of MZMs could still beexhibited even in the presence of an ABS.As shown in Fig. 2(a), the left arm of the cross-shapedjunction is tuned into the ABS region, while the otherthree arms are still in the topologically non-trivial re-gion supporting MZMs as before. At the beginning,the low-energy ABS can be regarded as a pair of cou-pled MZMs γ and γ which are spatially separated witha limited distance. In this point of view, the braid-ing in the presence of ABS could be equivalent to theexchange between one “free” MZM and another MZMbounded in the ABS. Therefore, a new ABS is formedafter such braiding operation by fusing (coupling) thenew pair of MZMs. Such fusion (coupling) process canbe generally described by the dynamic Hamiltonian as (a)(b) |< (6T)| (0)>| |< (6T)| (0)>| |< (6T)| (0)>| |< (6T)| (0)>| T(100/ )step t/T step t/T -0.03-0.02-0.0100.010.020.03-0.6-0.4-0.200.20.4-0.6 E / FIG. 3: (a) Energy spectrum of the bulk states and the ABSin the cross-shaped junction during the braiding process. Herewe set the Zeeman field in the left arm of the cross-junctionas V z = 0 . V z = 2∆. In such case, an ABS is presented at theleft arm while two pairs of MZMs are presented in the otherthree arms as before. (b) Numerical simulation of the braidingresults as functions of the braiding time-cost T . H ( t ) = i (cid:80) i,j (cid:15) k ( t ) γ i γ j , in which the corresponding time-evolution operator U ( t ) = ˆ T exp[ i (cid:82) t dτ H ( τ )] is equiva-lent to a unitary transformation on γ i and γ j [50]:˜ γ i = cos( θ k ) γ i + sin( θ k ) γ j , ˜ γ j = − sin( θ k ) γ i + cos( θ k ) γ j . (2)where θ k / (cid:82) (cid:15) k ( t ) dt is the dynamic phase induced bythe coupling energy between the two MZMs which formthe ABS.With such a dynamic phase being taken into account,the braiding in the presence of ABS can be clearly de-scribed as below. In the first step of the braiding oper-ation, one MZM in the ABS is moved to the top arm.The coupling strength between γ and γ varies duringsuch moving process. This will induce a unitary evolu-tion as γ → ˜ γ = cos( θ ) γ + sin( θ ) γ and γ → ˜ γ = − sin( θ ) γ + cos( θ ) γ , in which θ / (cid:82) T (cid:15) ( t ) dt isthe dynamic phase accumulated. Such dynamic phasewill significantly alter the braiding properties below. Inthe second step of the braiding operation, γ is moved tothe original position of γ . In the third step of the braid-ing operation, ˜ γ instead of γ is moved to the originalposition of γ , which gives rise to ˜ γ → γ and γ → − ˜ γ .Similarly, the dynamic phase induced by the coupling be-tween ˜ γ and γ will also alter the braiding results inthe second half of the complete braiding process that γ and γ are swapped twice in succession. Such completebraiding process can be effectively decomposed into fiveoperation steps as shown in Fig. 2(b). Operation steps 1,3 and 5 are the fusion (coupling) process with nonvanish-ing dynamic phase that forms an ABS, while operationsteps 2 and 4 are the exchange operation swapping one“free” MZM and another MZM bounded in the ABS.Therefore, the wavefunctions evolve as γ → − ˜ γ and γ → − ˜ γ after the complete braiding process, in which˜ γ = − sin( θ )˜ γ + cos( θ ) γ and θ / (cid:82) TT (cid:15) ( t ) dt isthe dynamic phase accumulated due to the fusion (cou-pling) energy between γ and ˜ γ .The numerical simulation results support the analy-sis above pretty well. Fig. 3(a) shows the energy spec-trum for both the bulk states and the ABS, in which thegap between the low-energy subgap states and the bulkstates keeps integrity during the braiding. Therefore,the scattering between the bulk states and the subgapstates including the ABS and the MZMs is prohibited inthe adiabatic condition. In addition, although the initial( t = 0) fusion energy between γ and γ is very small, amagnified view of the energy spectrum [right half of Fig.3(a)] shows that the ABS’s eigenenergy becomes rela-tively larger and cannot be neglected during the braid-ing. Such larger energy will significantly alter the braid-ing results. For example, ψ +2 (0) = ( γ + iγ ) / √ ψ +2 (6 T ) = ( − ˜ γ + iγ ) / √ ψ +2 (6 T )on ψ +2 (0) is (1 − cos( θ )) /
2, on ψ − (0) is (1 + cos( θ )) / ψ ± (0) is sin( θ ) /
2, in which θ is the dynamicphase accumulated during t ∈ [ T, T ] due to the fusion(coupling) energy [red curves in Fig. 3(a)]. The meanvalue of such fusion energy is about ¯ (cid:15) ≈ − ∆, whichgives rise to the oscillation period of ∆ T = π (cid:15) ≈ .The numerical results shown in Fig. 3(b) is fully consis-tent with these analytical predictions. Elimination of the dynamic phase — As a dynamicaleffect, such oscillation behaviour is expected to be re-moved by eliminating the dynamic phase. A direct wayis to decrease the ABS’s fusion energy during the braid-ing process. In Fig. 3 we set the chemical potential in thecross point of the junction as µ c = µ = − t . This meansthat the arms are perfectly connected to each other whenthe gate voltage is turned off. It is worth noting that thefusion energy significantly decreases if these arms are notperfectly connected to each other [52]. As shown in Fig.4(a), in the case of µ c = − t − . ∼ − ∆. Therefore, the braiding results oscillate veryslowly with respect to the braiding time T [Fig. 4(c)]. Inaddition, for relatively small braiding time T ∼ / ∆,the dynamic phase is much smaller compared with thegeometric phase so that the braiding results retrieve itsoriginal form.In the traditional GQC, the dynamic phase can beeliminated through the spin-echo technique [53–55] which T(100/ )
T(100/ ) |< (6T)| (0)>||< (6T)| (0)>||< (6T)| (0)>| +|< (6T)| (0)>| t/T -0.0100.010 1 2 3 t/T -0.0100.01 E / (a)(c) (b)(d) FIG. 4: (a) Energy spectrum for the ABS during the braidingprocess. Here µ c = − t − . µ c = − t − V z = [0 .
603 + 0 .
02 cos( t/T · π )]∆. (c)The braiding results for Fig. 4(a), which oscillate very slowlywith the braiding time T . For relatively small braiding time T ∼ / ∆, the braiding results retrieve its original form. (d)The braiding results for Fig. 4(b). The MZMs’ non-Abelianbraiding statistics is well exhibited for T (cid:38) / ∆. The bluedashed lines in both (c) and (d) denote T = 200 / ∆. reverses the sign of the eigenenergy at the middle of thesymmetric braiding protocol. Similar technique is also apowerful method to cancel out the ABS-related dynamicphase. Noticing that the spectrum of the ABS [Fig. 1(c)]is nearly symmetric about the zero-energy in the vicinityof V z = 0 . V z = 0 .
62∆ during t ∈ [0 , T / V z = 0 . t ∈ [ T / , T ]. Such dynamic phase eliminationworks better if the magnetic field is modulated in a moresmooth way. As shown in Fig. 4(b), we choose a sinu-soidal magnetic field as V z = [0 .
603 + 0 .
02 cos( t/T · π )]∆(since the spectrum is not perfectly symmetric, the meanvalue of V z will slightly deviate from 0 . t = T / T (cid:38) / ∆ to avoid the mixing between the subgapstates and the bulk states. Since the ∆ is typically in theorder of ∼ T should be in theorder ∼ . Discussion — We have shown that the Non-Abelianstatistics of MZMs can still be preserved in the presenceof ABS. It would suggest that ABS will provide moreadvantages in the following TQC. In the TQC, the in-formation stored in the qubit could be read out by mea-suring the parity of two combined MZMs [56–60]. SinceMZMs are usually non-locally distributed, it means thatthe reading out technique would be very difficult. Whilein the case of ABS, however, the reading out techniquecould be rather easy since a ABS consists of two little sep-arated MZMs. Moreover, since the ABS is deemed as twoweakly coupled MZMs with finite distance, the dynamicphase elimination method discussed above can also beperformed for the finite-size-induced partially overlappedMZMs. In one of our previous work [38], we have revealedthat the spectrum of such partially overlapped MZMswill cross at zero-energy with definite parity by modu-lating the Zeeman field or gate voltage. It implies thatthe dynamic phase can be canceled out by modulatingeither the Zeeman field or the gate voltage. Therefore,the TQC can be realized even in shorter TSC nanowire.
Acknowledgement. — Wenqin Chen and Jiachen Wangcontributed equally to this work. This work is fi-nancially supported by NSFC (Grants No. 11974271)and NBRPC (Grants No. 2017YFA0303301, and No.2019YFA0308403). ∗ Electronic address: [email protected][1] A. Kitaev, Phys. Usp. , 131 (2000).[2] C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. DasSarma, Rev. Mod. Phys. , 1083 (2008).[3] L. Fu, and C. L. Kane, Phys. Rev. Lett. , 096407(2008).[4] J. D. Sau, R. M. Lutchyn, S. Tewari, S. Das Sarma, Phys.Rev. Lett. , 040502 (2010).[5] S. Fujimoto, Phys. Rev. B. , 220501(R) (2008).[6] M. Sato, Y. Takahashi, and S. Fujimoto, Phys. Rev. B ,134521 (2010).[7] J. Alicea, Phys. Rev. B , 125318 (2010).[8] R.M. Lutchyn, J.D. Sau, and S. Das Sarma, Phys. Rev.Lett. , 077001 (2010).[9] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. , 177002 (2010).[10] A. C. Potter and P. A. Lee, Phys. Rev. B , 094525(2011).[11] M. Hell, M. Leijnse, K. Flensberg, Phys. Rev. Lett. ,107701 (2017).[12] F. Pientka, A. 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