An experimentally accessible quality factor for Majorana wires
AAn experimentally accessible quality factor for Majorana wires
David J. Clarke
1, 2, 3 Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park Joint Quantum Institute, University of Maryland, College Park Station Q Maryland
Spin-orbit coupled semiconducting nanowires with proximity-induced superconductivity are expected to hostMajorana zero modes at their endpoints when a sufficiently strong magnetic field is applied. The resulting phasewould be a one-dimensional topological superconductor. However, while a variety of experiments have beenperformed observing a zero bias conductance peak (suggestive of Majorana zero modes), the topological natureof these physical systems is still a subject of debate. Here we suggest a quantitative test of the degree to whicha system displaying a zero bias peak may be considered topological. The experiment is similar to previousmeasurements of conductance, but is performed with the aid of a quantum dot at the wire’s end. We arriveat the surprising result that the non-local nature of the topological system may be identified through a localmeasurement.
Non-Abelian anyons, quasiparticle excitations in two-dimensional systems that enact non-commuting unitary trans-formations on the ground state of a system when exchanged,are a fascinating consequence of low-dimensional physics,and are expected to provide a significant advantage in the fieldof quantum computation[1–7]. Non-Abelian statistics of thissort has yet to be demonstrated definitively, but recent yearshave shown a flurry of experimental [8–18] and theoretical[19–40] progress in the development of the simplest type ofnon-Abelian excitation, defects binding real fermionic zeromodes called Majorana zero modes or MZMs. Some of themost exciting of these results have been the observation ofzero bias conductance peaks in spin-orbit coupled semicon-ducting nanowires with proximity-induced superconductivity.Such systems are expected to host Majorana zero modes at theends of a region of an effective spinless p-wave ‘topological’superconducting region [19–22]. While experiments thus farare extremely suggestive of Majorana physics and tend to defyexplanations of a non-topological nature ( e.g.
Kondo physics)[16, 24–29], the observation of a zero-bias conductance peakis itself only a necessary and not a sufficient condition for de-termining that a system is behaving as a topological super-conductor. For this reason, a variety of experiments have beenproposed for probing the non-Abelian nature of the Majoranadefects as an indication of topological physics [30–34] and asa stepping stone toward quantum computation [35–40]. Theseproposals have in common that they involve non-local probesof the system ( e.g. measurement of the fermion parity of anentire wire segment). It is a unique feature of the measure-ment that we propose here that it is both local and sufficient todetermine whether a system is behaving topologically.The reason that observation of a zero-bias conductancepeak is not sufficient to determine that a nanowire is in a topo-logical phase is somewhat subtle. When the lead probing thesystem is a normal metal, a zero-bias peak in conductancedoes indicate that there is at least one pair of energy levels inthe system that differ in quantum numbers by an electron andthat are degenerate to within the resolution of the experiment.A zero-bias peak in conductance is therefore sufficient to de-termine that there exists a fermionic mode ˆ d in the nanowire FIG. 1: Schematic diagram of the system under consideration. Con-ductance spectroscopy is performed on a wire containing a singlelow-lying fermionic level ˆ d = ( γ + iγ ) by coupling to a lead atvoltage V bias relative to ground through a quantum dot with a singlefermionic level with annihilation operator ˆ c . The energy difference (cid:15) d between the occupied and unoccupied states of that dot is set bya side-gate voltage and varied to perform the experiment suggestedhere. The dot-wire system is characterized by three additional pa-rameters: the couplings η and η (cid:48) from the dot to the Majorana modes γ and γ respectively, and the Majorana hybridization (cid:15) w that setsthe energy difference between occupied and unoccupied states of thefermionic mode in the wire. If the wire is in the topological supercon-ducting phase, both η (cid:48) and (cid:15) w should be ∼ . A similar differentialconductance experiment without the dot present can only access thevalue of (cid:15) w . With the dot, all three parameters may be measured in-dependently. In particular, one can make measure of q = 1 −| η | / | η (cid:48) | ,a ‘topological quality factor’ quantifying the extent to which one mayexpect non-Abelian behavior from the Majorana bound states in thewire. at zero energy (within some bound given by the peak width).Such a degeneracy is predicted in the topological system[1–7], where the fermionic mode is shared among two Majoranazero modes separated by the length of the wire. (This sepa-ration is the source of the ‘topological protection’ of the de-generacy and the utility for quantum information applications[1–7].) However, it is important to realize that by taking realand imaginary parts of any fermionic zero mode, that modemay be formally divided into two Majorana modes γ and γ at zero energy (as d = iγ + γ , where γ = γ = 1 ).This alone does not suffice for the system to be topological.The hallmark of the topological phase is that these two Ma- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b jorana modes are separated in space, which cannot be deter-mined from an examination of the energy of the fermionicmode in and of itself (although it may be strongly suggested ifthe ground state energy splitting becomes exponentially smallas the wire length is increased) [1–7, 17]. Rather, to deter-mine that a system is topological, one must show that each ofthese Majorana modes is localized to a different position, andin particular that each mode may only couple to operators nearthat position.Here, we show that a modified transport experiment inter-posing a quantum dot between the lead and the nanowire canmeasure separately the coupling of the dot to each of the twoMajorana bound states that make up the fermionic mode inthe wire. If the wire is topological, the dot should only cou-ple to the Majorana bound state nearest the wire end next tothe dot. We therefore introduce a ‘topological quality factor’given by one minus the ratio of the two couplings. This qual-ity factor is 1 when the system is purely topological and zerofor an electronic (non-superconducting) bound state. A qual-ity factor near 1 is essential for the observation of non-Abelianbehavior from Majorana systems in future experiments [34].The quality factor may be obtained from differential con-ductance measurements by tuning the quantum dot through aresonance. One important property of the experiment we sug-gest is that it is eminently realizable in present day experimen-tal systems. In fact, a version of this experiment has alreadybeen performed in Ref. 18, though no systematic investigationof the quality factor was performed. We conclude our paperwith a comparison of the conductance spectra in that paperwith those predicted by the simple model presented here (SeeFig. 3). The qualitative similarly of the data and the results ofour model should allow the effective extraction of the modelparameters ( q in particular) from experiments on similar sys-tems. The Model : The bulk of this paper is devoted to the analysisof a simple model of the dot-wire system shown in Fig. 1. Wedescribe the quantum dot by a single fermionic mode ˆ c thatmay be occupied or unoccupied. The fermionic mode ˆ d in thewire is split into two Majorana modes γ = − i ( ˆ d − ˆ d † ) and γ = ˆ d + ˆ d † . These modes are not assumed to be of topologicalorigin. We note that charge is only conserved mod 2 withinthe superconducting nanowire, so the most general Hamilto-nian coupling the dot to the wire is given by H = (cid:15) d ˆ c † ˆ c − i (cid:15) w γ γ ˆ c + (cid:18) i η c † γ + η (cid:48) c † γ + h . c . (cid:19) (1)Here the couplings η and η (cid:48) may be taken to be real withoutloss of generality via gauge transformations on ˆ c and ˆ d . Wemay express this Hamiltonian in the occupation basis for thefermionic modes on the dot and the wire. We have H = (cid:18) | (cid:105)| (cid:105) (cid:19) T (cid:18) (cid:15) w ( η + η (cid:48) ) / η + η (cid:48) ) / (cid:15) d (cid:19) (cid:18) (cid:104) |(cid:104) | (cid:19) + (cid:18) | (cid:105)| (cid:105) (cid:19) T (cid:18) η − η (cid:48) ) / η − η (cid:48) ) / (cid:15) w + (cid:15) d (cid:19) (cid:18) (cid:104) |(cid:104) | (cid:19) (2) q = ϵ w / η = - - - - ϵ d / η V b i a s / η q = ϵ w / η = - - - - ϵ d / η V b i a s / η q = ϵ w / η = - - - - ϵ d / η V b i a s / η q = ϵ w / η = - - - - ϵ d / η V b i a s / η FIG. 2: Expected low-energy spectrum near the dot resonance forvarious values of the topological quality factor q and in the absenceof hybridization between the two Majorana modes. The distance be-tween the outer two mode peaks at resonance is twice the ‘topolog-ical’ coupling η . The distance between the two inner mode peaks atresonance is twice the ‘non-topological’ coupling η (cid:48) . For a perfectlytopological system ( q = 1 , (cid:15) w = 0 ), the zero-bias conductance peakdoes not split as it passes through the dot resonance. Note that we ignore any Coulomb interaction (such an interac-tion would add a term E C | (cid:105)(cid:104) | to the Hamiltonian). Againwithout loss of generality we take | η | > | η (cid:48) | > and definethe quality factor q = 1 − | η (cid:48) | / | η | . Measurement
We note that the Hamiltonian above is splitinto two subsectors with opposite fermion parity. A typicaltransport experiment measuring conductance through the dotinto the superconducting wire has conductance peaks when-ever the bias voltage is on resonance with transitions betweenthe ground state of the dot-wire system and an excited statein the opposite parity sector. In the absence of a significantCoulomb interaction between the wires, the transition spac-ings available from each of the four possible ground states ac-tually all coincide [43]. These conductance peaks will there-fore occur at the values V bias = ± (cid:112) ( (cid:15) w − (cid:15) d ) + ( η + η (cid:48) ) ± (cid:112) ( (cid:15) w + (cid:15) d ) + ( η − η (cid:48) ) . (3)If the dot is well off resonance ( | (cid:15) d | (cid:29) η, η (cid:48) ), then the tran-sitions between the lowest levels provides a direct measure of (cid:15) w , the ‘Majorana hybridization.’ This measurement has beenthe focus of much of the experimental effort thus far, whichfinds (cid:15) w ∼ over a wide range of parameters [8–15, 17, 18].However, by tuning the quantum dot to resonance [18], wemay achieve a direct measurement of the couplings betweenthe dot and the wire. In the ideal (topological) case, onlyone of the Majorana modes in the wire is coupled to the dot( η (cid:48) = 0 , or q = 1 ). In Fig. 2 we plot the transition spectrum ofthe dot-wire system for (cid:15) w = 0 and various values of q . Nearresonance, two conductance peaks break off from the contin-uum and have an avoided crossing at (cid:15) d = 0 . Their point ofclosest approach gives a measurement of η , the larger of thetwo Majorana couplings (the distance between the peaks is η ). Meanwhile, the states that are degenerate at large (cid:15) d splitnear the dot resonance by η (cid:48) . Comment on locality and Ising theory : It is somewhat re-markable that such a simple experiment, measuring effec-tively local properties of the hybridization between the dotand wire states, can determine the non-local nature of thefermionic mode on the wire that is shared between the twoMajoranas. It is somewhat revealing to approach this prob-lem from the point of view of the Ising anyon theory that is(approximately) realized by Majorana zero modes [1–7]. Thistheory is governed by the fusion rules Ψ ⊗ Ψ = I Ψ ⊗ σ = σσ ⊗ σ = I ⊕ Ψ , (4)which determine the possible resulting anyonic charges whentwo or more anyons in the theory are joined together. Thistheory is mapped to the MZM-containing system by equatingthe Majorana bound states with the σ charge, the occupiedstate of two Majorana modes by the Ψ charge, and the unoc-cupied state by the σ charge. The last line of Eq. (4) thereforeindicates that the combination of two Majorana modes mayeither contain a fermion ( Ψ ) or be empty ( I ). Importantly,any combination of anyon charges containing an odd numberof σ s will always fuse to a σ , while any even combination willfuse to I or Ψ . In the Majorana language, this means that anyodd number of Majorana modes, no matter how coupled , willalways have some Majorana mode left at zero energy.When the experimental device shown in Fig. 1 is con-sidered in these terms, it is clear that the topological case( (cid:15) w = η (cid:48) = 0 ) has a zero energy mode independent of thetuning of the dot energy (cid:15) d . (In this context it is useful tothink of the fermionic mode on the dot being made up of twoMajorana modes). When the last Majorana mode γ is cou-pled in ( (cid:15) w , η (cid:48) (cid:54) = 0 ), the number of Majorana modes (or σ charges) that are coupled becomes even, and an energy split-ting is allowed between the states with overall even ( I ) or odd( Ψ ) fermion parity. Non-zero Majorana hybridization : If the two Majoranabound states in the nanowire have non-zero overlap, the con-ductance spectrum near the dot resonance becomes asymmet-ric in (cid:15) d , and the two central peaks, separated by (cid:15) w off reso-nance, have a crossing that approaches the dot resonance as (cid:15) w increases. Such a crossing is observed in Ref. 18 (See Fig. 3,top). Consequences for topological operations : If we consider atopological operation, such as a step in a braid process [30–32, 35], there is a constraint on the quality factor required torecover the topological result of that computation [34]. Theconstraint stems from the requirement that the operation mustbe performed diabatically with respect to the non-topologicalcoupling η (cid:48) in order for the system to respond as if only thetopologically allowed couplings are present. In terms of the q = ϵ w / η = - - - - ϵ d / η V b i a s / η q = ϵ w / η = - - - - ϵ d / η V b i a s / η FIG. 3: Comparison of differential conductance color plots fromDeng et al. [18] (left) with transition plots from our simple model(right). Here we plot the transitions for a quality factor of q = 0 . and two values of (cid:15) w /η that lead to qualitatively similar traces. Thecircled point in the Deng et al. data (top left) indicates an anticross-ing between dot and wire states that is also present within our simplemodel (top right). In fact, the behavior is sufficiently alike that moreexact values for the model parameters might be extracted from simi-lar data in a systematic study. quality factor introduced here, the constraint is − q (cid:28) (cid:126) ητ , (5)where τ is the time taken to perform the operational step.However, in most applications there is an additional constraintthat the operation be performed adiabatically with respect tothe lowest excitation gap, which in this case is of the order of η itself [1–7, 31, 34, 35]. Likewise, in all cases the operationmust proceed quickly with respect to the hybridization (cid:15) w be-tween the two Majorana modes. This extends the requirementto − q, (cid:15) w η (cid:28) (cid:126) ητ (cid:28) . (6)If the operation is performed too slowly compared to the qual-ity factor and the excitation gap, or compared to the Majo-rana hybridization, the system will behave as if it is non-topological. If the operation is performed too quickly, dia-batic errors will result [1–7, 41]. We emphasize once morethat (cid:15) w and (1 − q ) η are two independent energy scales (al-though both are expected to be suppressed exponentially bythe Majorana separation in the topological case.) It is entirelypossible for a system to have a small Majorana hybridizationand still behave non-topologically. Comparison with experimental data : In Fig. 3, we showdata from the paper of Deng et al. measuring the conduc-tance through a quantum dot for situations in which the Majo-rana hybridization is either well resolved or near zero. Whilewe observe that a quality factor near . seems consistentwith the experimental data plotted here, these data sets werenot chosen to demonstrate a high quality factor, and sampleconfigurations with smaller splitting of the zero-bias conduc-tance peak at resonance have been observed [42]. Note thatthe curvature of the outer transitions in our simple modeldoes not match that of the experimental data, as the experi-mentally observed peak is bent toward the center horizon bylevel repulsion from the continuum states. Nevertheless, it isclearly within the realm of current experimental capabilitiesto perform the desired measurements, allowing direct extrac-tion of both the Majorana hybridization and the topologicalquality factor from experimental data. This experiment wouldtherefore be useful in determining whether a given systemwill behave topologically in more complicated experimentsinvolving braid operations [30–36] or non-local parity mea-surements [38–40]. ACKNOWLEDGEMENTS
This work is supported by Microsoft, Station Q, by the Lab-oratory for Physical Sciences (LPS-MPO-CMTC) and by theJoint Quantum Institute (JQI-NSF-PFC). We gratefully ac-knowledge fruitful discussions with Jay D. Sau and SankarDas Sarma. [1] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D.Sarma, Rev. Mod. Phys. , 1083 (2008).[2] J. Alicea, Rep. Prog. Phys. , 076501 (2012),arXiv:1202.1293.[3] M. Leijnse and K. Flensberg, Semiconductor Science Technol-ogy , 124003 (2012), 1206.1736.[4] C. W. J. Beenakker, Annu. Rev. Condens. Matter Phys. , 113(2013), arXiv:1112.1950.[5] T. D. Stanescu and S. Tewari, Journal of Physics CondensedMatter , 233201 (2013), 1302.5433.[6] S. Das Sarma, M. Freedman, and C. Nayak, NPJ Quantum in-formation , 15001 (2015).[7] S. R. Elliott and M. Franz, Reviews of Modern Physics , 137(2015), 1403.4976.[8] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M.Bakkers, and L. P. Kouwenhoven, Science , 1003 (2012).[9] A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrik-man, Nat. Phys. , 887 (2012), arXiv:1205.7073.[10] M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff, andH. Q. Xu, Nano Letters , 6414 (2012), arXiv:1204.4130.[11] L. P. Rokhinson, X. Liu, and J. K. Furdyna, Nat. Phys. , 795(2012), ISSN 1745-2473, arXiv:1204.4212.[12] A. D. K. Finck, D. J. Van Harlingen, P. K. Mohseni, K. Jung,and X. Li, Phys. Rev. Lett. , 126406 (2013).[13] H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen, M. T.Deng, P. Caroff, H. Q. Xu, and C. M. Marcus, Phys. Rev. B , 241401 (2013).[14] W. Chang, S. M. Albrecht, T. S. Jespersen, F. Kuemmeth,P. Krogstrup, J. Nyg˚ard, and C. M. Marcus, Nature Nanotech-nology , 232 (2015), 1411.6255. [15] H. Zhang, ¨O. G¨ul, S. Conesa-Boj, K. Zuo, V. Mourik, F. K. deVries, J. van Veen, D. J. van Woerkom, M. P. Nowak, M. Wim-mer, et al., ArXiv e-prints (2016), 1603.04069.[16] E. J. H. Lee, X. Jiang, M. Houzet, R. Aguado, C. M. Lieber,and S. de Franceschi, Nature Nanotechnology , 79 (2014),1302.2611.[17] S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuemmeth,T. S. Jespersen, J. Nyg˚ard, P. Krogstrup, and C. M. Marcus,Nature (London) , 206 (2016), 1603.03217.[18] M. T. Deng, S. Vaitiek˙enas, E. B. Hansen, J. Danon, M. Lei-jnse, K. Flensberg, J. Nyg˚ard, P. Krogstrup, and C. M. Marcus,Science , 1557 (2016), 1612.07989.[19] A. Y. Kitaev, Physics Uspekhi , 131 (2001), arXiv:cond-mat/0010440.[20] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys.Rev. Lett. , 040502 (2010), 0907.2239.[21] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. , 077001 (2010), 1002.4033.[22] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. ,177002 (2010), 1003.1145.[23] L. Fu, Physical Review Letters , 056402 (2010), 0909.5172.[24] L. Fidkowski, J. Alicea, N. H. Lindner, R. M. Lutchyn, andM. P. A. Fisher, Phys. Rev. B , 245121 (2012), 1203.4818.[25] ˙I. Adagideli, M. Wimmer, and A. Teker, Physical Review B ,144506 (2014).[26] S. Das Sarma, A. Nag, and J. D. Sau, Phys. Rev. B , 035143(2016), URL http://link.aps.org/doi/10.1103/PhysRevB.94.035143 .[27] J. Liu, A. C. Potter, K. T. Law, and P. A. Lee, Physical reviewletters , 267002 (2012).[28] C.-H. Lin, J. D. Sau, and S. Das Sarma, Phys. Rev. B ,224511 (2012), 1204.3085.[29] C. Moore, T. D. Stanescu, and S. Tewari, ArXiv e-prints (2016),1611.07058.[30] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher,Nature Physics , 412 (2011), 1006.4395.[31] J. D. Sau, D. J. Clarke, and S. Tewari, Phys. Rev. B , 094505(2011), 1012.0561.[32] B. I. Halperin, Y. Oreg, A. Stern, G. Refael, J. Alicea, and F. vonOppen, Phys. Rev. B , 144501 (2012), 1112.5333.[33] D. J. Clarke, J. D. Sau, and S. Tewari, Phys. Rev. B , 035120(2011), 1012.0296.[34] D. J. Clarke, J. D. Sau, and S. Das Sarma, ArXiv e-prints(2016), 1610.08958.[35] D. Aasen, M. Hell, R. V. Mishmash, A. Higginbotham,J. Danon, M. Leijnse, T. S. Jespersen, J. A. Folk, C. M. Mar-cus, K. Flensberg, et al., Physical Review X , 031016 (2016),1511.05153.[36] T. Hyart, B. van Heck, I. C. Fulga, M. Burrello, A. R.Akhmerov, and C. W. J. Beenakker, Phys. Rev. B , 035121(2013), 1303.4379.[37] D. J. Clarke, J. D. Sau, and S. Das Sarma, Physical Review X , 021005 (2016), 1510.00007.[38] S. Plugge, A. Rasmussen, R. Egger, and K. Flensberg, ArXive-prints (2016), 1609.01697.[39] T. Karzig, C. Knapp, R. Lutchyn, P. Bonderson, M. Hastings,C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, et al.,ArXiv e-prints (2016), 1610.05289.[40] S. Vijay and L. Fu, Phys. Rev. B , 235446 (2016),1609.00950.[41] C. Knapp, M. Zaletel, D. E. Liu, M. Cheng, P. Bonder-son, and C. Nayak, Phys. Rev. X , 041003 (2016), URL http://link.aps.org/doi/10.1103/PhysRevX.6.041003 ..