Approaching a topological phase transition in Majorana nanowires Ryan V. Mishmash,1,2 David Aasen,1 Andrew P. Higginbotham,3,4 and Jason Alicea1,2 1Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA 2Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA 3Department of Physics, Harvard University, Cambridge, Massachusetts, 02138, USA 4Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark (Dated: June 13, 2016) Recent experiments have produced mounting evidence of Majorana zero modes in nanowire- 6 superconductor hybrids. Signatures of an expected topological phase transition accompanying the 1 onset of these modes nevertheless remain elusive. We investigate a fundamental question concern- 0 ing this issue: Do well-formed Majorana modes necessarily entail a sharp phase transition in these 2 setups? Assuming reasonable parameters, we argue that finite-size effects can dramatically smooth n thisputativetransitionintoacrossover,eveninsystemslargeenoughtosupportwell-localizedMa- u joranamodes. Weproposeovercomingsuchfinite-sizeeffectsbyexaminingthebehavioroflow-lying J excitedstatesthroughtunnelingspectroscopy. Inparticular,theexcited-stateenergiesexhibitchar- 9 acteristic field and density dependence, and scaling with system size, that expose an approaching topologicalphasetransition. Wesuggestseveralexperimentsforextractingthepredictedbehavior. ] Asausefulbyproduct,theprotocolsalsoallowonetomeasurethewire’sspin-orbitcouplingdirectly l l in its superconducting environment. a h - s I. INTRODUCTION the predicted bulk phase transition even as a matter of e principle(say,bytunnelingintothemiddleofthewirein- m stead of its ends or by other proposed means19,29,31–38)? Tunneling spectroscopy provides a powerful probe of t. topological superconductivity1,2. Perhaps most notably, Below we argue that the previously examined supercon- a ducting wires may experience strong finite-size effects m Majorana zero modes hosted by such systems are pre- that quite drastically smooth the phase transition into dicted to mediate ‘perfect Andreev reflection’ in the d- asymptotic low-energy limit, generating quantized2e2/h a crossover. Indeed, with reasonable assumptions we es- n zero-biasconductanceastemperatureT 0.3–8 Inprox- timate that the gapless bulk modes at an infinite wire’s → phase transition acquire a sizable gap of order the in- o imitized nanowires (non-quantized) zero-bias peaks were c indeed observed9–13 in the presence of a modest applied duced pairing energy, even in the largest systems stud- [ ied to date. In this situation one should not expect to magneticfieldneededtodrivethesystemfromatrivialto 2 topological superconducting state14,15; see also Refs. 16– see sharp signs of a phase transition, though extended Majorana-induced zero-bias peaks can still occur pro- v 18 for similar measurements on ferromagnetic atomic 8 chains. These experiments offer tantalizing evidence of vided spin-orbit coupling is sufficiently strong. Subtler 0 methods are then called for to verify this key aspect of Majoranamodesandhavejustifiablysparkedagreatdeal 9 the theory. of activity. 7 0 For an infinite proximitized nanowire a sharp second- Weproposethattheonsetofatopologicalphasetran- . orderphasetransitionseparatesthetopologicalandtriv- sition can be detected by studying conductance spectra 1 ial states14,15,19,20. Consequently, as one ramps up the in clean, hard-gap wires40,41. Discrete low-lying sub-gap 0 6 field the bulk gap closes and then reopens21 in the topo- energylevelsforafinitewireshouldberesolvablethrough 1 logical phase, leaving end Majorana zero modes behind. conductance maps as nicely demonstrated, for example, : Thecollapseandrevivalofthebulkgapconcomitantwith in Ref. 42. Importantly, these levels exhibit universal v the onset of a Majorana-induced zero-bias peak consti- characteristics inherited from an infinite system’s bona i X tutesamorerefinedpredictionthatappearsverydifficult fide phase transition. We outline several specific exper- r tomimicinalternativezero-bias-anomalyscenarios22–26. imental protocols designed to probe the imprint of the a Gap revival at the putative phase transition has, how- putative topological phase transition on finite-size wires. ever, so far proven experimentally elusive27. Reference Inspired largely by the recent developments presented 28 suggested that observing this feature is difficult when in Refs. 42–44, these protocols rely on measurements for tunnelingintothesystem’sendssimplybecausethebulk systems with either (i) a fixed physical wire length L or wavefunctionsthatbecomegaplessatthetransitiontend (ii) the ability to systematically change L in a single de- tohavelittlesupportneartheboundaries. Similareffects vice. Incase(i),wemakedetailedpredictionsfortheevo- can appear in multi-channel wires where the ‘topological lution of the finite-system’s energy levels near the phase sub-band’ may couple weakly to the lead29,30. transition—notablytheirfieldanddensitydependence— Visibility issues aside, it is useful to pose a more gen- whichmaybecomparedwithexistingexperimentaldata. eral question: Could existing experiments have observed For(ii),weproposevaryingLusingeitherpinch-offgates 2 N SC SC SC SC N SC SC SC SC N SC SC SC SC L L L FIG. 1. Zeeman field scans at strong spin-orbit coupling: Local density of states (LDOS) [see Eq. (D1)] versus h/∆ and E/∆ with µ=0 in the strong spin-orbit coupling regime, mα2/∆=14. Panels correspond to system sizes L/(cid:96)=5,10,20 from left to right, where (cid:96) ≡ (cid:126)α/∆ [cf. Eq. (8)]. Data shown represents the usual definition of the LDOS averaged over the leftmost 5% of the system. Vertical dashed lines indicate the value h of the Zeeman field at the topological phase transition in the c thermodynamiclimit. HorizontallinesindicatethelevelspredictedbyEq.(4);foremphasis,weexplicitlylabelthetheoretical valueE =E giveninEq.(5). Atthesmallestsystemsize(leftpanel),weseeanappreciablefinite-sizebulkgap finite-sizegap n=1 at h=h , i.e., E ∼∆, with robust Majorana zero modes still forming deep in the topological phase; in the inset, c finite-sizegap we show the spatial profile of one of the (nearly) zero-mode wavefunctions at h/∆=3 whose envelope follows an exponential with correlation length ξ ≈ 0.8(cid:96), hence confirming Eq. (8). For the largest system size (right panel), we are approaching the true quantum critical behavior of the system. The inset in the middle panel extends for L/(cid:96)=10 the range of Zeeman fields out to h = 3mα2 = 42∆ (same data as in the main panel), showing the eventually resolvable splitting and oscillations39 of thezero-biaspeakforh(cid:38)mα2 (verticaldashedline). BottompanelsillustratetheexperimentalprotocoldescribedinSec.III in which system-size variation can be achieved in a single device via gate-tunable valves separating islands of a prescribed length. Experimentallydemonstratingthelengthdependenceshownherewouldconfirmtheapproachtocriticalityand reveal the spin-orbit strength α; recall Eqs. (3) and (5). between epitaxially grown mesoscopic superconducting reads islands (see Fig. 1 and Refs. 43–45), or via a more traditional approach46 of tuning nearby gates to selectively (cid:90) L (cid:20) (cid:18) (cid:126)2∂2 (cid:19) H = dx ψ† x µ i(cid:126)ασy∂ +hσx ψ deplete segments of a wire coupled to a single super- − 2m − − x 0 conductor. While more challenging, such experiments (cid:21) can reveal a universal 1/L scaling of excited-state en- +∆(ψ ψ +H.c.) , (1) ↑ ↓ ergylevelsatthephasetransition;confirmingthisbehav- ior would leave little doubt as to the topological nature where ψ describes electrons with spin α, effective mass of the system at higher fields. The methods we outline α m, and chemical potential µ; σx,y denote Pauli matri- furtherrevealtheproximitizedwire’sspin-orbitcoupling ces that act in spin space; α is the spin-orbit strength; strength, an important parameter that has not yet been h= 1gµ B 0 is the Zeeman energy, with g the wire’s determined directly in the hybrid structures relevant for 2 B ≥ effective g factor, µ the Bohr magneton, and B the Majorana physics. Overall, we hope that our work helps B field strength; and ∆ represents the induced pairing po- toinspirefurtherexperimentaleffortsgearedtowardsthe tential. (Parameters in H should be regarded as effec- unambiguous discovery of a topological phase transition tive couplings renormalized by hybridization with the in Majorana nanowires. parent superconductor.48) Consider first an infinite system. Withinthismodelthephasetransitionoccurswhen (cid:112) h=h ∆2+µ2.14,15 This condition can be satisfied c ≡ by tuning either the field or chemical potential, and our analysisofthelevelstructureatandnearthetopological quantum critical point holds independent of which con- II. FINITE-SIZE EFFECTS ON THE TOPOLOGICAL PHASE TRANSITION trolparameterisvaried. Inthecaseofchemical-potential tuning, the analysis applies equally well to either of the two critical points that border the topological phase on For simplicity we consider the minimal single-band its low- and high-density sides (see Fig. 3). By diagonal- model for the superconducting wire.47 The Hamiltonian izing Eq. (1) one finds that at criticality the low-energy 3 FIG.2. Zeemanfieldscansatweakspin-orbitcoupling: LDOSversush/∆andE/∆withµ=0intheweakspin-orbitcoupling regime, mα2/∆=0.3. Panels correspond to system sizes L/(cid:96)=10,20,40 from left to right. Conventions and annotations are thesameasinFig.1. Thisdataelucidatesthenecessity—intheweakspin-orbitregime—ofthepresenceofasharptopological phase transition if robust zero-modes appear for h > h . In the left panel at L/(cid:96) = 10, the phase transition is very much a c crossoverand the zero-bias peakneverfully forms. On theother hand, inthe right panelat L/(cid:96)=40, arobust zero-biaspeak appears on the topological side of a sharp phase transition. excitation spectrum is furtherassumefornowthatthechemicalpotentialsatis- fiesµ(cid:46)∆sothatthetopologicalregimeappearsinfairly E =(cid:126)v k (2) k | | low fields. The Rashba spin-orbit strengths have not with k the momentum and beenmeasuredinproximitizedwires—forwhichtheadja- centsuperconductorcancontributeappreciably—though ∆ v =α , (3) based on previous measurements for bare nanowires50–55 (cid:112) ∆2+µ2 we expect that α 104 105 m/s 0.07 0.7 eV˚A/(cid:126) ∼ − ∼ − is reasonable. With α near the middle of this range, the avelocityboundedbythespin-orbitstrength,i.e.,v α. ≤ residual bulk gap at the ‘phase transition’ is then Supposenowthatthesuperconductingwirehasafinite lengthL. Inthiscaseitisnaturaltoexpectthecontinu- E 1 K, (6) ousenergyspectruminEq.(2)tobecomediscretedueto finite-sizegap ∼ finite-size momentum quantization. Appendix A shows indeed comparable to the induced pairing energies de- thattheallowedmomentaindeedbecomekn = Lπ(n+21) duced experimentally9–13. for integer n. Inserting k into Eq. (2) yields quantized n If finite-size effects obscure the phase transition to the energy levels extent suggested by Eq. (6), can well-formed Majorana π(cid:126)v (cid:18) 1(cid:19) modes still appear? The answer depends sensitively on En = n+ , n=0,1,2, . (4) the spin-orbit strength, quantified by the energy scale L 2 ··· mα2. The lowest n = 0 bulk mode evolves into localized Ma- jorana end-states in the adjacent topological phase (see Fig. 1). We are interested primarily in the next excited A. Weak spin-orbit coupling: mα2 (cid:28)∆ state, whose energy at criticality is given by 3π(cid:126)v Weak spin-orbit coupling corresponds to mα2 ∆. (cid:28) Efinite-sizegap En=1 = . (5) In this regime the topological phase’s correlation length, ≡ 2L which determines the spatial decay of Majorana-zero- Thislevelgenerallycorrespondstothelowest-lyingmode mode wavefunctions, is approximately thatisabulkstateonbothsidesofthetransition. Inthe literature,sometimesgapclosureatthetopologicalphase (cid:18) h (cid:19)(cid:126)α ξ (weak spin-orbit). (7) transitionreferssimplytocontinuousformationofazero- ∼ mα2 ∆ bias peak upon increasing field, i.e., E 0. For n=0 → quantifying how well a finite-size system approximates This estimate—as well as Eq. (8) below—simply follows truequantumcriticalbehavior, however, itisveryuseful from the ratio of the Fermi velocity to the bulk gap and toconsiderthebehaviorofthisnextexcitedstateE — holds provided the system is not too close to the transi- n=1 which reveals not only the closing but also the crucial tion (where ξ formally diverges). Importantly, since the reopeningofthebulkgap(preciselytheE level)upon topologicalphaserequiresh>∆,theprefactorinparen- n=1 entering the topological regime49. thesisaboveislargeintheweakspin-orbitregime. Using Now, to get a sense of scales, consider a wire with Eqs. (3) and (5), we then see that having E finite-sizegap lengthL 1µmcomparabletothelargestsuperconduct- comparable to ∆ implies that ξ (cid:38) L. Robust Majorana ing segm∼ents studied in previous experiments9–13. We modes are then generically absent in this scenario since 4 theirspatialextentwouldexceedthesystemsize. Equiv- L/(cid:96) = 5,10,20 from left to right. For wurtzite InAs alently,observingaMajorana-inducedzero-biaspeakina wires with56 m = 0.05m (m denotes the bare elec- e e weaklyspin-orbit-coupledwirewouldnecessitatearather tron mass) and ∆ = 1.5 K = 130 µeV the param- sharp topological phase transition detectable by some eters specified above correspond approximately to sys- means. tems of length L = 2,4,8 µm with Rashba coupling57 We confirm this behavior numerically in Fig. 2 (see α=8 104 m/s=0.5 eV˚A/(cid:126). We believe these are rea- Appendix D for numerical details) where we plot the lo- sonabl·e numbers for Majorana experiments58 based on cal density of states (LDOS) averaged over the leftmost the technology developed in Refs. 40 and 41. 5%ofthewireversusZeemanstrength,h/∆,andenergy, In each panel of Fig. 1, the vertical dashed line indi- E/∆. In these simulations we choose parameters µ = 0 cates the location of the phase transition for an infinite andmα2/∆=0.3—deepintheweakspin-orbitregime— system, while the horizontal lines denote the levels pre- and use as a reference length (cid:96) (cid:126)α/∆. The three pan- dictedbyEq.(4). [Deviationsbetweennumericsandour ≡ els correspond to system sizes L/(cid:96) = 10,20,40 from left analytical prediction arise from higher-order terms not to right. For the shortest system, L/(cid:96) = 10, we indeed included in Eq. (2), which become progressively less im- haveEfinite-sizegap ∆withstronglysplitzeromodesfor portant as the wire length increases; see Fig. 6.] For ∼ h > h = ∆. It is not until the wire is sufficiently long, the shortest system size in Fig. 1 (left panel, L/(cid:96) = 5), c e.g., at L/(cid:96) = 40, that a robust zero-bias peak forms we see precisely the interesting scenario described above after crossing a nearly genuine thermodynamic topologi- where the phase transition is completely smoothed into cal phase transition with its associated pile-up of energy a crossover, i.e., E ∆, yet robust Majorana finite-sizegap ∼ levels at the critical point. modes develop in the topological regime. Similar plots appear in, for example, Refs. 59 and 60 where the evolu- tionoftheMajoranamodeinfinite-sizesystems—rather B. Strong spin-orbit coupling: mα2 (cid:29)∆ than excited states at the transition—was studied. It is worth stressing that while a mere two sub-gap Qualitatively different physics emerges at strong spin- states appear in the crossover region, the Majorana orbit coupling, i.e., mα2 ∆. Here the topological modes remain well localized. This feature is demon- (cid:29) phase enjoys a broad field range, extending from h ∆ strated explicitly in the inset of the left panel of Fig. 1, to h mα2, where the modes gapped by Cooper p∼air- where we plot the wavefunction amplitude ψ (x)2 = 0 ing ca∼rry nearly antiparallel spins—thereby maximizing (cid:80) (cid:2)us(x)2+ vs(x)2(cid:3) at h/∆ = 3 for t|he low|est- s=↑,↓ | 0 | | 0 | the bulk gap. Within this field window a parametrically lying eigenstate of Eq. (1); here us and vs respectively 0 0 shorter correlation length arises, denote components in the particle and hole sectors (see (cid:126)α also Appendix D). Indeed, the probability weight expo- ξ (strong spin-orbit), (8) nentiallylocalizestotheedgesandfitsverywelltoaform ∼ ∆ ψ (x)2 e−2x/ξ +e−2(L−x)/ξ, with correlation length 0 which again holds not too close to the transition. [For ξ| 0.|8(cid:96)∼= 0.8(cid:126)α/∆ that agrees well with the estimate h(cid:38)mα2 theZeemanfieldbeginstooverwhelmthespin- fro≈m Eq. (8). Furthermore, we have verified that the re- orbitenergy,yieldingasmallerbulkgapandcorrespond- solved zero-mode splitting at h/∆ = 3 in the left panel ingly larger correlation length that eventually recovers of Fig. 1 is fully consistent with this correlation length Eq. (7)—see inset of Fig. 1, middle panel.] Systems inserted into Eq. (9). for which Efinite-sizegap approaches ∆ can consequently Asweapproachthethermodynamiclimitbyincreasing still support localized Majorana modes with ξ L the system size to L/(cid:96)=10,20 (middle and right panels (cid:28) over an extended field interval. For a quantitative esti- of Fig. 1), the finite-size energy levels at the crossover mate,thesplittingthatarisesfromoverlapofMajoranas decrease as expected from Eqs. (4) and (5), and the ap- boundtooppositeendsofthewireis(modulooscillatory proach to true criticality becomes evident. At L/(cid:96) = 20 corrections)39 we obtain very good agreement with the theoretical prediction of Eq. (4). E ∆e−L/ξ. (9) splitting ∼ Taking E = ∆ and µ = 0 then yields finite-sizegap E ∆e−3π/2 0.01∆—well below the bulk gap. C. Chemical potential scans at fixed Zeeman field splitting ∼ ≈ Thusstrongspin-orbitcouplingallowsratherrobustlocal- ized Majorana modes to appear even in ‘small’ systems FollowingmeasurementsinRef.42,itisalsointeresting where the topological phase transition is severely obliter- to contemplatefinite-size effects atthe topologicalphase ated into a crossover. transitionexhibitedbyEq.(1)uponvaryingthechemical To drive home this key point in more detail, we now potential, µ, and hence the electron density, n, at fixed present numerical simulations of Eq. (1) in the strong magnetic fields. spin-orbit regime. We again set µ = 0 but now as- Wefirstbrieflyreviewthestructureofthefree-fermion sume strong spin-orbit coupling with mα2/∆=14. Fig- band structure of Eq. (1) at ∆=0. For concreteness we ure 1 displays the end-of-wire LDOS for system sizes focus on h < mα2—the case of interest in the strong 5 FIG.3. Chemicalpotentialscansatstrongspin-orbitcoupling: LDOSversusµ/∆andE/∆onasystemoflengthL/(cid:96)=5inthe strong spin-orbit coupling regime, mα2/∆=14. Different panels correspond to different values of the Zeeman energy h/∆ as indicatedaboveeachplot. Theinducedgap∆isindicatedwithabluehorizontallineinthefirstpanel. Theoutermostvertical √ dashed lines, if they exist, indicate the boundaries, µ = ±µ = ± h2−∆2, of the topological phase in the thermodynamic c limit. Finally, the horizontal dashed lines highlight the finite-size energy levels at the critical points as predicted by theory through Eqs. (3) and (4). spin-orbit regime (see inset of Fig. 1, middle panel). 42, it is thus reasonable to interpret our approach here Here the wire has no electrons when µ < µ = as putting in by hand the energy ∆ of the lowest-lying bottom mα2/2 h2/(2mα2); hosts two sets of Fermi points for (extended in our model) Andreev bound states at zero − µ < µ < h or µ > h; and most interestingly re- field arising from a more exact treatment of the proxim- bottom alizes a spinless−regime with one pair of Fermi points for ity effect62. For eventual Majorana physics at finite h, it h<µ∆and hasbeenargued63thatthis∆shouldonlybeafractionof − thewireissufficientlylong,thesystemadmitsatopologi- theparentsuperconductor’sgapgivingrisetoproximity- calphasewithinasubregion µ <µ<µ ofthespinless induced pairing. With the above interpretation in mind, c c − regime, whereµ =√h2 ∆2. Wethushavetopological this indeed appears to be the case in the zero-field data c − phase transitions at critical values µ = µ . At these of Ref. 42. c ± critical points, our analysis of the spectra for finite-size As we turn on the Zeeman field, a state centered at systems from Sec. II (see also Appendix B) carries over µ = 0 begins to descend. This state eventually evolves completely. intoaMajoranazeromodeonceinthetopologicalphase. Toexposetheresultingphysics,Fig.3presentsthenu- For h < ∆, there is no topological phase for any value mericallycalculatedend-of-wireLDOSatfixedh/∆ver- of µ, while for h > ∆, the (infinite system) topologi- susµ/∆. Weagainfocusonthestrongspin-orbitregime cal regime in the interval µc < µ < µc lies between − (mα2/∆ = 14) for a relatively ‘short’ wire (L/(cid:96) = 5); the pair of vertical dashed lines in the bottom panels of cf. Fig. 1, left panel. At zero field (h/∆ = 0), the Fig. 3. (At h/∆ = 1, the topological phase shrinks to a levels form shifted parabolas with minimum energy ∆. point at µ = √h2 ∆2 = 0; this situation appears in c − This structure reflects a combination of (i) the momen- the upper-right panel.) For cases in which a topological tum quantization in our finite-size wire and (ii) the regimeexists,wealsoshowthefinite-sizeenergylevelsat quadratic energy dispersion—arising from the supercon- the critical points µ as given by Eqs. (3) and (4) via c ± ducting gap—near the Fermi wavevectors. The latter horizontal dashed lines as in Figs. 1 and 2. The agree- varyapproximatelylinearlywithµoversufficientlyshort ment between theory and numerics is quite good, espe- intervals centered at the bottom of the parabolas61. For cially for larger h/∆ where the zero modes span larger longwires,werecoveracontinuumofstatesaboveenergy rangesofµ. Collectively,theseplotsfurtherdemonstrate ∆withalevelspacingwhichdecreasesas1/L. Inmodel- the robustness of Majorana zero-mode physics in ‘short’ inganexperimentalsystemsuchasthoseinRefs.41and wires with strong spin-orbit coupling. 6 In the weak spin-orbit regime, robust Majorana zero modesinthetopologicalphasestillrequireasharpphase transition at both critical points µ = µ , and the level c ± structuregivenbyEq.(4)stillapplies. However,forweak spin-orbitcoupling,thephasetransitionsonthelow-and high-densitysidescandifferintermsofvisibilityforend- of-wireconductanceprobes. Notably,thelow-lyingstates on the trivial low-density side have very poor visibility, whileonthetrivialhigh-densitysideapairofend-of-wire localized Andreev bound states appear that are energet- ically separated from the bulk states. Upon entering the topologicalphase,oneoftheseAndreevboundstatesbe- comes the n=0 level that evolves into Majorana modes while the other becomes the n = 1 bulk mode, with energies at the transition still given by Eq. (4). Finally, we have thus far only considered the situation with a constant pairing amplitude ∆. It is well known, however, that ∆ renormalizes downwards as the field increases. ThiseffectisparticularlyimportantforAl/InAs experiments in, for example, Refs. 41 and 42 where the critical magnetic field of the parent superconductor B SC mayonlybeafactoroftwoorthreegreaterthanthemag- neticfieldB atthetopologicalphasetransitionitself. In c Appendix C, we include the effects of reasonable pairing FIG.4. Leveltrackingatthecriticalpoint: LDOSversush/∆ suppression due to the Zeeman field and show that the and E/∆ at µ=0 (top panel; cf. Fig. 1) and µ/∆ and E/∆ resultsinFigs.1and3remainqualitativelyintact,albeit at h = ∆ (bottom panel; cf. Fig. 3) in the strong spin-orbit with some quantitative differences. regime,mα2/∆=14,onasystemoflengthL/(cid:96)=10. These plotsdemonstratetheaccuracyoftheanalyticallevel-tracking formulas,Eqs.(10)and(11),presentedinthetext(solidgreen curves) for the lowest-lying three levels (n = 0,1,2) at the III. EXPERIMENTAL PROTOCOLS transition. With the advent of clean, hard-gap wires40,41 that largely eliminate unwanted background conductance, tothisµ 0limitbyadjustingside-gatevoltagestomin- ≈ mapping the low-lying level structure in fields should be imize the Zeeman field necessary to produce Majorana- possible through NS tunneling spectroscopy9,10,12,13. In induced zero-bias peaks. Transitions at finite µ yield en- fact, our study was strongly informed by recent mea- ergiesthataremorecomplicatedyeteasilyobtainableas surements that clearly resolve multiple sub-gap states— outlined in Appendix B. including a level that evolves into a zero-bias peak42. Forasystematconstantµ=0withvariablemagnetic One relatively simple way to quantify the approach of field B (cf. Figs. 1 and 2), we find that for fields close a topological phase transition in such experiments is to the critical magnetic field Bc, the discrete energies in through the field or density dependence of the finite-size Eq. (4) are modified to energy levels near the critical point [e.g., the behavior of the bottom two levels near the vertical dashed line in π(cid:126)α(cid:18)n+ 1(cid:19) c 1gµ δB+c (cid:2)21gµBδB(cid:3)2, Fig. 1(a)]. Conveniently, this characterization relies on En ≈ L 2 − n12 B n2π(cid:126)α(cid:0)n+ 1(cid:1) L 2 measurements of a system with fixed length L that can (10) besufficientlysmallthatthetransitionsupportsveryfew an expression good to O(δB2) where δB =B B . The c − sub-gap states. numerical factors c are given in Eq. (B6). This form n1,2 We describe in Appendix B a very general procedure assumesthatthemagneticfieldalterstheZeemanenergy foranalyticallytrackingtheevolutionofthesystem’sen- but not the pairing potential ∆ or chemical potential. ergy levels as a function of a tuning parameter—such as The levels depend on parameters g, B , and α/L. One c magnetic field or chemical potential—in the vicinity of can roughly estimate g through the slope of the lowest- the topological phase transition. While the procedure lying level on the trivial side of the transition; B n=0 c E should be applicable to any microscopic model, e.g., one through the field that minimizes ; and α/L from n=1 E with multiple bands, we here specialize it to Eq. (1). In the measured difference at B . These rough n=1 n=0 c E −E this section, we further focus on the scenario where the estimates can be refined through a more careful fit to phase transition occurs at µ 0, an experimentally rel- Eq. (10) for the lowest experimentally resolved sub-gap ≈ evant limit which gives particularly simple and universal levels (whose properties should be most accurately cap- expressionsfortheenergylevels. Inpracticeonecantune tured by this expression). Importantly, if L is known 7 then such fits yield the spin-orbit strength α. bygate-tunable‘valves’43–45thatcontrolthecouplingbe- InthetoppanelofFig.4,weoverlaythepredictionsof tweenadjacentislands. Theconductance ismeasuredby Eq. (10) on top of the numerical data from Fig. 1 at the sending in current from the normal lead to the leftmost intermediate length L/(cid:96) = 10 for the lowest three levels island,whichisgrounded. Successivelyopeningandclos- (n = 0,1,2). We see that the agreement is quite good ing valves as in the figure effectively changes the length given the discrepancy already apparent at δB = 0 on L of the region probed by the lead and allows one to thissystemsize(seeFig.6). Thequantitativeagreement track the finite-size energy levels at the transition. Ob- between Eq. (10) and the numerics continues to improve serving the characteristic 1/L scaling of the bulk gap— as we increase L (not shown), thereby confirming the i.e., E —would provide additional sharp evidence for n=1 validity of the approach spelled out in Appendix B. the expected critical behavior. Here too such measure- It would also be interesting to tune through the topo- ments constrain the system’s Rashba spin-orbit coupling logical phase transition by changing µ via nearby side- by virtue of Eqs. (3) and (5). As an independent check, gate voltages64, although obtaining such large sweeps (cid:96)=(cid:126)α/∆ can be separately inferred from the (exponen- as in Fig. 3 may be practically difficult. Tracking of tial) L dependence of the Majorana mode splitting that the finite-size levels near the critical values µ = µ = occurs in the topological phase. c ± √h2 ∆2 on a single, fixed-length device is possible Another strategy using more traditional technology ± − here too. We again focus on the experimentally relevant would be to take a single, long superconducting island case of µ 0 which occurs for fields h ∆. Taking as in the device in Ref. 42, but place several side gates c ≈ ≈ µ = 0, h = ∆ exactly (cf. Fig. 3, upper-right panel) so of known lengths nearby. Selectively tuning the side- c that the topological phase has shrunk to a point upon gate voltages into and out of the topological regime sys- scanning µ, we find that the energy levels near the tran- tematically alters the length of the topological segment sition evolve as of the wire, thereby effectively changing L. (A simi- π(cid:126)α(cid:20) δµ2(cid:21)(cid:18) 1(cid:19) δµ2 lar gate setup was already realized in the original Delft 1 n+ +c , (11) experiment9.) Webelievebothschemesarequitereason- En ≈ L − 2∆2 2 n12∆ able with present technology and hope that they may be which is good to O(δµ2) where δµ = µ µ = µ. In pursued in the near future. c the bottom panel of Fig. 4, we overlay t−he levels from Finally, we remark that the issue of poor visibility of Eq. (11) onto the corresponding numerical data, again bulk states in end-of-wire tunneling measurements28–30 at L/(cid:96) = 10. The agreement is reasonable and indeed is expected to be alleviated in small systems for which improvesasweincreaseL. Notethatthecurvatureofthe thephasetransitionisverymuchacrossover. Variations levels decreases with increasing n. Similarly to the field- whereintunnelingoccursinthemiddleofthewirewould, scanprotocoldiscussedabove, itwouldbeworthwhileto however, skirt this issue entirely. fitpredictionssuchasEq.(11)totheexperimentaldata. Useful device information can be extracted from a fixed-length wire even away from the µ 0 limit con- IV. DISCUSSION ≈ sidered above. For instance, one can quite generally estimate Bc (and hc if the g-factor is known) by read- WiththefieldofMajoranananowirespresentlypoised ing off the field that minimizes the n = 1 level. We to move beyond zero-mode detection, we have exam- point out that for short, strong-spin-orbit systems this ined several fundamental questions regarding the puta- method provides a more reliable way of determining Bc tive topological phase transition that accompanies the than by the field at which zero modes form60,65 (see formation of these zero modes. We have shown that— Fig. 1, left panel). Assuming ∆ is inferable from tun- rathersurprisingly—robustzeromodescaneasilyexistin (cid:112) neling spectroscopy66, identifying hc = ∆2+µ2 corre- systemstoosmalltoexhibitanythingclosetoatruetopo- spondingly determines µ—which may be otherwise dif- logicalphasetransition,providedthatthespin-orbitcou- ficult to estimate. Reading off Efinite−size gap and using pling is strong. However, it should be emphasized that Eq. (5) with v = α∆/hc, one can then estimate α— here there is no parametric suppression of the Majorana or more conservatively α/L—at different gate voltages. splitting once the finite-size gap at the phase transition Even more simply, identifying Efinite−size gap at a single, becomescomparabletotheinducedpairinggap. Numer- fixed gate voltage gives a very useful lower bound on the ical factors instead conspire to make this splitting quite spin-orbit strength since α v = 2LEfinite−size gap. In- small in practice. In fact, there may actually be as few ≥ 3π(cid:126) terestingly, the only parameter required to establish this as two sub-gap states visible in the conductance spectra lower bound is the length L. even if robust Majorana-induced zero-bias peaks appear Asamoreambitiousexperiment,onecouldexperimen- in the topological phase. This point is particularly note- tally implement finite-size scaling to probe the predicted worthy in light of recent experimental data on epitaxial evolutionofstatessketchedinFig.1asLvaries. Thebot- Al/InAshard-gapdevicesfromRef.42whichindeedmay tompanelsillustrateonepossiblewaytoperformtheex- reside in this regime. perimentusingasingledevice. Hereawire(e.g.,InAs)is Since the original Majorana nanowire proposals in coated with superconducting islands (e.g., Al) separated Refs.14and15,therehavebeenaproliferationofpapers 8 overthepastseveralyearsonEq.(1)anditsrefinements, (a) (b) γR yetthefateofthefirst-excitedstaten=1levelonfinite- gapped ‘critical’ gapped sthizeelsoywsetsetm-lsyihnagslgevoenlewrehliacthiviselgyaupnpeexdpolonrbeodt.hSsiindceestohfitshies γL γR ~ Λ=0 Λ=0 topological phase transition, understanding its behavior (cid:31) L (cid:31) isoffundamentalimportance,especiallywhendiscussing γL physics of the phase transition itself. We hope that our work will help to bring investigation of such sub-gap FIG.5. (a)SchematicoftheeffectiveHamiltonianinEq.(A3) describingalength-Lwiretunedtothecriticalpointbetween states into greater prominence. Indeed studying the be- topological and trivial phases. Right- and left-moving Majo- havioroftheselevelsaswehaveproposedinthiswork— rana fields γ remain uncoupled in the central region but on both fixed- and variable-length finite-size wires—can R/L are gapped out by a hybridization Λ elsewhere. (b) In the give very valuable universal information about the even- Λ → ∞ limit the system maps to the chiral edge of a p+ip tual topological phase transition expected at L . superconductor. → ∞ Furthermore, such analysis can provide nontrivial information about parameters of the hybrid device, most notably its effective spin-orbit coupling strength α. Ex- Hamiltonian for the transition then arises: perimentally probing the finite-size scaling and tracking (cid:90) offinite-sizeenergylevelsatthetopologicalphasetransi- H = ( i(cid:126)vγ ∂ γ +i(cid:126)vγ ∂ γ ). (A2) eff R x R L x L tioninMajoranananowiresthusconstitutesaworthwhile x − pre-braiding endeavor in the Majorana problem. The spectrum for a finite-size system of length L can beefficientlyderivedbymodifyingthelow-energyHamil- tonian above to ACKNOWLEDGMENTS (cid:90) H [ i(cid:126)vγ ∂ γ +i(cid:126)vγ ∂ γ +2iΓ(x)γ γ ] We are indebted to M. Deng, K. Flensberg, L. Glaz- eff → − R x R L x L R L x man, M. Hell, and C. Marcus for helpful discussions on (A3) this work, and especially thank M. Deng and C. Mar- where cus for sharing unpublished results. We also gratefully (cid:26) acknowledge support from the NSERC PGSD program 0, x 0, x >L/2 | | DMR-1341822 (J. A.); the Alfred P. Sloan Foundation introduces a boundary to the ‘critical’ wire by gap- (J. A.); the Caltech Institute for Quantum Information ping the adjacent regions; see Fig. 5(a). Solving for and Matter, an NSF Physics Frontiers Center with sup- the wavefunctions in each piecewise-uniform region and portoftheGordonandBettyMooreFoundationthrough matchingboundaryconditionsyieldsquantizedmomenta Grant GBMF1250; and the Walter Burke Institute for k = π(n+ 1) in the Λ limit (n is an integer). Theoretical Physics at Caltech. Part of this work was n L 2 → ∞ performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY- 1 1066293 (R. V. M.). 0.8 Appendix A: Derivation of finite-system energies at ∆ the topological phase transition )/ hc0.6 = This Appendix derives an effective low-energy Hamil- h tonian from Eq. (1) at the critical magnetic field h=h , (=10.4 c n in particular to assess the influence of finite-size effects E on the spectrum. In an infinite system the energies at 0.2 Numerics criticalityaregivenbyE =(cid:126)v k [Eq.(2)]withmomen- k Theory [Eq. (5)] | | tum k a continuous parameter. Projection onto these low-lying excitations follows by sending 00 0.05 0.1 0.15 0.2 ℓ/L 1 ψ ( ieiθγ +ie−iθγ ) ↑ → √2 − R L FIG. 6. Bulk gap En=1 in units of ∆ at the critical Zeeman 1 strength h = hc with chemical potential µ = 0. Solid and ψ↓ (e−iθγR+eiθγL), (A1) dashed curves respectively correspond to numerical results → √2 and analytical predictions, while the horizontal axis repre- whereγ areright/left-movinggaplessMajoranafields sents(cid:96)/L=(cid:126)α/(∆L). Asthelengthincreases,thetwocurves R/L nicely converge. and tan(2θ) = µ/∆. The following elegant effective 9 FIG. 7. Zeeman field scans at strong spin-orbit coupling with a field-suppressed induced gap: LDOS versus h/∆ and E/∆ 0 0 with µ = 0 in the strong spin-orbit coupling regime, mα2/∆ = 14, including a field-suppressed superconducting gap given 0 by Eq. (C1) with h = 3.5∆ ; we show ∆(h) as a solid blue curve in all plots. As in Fig. 1, different panels correspond to SC 0 system sizes L/(cid:96) = 5,10,20 from left to right, where here (cid:96) ≡ (cid:126)α/∆ . The dashed vertical and horizontal lines carry the 0 0 0 same meaning as in Figs. 1 and 2, now taking into account Eq. (C1). One can intuitively understand this result as follows. where we have added a mass M coupling right- and left- Figure 5(b) illustrates that the system maps to a sin- moving Majorana fields in the wire region of length L. gle chiral Majorana fermion on a ring of circumference We will again take Λ in the adjacent outer regions L˜ = 2L—precisely as in the edge of a two-dimensional to impose hard-wall b→oun∞dary conditions on the wire. A spinless p+ip superconductor. The chiral fermion must positive mass M > 0 (corresponding to the same sign exhibit anti-periodic boundary conditions, since peri- massinthewireandouterregions)movesthesystemoff odic boundary conditions would yield a single Majorana criticalityintothetrivialstate;forM <0(corresponding zero mode with no partner (which is impossible). From to opposite-sign masses) the topological phase instead this perspective the momenta are immediately given by appears. k = 2π(n+1),inharmonywiththeresultquotedabove. Eigenstates of the effective Hamiltonian can again n L˜ 2 Insertingthesequantizedmomentaintothecontinuum be obtained straightforwardly by solving for the wave- energy E = (cid:126)v k yields the discrete spectrum speci- functions in each piecewise-uniform region and imposing k | | fied in Eq. (4)—in particular with a finite-size bulk gap boundary conditions. Carrying out this exercise, we find E = 3π(cid:126)v [Eq. (5)] corresponding to the k mode. that the energies must satisfy n=1 2L n=1 E (As noted in the main text the k mode is special n=0 (cid:18) (cid:19)2 because it evolves into a Majorana zero mode on the ei2kL = 1+iA (B2) topological side of the transition.) It is useful to sys- 1 iA − tematicallycomparetheE bulkgapderivedfromthe n=1 low-energy effective Hamiltonian with that obtained nu- with merically from the more microscopic model of Eq. (1). 1(cid:112) √ 2 M2 Figure 6 illustrates the length dependence for these an- k = 2 M2, A= E − . (B3) v E − M alytical and numerical values using the same parameters E − asforFig.1. Bothfiguresshowthattheanalyticalresult When M =0 the energies are indeed converges well with numerics as the system size increases. π (cid:18) 1(cid:19) E =(cid:126)vk , k = n+ (B4) n n n L 2 Appendix B: Field and chemical potential for non-negative integers n, as obtained in Appendix A. dependence of finite-size energies near the CorrectionsarisingfromafinitemassM maybeobtained topological phase transition by assuming a power-series: OurgoalhereistoextendtheanalysisfromAppendix M2 =E +c M +c + . (B5) Atoextractthefinite-sizeenergylevelsforsystemstuned En n n1 n2E ··· n slightly away from criticality. Sufficiently close to the transition,thestructureoftheenergiesisexpectedtore- Inserting this ansatz into Eq. (B2) yields main universal and well-captured by the effective Hamil- 1 1 1 tonian in Eq. (A3). To model the system off criticality c = , c = . (B6) n1 π(n+1/2) n2 2 − [π(n+1/2)]2 we now take (cid:26) M, x 0, x >L/2 and the mass M, both inputs to the effective model | | 10 FIG. 8. Chemical potential scans at strong spin-orbit coupling with a field-suppressed induced gap: LDOS versus µ/∆ and 0 E/∆ on a system of length L/(cid:96) = 5 in the strong spin-orbit coupling regime, mα2/∆ = 14, including a field-suppressed 0 0 0 superconducting gap given by Eq. (C1) with h =3.5∆ ; the value of ∆(h) used in each plot is marked by a horizontal blue SC 0 line. As in Fig. 3, different panels correspond to different values of the Zeeman energy h/∆ , and the meaning of the dashed 0 vertical and horizontal lines is the same. [Eq. (A3)]. To relate these quantities to parameters in a levelstructureoverthefieldintervalsdisplayedinFigs.1, given microscopic model such as Eq. (1), we can employ 2, and 3. Such effects are, for example, present in the thefollowingprocedure. Weknowthatforasystemwith Al/InAs devices studied in Ref. 42. We now incorpo- periodic rather than hard-wall boundary conditions, the rate pairing suppression by assuming a Zeeman field- energy dispersion of our effective Hamiltonian at finite dependent pairing amplitude M reads (cid:115) Eeff(k)=(cid:112)((cid:126)vk)2+M2. (B7) ∆(h)=∆0 1 (cid:18) h (cid:19)2 (C1) − h SC We can thus expand the square of the dispersion for our microscopic model, E2 (k), about k = 0 and identify in our simulations of Eq. (1). Here ∆ is the induced micro 0 theO(k2)termwith((cid:126)vk)2andtheO(k0)termwithM2. pairing amplitude at zero field and h = 1gµ B is SC 2 B SC Applying this algorithm to Eq. (1), we find the Zeeman energy associated with the parent superconductor’s critical magnetic field B ; h should not be µ (cid:20) h (cid:21) (cid:34) µ2 (cid:35) confused with the critical value hSCat wShCich the topolog- v2 = 1 +α2 1 , c m √∆2+h2 − − h(cid:112)∆2+µ2 ical phase transition arises. [In the main text, we took a field-independent ∆(h) = ∆ = ∆ , corresponding to (B8) 0 h in Eq. (C1).] SC M2 =h2+∆2+µ2 2h(cid:112)∆2+µ2. (B9) W→e p∞resent in Fig. 7 scans of the end-of-wire LDOS − versus Zeeman field as in Fig. 1, still at strong spin- Next, we expand these expressions in a power series in orbit coupling with mα2/∆0 =14, but now with a field- thedeviationsawayfromthecriticalpoint(e.g., inδhor renormalizedinducedpairinggapgivenbyEq.(C1)with δµ) and insert the result into Eq. (B5) to derive leading- hSC = 3.5∆0. For units we use the zero-field gap ∆0 order level tracking formulas such as Eqs. (10) and (11) and the length scale (cid:96)0 (cid:126)α/∆0. Figure 8 shows corre- ≡ in the main text. Expanding about a critical point with sponding scans versus chemical potential at a few values µ=0yieldsparticularlysimpleresults;seeEqs.(10)and of h/∆0 for the shortest wire, L/(cid:96)0 =5. (11) and Fig. 4. The procedure is, however, still valid at Overall, the physics is qualitatively similar to the finite µ; for instance, the energies so obtained describe constant-∆ results of Figs. 1 and 3. The finite-size level well the low-lying levels near the critical points in the structure at the eventual topological phase transition is bottom panels of Fig. 3. basically unaffected. However, now the zero mode in the Finally, we note that Eq. (B9) determines M only up shortest wire (L/(cid:96) = 5) does begin to experience no- 0 to a sign. The sign can be easily fixed, however: M ticeable splitting as we approach h = h . Still, in this SC is positive (negative) if the tuning parameter takes the strong spin-orbit regime, this zero mode remains reason- system into the trivial (topological) phase. ably robust over an appreciable field and chemical potential range (see Fig. 7, left panel, and Fig. 8). On the other hand, for the longer wires (L/(cid:96) =10,20) the zero 0 Appendix C: Effect of pairing suppression by the modes remain intact essentially right up to h=h . SC magnetic field Finally, wepointoutthedensesetoflevelsdeveloping above ∆(h) in the topological regime for the larger sizes Suppression of the pairing energy ∆ by the magnetic in Fig. 7—strong spin-orbit coupling optimizes the exci- field was so far ignored but can quantitatively effect the tationgaptoverynear∆(h)asexpected. Inarealexper-

Approaching a topological phase transition in Majorana nanowires

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