Majorana Zero Modes Protected by Hopf Invariant in Topologically Trivial Superconductors

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by  Zhongbo Yan

Majorana Zero Modes Protected by HopfInvariantinTopologicallyTrivialSuperconductors
Zhongbo Yan,1 Ren Bi,1 and Zhong Wang1,2,
∗
1Institute for Advanced Study, Tsinghua University, Beijing, 100084, China
2Collaborative Innovation Center of Quantum Matter, Beijing, 100871, China
Majorana zero modes are usually attributed to topological superconductors. We study a class of two-
dimensional topologically trivial superconductors without chiral edge modes, which nevertheless host robust
Majoranazeromodesintopologicaldefects. Theconstructionofthisminimalsingle-bandmodelisfacilitated
7
bytheHopfmapandtheHopf invariant. ThisworkwillstimulateinvestigationsofMajorana zeromodesin
1
superconductorsinthetopologicallytrivialregime.
0
2
r Majorana zero modes (MZMs) or Majorana bound states (a) y
p (b)
areexoticexcitationspredictedtoexistinthevortexcores[1, 1.1
A
2]oftwo-dimensional(2D)topologicalsuperconductors[3–7]
6 and at the ends of 1D topological superconductors[8]. Spa- A 1Nh.0
tially separatedMZMs giverise to degenerategroundstates, x
] o 0.9
n whichencodequbitsimmunetolocaldechoerence[8,9]. Fur-
o thermore, unitary transformations among the ground states
0.8
c canbeimplementedbybraiding[10–14]ormeasurements[15,
-
r 16] of these modes, indicating that such qubits may be- 0.7
p come building blocks in topological quantum computation 8 12 16 20 24N28 32 36 40
u
and information[17–22]. Therefore, MZMs have been vig-
s FIG. 1. (a) Sketch. The Hamiltonian varies as a function of θ,
. orouslypursuedincondensedmatterphysics[23–29].
at There have been a great variety of proposals for acrrcetaatnin(gy ayde)/fe(cxt axt O).(b=)T(hx0e,Hy0o)p.fiInnvatrhieanptofloarrηc=oo1rdainndateη,=θ0.≡5
m − 0 − 0
topological superconductors, including 2D semicon- (n = 1) in discretized-zone calculation, with N3 grid points in the
d- ductor heterostructures[30, 31], topological insulator- (k,θ)space.Asthegridbecomesfiner,Nhconvergesrapidlyto1.
n superconductor proximity[32–36], 1D spin-orbit-coupled
o quantum wires[37–45], spiral magnetic chains on
c superconductors[46–50], Shockley mechanism[51, 52], eryfixedθistopologicallytrivial. Stimulatedbythismecha-
[ nism,whichsignificantlydiffersfromthemagnetic-vortexori-
and cold atom systems in 2D[53–56] and 1D[57, 58], etc.
ginofzeromodeintopologicalp-wavesuperconductor[1,2],
3 Experimentally, suggestive signatures of MZMs in both
v 1D[59–69]and2D[70–75]topologicalsuperconductorshave we design trivial-superconductor-based (and vortex-free) T-
8 junctionsharboringMZMs.
beenfound.
5 Zeromodes.–Beforestudyingtopologicaldefects,wecon-
5 It is often implicitly assumed that topological supercon-
siderspatiallyuniform2Dsingle-bandBogoliubov-deGennes
2 ductivity is a prerequisite for MZMs, accordingly, the chiral
(BdG)Hamiltoniansparameterizedbyλ:
0 edge states go hand in hand with the vortex zero modes in
.
1 2Dsuperconductors.InthisLetterweshowthatcertaintopo- ξ (λ) ∆ (λ)
H(k,λ)= k k (1)
70 ldougcitcoarlsdceafnecstus[p7p6o–r8t1r]obinus2tDMtZoMposl.ogSicoamlleywthriavtiaslurspurpiseirncgolny-, ∆∗k(λ) −ξ−k(λ) !
1 single-bandsuperconductorssufficethispurpose. Themodel wherek = (kx,ky), ξk = Ek µ, Ek andµ is theenergyand
: −
v HamiltonianisrelatedtotheHopfmaps,whichoriginallyre- chemicalpotential,respectively,and∆kistheCooperpairing.
Xi fertonontrivialmappingsfroma3DsphereS3toa2Dsphere Itdescribessingle-bandspinless(orspin-fully-polarized)su-
S2,characterizedbytheintegerHopfinvariant[82,83]. Map- perconductors. This Hamiltonian can be written in terms of
r
a pings from a 3D torus T3 to S2 inherit the nontrivial topol- thePaulimatricesτi as
ogy from the mappings S3 S2. The Hopf invariant has
→ H(k,λ)= di(k,λ)τi, (2)
foundinterestingapplicationsinnonlinearσmodelsandspin
i=Xx,y,z
systems[82, 84], Hopf insulators[85–90], liquid crystals[91],
andquenchdynamicsofCherninsulators[92,93]. withdx = Re∆k, dy = Im∆k, dz = ξk (wehaveξk = ξ k in
− −
ourmodel).Forreasontobecomeclearshortly,wetake
Our model describes topologicallytrivial superconductors
with zero Chernnumberand no chiraledge state. Neverthe-
d =z τz, (3)
i † i
less,atopologicalpointdefectischaracterizedbyaHopfin-
variant defined in the (kx,ky,θ) space, where kx,ky are crys- wherez=(z1,z2)T and
tal momenta and θ is the polar angle[94](Fig.1a). The par-
ity(even/odd)ofHopfinvariantdeterminesthepresence(ab- z1 =sinkx+isinky,
sence) of robust MZMs, though the superconductor for ev- z =sinλ+i(cosk +cosk +cosλ m ), (4)
2 x y 0
−
2
with m = 3. We can check that ∆ = ∆ and ∆ =
i∆ ,0thus2thepairingis p-wave. −Gkiven−Eqk.(4),the(−pkya,kirx)ing
(kx,ky)
∆ isofthesameorderasthehoppingξ .Todescribeweakly-
k k
pairingsuperconductors,onemayconsider
H (k,λ)=η(d τ +d τ )+d τ , (5)
η x x y y z z
withasmallbutnonzeroη. Nevertheless,tuningthevalueof
ηdoesnotclosetheenergygap,henceitdoesnotqualitatively
changetheresults.Thuswewillsimplytakeη=1below.The
mathematicalformofEq.(4)hasbeenintroducedfor3DHopf
insulators[85–90],withλreplacedbythethirdmomentumk .
z
Thephysicalsystemwewillstudyisneverthelessnotdirectly
relatedtoHopfinsulator.
ThefamiliarChernnumber[1,33,95]C thatcharacterizes
2Dtopologicalsuperconductorscanbeobtainedbyastraight-
forwardnumericalcalculation,whichyieldsC(λ) = 0forev-
eryλ.
A topological defect can be generated if the parameter λ
depends on spatial coordinates, in a manner that the config-
uration cannot be smoothly deformed to a spatially uniform
one. Letusfocusonthedefectswithλdependingonthepo-
larangleθ(Fig.1a)as
λ=nθ, (6)
wherenisaninteger. Theseconfigurationsaretopologically
nontrivial due to a nonzero Hopf invariant, as we now ex-
plain. The unit vector dˆ(k,θ) 1 (ηd ,ηd ,d )
≡ √η2(d2x+dy2)+dz2 x y z FIG.2. (a)ProfilesoftwoMZMsinasquaresamplewithaHopf
maps the 3D torus T3 (k ,k ,θ are defined modulo 2π) defect (n = 1). The main figure shows the particle component u,
x y
to the 2D unit sphere S2. For nonzero n, the inverse- andtheleftinsetshowstheholecomponentv.Therightinsetshows
image circles of two points on S2 are linked[96]. To quan- severalenergiescloseto0,withtwozeroenergiescoloredinred.(b)
isthesameas(a)exceptthatanimpuritypotentialatthesinglesite
tify such linking, the Hopf invariant can be defined[82,
A(indicatedinFig.1a)isadded.
85]: N = 1 dθd2kǫµνρa ∂ a , where the integrating
h −4π2 µ ν ρ
range is the BrilloRuin zone for k and [0,2π] for θ, a =
µ
i ψ(k,θ)∂ ψ(k,θ) , with µ,ν,ρ = k ,k ,θ, and ψ is the
µ x y
−nehgative-e|ner|gyeigeinfunctionof H (k,θ). Alternat|iviely, we TochecktherobustnessofMZMs,weaddanimpuritypo-
η
candefineAµ = aµ/2π, jµ = ǫµνρ∂νAρ = (1/8π)ǫµνρdˆ (∂νdˆ tential Uc†AcA (c is the fermion operator) at a single site A
∂ρdˆ),then[82,85] · × (specifiedinFig.1a),whichamountstoaddingaUτz termat
site Ainthereal-spaceBdGHamiltonian. Thenumericalre-
N = d2kdθj A. (7) sultforU = 1.0 is shownin Fig.2b. TheenergiesofMZMs
h
−Z · remainpinnedto E = 0, thoughthemodeprofileischanged
comparedtoFig.2a.
ItcanbecalculatednumericallybydiscretizingtheBrillouin
zone[96]. Thenumericalresultforn = 1isshowninFig.1b. For n = 2, we find two localized modesin the defectand
Moregenerally,wehaveN =n. Wewillcallthetopological two at the boundary (near θ = 0 and θ = π). Unlike the
h
defectsdefinedbyEq.(6)asHopfdefects. n = 1 case, the energies of defect modes are not pinned to
To obtain energy spectra, we Fourier-transform the BdG zero.Highern’sarealsocalculated,andtheresultssupportthe
Hamiltonian to real-space lattice, then numericallysolve the conclusionthatthereisasinglerobustMZMforodd-integer
Hamiltonian[96]. For n = 1, two zero modes are found n, andno robustMZMforeven-integern, therefore,itis the
(Fig.2a),oneofwhichissharplylocalizedaroundthedefect, Z Hopfinvariant(even/odd)thatdeterminestheexistenceof
2
the other is localized at the sample boundary. The profile of MZM in the defect. One may notice that the Hopf invariant
particle component (τ = 1 component, denoted by u) and takestheformofaChern-Simonsinvariant[76,97],whichis
z
holecomponent(τ = 1component,denotedbyv)isshown notaccidental,becausethelatterisindeedageneraltopologi-
z
−
inthemainfigureandtheinset,respectively.Itisapparentthat calinvariant,nevertheless,ithasbeenapplied[76,77]onlyto
thezeromodesareequal-weightsuperpositionsofparticleand topologicallynontrivial superconductors,for which it is just
holecomponents,whichisafeatureofMZMs(inspectionof the productofthe Chern numberandthe vorticityof pairing
thewavefunctionconfirmsthatu=v ). phase. OurmodelshowsthatnonzeroChernnumberisnota
∗
3
y therefore, we focus on the neighborhoodof θ = 0. Suppose

(a) (b) thatboththeouterandinnerradiusesR,R′ arelarge. On the
outer boundary, we have λ = θ = y/R, thus the edge state
1
spectra are given by solving H (k i∂ ,λ y/R)ψ =
E eff y → − y →
x Eψ. Moreexplicitly,itreads
0
R R'
[ivσ ∂ +(M/R)yσ ]ψ= Eψ, (9)
-1 x y y
whichsquaresto( v2∂2+M2y2/R2 vM/Rσ )ψ=E2ψ. This
-2 0 ky 2 equationresemble−stheySchro¨dinger−equationzofharmonicos-
cillators, though E is replaced by E2, and there is a cru-
FIG. 3. (a) The energy bands for a ribbon with size L L = cial additional vM/Rσ term. The eigenfunctions are σ -
x × y − z z
40 ;λ=0(solidcurves).Thetwodashedcurvesshowthegapped eigenvectors(eigenvaluesaredenotedbys = 1),withener-
edg×e∞modes forλ = π/20asacomparison. Eachedge-mode band giesgivenby z ±
isdoublydegeneratebecausearibbonhastwoboundaries(ignoring
AaslmaraglelshpoliltltoiwngdtihsaktwexipthontwenotiaMllZyMdescialylussatrsataedfu.nTcthioeninonfeLrxM).Z(Mb) Eo2uter(sz,n)=2(n+1/2)vM/R−szvM/R, (10)
persists as the inner radius R′ → 0, evolving to the defect MZM where n = 0,1, . There is a MZM in the sz = +1 sector,
protectedbyHopfinvariant. ···
with n = 0, which is illustrated as the blue bump in Fig.3b.
Since this mode is the eigenfunction of τ , it is an equal-
x
weightsuperpositionofparticleandholecomponents.
necessaryconditionforMZM.
For a semi-infinite geometry with sample occupying the
Edgetheory.–Itisdesirabletohaveanintuitiveunderstand-
x > 0 region, the effective edge theory is almost the same
ingof theMZM fromthe perspectiveofedgetheory,which,
as Eq.(8), exceptthat the sign of the first term reversed[96].
as we will show, differs significantly from that of the chiral
Ontheinnerboundaryofthehollowdisk(againnearθ = 0),
topologicalsuperconductor[1, 98,99]. First, we numerically
we haveλ = θ = y/R, thusthe edge-modespectrumcan be
solved the edge states of open-boundarysystems for various ′
obtainedfrom[ ivσ ∂ +(M/R)yσ ]ψ = Eψ, analogousto
valuesofλ,andfoundthatgaplessedgemodesexistonlyfor x y ′ y
−
Eq.(9). Theenergiesaregivenby
λ=0.InFig.3a,weshowtheenergybandsforaribbonalong
y direction. The gapless edge modes for λ = 0 are shown
E2 (s ,n)=2(n+1/2)vM/R +s vM/R, (11)
asthesolidbluelines. Theyarenon-chiral,andareimmedi- inner z ′ z ′
ately gappedout when λ is tuned away from 0 (edge modes whichfeaturesaMZMinthe s = 1sector. Thezero-mode
z
ofλ = π/20areshownindashedcurves),inotherwords,the wavefunctionis −
edge modes are not topologically robust. This is consistent
withthevanishingofChernnumber. ψinner exp( My2/2vR′)sz = 1 , (12)
∼ − | − i
This numericalobservation is confirmed by analytic solu-
whichisexponentiallylocalizedneary=0,namelyθ=0(il-
tions. We considera semi-infinitegeometrywiththe sample
lustratedbythegreenbumpinFig.3b). Allnonzeroenergies
occupyingx<0region,k beingagoodquantumnumber.For
y
grow as 1/R as R is decreased, while the MZM remains at
λ=0,weobtaintwodegenerateedgemodesatk =0,bothof ′ ′
y
zeroenergy,evolvingtothedefectmodeshowninFig.2. For
which are eigenfunctionsof τ with eigenvalue 1[96], thus
x
− ahollowdiskwithλ = 2θ, therearetwoMZMsontheinner
theyareequal-weightsuperpositionsofparticleandholecom-
boundary for large R, near θ = 0 and θ = π, respectively.
ponents. We introducePauli matricesσ (unrelatedto the ′
x,y,z
ShrinkingR causes overlappingbetween them, which splits
τ matrices)inthistwo-dimensionalspace,sothatthetwo ′
x,y,z
the two zero energies to nonzero values. This is consistent
eigenfunctionshaveσ = 1,respectively.Includingsmallk
z y
± withtheabsenceofMZMinthen=2defect.
andλasperturbations,wederiveaneffectivetheory[96]:
It is useful to compare our systems with the chiral topo-
Heff(ky,λ)= vkyσx+Mλσy, (8) logical superconductor,for which a magnetic vortex with π-
−
fluxhostsaMZM[1,2,32,100]. Inahollow-diskgeometry,
wheretheeffectiveparametersv,M arefoundtobeboth3/4
this MZM comes from the chiral edge states on the bound-
in our specific model[96]. Thus the edge-state spectra are
ary circle[1, 98, 99]. The MZM wavefunctionis evenly dis-
E±(ky) = ±qv2ky2+M2λ2. It is immediately clear that the tributed on the circle, which implies its sensitiveness to the
edgestatesbecomegappedwhenλmovesawayfrom0,which magnetic flux. In contrast to this picture, the MZM in our
isconsistentwiththenumericalfindinginFig.3a. Asacom- modelis notderivedfrom chiraledge state, which is simply
parison,wenotethattheedgespectrumofachiraltopological absent here, moreover, the MZM is exponentially localized
superconductor[1],E(k )=vk ,cannotbegappedout. nearθ = 0(Fig.3b),thusitisinsensitiveifamagneticfluxis
y y
Based on this effective edge theory, we proceed to study inserted.
a hollow disk with polar-angle-dependentparameter, λ = θ T-junctions.–So far, we have only studied configurations
(Fig.3b). We are only concerned with low-energy modes, withλcontinuouslyvaried. Itisconceivablethatthesmooth
4
the numerical results thus obtained indicate that MZMs do
exist in such T-junctions, as shown in Fig.4a for λ = 0.1π,
|u|2 1
λ = π. There is certain arbitrariness in choosing the hop-
(a) 0.1 2
pingat the boundarybetween the λ regions, forwhich we
1,2
keeponlythenearest-neighborhopping(discardingthenext-
0.05 nearest-neighborhoppingand the pairing)[101]. The energy
eigenvalues near zero are shown in Fig.4c (we show 12 of
them),fromwhichitisclearthatthezero-modelevelsaresep-
0
40 aratefromallotherenergylevelsbyafinitegapintheL
30 20 30 40 limit. →∞
20
10 y x 10
We have also studied superconductor-insulator-vacuumT-
junctions.Tothisend,weconsidertheH inEq.(5),inwhich
η
taking η = 0 amounts to removing the Cooper pairing. We
|u|2
noticethatH (k,λ = π)describesaninsulatorwithoutany
(b)0.15 η=0
Fermi surface [In contrast, H (k,λ = 0.1π) describes a
η=0
0.1 metal].Nowwecandesignsuperconductor-insulator-vacuum
T-junctionsbytakingη=1intheλ =0.1πregion,andη=0
1
0.05 inthe λ = πregion. We findoneMZM foreachT-junction
2
(showninFig.4b),andtheenergygapbetweenthezero-mode
0
40 levelsandotherenergylevelsis apparentin Fig.4d. We em-
30 40
20 20 30 phasize that each region by itself is topologically trivial, in
10 y x 10 particular, the superconductor(with λ = 0.1π)is a topologi-
callytrivialonewithoutgaplessedgestate.
(c) (d)
0.5 Conclusions.–Wehaveinvestigatedtheintriguingpossibil-
0.5
ity of creating MZMs in 2D topologically trivial supercon-
E E
ductors. TheHopf defectis constructedas a minimalmodel
0.0 0.0 for this purpose. Furthermore, we studied the more accessi-
ble T-junctions constructed from topologically trivial super-
-0.5 conductors. Hopefully, the trivial-superconductor-basedap-
-0.5
proachwillbroadenthescopeofsearchingMZMsinvarious
8 16 24 32 40 48 56 8 16 24 32 40 48 56
L L superconductors. In particular, absence of chiral Majorana
edge state in a 2D superconducting sample does not neces-
FIG. 4. (a) Superconductor-superconductor-vacuum T-junctions.
The system size is L2, with L = 40. The x > L/2 and x L/2 sarilyimplyabsenceofrobustMZMinitspointdefects.
≤
region,shownindifferentcolors,isdescribedbyEq.(2)withλtaking
λ =0.1πandλ =π,respectively. Thepeaksaretheprofilesofthe We conclude with several remarks. First, we have fo-
1 2
two MZMs localized around the two T-junctions (each T-junction cused on a single-band model, while many materials have
hostsonemode). (b)Superconductor-insulator-vacuumT-junctions. multi-bands. We emphasize that the MZMs found here are
Theparametersarethesameas(a)exceptthatdxanddyaretunedto nevertheless robust to small mixing with other bands, be-
0intheλ region,sothatλ regionisaninsulator.(c)and(d)shows
2 2 cause a single localized MZM cannotmove away from zero
12energiesclosestto0,forsystem(a)and(b),respectively.
energy, as required by the intrinsic particle-hole symme-
try of the BdG Hamiltonian[3]. Second, we have taken a
simple model BdG Hamiltonians as our starting point (like
HopfdefectdefinedbyEq.(6)canbeimitatedbyadiscontin-
Ref.[1]).MorerealisticHamiltoniansshouldbeadoptedwhen
uousone,forinstance,wemayconsideraT-junction:
dealing with real materials, for instance, the semiconductor-
λ , θ [0,π/2] superconductor heterostructures[30, 31], for which our the-
1
∈
λ(θ)= λ2, θ [π/2,π] (13) ory implies that robust MZMs can exist in certain defects
 λ3, θ∈ [π,2π]. (e.g. judiciously constructed T-junctions), without requiring
∈
λ1,2,3 being three unequal constants. Such T-junctions will trhegeiumneif.oTrmhissywsitlelmbebleeifntgfoturnfeudtutroetwheortokps.ologicallynontrivial
presumably be easier to realize than configurations with λ
smoothlyvaryinginspace. Acknowledgements.–We would like to thank Suk Bum
Wewillstudythesimplersuperconductor-superconductor- Chung for helpful suggestions on the manuscript. This
vacuumT-junctionbyreplacingtheλ -regionbythevacuum. work is supported by NSFC (No. 11674189). Z.Y. is sup-
3
Since the value of λ in the vacuum is not well defined, this portedinpartbyChinaPostdoctoralScienceFoundation(No.
replacement is not fully justified in advance. Nevertheless, 2016M590082).
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8
SupplementalMaterial Forthespecialcasem = 3,theybecome
0 2
d =2sink sinλ+2sink cosk +sin2k +2sink cosλ
x x y x y y
3sink ,
y
−
d = 2sink sinλ+2sink cosk +sin2k +2sink cosλ
y y x y x x
−
3sink ,
x
−
13
d = cos2k cos2k 2cosk cosk 2cosk cosλ
I.EXPLICITEXPRESSIONSOFdx,y,z z − 4 − x− y− x y− x
2cosk cosλ+3(cosk +cosk +cosλ). (16)
y x y
−
In the main article, we have taken d = z τz with z =
i † i
(z ,z )T and II.EXPLICITEXPRESSIONSOFBDGHAMILTONIANIN
1 2
THEREALSPACE:UNIFORMSYSTEM
Inthemainarticle,theHamiltonianiswritteninacompact
forminthekspace. Moreexplicit(lesscompact)expressions
z =sink +isink ,
1 x y
forthekineticenergyξ andthepairinggap∆ are
z =sinλ+i(cosk +cosk +cosλ m ). (14) k k
2 x y 0
−
ξ = cos2k cos2k 2cosk cosk 2cosk cosλ
k x y x y x
− − − −
13
2cosk cosλ+3(cosk +cosk +cosλ) ,
y x y
− − 4
Forgeneralm0,theexplicitformofthesedi’sread ∆k = 2iη(sinkx+isinky)
−
3
[eiλ+(cosk +cosk )]. (17)
x y
× − 2
Formanypurposes,itisusefultodoaFouriertransformation
d =2sink sinλ+2sink (cosk +cosk +cosλ m ),
x x y x y − 0 totherealspace.Forauniformsystem(i.e.spatiallyindepen-
dy =−2sinkysinλ+2sinkx(coskx+cosky+cosλ−m0), dentλ),theBdGHamiltonianisgivenbyHˆ = Hˆ0+Hˆ∆ with
d =sin2k +sin2k sin2λ (cosk +cosk +cosλ m )(21.5) thekineticenergyterm
z x y x y 0
− − −
1
Hˆ = (2cosλ 3) c c +c c + c c +c c +
0 −2 − †x,y x+1,y †x,y x,y+1 †x,y x+2,y †x,y x,y+2
Xx,y nh (cid:16) (cid:17) (cid:16) (cid:17)
13
c c +c c +h.c. ( 3cosλ)c c , (18)
(cid:16) †x,y x+1,y+1 †x,y x+1,y−1(cid:17)i o−Xx,y 4 − †x,y x,y
andthepairingterm
3 3
Hˆ∆ =η ("(2 −cosλ−isinλ)c†x,yc†x+1,y+h.c.#+"(sinλ−icosλ+ 2i)c†x,yc†x,y+1+h.c.#
Xx,y
1 1+i 1 i
−2 c†x,yc†x+2,y+ic†x,yc†x,y+2+h.c. −" 2 c†x,yc†x+1,y+1+ 2− c†x,yc†x+1,y 1+h.c.#), (19)
h i −
wherex,yarereal-spacecoordinatestakingintegervalues.As thus
explainedin the main article, λ is simply a parameter of the
Hamiltonian.Wecanseethatthepairingbetweenthenearest-
neighborsites(x,y)and(x+1,y)is ∆ =i∆ , (22)
(x,y);(x,y+1) (x,y);(x+1,y)
3
∆ =η( cosλ isinλ), (20)
(x,y);(x+1,y)
2 − −
showing a p-wave character in real space. Since the Chern
andthepairingbetweensites(x,y)and(x,y+1)is
numberiszero(foranyvalueofparameterλ), thesupercon-
3 ductor is topologically trivial, as we have explained in the
∆(x,y);(x,y+1) =η(sinλ icosλ+ i), (21) mainarticle.
− 2
9
III.EXPLICITEXPRESSIONSOFBDGHAMILTONIANIN instead of sites, it is natural to take the middle point of
THEREALSPACE:WITHADEFECT the link to define θ, for instance, cosθc†x,ycx+1,y is taken as
cosθx+1/2,yc†x,ycx+1,ywithθx+1/2,y =arctan[(y y0)/(x+1/2
− −
In the previous section, we have focused on uniform sys- x )]. It is also viable to take a different convention, say
0
temswith spatiallyindependentλ. Asexplainedin themain cosθx,yc†x,ycx+1,y,whichdoesnotaffecttheexistenceofMajo-
article,atopologicaldefectcanbecreatediftheHamiltonian ranazeromode,asverifiedinournumericalcalculations.The
parameterλdependsonthepolarangleθ,whichismeasured reasonisthat θ θ 0as(x x )2+(y y )2 ,
x,y x+1/2,y 0 0
| − |→ − − →∞
fromthedefectcenter(x0,y0),i.e. thusdifferentchoicesmerelydifferinthenear-defect-corere-
gion. The topologicalcharacter of the defect is fixed by the
y y
θx,y =arctan − 0, (23) Hamiltonianfarfromthedefectcenter,whichisnotaffected
x x
0
− bymodifyingthenear-defect-coreregion.
such that the configuration cannot be smoothly deformed to
auniformone. Thespatialdependenceoftheparameterλin
ourtopologicaldefectsisgivenby
λ =nθ . (24)
x,y x,y
Withtheabovetechnicalaspectexplained,wearereadyto
Before proceeding, it is useful to point out that, since the write down the real-space BdG Hamiltonian in the presence
hoppingtermsandthe pairingtermsare definedonthe links ofadefect:
1
Hˆ = (2cosnθ 3)c c +(2cosnθ 3)c c + c c +c c +
0 −2Xxy nh x+12,y− †x,y x+1,y x,y+21 − †x,y x,y+1 (cid:16) †x,y x+2,y †x,y x,y+2(cid:17)
13
c c +c c +h.c. ( 3cosnθ )c c ,
(cid:16) †x,y x+1,y+1 †x,y x+1,y−1(cid:17)i o−Xxy 4 − x,y †x,y x,y
3
Hˆ∆ =ηXxy ("(2 −cosnθx+21,y−isinnθx+12,y)c†x,yc†x+1,y+h.c.#
3
+"(sinnθx,y+21 −icosnθx,y+21 + 2i)c†x,yc†x,y+1+h.c.#
1 1+i 1 i
−2 c†x,yc†x+2,y+ic†x,yc†x,y+2+h.c. −" 2 c†x,yc†x+1,y+1+ 2− c†x,yc†x+1,y 1+h.c.#), (25)
h i −
whichsimplyreplacetheparameterλinEq.(18)andEq.(19) thereal-spaceBdGHamiltonianbecomes
bynθ.
Hˆ = Ψ (x,y)H Ψ(x,y), (26)
† x,y;x,y ′ ′
′ ′
x,Xy;x,y
′ ′
ItisaconventionalsteptodefineΨ(x,y)=(cx,y,c†x,y)T,and withnonzeroelementsofHx,y;x′,y′ givenby
13 +3cosnθ 0
Hx,y;x,y = −4 0 x,y 143 −3cosnθx,y !,
3 cosnθ isinnθ cosnθ 3
Hx,y;x±1,y =∓isinnθ23x±−12,yco±scnoθsx±n12θ,yx±12,y∓ 32 ∓icos−nθx23±+21,yc∓osn3θix±21sx,yi±n21n,yθ± 2 ,
Hx,y;x,y±1=∓i1cos1nθ2x,y−±21 ± 32ix∓,y±si21nnθx,y±121 ∓1i −x32,y±+12c±os2nθ±x,y±21 x,y±21 ,
Hx,y;x±2,y = −±221 ∓122 !, Hx,y;x,y±2 = ∓−122i ∓212 !,
1 1i 1 1 1i 1
Hx,y;x±1,y±1= ∓12−i±2 12 ∓2 21∓ 2 !, Hx,y;x±1,y∓1 = ±12−i±2 12 ±2 12∓ 2 !. (27)
10
whereNisthenumberoflatticesites(thelatticeconstantisset
tounit)andk ,θtakediscretevaluesin π, π+ 2π,...,π
x,y {− − N −
2π .
N}
ToobtaintheexpressionofAfromj, wedothefollowing
Fouriertransformation:
1
A(k,θ)= A(q)e−i(qxkx+qyky+qθθ),
N3/2
Xq
1
k kx j(k,θ)= N3/2 j(q)e−i(qxkx+qyky+qθθ), (33)
y Xq
FIG.5.Theinverseimagesof(0,0,1)(lightcyan)and(1,0,0)(dark whereq = (qx,qy,qθ), whose componentsqx,y,θ take discrete
blue)ofthemappingdˆ(k,θ):T3 S2. valuesin N, N +1,...,N 1 . Underthegaugeq A=0,
→ {−2 −2 2 − } ·
itisreadilyfoundthat
TherankofthematrixH is2L2,Lbeingthelinearsize
x,y;x′,y′ q j(q)
ofasquaresample. A(q)= i × , (34)
− q2
IV.NUMERICALCALCULATIONOFHOPFINVARIANT thustheHopfinvariantbecomes
Westartfromthek-spaceBdGHamiltonian: (2π)3
N = j( q) A(q)
h − N3 − ·
H (k,θ) d˜τ =η(d τ +d τ )+d τ . (28) Xq
η i i x x y y z z
≡Xi =i(2π)3 j(−q)·(q×j(q)). (35)
N3 q2
Forη 1, Hη describesasuperconductorwithweakpairing Xq
≪
(inthemainarticle,wefocusedontheη=1case).
Asweexplainedinthemainarticle,theunitvector The numerical integration convergesquite rapidly as we in-
crease N. For λ = nθ, we find N = n. The n = 1 case is
1 h
dˆ(k,θ) (ηd ,ηd ,d ) (29) showninFig.1bofthemainarticle.
x y z
≡
η2(d2+d2)+d2
x y z
q
mapsthe3DtorusT3 (k ,k ,θaredefinedmodulo2π)tothe
x y
two-dimensionalunitsphereS2. For nonzeron, the inverse- V.MORESUPPORTINGDATAONTHEEFFECTSOF
image circles of two points on S2 are linked, which is illus- IMPURITYPOTENTIAL
tratedinFig.5forthen=1case.
The Hopf invariant, which characterizes the topological
Inthemainarticle,wehaveshownthattheMajoranazero
propertyoftheHamiltonian,isgivenby
mode for n = 1 is robust to an impurity potential, and re-
marked that robust Majorana zero mode is absent for even-
Nh = d2kdθj(k,θ) A(k,θ), (30) integer Hopf invariant. Here, more data on this is shown in
−Z ·
Fig.6. Asexplainedinthemainarticle,animpuritypotential
wherej(k,θ)=(jx, jy, jθ)(Notethatthesuperscriptsx,ystand Uc†AcA atasinglesite A(indicatedinFig.1ainthemainarti-
forkx,ky)takestheformof cle) is added to test the robustness of Majorana zero modes.
For odd-integerHopf invariant, Fig.6a shows that the impu-
1
jµ = ǫµνρdˆ (∂ dˆ ∂ dˆ) (31) ritypotentialcannotmovethezeromodeenergiesawayfrom
ν ρ
8π · × E = 0 (in consistent with the results given in the main arti-
and A = (Ax,Ay,Aθ) (again, the superscripts x,y stand for cle). Foreven-integerHopfinvariant,asshowninFig.6band
k ,k ) is the gauge potential satisfying A = j, where Fig.6c, zero modes are generally absent in the defect in the
x y
=(∂/∂k ,∂/∂k ,∂/∂θ).Itisdifficulttod∇o×theintegrationin presence of impurity potential (the blue lines in Fig.6b and
x y
∇Eq.(30)analytically. Todoitnumerically,werewriteEq.(30) Fig.6caretheenergiesformodesneartheboundary,notnear
intoadiscreteform: the defect at the system center). Note that several nonzero
energy eigenvalues are almost independent of U in Fig.6a,
(2π)3
N = j(k,θ) A(k,θ), (32) Fig.6b,andFig.6c,becausetheireigenfunctionsarequitefar
h − N3 Xk,θ · fromtheimpuritysiteA.