Non-Abelian statistics and topological quantum information processing in 1D wire networks

23 Pages·2010·5.19 MB·English

Non-Abelianstatisticsandtopologicalquantuminformationprocessingin1Dwirenetworks
Jason Alicea,1,2 Yuval Oreg,3 Gil Refael,1 Felix von Oppen,4 and Matthew P. A. Fisher1
1Department of Physics, California Institute of Technology, Pasadena, California 91125
2Department of Physics and Astronomy, University of California, Irvine, California 92697
3DepartmentofCondensedMatterPhysics, WeizmannInstituteofScience, Rehovot, 76100, Israel
4DahlemCenterforComplexQuantumSystemsandFachbereichPhysik,FreieUniversita¨tBerlin,14195Berlin,Germany
(Dated:September21,2010)
Topological quantum computation provides an elegant way around decoherence, as one encodes quantum
informationinanon-localfashionthattheenvironmentfindsdifficulttocorrupt. Hereweestablishthatoneof
thekeyoperations—braidingofnon-Abeliananyons—canbeimplementedinone-dimensionalsemiconductor
0
wirenetworks. Previouswork1,2 providedarecipefordrivingsemiconductingwiresintoatopologicalphase
1
0 supportinglong-soughtparticlesknownasMajoranafermionsthatcanstoretopologicallyprotectedquantum
2 information.Majoranafermionsinthissettingcanbetransported,created,andfusedbyapplyinglocallytunable
gatestothewire.Moreimportantly,weshowthatnetworksofsuchwiresallowbraidingofMajoranafermions
p
andthattheyexhibitnon-Abelianstatisticslikevorticesinap+ipsuperconductor. Weproposeexperimental
e
setupsthatenabletheMajoranafusionrulestobeprobed,alongwithnetworksthatallowforefficientexchange
S
ofarbitrarynumbersofMajoranafermions. Thisworkpavesanewpathforwardintopologicalquantumcom-

9 putationthatbenefitsfromphysicaltransparencyandexperimentalrealism.
1
PACSnumbers:

]
l
l
a Theexperimentalrealizationofaquantumcomputerranks perimentallyrealizingthisphase,weexaminetheprospectof
h
amongtheforemostoutstandingproblemsincondensedmat- exploiting1Dsemiconductingwiresfortopologicalquantum
-
s ter physics, particularly in light of the revolutionary rewards computation.
e the achievement of this goal promises. Typically, decoher-
m Thesuitabilityof1Dwiresforthispurposeisbynomeans
ence presents the primary obstacle to fabricating a scalable
obvious. Oneoftheoperationsdesiredfortopologicalquan-
. quantumcomputer. Inthisregardtopologicalquantumcom-
t tum computation is braiding (though measurement-only ap-
a putingholdsconsiderablepromise,ashereoneembedsquan-
m proaches sidestep the need to physically braid particles7),
tuminformationinanon-local,intrinsicallydecoherence-free
and here non-Abelian statistics is critical. While Majorana
- fashion3–8. Thecoreideascanbeillustratedwithatoymodel
d fermionscanbetransported,created,andfusedinaphysically
of a spinless, two-dimensional (2D) p + ip superconductor.
n transparent fashion by applying independently tunable gates
Vortices in such a state bind exotic particles known as Ma-
o tothewire,braidingand,inparticular,non-Abelianstatistics
c jorana fermions, which cost no energy and therefore gener- poses a serious puzzle. Indeed, conventional wisdom holds
[ ate a ground state degeneracy. Because of the Majoranas,
thatbraidingstatisticsisill-definedin1D,sinceparticlesmust
vortices exhibit non-Abelian braiding statistics9–13: adiabat-
2 passthroughoneanotheratsomepointduringtheexchange.
ically exchanging vortices noncommutatively transforms the
v Thisproblemcanbesurmountedbyconsideringnetworksof
5 systemfrom one groundstate toanother. Quantuminforma- 1Dwires,thesimplestbeingtheT-junctionofFig.3. Evenin
9 tion encoded in this ground state space can be controllably
such networks, however, non-Abelian statistics does not im-
3 manipulatedbybraidingoperations—somethingtheenviron-
mediately follow as recognized by Wimmer et al.24 For ex-
4 mentfindsdifficulttoachieve.
. ample, in a 2D p + ip superconductor, vortices binding the
6
Majoranas play an integral role in establishing non-Abelian
0 Despite this scheme’s elegance, realizing suitable topo-
statistics12,13. Wedemonstratethatdespitetheabsenceofvor-
0 logical phases poses a serious challenge. Most effort has
ticesinthewires,Majoranafermionsinsemiconductingwires
1 focused on the quantum Hall state at filling fraction12,14
v: ν = 5/2, though very recently the list of candidate ex- exhibit non-Abelian statistics and transform under exchange
exactlylikevorticesinap+ipsuperconductor.
i perimental systems has rapidly expanded. Indeed, topolog-
X ical insulators15,16, semiconductor heterostructures17,18, non- Wefurtherproposeexperimentalsetupsrangingfrommin-
r centrosymmetricsuperconductors19–21,andquantumHallsys- imal circuits (involving one wire and a few gates) that can
a
tems at integer plateau transitions22 can all be engineered probetheMajoranafusionrules,toscalablenetworksthaten-
into non-Abelian topological phases similar to a spinless able efficient exchange of many Majoranas. The ‘fractional
p + ip superconductor. More recently, two groups1,2 rec- Josephson effect’1,2,15,23,25, along with Hassler et al.’s recent
ognized that topological superconductivity can be perhaps proposal26enablereadoutofthetopologicalqubitsinthisset-
mosteasilyengineeredinone-dimensional(1D)semiconduct- ting.Therelativeeasewithwhich1Dwirescanbedriveninto
ing wires deposited on an s-wave superconductor. These atopologicalsuperconductingstate,combinedwiththephys-
proposals provide the first realistic experimental setting for icaltransparencyofthemanipulations,renderthesetupsdis-
Kitaev’s231Dtopologicalsuperconductingstate.Remarkably, cussedhereextremelypromisingvenuesfortopologicalquan-
theendsofsuchwiressupportalocalized,zero-energyMajo- tuminformationprocessing. AlthoughbraidingofMajoranas
ranafermion1,2,23. Motivatedbytheexcitingpossibilityofex- alone does not permit universal quantum computation8,27–30,
2
implementationoftheideasintroducedherewouldconstitute (a)
acriticalsteptowardsthisultimategoal.
γA,1 γB,1 γA,2 γB,2 γA,3 γB,3 γA,N γB,N
I. MAJORANAFERMIONSIN‘SPINLESS’p-WAVE
(b) (c)
SUPERCONDUCTINGWIRES
E
Webeginbydiscussingthephysicsofasinglewire. Valu- B
able intuition can be garnered from Kitaev’s toy model for a z
y
spinless,p-wavesuperconductingN-sitechain23: semiconducting wire µ
x
N N−1
(cid:88) (cid:88)
H = µ c†c (tc†c + ∆eiφc c +h.c.) s-wave superconductor
− x x− x x+1 | | x x+1 k
x=1 x=1
(1)
where c is a spinless fermion operator and µ, t > 0, and FIG.1: (a)PictorialrepresentationofthegroundstateofEq.(1)in
x
∆eiφ respectively denote the chemical potential, tunneling the limit µ = 0, t = |∆|. Each spinless fermion in the chain is
|stre|ngth,andpairingpotential. Thebulk-andend-statestruc- decomposedintermsoftwoMajoranafermionsγA,xandγB,x.Ma-
ture becomes particularly transparent in the special case23 joranasγB,xandγA,x+1combinetoformanordinary,finiteenergy
fermion,leavingtwozero-energyendMajoranasγ andγ as
µ=0,t= ∆. Hereitisusefultoexpress A,1 B,N
| | shown23.(b)Aspin-orbit-coupledsemiconductingwiredepositedon
ans-wavesuperconductorcanbedrivenintoatopologicalsupercon-
cx = 21e−iφ2(γB,x+iγA,x), (2) dexutcetrinnaglsmtaatgeneextihcibfiietilndg1,2s.u(cch)eBnadndMsatrjuocratunraemofodtheessbeymaicpopnlydiuncgtinang
wirewhenB = 0(dashedlines)andB (cid:54)= 0(solidlines). Whenµ
with γ = γ† Majorana fermion operators satisfying liesinthebandgapgeneratedbythefield,pairinginheritedfromthe
α,x α,x
γα,x,γα(cid:48),x(cid:48) = 2δαα(cid:48)δxx(cid:48). These expressions expose the proximatesuperconductordrivesthewireintothetopologicalstate.
{ }
definingcharacteristicsofMajoranafermions—theyaretheir
ownantiparticleandconstitute‘half’ofanordinaryfermion.
simplestHamiltoniandescribingsuchawirereads
InthislimittheHamiltoniancanbewrittenas
(cid:90) (cid:20) (cid:18) (cid:126)2∂2
N(cid:88)−1 H = dx ψx† − 2mx −µ−i(cid:126)uˆe·σ∂x
H = it γ γ . (3)
− B,x A,x+1 gµ B (cid:19) (cid:21)
x=1 B zσz ψ +(∆eiϕψ ψ +h.c.) . (4)
x ↓x ↑x
− 2 | |
Consequently, γ and γ combine to form an ordi-
B,x A,x+1 Theoperatorψ correspondstoelectronswithspinα,effec-
nary fermion d = (γ + iγ )/2 which costs en- αx
x A,x+1 B,x tivemassm,andchemicalpotentialµ. (Wesuppressthespin
ergy 2t, reflecting the wire’s bulk gap. Conspicuously ab-
indicesexceptinthepairingterm.)Inthethirdterm,udenotes
sent from H, however, are γ and γ , which represent
A,1 B,N the (Dresselhaus31 and/or Rashba32) spin-orbit strength, and
end-Majorana modes. These can be combined into an ordi-
σ =(σx,σy,σz)isavectorofPaulimatrices. Thiscoupling
nary (though highly non-local) zero-energy fermion d =
end favorsaligningspinsalongoragainsttheunitvectorˆe,which
(γ + iγ )/2. Thus there are two degenerate ground
A,1 B,N weassumeliesinthe(x,y)plane. Thefourthtermrepresents
states 0 and 1 = d† 0 ,whered 0 = 0,whichserve
| (cid:105) | (cid:105) end| (cid:105) end| (cid:105) theZeemancouplingduetothemagneticfieldBz < 0. Note
astopologicallyprotectedqubitstates. Figure1(a)illustrates thatspin-orbitenhancementcanleadto33 g 2. Finally,the
thisphysicspictorially. (cid:29)
last term reflects the spin-singlet pairing inherited from the
Away from this special limit the Majorana end states no s-wavesuperconductorviatheproximityeffect.
longer retain this simple form, but survive provided the bulk To understand the physics of Eq. (4), consider first B =
z
gap remains finite23. This occurs when µ < 2t, where a ∆ = 0. The dashed lines in Fig. 1(c) illustrate the band
| |
partiallyfilledbandpairs. Thebulkgapcloseswhen µ =2t, structure here—clearly no ‘spinless’ regime is possible. In-
| |
andforlarger µ atopologicallytrivialsuperconductingstate troducing a magnetic field generates a band gap B at
withoutendM|aj|oranasemerges.Herepairingoccursineither zero momentum as the solid line in Fig. 1(c) depic∝ts.|Wzh|en
afullyoccupiedorvacantband. µliesinsideofthisgapthesystemexhibitsonlyasinglepair
RealizingKitaev’stopologicalsuperconductingstateexper- ofFermipointsasdesired. Turningon∆whichisweakcom-
imentally requires a system which is effectively spinless— paredtothegaptheneffectivelyp-wavepairsfermionsinthe
i.e.,exhibitsonesetofFermipoints—andp-wavepairsatthe lower band with momentum k and k, driving the wire into
Fermi energy. Both criteria can be satisfied in a spin-orbit- Kitaev’stopologicalphase1,2. [Singl−etpairinginEq.(4)gen-
coupled semiconducting wire deposited on an s-wave super- erates p-wave pairing because spin-orbit coupling favors op-
conductorbyapplyingamagneticfield1,2[seeFig.1(b)]. The posite spins for k and k states in the lower band.] Quan-
−
3
γ γ
titatively, realizing the topological phase requires1,2 ∆ < 1 2
| |
gµ B /2, which we hereafter assume holds. The opposite (a)
B z
| |
limit ∆ > gµ B /2effectivelyviolatesthe‘spinless’cri-
B z
| | | |
terionsincepairingstronglyintermixesstatesfromtheupper
band, producing an ordinary superconductor without Majo-
ranamodes. γ1 γ2
Inthetopologicalphase,theconnectiontoEq.(1)becomes (b)
more explicit when gµ B mu2, ∆ where the spins
B z
| | (cid:29) | |
nearly polarize. One can then project Eq. (4) onto a sim-
plerone-bandproblembywritingψ u(ey+iex)∂ Ψ and
↑x ∼ gµB|Bz| x x γ γ γ γ γ γ
ψ Ψ , with Ψ the lower-band fermion operator. To 3 4 1 5 6 2
↓x x x
∼
leadingorder,oneobtains (c)
(cid:90) (cid:20) (cid:18) (cid:126)2∂2 (cid:19)
dx Ψ† x µ Ψ
Heff ∼ x − 2m − eff x
(cid:21) γ3 γ4γ1 γ5 γ6γ2
+ (cid:0)∆ eiϕeffΨ ∂ Ψ +h.c.(cid:1) , (5)
| eff| x x x (d)
where µ = µ+gµ B /2 and the effective p-wave pair
eff B z
| |
fieldreads FIG.2: Applyinga‘keyboard’ofindividuallytunablegatestothe
wireallowsonetolocallycontrolwhichregionsaretopological(dark
u∆
∆ eiϕeff | | eiϕ(e +ie ). (6) blue)andnon-topological(lightblue),andhencemanipulateMajo-
eff y x
| | ≈ gµ B ranafermionswhilemaintainingthebulkgap. As(a)and(b)illus-
B z
| |
trate,sequentiallyapplyingtheleftmostgatesdrivestheleftendof
The dependence of ϕ on ˆe will be important below when thewirenon-topological,therebytransportingγ rightward. Nucle-
eff 1
weconsidernetworksofwires.Equation(5)constitutesanef- atingatopologicalsectionofthewirefromanordinaryregionorvice
fectivelow-energyHamiltonianforKitaev’smodelinEq.(1) versacreatespairsofMajoranafermionsoutofthevacuumasin(c).
inthelow-densitylimit. Fromthisperspective,theexistence Similarly,removingatopologicalregionentirelyorconnectingtwo
topologicalregionsassketchedin(d)fusesMajoranafermionsinto
ofend-Majoranasinthesemiconductingwirebecomesmani-
eitherthevacuumorafinite-energyquasiparticle.
fest. Weexploitthiscorrespondencebelowwhenaddressing
universalpropertiessuchasbraidingstatistics,whichmustbe
sharedbythetopologicalphasesdescribedbyEq.(4)andthe
0.1µmwidegateisthenoforder1K.Weconsiderthisacon-
simplerlatticemodel,Eq.(1).
servativeestimate—heavy-elementwiressuchasInSband/or
We now seek a practical method to manipulate Majorana
narrowergatescouldgeneratesubstantiallylargergaps.
fermionsinthewire. Asmotivation,considerapplyingagate
LocalgatesallowMajoranafermionstobetransported,cre-
voltagetoadjustµuniformlyacrossthewire. Theexcitation
ated,andfusedasoutlinedinFig.2. Asonegerminatespairs
gapobtainedfromEq.(4)atk =0varieswithµvia
ofMajoranafermions, thegroundstatedegeneracyincreases
Egap(k =0)=(cid:12)(cid:12)(cid:12)gµB|Bz| (cid:112)∆2+µ2(cid:12)(cid:12)(cid:12). (7) atisondoiensthoeurwciarpe.acSitpyectoifitcoaplolylo,g2incaMllyajsotroarneaqsugaennteurmateinnfoormrdai--
(cid:12) 2 − | | (cid:12)
naryzero-energyfermionswhoseoccupationnumbersspecify
(cid:112) topologicalqubitstates. AdiabaticallybraidingtheMajorana
For µ <µ = (gµ B /2)2 ∆2 thetopologicalphase
c B z
| | −| | fermionswouldenablemanipulationofthequbits, butisnot
withendMajoranasemerges, whilefor µ > µ atopologi-
c
| | possibleinasinglewire. Thuswenowturntothesimplestar-
callytrivialphaseappears. Applyingagatevoltageuniformly
rangementwhichallowsforexchange—theT-junctionofFig.
thus allows one to create or remove the Majorana fermions.
3.
However, when µ = µ the bulk gap closes, and the exci-
c
| |
tation spectrum at small momentum behaves as E (k)
gap
(cid:126)v k ,withvelocityv = 2u∆/(gµ B ). Thegapclosu≈re
B z
| | | | | |
isclearlyundesirable,sincewewouldliketomanipulateMa- II. MAJORANABRAIDINGANDNON-ABELIAN
joranafermionswithoutgeneratingadditionalquasiparticles. STATISTICS
This problem can be circumvented by employing a ‘key-
board’ of locally tunable gates as shown in Fig. 2, each First,weexplorethephysicsatthejunctionwherethewires
of which impacts µ over a finite length L of the wire. in Fig. 3 meet (see the Supplementary Material for a more
gate
Whenagivengatelocallytunesthechemicalpotentialacross detailed discussion). It will be useful to view the T-junction
µ = µ ,afiniteexcitationgapE (cid:126)vπ/L remains. as composed of three segments whose ends meet at a point.
c gap gate
| | ∼
(Roughly, thegatecreatesapotentialwellthatsupportsonly Whenonlyonesegmentrealizesatopologicalphase,asingle
klargerthan π/L .) Assuminggµ B /2 2∆ and zero-energy Majorana fermion exists at the junction. When
gate B z
(cid:126)u 0.1eVA∼˚ yields a velocity v 104|m/s|; th∼e ga|p |for a twotopologicalsegmentsmeetatthejunction,asinFigs.3(a)
∼ ∼
4
γ1 (a) γ2 γ1 (e) γ2 semiconductorcontextisgovernedbythesameφL φR de-
pendenceasinEq.(8)1,2. −
Finally,whenallthreesegmentsaretopological,againonly
a single Majorana mode exists at the junction without fine-
tuning. ThreeMajoranamodesappearonlywhenallpairsof
wiressimultaneouslyformmutualπ junctions(whichispos-
(b) γ (f) γ sible as described in the Supplementary Material, since the
2 2
superconductingphasesaredefinedwithrespecttoadirection
in each wire). Recall from Eq. (6) that the spin-orientation
favoredbyspin-orbitcouplingdeterminestheeffectivesuper-
conducting phase of the semiconducting wires. Two wires
at right angles to one another therefore exhibit a π/2 phase
γ γ
1 1 difference,wellawayfromthepathologicallimitsmentioned
(c) (g) above.
γ γ
2 2 TheT-junctionpermitsexchangeofMajoranasfromeither
the same or different topological regions. First, consider the
configurationofFig.3(a)wherethehorizontalwireresidesin
atopologicalphasewhiletheverticalwireisnon-topological.
Counterclockwiseexchangeofγ andγ canbeimplemented
1 2
γ γ as outlined in Figs. 3(b)-(d). Here, one shuttles γ to the
1 1 1
junction by making the left end non-topological; transports
γ2 (d) γ1 γ2 (h) γ1 γ1 downward by driving the vertical wire into a topological
phase; transportsγ leftwardinasimilarfashion; andfinally
2
directs γ up and to the right. Exchange of two Majorana
1
fermionsconnectedbyanon-topologicalregionasinFig.3(e)
canbesimilarlyachieved—counterclockwiseexchangeofγ
1
andγ proceedsassketchedinFigs.3(f)-(h).
2
While the Majoranas can now be exchanged, non-Abelian
FIG.3:AT-junctionallowsforadiabaticexchangeoftwoMajorana statistics is not obvious in this context. Recall how non-
fermionsbridgedbyeitheratopologicalregion(darkbluelines)as
Abelian statistics of vortices arises in a spinless 2D p + ip
in(a)-(d),oranon-topologicalregion(lightbluelines)asin(e)-(h).
superconductor12,13,followingIvanov’sapproach.Ultimately,
TransportofMajoranafermionsisachievedbygatesasoutlinedin
this can be deduced by considering two vortices which bind
Fig.2.Thearrowsalongthetopologicalregionsin(a)-(d)areuseful
Majorana fermions γ and γ . Since the spinless fermion
forunderstandingthenon-Abelianstatisticsasoutlinedinthemain 1 2
text. operators effectively change sign upon advancing the super-
conductingphaseby2π, oneintroducesbranchcutsemanat-
ingfromthevortices; crucially, aMajoranafermionchanges
and (b), generically no Majorana modes exist there. To see sign whenever crossing such a cut. Upon exchanging the
this,imaginedecouplingthetwotopologicalsegmentssothat vortices, γ (say) crosses the branch cut emanating from the
2
twoMajoranamodesincloseproximityexistatthejunction; other vortex, leading to the transformation rule γ γ
1 2
→
restoringthecouplinggenericallycombinestheseMajoranas and γ γ which is generated by the unitary operator
2 1
→ −
intoanordinary,finite-energyfermion. U = exp(πγ γ /4). With many vortices, the analogous
12 2 1
As an illustrative example, consider the setup of Fig. 3(a) unitaryoperatorsU correspondingtotheexchangeofγ and
ij i
andmodeltheleftandrighttopologicalsegmentsbyKitaev’s γ donotgenerallycommute,implyingnon-Abelianstatistics.
j
model with µ = 0 and t = ∆ in Eq. (1). [For simplic- FollowinganapproachsimilartothatofSternetal.34, we
| |
ity we will exclude the non-topological vertical wire in Fig. now argue that Majorana fermions in semiconducting wires
3(a).] Supposefurthermorethattheφ=φ intheleft/right transformexactlylikethoseboundtovorticesunderexchange,
L/R
chains and that the fermion c at site N of the left chain and hence also exhibit non-Abelian statistics. This can be
L,N
couplesweaklytothefermionc atsite1oftherightchain established most simply by considering the exchange of two
R,1
vjoiaraHnaΓsa=tt−heΓj(ucn†Lc,tNiocnRc,1ou+plhe.ca.s).foUllsoiwngs,Eq.(2),theendMa- MAtaejoarcahnsatefperomfitohneseγxc1haanndgeγ,2thaesreilaluresttrwatoeddeingeFniegrsa.te3(gar)o-u(dn)d.
states 0 and 1 = f† 0 , where f = (γ +iγ )/2 annihi-
1 2
(cid:18) (cid:19) | (cid:105) | (cid:105) | (cid:105)
iΓ φ φ lates 0 . Inprinciple,onecandeducethetransformationrule
HΓ ∼− 2 cos L−2 R γBL,NγAR,1 (8) from|th(cid:105)eBerryphasesχn Im(cid:82) dt n∂t n acquiredbythe
≡ (cid:104) | | (cid:105)
many-bodygroundstates n = 0 and 1 ,thoughinpractice
| (cid:105) | (cid:105) | (cid:105)
and therefore generally combine into an ordinary fermion23. thesearehardtoevaluate.
An exception occurs when the regions form a π-junction— Since exchange statistics is a universal property, however,
thatis,whenφ φ = π—whichfine-tunestheircoupling we are free to deform the problem to our convenience pro-
L R
−
tozero. Importantly,couplingbetweenendMajoranasinthe videdtheenergygapremainsfinite. Asafirstsimplification,
5
γ γ γ γ γ γ γ γ
since the semiconductor Hamiltonian and Kitaev’s model in 1 2 3 4 5 6 7 8
Eq. (1) can be smoothly connected, let us consider the case
(a)
where each wire in the T-junction is described by the latter.
Moreimportantly,wefurtherdeformKitaev’sHamiltonianto
be purely real as we exchange γ . The states 0 and 1
1,2
| (cid:105) | (cid:105)
canthenalsobechosenreal, leadingtoanenormoussimpli-
fication: while these states still evolve nontrivially the Berry
phaseaccumulatedduringthisevolutionvanishes.
For concreteness, we deform the Hamiltonian such that
µ < 0 and t = ∆ = 0 in the non-topological regions of (b)
Fig. 3. For the topological segments, reality implies that the ϕi ϕi
L R
superconductingphasesmustbeeither0orπ. Itisusefulto
visualizethesignchoiceforthepairingwitharrowsasinFig.
3. (To be concrete, we take the pairing ∆eiφc c such
j j+1
| |
thatthesiteindicesincreasemovingrightward/upwardinthe γ γ
1 2
horizontal/vertical wires; the case φ = 0 then corresponds
torightward/upwardarrows,whileleftward/downwardarrows
indicateφ = π.) Toavoidgeneratingπ junctions, whentwo
topological segments meet at the junction, one arrow must
γ γ γ γ γ γ γ γ
point into the junction while the other must point out. With 1 3 4 2 1 2 3 4
thissimpleruleinmind,weseeinFig.3thatalthoughwecan
successfullyswaptheMajoranaswhilekeepingtheHamilto-
nianreal,weinevitablyendupreversingthearrowsalongthe
γ γ γ γ γ γ γ γ
topologicalregion. Inotherwords,thesignofthepairinghas 1 3 4 2 1 2 3 4
flippedrelativetoourinitialHamiltonian.
To complete the exchange then, we must perform a gauge FIG. 4: (a) Example of a semiconductor wire network which al-
transformationwhichrestorestheHamiltoniantoitsoriginal lows for efficient exchange of many Majorana fermions. Adjacent
form. This can be accomplished by multiplying all fermion MajoranascanbeexchangedasinFig.3,whilenon-adjacentMajo-
creation operators by i; in particular, f† = (γ iγ )/2 ranascanbetransportedtothelowerwiretobesimilarlyexchanged.
if† = (γ +iγ )/2. Itfollowsthatγ γ a1nd−γ 2 γ→, (b)Minimalsetupdesignedtodetectthenon-trivialMajoranafusion
which the2unita1ry transformation U 1=→ex2p(πγ γ2/→4)−gen1- rules. Majoranasγ1,2arefirstcreatedoutofthevacuum. Intheleft
12 2 1 path,γ isshuttledrightward,andMajoranasγ alwayscombine
2 3,4
erates as in the 2D p + ip case. We stress that this result
toformafinite-energystatewhichisunoccupied. Intherightpath,
appliesalsointhephysicallyrelevantcasewheregatestrans- γ arealsocreatedoutofthevacuum,andthenγ andγ fusewith
3,4 2 3
porttheMajoranaswhilethesuperconductingphasesremain 50%probabilityintoeitherthevacuumorafinite-energyquasiparti-
fixed;wehavemerelyusedourfreedomtodeformtheHamil- cle.TheJosephsoncurrentflowingacrossthejunctionallowsoneto
tonian to expose the answer with minimal formalism. Ad- deducethepresenceorabsenceofthisextraquasiparticle.
ditionally, since Figs. 3(e)-(h) also represent a counterclock-
wiseexchange, itisnaturaltoexpectthesameresultforthis
case. TheSupplementaryMaterialanalyzesbothtypesofex- outtheexchangeusingthemethodsofFig.3.) However,net-
changesfromacomplementaryperspective(andwhenthesu- worksconsistingofseveralT-junctions—suchasthesetupof
perconductingphasesareheldfixed),confirmingtheirequiva- Fig. 4(a)—enable more efficient Majorana exchange. In the
lence.Therewealsoestablishrigorouslythatinnetworkssup- figure, all adjacent Majorana fermions can be immediately
portingarbitrarilymanyMajoranasexchangeisimplemented swapped using Fig. 3, while non-adjacent Majoranas can be
by a set of unitary operators Uij analogous to those in a 2D shuttled down to the lower wire to be exchanged. This ‘lad-
p+ip superconductor. Thus the statistics is non-Abelian as der’configurationstraightforwardlyscalesupbyintroducing
advertised. additional‘rungs’and/or‘legs’.
As Fu and Kane suggested in the topological insulator
context15,fusingMajoranafermionsacrossaJosephsonjunc-
III. DISCUSSION tionprovidesareadoutmethodforthetopologicalqubitstates.
WeillustratethephysicswiththeschematicsetupofFig.4(b),
The keyboard of gates shown in Fig. 2 and the T-junction whichextendstheexperimentsproposedinRefs.1,2toallow
of Fig. 3 provide the basic elements allowing manipulation theMajoranafusionrulestobedirectlyprobed. Hereasemi-
oftopologicalqubitsinsemiconductingwires. Inprinciple,a conductingwirebridgestwos-wavesuperconductorswithini-
singleT-junctioncansupportnumerouswell-separatedMajo- tial phases ϕi ; we assume ∆ϕi ϕi ϕi = π. Three
L/R ≡ L − R (cid:54)
ranamodes,eachofwhichcanbeexchangedwithanyother. gatesdrivethewirefromaninitiallynon-topologicalground
(First, create many Majoranas in the horizontal wire of the stateintoatopologicalphase. Importantly,theorderinwhich
T-junction. To exchange a given pair, shuttle all intervening one applies these gates qualitatively affects the physics. As
Majoranasdowntotheendoftheverticalwireandthencarry wenowdiscuss,onlyintheleftpathofFig.4(b)canthequbit
6
stateatthejunctionbedeterminedinasinglemeasurement. ity to realize non-Abelian statistics in this setting is remark-
ConsiderfirstgerminatingMajoranafermionsγ andγ by able. Perhaps even more appealing, however, are our pro-
1 2
applyingtheleftgate. Assumingf =(γ +iγ )/2initially posal’sphysicaltransparencyandexperimentalpromise,par-
A 1 2
costsfiniteenergyasγ andγ separate,thesysteminitializes ticularly given the feats already achieved in Ref. 35. While
1 2
intoagroundstatewithf unoccupied. Applyingthecentral topologicalquantuminformationprocessinginwirenetworks
A
and then right gates shuttles γ to the other end [see the left requires much experimental progress, observing the distinct
2
pathofFig.4(b)]. Sinceanarrowinsulatingbarrierseparates fusion channels characteristic of the two paths of Fig. 4(b)
thesuperconductors,anordinaryfermionf =(γ +iγ )/2 would provide a remarkable step en route to this goal. And
B 3 4
arisesfromtwocoupledMajoranasγ atthejunction. Simi- ultimately, ifbraidinginthissettingcanbesupplementedby
3,4
lartoEq.(8),theenergyofthismodeiswell-capturedby1,2,23 aπ/8phasegateandtopologicalchargemeasurementoffour
H i(cid:15)iγ γ = (cid:15)i(2f†f 1) where (cid:15)i = δcos(∆ϕi/2) Majoranas,wirenetworksmayprovideafeasiblepathtouni-
wiJth∼non-u3niv4ersal δ. BThBe s−ystem has been prepared in a versalquantumcomputation8,27–30.
ground state, so the f fermion will be absent if (cid:15)i > 0 but
B
occupiedotherwise.
Supposewenowvarythephasedifferenceacrossthejunc- IV. SUPPLEMENTARYMATERIAL
tionawayfromitsinitialvalueto∆ϕ. ThemeasuredJoseph-
son current (see Supplementary Material for a pedagogical A. PropertiesoftheT-junction
derivation)willthenbe
Here we investigate in greater detail the properties of the
2e dE eδ
I = = sgn((cid:15)i)sin(∆ϕ/2)+I , (9) junctioninFig.3wherethethreewiresegmentsmeet. There
(cid:126) d∆ϕ (cid:126) 2e are three cases to consider, corresponding to the situations
where one, two, or all three of the wire segments emanat-
whereE istheground-stateenergyandI denotestheusual
2e ing from the junction reside in a topological superconduct-
Cooper-pair-tunnelingcontribution.Thefirsttermontheright
ing state. It is conceptually simplest to address each case by
reflects single-electron tunneling originating from the Majo-
viewingtheT-junctionascomposedofthreeindependentwire
ranas γ . This ‘fractional’ Josephson current exhibits 4π
3,4 segments as in Fig. 5, which initially decouple from one an-
periodicityin∆ϕ, but2π periodicityintheinitialphasedif-
other. InthislimitasingleMajoranaexistsattheendofeach
ference∆ϕi.
topological segment. One can then straightforwardly couple
TherightpathinFig.4(b)yieldsverydifferentresults,re-
the wire segments at the junction and explore the fate of the
flectingthenontrivialMajoranafusionrules. Here,aftercre-
Majoranaendstates.
ating γ one applies the rightmost gate to nucleate another
1,2 Suppose that the phases of the p-wave pair fields in each
pair γ . Assuming f and f defined as above initially
3,4 A B region are φ as shown in Fig. 5 [in the semiconduc-
costfiniteenergy, thesysteminitializesintothegroundstate A/B/C
torwirecontext,thesephasescorrespondtoϕ inEq.(6)of
0,0 satisfying f 0,0 = 0. Applying the central gate eff
| (cid:105) A/B| (cid:105) the main text]. To be precise, if the wires are described by
then fuses γ and γ at the junction. To understand the out-
2 3 a lattice model, we define these phases relative to a pairing
come, it is useful to re-express the ground state in terms of
f(cid:48) = (γ + iγ )/2 and f(cid:48) = (γ + iγ )/2. In this ba- term|∆|eiφαcjcj+1suchthatthesiteindicesincreasemoving
A 1 4 B 2 3 rightward in the horizontal wires and upward in the vertical
sis 0,0 = (0(cid:48),0(cid:48) i1(cid:48),1(cid:48) )/√2, where f(cid:48) annihilate
| (cid:105) | (cid:105)− | (cid:105) A,B wire. Asimilarconventioncanbeemployedinthesemicon-
0(cid:48),0(cid:48) and 1(cid:48),1(cid:48) = f(cid:48)†f(cid:48)† 0(cid:48),0(cid:48) . Following our previous ductor wire context. Now suppose we allow single-electron
d|iscus(cid:105)sion, |f(cid:48) ac(cid:105)quiresAfinBit|e ener(cid:105)gy at the junction, lifting tunnelingbetween theends ofeachsegment, withamplitude
B
the degeneracy between 0(cid:48),0(cid:48) and 1(cid:48),1(cid:48) . Measuring the Γ as shown schematically in Fig. 5. (Pairing between elec-
| (cid:105) | (cid:105)
Josephsoncurrentthencollapsesthewavefunctionwith50% trons residing at the ends of each region is also generally al-
probability onto either the ground state, or an excited state lowed, butdoesnotchangeanyqualitativeresultsbelowand
with an extra quasiparticle localized at the junction. In the willthereforebeneglected.) Forconveniencewewillassume
former case Eq. (9) again describes the current, while in the thatthetunnelingstrengthisweakcomparedtothebulkgaps
latterthefractionalcontributionsimplychangessign. in the wires, which will allow us to focus solely on the Ma-
InmorecomplexnetworkssuchasthatofFig.4(a),fusing joranaendstates;ourconclusions,however,aremoregeneral
the Majoranas across a Josephson junction—and in particu- anddonotrequirethisassumption.
lar measuring the sign of the fractional Josephson current— In the setup of Fig. 5(a) with only one topological region,
similarly allows qubit readout. Alternatively, the interesting the Majorana γ is qualitatively unaffected by the coupling
1
recentproposalofHassleretal.26 forreadingqubitstatesvia to the non-topological wires. At most its wavefunction can
ancillary non-topological flux qubits can be adapted to these be quantitatively modified, but it necessarily remains at zero
setups(andindeedwasoriginallydiscussedintermsofaniso- energy. This reflects the familiar topological protection of a
latedsemiconductingwire26). singleisolatedMajoranamodeinagappedsystem.
To conclude, we have introduced a surprising new venue With two of the three wires topological as in Fig. 5(b),
forbraiding,non-Abelianstatistics,andtopologicalquantum the end Majoranas γ and γ generally combine into an or-
1 2
information processing—networks of one-dimensional semi- dinary finite-energy fermion, except with fine-tuning. To
conducting wires. From a fundamental standpoint, the abil- a good approximation, the Majoranas couple through a
7
γ
φA 1 Γ φB Equation(10)demonstratesthatγ1andγ2remainzero-energy
modesonlywhenthetopologicalwiresformaπjunction,i.e.,
whenφ =φ +π.
A B
Γ Γ Finally,considerthecaseshowninFig.5(c)whereallthree
(a) segments are topological. Here the Majoranas γ1,2,3 couple
via
φ (cid:20) (cid:18) (cid:19)
C φA φB
H iΓ cos − γ γ
123 1 2
∝ − 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
φ φ φ φ
C B A C
+ cos − γ γ +sin − γ γ .
3 2 1 3
γ γ 2 2
φ 1 Γ 2 φ
A B (11)
Note the sine function determining the coupling between γ
Γ Γ 1
andγ ,whicharisesbecauseoftheconventionswechosefor
(b) 3
defining φ above. Recall that to make the problem well-
α
defined,weneededtodefinethephaseswithrespecttoapar-
φ ticular direction in each wire; otherwise there is an ambigu-
C ity of π in the definition, since for instance ∆eiφc c =
j j+1
| |
∆ei(φ+π)c c .Wedefinedthephasessuchthatthesitein-
j+1 j
| |
dicesincreaseuponmovingrightwardorupwardinthewires.
γ γ But this implies that the site indices in both the left and bot-
φ 1 Γ 2 φ tomwiresincreaseuponmovingtowardsthejunction,incon-
A B
γ trast to all other pairs of wires. It follows that the splitting
3
of γ and γ is proportional to cos[(φ φ π)/2] =
1 3 A C
− −
Γ Γ sin[(φA φC)/2]. Hencewithourconventionsaπ junction
−
between these two regions actually corresponds to the case
(c)
φ =φ .
A C
The Hamiltonian H implies that γ all remain zero-
123 1,2,3
φ
C energy modes only when φA = φC = φB + π, where
all pairs of wires form mutual π junctions. (This remains
true even when coupling to the ordinary gapped states is in-
cluded.) Aside from this fine-tuned limit, however, H al-
123
ways supports one zero-energy Majorana mode and one or-
FIG.5:T-junctionviewedasthreewiresegmentswithp-wavesuper- dinary finite-energyfermion. As an illustration, consider the
conductingphasesφ . Theendsofeachsegmentarecoupled
A,B,C caseφ =φ +πandφ =φ ,sothatonlythehorizontal
viatunnelingwithamplitudeΓasshown. (a)Whenonlyoneseg- A B C (cid:54) A
wiresformaπjunction. HeretheHamiltoniansimplifiesto
mentistopological, thetunnelingcannotdestroytheMajoranaγ
1
atthejunction. (b)Twotopologicalregionsmeetingatthejunction
(cid:18) (cid:19)
oleraddinsatroy,thfienietne-denMeragjoyrafenramsiγo1n,aunndleγs2sgtheenetoraplolylocgoicmalbiwniinregsifnotromana H123 iΓcos φC −φB γ3(γ2 γ1). (12)
∝− 2 −
π junction. (c)Whenallthethreewiresaretopological,theMajo-
ranasγ generallycombinetoformafinite-energyfermionand
1,2,3
asingletopologicallyprotectedMajorana. AllthreeMajoranasre- Itfollowsthatthelinearcombination(γ +γ )/√2remains
1 2
mainatzeroenergyonlywhenallthreewiresegmentsformmutual a zero-energy Majorana mode, while γ and (γ γ )/√2
3 1 2
πjunctions. −
combine into a finite-energy fermion. While here the zero-
energy Majorana carries weight only on the horizontal wires
Hamiltonian1,2,23 whichformedtheπjunction,ingeneralitswavefunctionwill
haveweightonallthreesegments.
(cid:18) (cid:19)
φ φ
H iΓcos A− B γ γ . (10) As we braid Majorana fermions using the methods de-
12 1 2
∝− 2
scribed in the main text, it is imperative that we avoid gen-
Thiswasdiscussedinthemaintextinthecontextoftwowires eratingspuriouszero-modesattheT-junction. Theabovedis-
described by Kitaev’s toy model in a particular limit, but is cussion implies that we are safe in this regard so long as we
qualitatively rather general—the 4π periodicity in φ φ avoidπ junctions. Fortunately, thesemiconductingwireswe
A B
has a topological origin23. For instance, end Majoran−as in considerednaturallyavoidsuchsituations,sincetwowiresat
twotopologicalsemiconductingwirescoupledthroughanor- right angles to one another exhibit effective p-wave phases
dinaryregionexhibitthesamephasedependenceasabove1,2. thatdifferbyπ/2asdiscussedinthemaintext.
8
B. WavefunctionapproachtoMajoranafermionexchange γ byforminganotherfar-awaytopologicalregion,etc.) One
4
canconstructM fermionoperatorsfromthesevia
In this section we explore in much greater detail the ex-
1
change of Majorana fermions in a 1D wire networks. We fj = (γ2j−1+iγ2j), (13)
2
once again emphasize the nontrivial nature of the problem
since non-Abelian statistics in a 2D p + ip superconductor which correspond to zero-energy modes (up to corrections
is typically understood as arising because of superconduct- whichareexponentiallysmallintheseparationbetweenMa-
ing vortices. One might worry that the Majoranas in the joranas). These modes give rise to 2M degenerate ground
wiresperhapsbindvorticesintheneighboringparents-wave stateswhichcanbelabeledbytheoccupationnumbersn =
j
superconductor, but this is certainly not the case. This be- 0,1forthef fermions. Wewouldliketoconstructthesede-
j
comes apparent when one recalls how the effective super- generate ground state wavefunctions and understand the ex-
conducting Hamiltonian for the wire is derived (see, e.g., change of Majorana fermions in 1D wire networks from this
Ref. 36). Namely, one considers a Hamiltonian of the form perspective.
H =H +H +H ,whereH andH describethe Let us denote the positive-energy Bogoliubov-de Gennes
wire SC t wire SC
wireandsuperconductorinisolation,andH encodessingle- quasiparticle operators by d , each of which must annihilate
t α
electrontunnelingbetweenthetwo. Uponintegratingoutthe thegroundstates. Asusual,theexplicitformoftheseground
gappedsuperconductingdegreesoffreedomassumingauni- states is nontrivial because both the Majorana operators and
formpairfield∆,onearrivesataneffectiveHamiltonianfor d representlinearcombinationsoftheoriginalfermioncre-
α
thewirewhichincludesproximityinducedpairingterms.Any ationandannihilationoperatorsforthe1Dwirenetwork. By
phase variations in the parent superconductor’s order param- constructionthewavefunction
eter are ruled out by assumption, yet Majoranas can never-
1 (cid:89)
thelessexistinthewires. Thusdevelopingaphysicalpicture ψ = d vac , (14)
α
| (cid:105) √ | (cid:105)
fortheexchangeinthissettingposesanextremelyimportant N α
issue.
with vac the vacuum of the original fermion operators and
Ouraimhereistoprovidegreaterrigorandacomplemen- | (cid:105)
the normalization, must constitute one of the degenerate
tary picture for the discussion presented in the main text for N
ground states since any d clearly annihilates this state. Be-
howtheMajoranastransformunderexchange. Wewillbegin α
causewepulledγ andγ outofthevacuum,weareguar-
by constructing the many-body wavefunctions for a general 2j−1 2j
anteedthat ψ willbeaneigenvectoroff†f witheigenvalue
1DwirenetworksupportinganarbitrarynumberofMajorana | (cid:105) j j
n .
fermions. We will then establish some important general re- j
Howwelabelthisstateintermsofthen occupationnum-
sultsforbraidingthatrelyonminimalassumptionsaboutthe j
bersisamatterofconventionbecausethereisasignambigu-
underlyingHamiltonian.Heretherewillbesomeoverlapwith
the approach followed by Bonderson et al.37, who recently ity in the Majorana wavefunctions. Specifically, if φj is the
Majorana wavefunction corresponding to γ , then φ also
revisited the issue of non-Abelian statistics in the fractional j j
−
denotesalegitimateMajoranawavefunctionthatpreservesthe
quantum Hall context. Following this general analysis, we
willconsideragaintheexchangeofFigs.3(a)-(d)andexplore relationsγj = γj† andγj2 = 1. Asanexample, supposethat
howtheMajoranastransformwhenthebraidisimplemented uponmakingsomeparticularoverallsignchoicesforφ1,2we
bykeepingthesuperconductingphasesfixed,aswouldbethe findthatf1 ψ =0;wewouldthenidentify ψ withn1 =0.
| (cid:105) | (cid:105)
case in practice. Next, we will turn to an analysis of the ex- Hadwechosentheoppositesignforφ2, however, wewould
changeoutlinedinFigs.3(e)-(h)andshowthat,asclaimedin findinsteadthatf† ψ =0andidentifythisstatewithn =1
1| (cid:105) 1
themaintext,thisbraidtransformstheMajoranasinanidenti- (sendingφ φ isequivalenttosendingγ γ and
2 2 2 2
calfashiontothebraidofFigs.3(a)-(d). Finally,wewillcon- f f†). W→ew−illassumeforconcretenessthat→the−signsof
1 → 1
sidersomespecialexampleswherethefullmany-bodywave- the Majorana wavefunctions have been chosen such that ψ
| (cid:105)
functionscanbeanalyzedduringtheexchange,allowingusto correspondstothegroundstatewithn =1,i.e.,
j
exploreimportantissuessuchastheoverallphaseacquiredby
thegroundstatesunderbraiding. 11 1 = ψ . (15)
| ··· (cid:105) | (cid:105)
Thegroundstatewithn =0canthenbewritten
j
1. Constructionofdegenerategroundstatewavefunctions
00 0 =f f f ψ , (16)
1 2 M
| ··· (cid:105) ··· | (cid:105)
Consider a 1D wire network with 2M well-separated, lo- which is manifestly annihilated by all fj. All other ground
calizedMajoranamodescorrespondingtooperatorsγ1,...,2M states can be obtained by applying creation operators fj† to
that satisfy γ = γ† and γ2 = 1. By ‘well-separated’, we 00 0 or,equivalently,annihilationoperatorsf tothestate
j j j | ··· (cid:105) j
mean that different Majorana wavefunctions overlap negligi- 11 1 .
| ··· (cid:105)
bly with each other. Suppose moreover that the pairs γ Beforemovingon,wenotethatthewaywecombinedMa-
2j−1
andγ weregerminatedfromthevacuum.(Forexample,one joranasinEq.(13)toconstructtheordinaryf fermionoper-
2j j
couldstartwithanon-topologicalnetwork,generateγ andγ ators was convenient because of how we assumed the Majo-
1 2
by nucleating a single topological region, then create γ and ranasweregerminated,butisbynomeansunique. Onecould
3
9
alwayschoosetocombinepairsofMajoranasdifferentlyand modifications only impact the Majoranas γ which are be-
1,2
constructoperatorsf(cid:48) andgroundstates n(cid:48)n(cid:48) n(cid:48) . The ingbraided.Thiswillcertainlybethecase,forexample,inthe
j | 1 2··· M(cid:105)
ground states in this representation would then be related to eight-MajoranaconfigurationfromFig.4(a)ofthemaintext;
thegroundstateswedefinedabovesimplybyachangeofba- in fact, there any pair can be exchanged without disturbing
sis. anyotherMajoranasinthesystem. However,oneisnotguar-
anteed that this is always immediately possible, since there
may be intervening Majoranas that prevent γ and γ from
1 2
2. GeneralresultsforMajoranaexchange beingsoexchanged. Suchcasescanbetreatedinoneoftwo
ways.First,onecanalwaysfirsttransporttheMajoranasinthe
system(butimportantlywithoutbraidinganyofthem)topro-
We now proceed to obtain some important generic results
duce an arrangement where γ can be directly exchanged.
for Majorana exchange that rely on very minimal assump- 1,2
Alternatively,onecanalwayspairwisebraidMajoranasinthe
tions about the underlying Hamiltonian for the 1D wire net-
systemuntiltheinterveningMajoranasareremoved. Wewill
work. Supposethatwewishtoexchangeγ andγ . [Below
1 2 proceedbelowbyassumingthatinsuchcases,theprerequisite
it will prove extremely convenient to work in a basis where
translationsorexchangeshavealreadybeencarriedout.
the Majoranas we braid combine into an ordinary fermion.
Wewillfirstdemonstratethatalloff-diagonalBerryphase
This is the case for the basis we introduced above, since
elementsvanishtriviallyinthebasiswehavechosen,andthat
f = (γ +iγ )/2. If instead we wanted to exchange, say,
1 1 2 theexchangedependsonlyontheoccupationnumbern for
γ andγ ,onewouldfirstwanttochangebasisandwritethe 1
gr1oundst3atesintermsofoperatorssuchasf(cid:48) =(γ +iγ )/2 thef1 =(γ1+iγ2)/2fermion.Thisfollowsfromourassump-
1 1 3 tion above that the exchange can be implemented by simply
and f(cid:48) = (γ +iγ )/2, then proceed as we outline below.]
2 2 4 adjusting local terms in the Hamiltonian that affect only the
Let λ be the parameter in the Hamiltonian that varies to im-
regions near γ . Assuming all other Majoranas are well-
plement the exchange of γ and γ . If λ varies from λ 1,2
1 2 init localizedfarfromγ ,theirwavefunctionswillbeunaffected
to λ during the course of the exchange, then we require 1,2
final bytheselocalchangestotheHamiltonian(uptoexponentially
that H(λ ) = H(λ ) so that the Hamiltonian returns
final init smallcorrections). Consequently,toanexcellentapproxima-
to its original form after the braid. (If the Hamiltonian does
tionf canbeassumedλ-independent.
notreturntoitsoriginalform,thenwecannotmakerigorous 2,...,M
In contrast, f and the bulk quasiparticle operators d do
statementsabouthowthewavefunctionstransformunderthe 1 α
depend on λ. Explicitly displaying the λ-dependence, the
exchange.)
groundstate 00 0 canthenbewrittenas
Therearetwoimportantcontributionsthatonemustunder- | ··· (cid:105)
standtoanalyzetheexchange. ThefirstaretheBerryphases
00 0 =f (λ)f f ψ(λ) . (19)
1 2 M
acquiredbythedegenerategroundstates,whichfollowfrom | ··· (cid:105) ··· | (cid:105)
Again, all other ground states can be obtained from this by
(cid:104)n1n2···nM|∂λ|m1m2···mM(cid:105). (17) application of f1(λ)† and f2†,...,M. Because f2,...,M are λ-
independent,differentiatingthestates n n n withre-
Wehaveincludedthepossibilitythatnon-trivialoff-diagonal | 1 2··· M(cid:105)
spect to λ does not change the occupation numbers for the
Berry phases occur, since a priori these need not vanish. If
correspondingzero-energymodes. Itthereforefollowsthat
we assume that the degenerate ground states return to their
preciseoriginalformatλ ,theBerryphasesencodeallinfor-
f n n n ∂ m m m δ δ .(20)
mationabouttheexchange.However,ifwerelaxthisassump- (cid:104) 1 2··· M| λ| 1 2··· M(cid:105)∝ n2,m2··· nM,mM
tion(whichwillindeedbeusefulbelow),thenweadditionally One can see this relation formally by inserting the ground
need to compare the explicit changes between the initial and states obtained from Eq. (19) into the left-hand side of the
finalstates. Ingeneral,thefinalgroundstatescouldrepresent aboveequation. Wefurthermorehave
anontriviallinearcombinationoftheinitialgroundstates,so
wewrite n n n ∂ m n n δ (21)
(cid:104) 1 2··· M| λ| 1 2··· M(cid:105)∝ n1,m1
(cid:88)
|n1n2···nM(cid:105)f = ρ({mj};{nj})|m1m2···mM(cid:105)i, soinnaclebweictahunse1 t(cid:54)=hemy1exthheibaibtodvifefebrreanatnfderkmetioanreptrairvitiya.llyTohruthsoagl-l
{mj}
off-diagonalBerryphasesindeedvanish, andtheonlypossi-
(18)
blynon-zeroBerryphasesfollowfrom
where the subscripts i/f denote the initial/final states.
Λ (λ) 0n n ∂ 0n n
The combination of the Berry phases and the coefficients 0 ≡ (cid:104) 2··· M| λ| 2··· M(cid:105)
= ψ(λ)f (λ)†∂ f (λ)ψ(λ) , (22)
ρ( m ; n )fullyspecifytheoutcomeoftheexchange. 1 λ 1
j j (cid:104) | | (cid:105)
{ } { }
Whiletheproblemappearsdaunting,aremarkableamount Λ (λ) 1n n ∂ 1n n
1 2 M λ 2 M
≡ (cid:104) ··· | | ··· (cid:105)
ofprogresscaninfactbemadeonverygeneralgrounds. We = ψ(λ)∂ ψ(λ) , (23)
λ
will make only one additional assumption about the physi- (cid:104) | | (cid:105)
cal system. Specifically, to carry out the exchange we will whichmanifestlydependonlyonn .Asonewouldintuitively
1
assumethatonlylocaltermsintheHamiltonian—suchaslo- expect,theadditionalfar-awayMajoranamodesdonotimpact
cal chemical potentials—need to be modified, and that such the outcome of the exchange of γ . These modes trivially
1,2
10
factoroutintheabovesense,andtheexchangesimplysends Thisdoesnotimplythattheexchangeistrivial,sincenon-
γ γ fori=1or2. trivial effects can still arise due to explicit differences be-
i i
→ (cid:54)
We can actually evaluate the Berry phases even further. tween the initial and final states. Recall that by assumption
FromEqs.(22)and(23),weobtainthefollowingrelation: theHamiltonianreturnsbacktoitsoriginalformaftertheex-
change. Therefore we must have f1(λfinal) = eiθff1(λinit)
Λ0(λ)=Λ1(λ)+(cid:104)ψ(λ)|f1(λ)†[∂λf1(λ)]|ψ(λ)(cid:105), (24) and dα(λfinal) = eiθαdα(λinit), for some unspecified phase
factors θ and θ . By choosing the bulk quasiparticle wave-
whereontherightsidethederivativeactsonlyonf (λ). One f α
1 functions appropriately, we can always set without loss of
canalwaysdecompose∂ f (λ)as
λ 1 generality θ = 0, though we must allow for nontrivial θ
α f
∂ f (λ) = A(λ)f (λ)+A(cid:48)(λ)f (λ)† since we restricted f1(λ) to satisfy Eqs. (28) for all λ. The
λ 1 1 1
initialandfinalgroundstatesarethenrelatedby
+ (cid:88)(cid:2)B (λ)d (λ)+B(cid:48)(λ)d (λ)†(cid:3). (25)
α α α α 0n n = eiθf 0n n
α | 2··· M(cid:105)f | 2··· M(cid:105)i
1n n = 1n n . (32)
(Theright-handsidedoesnotinvolvef2,...,M becausebyas- | 2··· M(cid:105)f | 2··· M(cid:105)i
sumption the corresponding wavefunctions have negligible The most important implication of these results is that the
weightnearγ1andγ2.)InsertingthisexpressionintoEq.(24), outcomeoftheexchangefollowssolelyfromtheevolutionof
onefindsthatonlytheA(λ)termcontributes: the localized Majorana wavefunctions for γ , which deter-
1,2
mines θ . [This is true up to an overall Berry phase arising
f
Λ (λ)=Λ (λ)+A(λ). (26)
0 1 from Λ (λ) which, by contrast, depends additionally on the
j
evolutionofthebulkquasiparticlestates.Butthiscontribution
ItisusefultoisolateA(λ)inEq.(25)byconsideringthefol-
isnon-universalinanyeventaswewillseelater.] Itisworth
lowinganticommutator,
emphasizingthatthewavefunctionscorrespondingtoγ de-
1,2
f (λ)†,∂ f (λ) =A(λ). (27) pend neither on the presence of the additional Majoranas in
1 λ 1
{ } the system, nor on changes to the bulk quasiparticle wave-
Supposenowthatthroughouttheexchangethefollowingstan- functions far from γ . This leads to a great simplification:
1,2
dardrelationshold if we know how γ transform in some minimal configura-
1,2
tion,thenthesametransformationmustholdwhenarbitrarily
1
f (λ) = [γ (λ)+iγ (λ)] manyadditionaldistantMajoranasareintroducedby,say,nu-
1 1 2
2
cleatingothertopologicalregionsintothenetwork. Oncethis
γj(λ) = γj(λ)† (28) isknownwecandeterminetheunitaryoperatorU12 thatacts
γj(λ)2 = 1. onγ1,2 toimplementtheirexchange. Importantly,U12 isba-
sisindependentasitonlyactsontheMajoranaoperators,and
Note that this does not constitute an additional physical as- thereforeallowsonetodeducetheevolutionofthemany-body
sumption,butratheraconvention;onecanalwayschoosethe ground states under the exchange (up to an overall phase) in
operatorsinthisway. TheanticommutatorinEq.(27)isthen anybasisonechooses.
In the more general situation where one exchanges γ and
f1(λ)†,∂λf1(λ) = 1(cid:2) γ1,∂λγ1 + γ2,∂λγ2 γj, exactly thesame analysisholds whenone worksiniaba-
{ } 4 { } { } sis where one of the occupation numbers corresponds to the
(cid:3)
+ i{γ1,∂λγ2}−i{γ2,∂λγ1} ,(29) ordinaryfermionoperatorf(cid:48) =(γi+iγj)/2,asbrieflymen-
tioned at the beginning of this subsection. The implications
whereforbrevityontheright-handsidewesuppressedtheλ
discussed in the preceding paragraph that follow from these
dependence. Thefirsttwotermsvanishtriviallybecause
results also carry over to this more general case. We stress
thatalthoughitiseasiesttodrawtheseconclusionsbychoos-
γ ,∂ γ = γ (∂ γ )+(∂ γ )γ
j λ j j λ j λ j j
{ } ingadifferentbasisdependingonwhichMajoranasarebeing
= ∂ (γ γ )=∂ (1)=0. (30)
λ j j λ braided, one again ultimately deduces basis-independent op-
erators U that act on γ to implement their braid. The set
Interestingly,thelasttwotermsinEq.(29)alsovanish(upto ij i,j
of unitary operators U are sufficient to deduce how the
exponentiallysmallcorrections)becauseγ1 andγ2 arelocal- degenerate ground sta{tesijt}ransform in a fixed basis under ar-
ized far apart from one another. (To see this explicitly, one
bitraryexchanges(uptoanoverallnon-universalphase); see
canexpandγ intermsoftheoriginalfermioncreationand
1,2 forexampleEq.(9)inRef.13.
annihilation operators for the wire network, evaluate ∂ γ
λ 1,2 We have now distilled the problem of demonstrating non-
intermsofthisexpansion,andthenevaluatetheanticommu-
AbelianstatisticsdowntothatofdetermininghowtheMajo-
tators.) Thus we obtain the remarkable result that as long as
ranasγ andγ transformundertheexchangesshowninFig.
Eqs.(28)holdthroughouttheexchange, 1 2
3. Thesetwoelementaryoperationsaloneindeedallowoneto
Λ (λ)=Λ (λ) (31) braid arbitrary pairs of Majorana fermions in networks com-
1 0
posed of trijunctions, such as that shown in Fig. 4(a) of the
and the states with n = 0 and 1 acquire identical Berry main text. We proceed now by analyzing these two cases in
1
phasesundertheexchange. turn.