Majorana Fermions and Multiple Topological Phase Transition in Kitaev Ladder Topological Superconductors

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by  Ryohei Wakatsuki

Majorana Fermions and Multiple Topological Phase Transition
in Kitaev Ladder Topological Superconductors
Ryohei Wakatsuki1, Motohiko Ezawa1 and Naoto Nagaosa1,2
1Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan and
2Center for Emergent Matter Science (CEMS), ASI, RIKEN, Wako 351-0198, Japan
Motivated by the InSb nanowire superconductor system, we investigate a system where one-dimensional
4 topologicalsuperconductorsareplacedinparallel.ItwouldbesimulatedwellbyaladderoftheKitaevchains.
1 Thesystemundergoesmultipletopologicalphasetransitions,wherethenumberofMajoranafermionschanges
0 asafunctionoftheinterchainsuperconductingpairings. Weanalyticallydeterminethetopologicalphasedia-
2 grambyexplicitlycalculatingthetopologicalnumberandthebandstructure.Theyshoweven-oddeffectswith
y respecttothenumber oflegsoftheladder. Whentherelativephasebetweentheinter-andintrachainsuper-
a conductingpairingsis0orπ,thesystembelongstotheclassBDIcharacterizedbytheZindex,andotherwise
M it belongs to the class D characterized by the Z2 index. This topological class change would be caused by
applyingtheJosephsoncurrentoranexternalmagneticfield,andcouldbeobservedbymeasuringthezero-bias
3 differentialconductance.
2

] I. INTRODUCTION
n
o
c The Majorana fermion is one of the hottest topics in con-
- densed matter physics1–3. Majorana fermions are particles
r
p which are their own antiparticles. Because of their nonlo-
u cality and non-Abelian statistics, they are considered to re-
s alize exotic phenomenaand there are possibilities to encode
.
t fault tolerant topological quantum computations. There are
a
m several suggestions for physical systems that support Majo- FIG.1: (Coloronline)Illustrationofaladdertopologicalsupercon-
ranazero-energystates(MZES)4–10. TheKitaevmodelisthe ductor. It is decomposed into the superconducting region (purple)
d- simplest model that realizes the MZES11. It well describes andthelead(blue). Thepurpleregionistheladdertopological su-
perconductor. Thick lines indicate the chains, while the thin lines
n a one-dimensional nanowire with strong Rashba spin-orbit
o interaction5,6. Thereare generalizationsof the Kitaev model indicatetheinterchaincouplings. Listhenumberofsitesalongthe
c byseveralauthors12–21. chains,whileN isthenumberoftheKitaevchains. Theblueregion
[ isthenormallead,whichisattachedwhenwecalculatethedifferen-
Thin films and multichannel nanowires with p+ip-wave tialconductanceoftheNSjunction.
4 superconducting pairing have been investigated12–21. In the
v
quasi-one-dimensionalp+ip-wavepairingsystem, onepair
2
9 ofMajoranafermionsappearsifanoddnumberofsubbands experimental results are still controversial because there are
1 are filled. This is because the class of the system is D, and manypossiblereasonsforthezero-biasanomaly,suchasthe
5 is characterized by the Z2 index. Recently, it was pointed Kondoeffect,disorder,ortunnelingviaAndreevboundstates,
1. out22 that the realistic spin-orbit nanowire system approxi- whichdonotoriginateinMajoranafermions.27–30
0 matelypossesses chiralsymmetryand thesystem belongsto In this paper, motivatedby the InSb nanowire system, we
4 theBDIclass,wherethenumberofMZEScanhavearbitrary investigate a system where one-dimensional topological su-
1 integervalues.
perconductorsareplacedinparallel,wherethesuperconduct-
:
v To realize the Kitaev model, there is a proposal to use a ing pairing phase can be changed arbitrarily. The system
i nanowirewhichhasastrongspin-orbitinteraction(SOI)with would well be simulated by the ladder of topologicalsuper-
X
anordinarys-wavesuperconductorandtheZeemanfield.Be- conductorsshowninFig. 1. Thismodelbelongstotheclass
r
cause of the strong SOI, the system becomes a helical state, BDI or D, dependingon the phase of the interchain pairing.
a
inwhichtheelectronspinandthemomentumcorrespondone We first analyze the system where the interchain supercon-
toone. Themagneticfieldperpendiculartothespin-orbitef- ducting coupling is absent. The system belongs to the class
fectivemagneticfieldbreaksthetime-reversalsymmetryand BDI,wherethetopologicalnumberischaracterizedbytheZ
opensa gapatk = 0. Then,thesystem effectivelybecomes index.WefindthatthenumberoftheMZESchangesfromN
spinlesswhentheFermienergyiswithinthisenergygap.Fur- to0bychangingthemagnitudeoftheinterlayerhopping.We
thermore,thes-wavesuperconductorproximityisinducedin determinethetopologicalphasediagrambycalculatingtheZ
the nanowire generates p-wave pairings. A most promising index,whichisatopologicalnumberofthesystem. Nextwe
candidate for a strong SOI nanowire is that of InSb, which introducetheinterchainsuperconductingpairings. Whenthe
has a large SOI (∆ = 0.3meV), a large electron g factor relativephasebetweentheinter-andintrachainsuperconduct-
SOI
(g 50),andasmalleffectivemass(m∗ = 0.015m ). Re- ing pairings is 0 or π, the system belongs to the class BDI
e
c|en|tl∼y,thesignatureofzero-energyboundstateswasobserved and is characterized by the Z index. Otherwise, the system
in the system using InSb or InAs nanowire23–26. However, becomes class D and is characterized by the Z index: The
2
2
number of the MZES changes alternately between 0 and 1 of generality we may assume that d is real in the effective
x
due to the level repulsion by the interchain superconducting Hamiltonian,whiled iscomplexingeneral. We defineθ as
y
pairings. thephaseofd ,i.e.,d = d eiθ.
y y y
| |
Therestofthepaperisorganizedasfollows. InSec.IIwe Theenergyspectrumissymmetricaroundzeroenergydue
introducethemodelanddiscusstherelationshipbetweenour totheparticle-holesymmetryinducedbythesuperconductiv-
modeland the Kitaev model. We also discuss the symmetry ity. There are N legs in the ladder (Fig. 1). We refer to L
andthetopologicalclassificationoftheHamiltonian. InSecs asthenumberofsitesalongtheladder. Wenotethatthesys-
.III and IV, we study the model without and with interchain temisreducedtotheKitaevchainwhenN = 1. Ifthephase
superconductingpairing. We calculate the Z and Z indices difference between d and d is π/2 and L N, it is an
2 x y
≈
and determine the phase diagram. We also discuss the en- anisotropicp+ipsuperconductorsystem. Weanalyzemainly
ergyspectrum. In Sec. V we derive the lowenergyeffective thesmall-N region.
Hamiltonianwritten in termsofMajoranaoperatorsanddis- We clarify the topological class31 of the Hamiltonian (1).
cussitslowenergybehavior.InSec. VI,examiningthetrans- Inthemomentumrepresentation,theBogoliubov–deGennes
port property of the model, we study how the conductance Hamiltonianisgivenby
dependsonthephasedifferencebetweenintra-andinterchain
superconductingpairings. ξ ∆
H(k)= , (2)
∆† ξ
(cid:18) − (cid:19)
II. KITAEVLADDERMODEL whereξand∆areN N tridiagonalmatriceswiththematrix
×
elements
Majorana fermions emerge at the edges of an InSb
ξ =( µ 2t cosk)δ t δ t δ , (3)
nanowireinthepresenceofthesuperconductingorderandthe i,j x i,j y i,j+1 y i+1,j
− − − −
externalmagneticfieldperpendiculartoit. Itiswelldescribed ∆ =2id sinkδ +d δ d δ , (4)
i,j x i,j y i,j+1 y i+1,j
−
bytheKitaevchain11,whereaone-dimensionalp-wavetopo-
logical superconductivity is realized. A natural question is where k is the crystal momentum along the chain direction,
whathappenswhenweplaceseveralInSbnanowiresinparal- i.e.,thexaxis.
lelonorunderthes-wavesuperconductor.Thissetupwillbe The explicit representations of the time reversal (Θ), the
easilyrealizedusingrecentnanofabricationtechnology. particle-holereversal(Ξ), andthechiral(Π)operatorarede-
To simulate this system, we propose to investigate the finedby
HamiltoniandescribingaladderoftheKitaevchains,
Θ=K, Ξ=τ K, Π=τ , (5)
x x
H = µ c† c
− i,j i,j whereK representsthecomplexconjugateoperator. Wecan
i,j
X showtheHamiltoniansatisfiestherelation
(t c† c +t c† c +H.c.)
− x i,j i+1,j y i,j i,j+1
i,j ΞH(k)Ξ−1 = H( k), (6)
X
− −
− (dxc†i,jc†i+1,j +dyc†i,jc†i,j+1+H.c.), (1) ΠH(k)Π−1 =−H(k). (7)
i,j
X
WhenImd =0,theHamiltoniansatisfiestherelation
whereiandj arethelatticecoordinatesforthexandyaxes. y
The intrachain transfer integral tx and the intrachain super- ΘH(k)Θ−1 =H( k). (8)
conducting pairing amplitude d are present in the Kitaev −
x
model, while we have newly introduced the effective inter- Itiseasily seenthatΘ2 = Π2 = Ξ2 = 1. Then,theHamil-
chaintransferintegralt andthesuperconductingpairingam-
y tonian belongs to the class BDI, which is characterized by
plitudedy. the Z index. On the other hand, when Imd = 0, the time-
y
Physically, t andd will be derivedbythe couplingsbe- 6
y y reversalsymmetryisbroken,andtheclasschangestotheclass
tweentheInSbnanowiresviathesubstratesuperconductorby
DwhereonlyΞsymmetryispresent,beingcharacterizedby
thesecond-orderprocess. Thesuperconductingpairingphase theZ index.
2
maybecontrolledbyasuperconductingquantuminterference
Itis convenientto use the Majoranarepresentation,which
devide(SQUID)configuration.Thecontrollablephasediffer-
isconstructedwithuseoftheunitarytransformation,
encebetweend andd isthenewfeaturecomparedwiththe
x y
p+ip-wave pairing system. We shall soon see that this su- 1 I iI
perconductingpairingphasedifferencecausesthetopological U = √2 I iI , (9)
classchangebetweenBDIandD. (cid:18) − (cid:19)
Thereis an importantcommenton the coefficientsdx and whereI is the unitmatrix. IttransformsDirac fermionsinto
dy. Bymakingaphasetransformationofci,j,wecanchange Majoranafermions,
the overall phase of the superconducting pairings. Namely,
athbesoplhuatesephdaifsfeesreonfcdexbeatnwdedeynadrxeamnedandiyngislepshs.ysWiciatlh,obuuttlothses c†i,j = γiA,j +2iγiB,j, ci,j = γiA,j −2iγiB,j, (10)
3
obeying γiα,j,γiα′,′j′ = δi,i′δj,j′δα,α′. The unitary trans-
formedHnamiltonianoisgivenby
1 ∆+∆† 2iξ i ∆ ∆†
U†HU = − − .
2 2iξ i ∆ ∆† ∆+∆†
(cid:18)− − − − (cid:0) (cid:1)(cid:19)
(11)
(cid:0) (cid:1) (cid:0) (cid:1)
OuranalysisiscarriedonbasedonthisMajoranarepresenta-
tionoftheHamiltonian.
III. TOPOLOGICALPHASEDIAGRAMWITHOUT
INTERCHAINSUPERCONDUCTINGPAIRINGS
We firstinvestigatethecase ofd = 0forsimplicity. The
y
transformedHamiltonianiswrittenas
0 v
U†HU =i , (12)
vT 0
(cid:18)− (cid:19)
whichhasonlyoff-diagonalelements,with
1
v =ξ ∆ ∆†
− 2 −
=( µ (cid:0)2t cosk(cid:1) 2id sink)I t M, (13)
x x y
− − − −
whereMrepresentsa tridiagonalmatrixwhichonlyhasele- FIG.2: (Color online) Phase diagrams of the four-leg and five-leg
ments1at(n,n+1)and(n+1,n). Itisstraightforwardto lnaudmdebresr.sTinhetheenefirgguyreunrietpirsestexn,tatnhdewZeinhdavexe.taTkehnedZx2 =ind0e.x5.isTthhee
diagonalizethematrixv. Asweseeattheendofthissection,
parityofthenumber.
theeigenvaluesare
nπ
z = µ 2t cos 2t cosk 2id sink, (14) otherhand,alongthet axiswithµ=0,multipletopological
n y x x y
− − N +1 − − phase transitions occur: The band gap closes at t = 2tx,
whiletheeigenfunctionsare and the Z index is 0 for even N and 1 for odd Ny for±suzfnfi-
cientlylarget .
y
2 nmπ We show the energyspectrumof the ladderof chainsas a
Ψn = sin (n=1,2, ,N), (15) function of t for N = 4,5 in Fig. 3. We find that the Z
N +1 N +1 ··· y
r
indexgivesthe numberof MZES. The wave functionsof all
wheremrepresentstheposition(m=1,2, ,N). theMZESarelocalizedattheedges.
TheZindexW isdefinedbythechiralin·d·e·x The above behaviors can be understood as follows. The
systemisequivalenttoN independentKitaevchainswiththe
dk
W ∂ lndetv(k) renormalizedchemicalpotential:
k
≡ 2πi
Z N
dk dk
=tr ∂ lnv(k)= ∂ lnz (k), (16) H = H , (18)
k k n n
2πi 2πi
Z n Z n=1
X X
whereznaretheeigenvaluesofv. Itdescribesthesumofthe Hn =−µ′n c†i,nci,n
winding numbers of v(k) around the origin of the complex i
X
plane.TheZindexiscalculablebysubstituting(14)into(16). t c† c +d c† c† +H.c. , (19)
Weinvestigatethetopologicalphasediagraminthe(t ,µ) − x i,n i+1,n x i,n i+1,n
plane. Thephaseboundariesaredeterminedbydetv =y0. It Xi (cid:16) n (cid:17)
µ′ =µ+2t cos π. (20)
followsfrom(14)thattheyaregivenby n y N +1
nπ Each renormalized Kitaev chain has MZES when µ′ <
−µ−2tycosN +1 =±2tx. (17) 2t . The total number of the MZES is given by t|hen|sum
x
| |
oftherenormalizedKitaevchains.
ThetopologicalphasediagramisillustratedinFig. 2. Wesee
For the sake of completeness we show how to derive the
thefollowingcharacteristicfeatures: TheZindexisW = N
eigenvalues(14).Theeigenvaluesz canbeobtainedbysolv-
at (ty,µ) = (0,0). Along the µ axiswith ty = 0, the topo- ingthecharacteristicequationfortnM,
logical phase transition occurs from W = N to W = 0 at y
µ = 2t irrespectiveof the numberN of the legs. On the f (x)=det(xI t M)=0. (21)
x N y
± −
4
FIG.3: (Color online) Energyspectraof anN = 4,5ladder with FIG.4:(Coloronline)EnergyspectraofN =4,5ladderwithdy =
dy =0.Theenergyunitistx,andwehavetakenµ=0.7,dx =0.5. 0.3ty. Theenergyunitistx,andwehavetakenµ=0.7,dx =0.5.
Thenumberofzero-energystatesdecreasesby2ateveryphasetran- Notethatbothdxanddy arereal. Thenumberofzero-energystates
sitionasty increases,sinceeachzero-energystateperchaindoubly decreasesbytwoateverytopological phasetransitionwiththegap
degeneratesduetothetwoendpointsofthechain.Inthestrongcou- closingofthebulkstates,becauseeachzero-energystateperchain
plinglimitty → ∞,thereisno(one)MZESpereachedgeincase doubly degenerates due to the two endpoints of the chain. In the
ofN =4(5).WehavesetL=64. strong coupling limit ty → ∞, there is no (one) MZES per each
edgeincaseofN =4(5).WehavesetL=64.
To solve this we write down the recurrence relation of the
characteristicequation, Weproceedtoinvestigatetheeffectsofcomplexinterchain
superconductingpairings (Imd = 0). The Hamiltonian no
y
f (x)=xf (x) t2f (x), (22) longersatisfiesthetime-reversals6ymmetry(8). Thetopolog-
N+2 N+1 − y N
icalclass ofthe Hamiltonianchangesto the classD, and the
fromwhichitfollowsthat topologicalnumberis theZ index. Thesystem allowsonly
2
thetwocases,whetherornotapairofMajoranafermionsex-
[N/2]
x N n ist.
f (x)=tNU = ( 1)n − t2nxN−2n
N y N(cid:18)2ty(cid:19) nX=0 − (cid:18) n (cid:19) y (23) preWseencsehoowf dthye(Fenige.rg5y).spTehcterubmehaavsioarfoufncthtieonMoZfEtSy dinrasthtie-
cally changes from the system with d = 0 to the one with
with Chebyshev polynomials of the second kind. We now y
Imd = 0. When we change the phase of d and the class
solvetheeigenequationas y 6 y
changesfrom BDI to D, the numberof MZES reducesfrom
nπ even (odd) number to zero (one). This is due to the inter-
x =2t cos (n=1,2, ,N), (24)
n y N +1 ··· ferenceofMZESpairsduetotheinterchainsuperconducting
pairings. Then,onlyoneMZESpaircanexist,whichimplies
from which we find the eigenvalues (14) and the eigenfunc- that the topologicalnumber is the Z index. The numberof
2
tions(15)becauseznisconstantminusxn. MZESpairsisgivenbymod2N.
The Z index is obtained by the sign of the Pfaffian at
2
the time-reversal-invariantmomenta k = 0 and π. At these
IV. TOPOLOGICALPHASEDIAGRAMWITH points,sinced disappearsfrom∆definedby(4),itisconve-
x
INTERCHAINSUPERCONDUCTINGPAIRINGS nientto make a phasetransformationc eiθ/2c, so thatd
y
7→
becomesreal.Then,theMajoranarepresentedHamiltonianat
We nextinvestigatethecasedy = 0. We firstassumethat thesepointsisgivenby(12),i.e.,
6
itisrealaswellasd . Then,thesystembelongstotheclass
x
BDI,andthesystemischaracterizedbytheZindex.Weshow 0 v
U†HU =i ± (25)
theenergyspectruminFig. 4. vT 0
(cid:18)− ± (cid:19)
5
FIG. 5: (Color online) Energy spectra of N = 4,5 ladder with FIG. 6: (Color online) Energy spectra of N = 4,5 ladder with
|dy| < ty. The energy unit is tx, and we have taken µ = 0.7, |dy| > ty. The energy unit is tx, and we have taken µ = 0.7,
dx = 0.5,anddy = 0.3ity. Eachsubgapstatedoublydegenerates. dx =0.5,anddy =0.3ity. Eachsub-gapstatedoublydegenerates.
With Imdy 6= 0, MZES disappear in the parameter region where WhenN =4(5),thesystemisalwaystrivial(topological).Wehave
thereareevenMZESfordy =0,whiletwoMZESremaininthepa- setL=64.
rameterregionwherethereareoddMZESfordy =0. Inthestrong
coupling limitty → ∞, thereisno(one) MZESper eachedge in
caseofN =4(5).WehavesetL=64.
whichis nothingbut(17) with the replacementof t by t′ =
y
t2 d 2. Thephasediagramdependsononly d .
y−| y| | y|
butnowwith
qWeobtainthephasediagramasfollows. When t > d ,
y y
| | | |
it is simply given by Fig. 2 with the understanding that the
v =(ξ ∆)
± ±− horizontal axis is t′. Recall that the energy spectrum as a
=( µ 2t )δ (t + d )δ (t d )δ , y
− ± x i,j − y | y| i,j+1− y −| y| (i2+61),j function of ty is given by Fig. 5. On the other hand, when
d >t ,t′ becomespureimaginary,andthephasediagram
| y| y y
becomesverysimple: WhenN isodd,thegapclosingcondi-
whereξ =ξ(0)= µ 2t andξ =ξ(π)= µ+2t .
TheP−faffianissim−ply−givexnby + − x tionisgivenonlybyµ=±2tx. WhenN iseven,ontheother
hand, the system is trivial in the whole range. The energy
spectrumasafunctionoft isnowgivenbyFig. 6.
0 v y
Pf α = detv , (27)
vT 0 ± α
(cid:18)− α (cid:19)
with α = , since there are no diagonal elements. Then
Z -invariant±isthesignoftheproductofthePfaffiansinEq. V. ENERGYSPECTRUMASAFUNCTIONOFTHE
2
(27), i.e., detv detv . We note that when d is real, the SUPERCONDUCTINGPHASEDIFFERENCE
+ − y
Z -invariantis theparityofthe Z-invariant. The topological
2
phaseboundariesaredeterminedbydetv =0asbefore. It is interesting to investigate how the energy spectrum
Weobtaintherecursionrelationofthecharacteristicpoly-
changesduringthetopologicalclasschangebetweentheBDI
nomialfN =detvas class and the D class. Therefore, we show the energy spec-
trumasafunctionofθdefinedbyd = d eiθ inFig. 7. One
fN+2 =(−µ±2tx)fN+1− t2y −|dy|2 fN. (28) canclearlyseetheMZESisclearlyyseen|iny|thebulkbandgap.
Thisrelationisobtainedbyreplacin(cid:0)gt2 tot2 (cid:1) d 2 in(22). Here,wederivethesinθdependenceofthesubgaplevels. In
y y −| y| the Kitaev model, if we define the Majorana operatorsas in
Then,theconditionf =0yields
N Eq.(10),thezero-energyMajoranafermionontheleft(right)
nπ edge contains only γA (γB) as in Kitaev’s original paper11.
µ 2 t2 d 2cos = 2t , (29)
− y−| y| N +1 ± x Therefore, at low energy, the dy term in our Hamiltonian is
q
6
laddernumerically. IntheMatsumoto-Shibaformalism42,we
expressthe semi-infinite wire by the delta-functionpotential
withinfinitestrength.Then,theGreen’sfunctionofthesemi-
infinitewireGi,i′ isgivenby
Gi,i′ =G0i,i′ −G0i,0 G00,0 −1G00,i′, (33)
where G0 is the bulk Green’s f(cid:0)unctio(cid:1)n. The bulk Green’s
i,i′
functionisgivenanalyticallyink-space. Thenthereal-space
representationisobtainedbyperformingFouriertransforma-
tionnumerically:
G0i,i′ = 21π dkeik(i−i′)G0(k). (34)
Z
FIG.7: (Coloronline)Energyspectrumasafunctionofthesuper-
On the otherhand, the surfaceGreen’sfunctionof the semi-
conductingpairingphaseθ.Theenergyunitistx,andwehavetaken
infinitenormalleadcanbegivenanalytically:
µ=0,dx =0.5,ty =0.5,dy =0.25,andL=64. Therearefour
MZESinthecaseofN =4,whichareproportionaltosinθ.
A(k) E i (2t)2 E2 (E <2t)
J − − J | J|
g (E)=A(k)(cid:16)E + pE2 (2t)2 (cid:17) (E < 2t),
reducedto m,n A(k)(cid:16)EJ pEJ2 −(2t)2(cid:17) (EJ >−2t)
d eiθc† c† +H.c. J − J − J
| y| i,jii,dj+1  (cid:16) p (cid:17) (35)
= | y|cosθ γAγB +γBγA
− 2 i,j i,j+1 i,j i,j+1 where m and n are the chain labels; A(k) =
i|dy|sinθ(cid:0)γAγA γBγB (cid:1) (30) N2+121t2 ksin(mka)sin(nka); and EJ = E + µ −
− 2 i,j i,j+1− i,j i,j+1 2tycoska, where t(> 0) and µ are the hopping and the
P
chemicalpotentialsoftheleads.
i|dy|sinθ(cid:0)γAγA (i leftha(cid:1)lf)
− 2 i,j i,j+1 ∈ . (31) Then,weconstructthe Green’sfunctionofthewholesys-
≈(+i|d2y|sinθγiB,jγiB,j+1 (i∈righthalf) tembythe recursionrelations. Expressingthe Green’sfunc-
tion of the left(right) semi-infinite wire as G (G ) and the
L R
Namely,theMajoranafermionsatchainj andj+1arecou- Green’sfunctionofthewholesystemasG,weobtain
pledbytheimaginarypartofthe interchainpairing,andthis
causesthesinθdependenceoftheenergy: G−1 =g−1 H G H , (36)
L,i,i i − i,i−1 L,i−1,i−1 i−1,i
G−1 =g−1 H G H , (37)
E dy sinθ. (32) R,i,i i − i,i+1 R,i+1,i+1 i+1,i
∝| |
G−1 =g−1 H G H
Inotherwords,breakingofthetime-reversalsymmetrybythe i,i i − i,i−1 L,i−1,i−1 i−1,i
H G H , (38)
imaginary part of the interchain coupling correspondsto the i,i+1 R,i+1,i+1 i+1,i
−
creationofthecouplingbetweenMajoranafermions. Bynu- G =G H G , (39)
i,i+1 i,i i,i+1 R,i+1,i+1
mericallydiagonalizingtheHamiltonian(1),weobtaintheen- G =G H G , (40)
i+1,i i+1,i+1 i+1,i L,i,i
ergyspectrumasafunctionofθford /t =0.5asshownin
x x g−1 =E H , (41)
Fig.7. Thesubgaplevelsareingoodagreementwiththesinu- i − i
soidalbehavior.Wehavethusdiscussedthesinθdependence
ofthesubgaplevels. AneffectiveHamiltonianonthesubgap whereEistheenergy,Hiistheon-siteHamiltonian,andHi,i′
isthehoppingbetweensitesi,i′.Ontheotherhand,weobtain
states will be derived in a manner similar to that in Ref.15.
theretardedGreen’sfunctionGRbyreplacingEwithE+iε,
Wenotethatthetopologicalclasschangewithoutgap-closing
whereεisaninfinitesimalpositivenumber.
occurswhenwechangeθ32.
Next,wecalculatethedifferentialconductancebytheLee-
Fisherformula35,37:
VI. TRANSPORTPROPERTY 2e2
G= Tr[P (G′′ G′′ +G′′ G′′
h e i,i+1 i,i+1 i+1,i i+1,i
It is well known33,34 that the local Andreevreflection rate G′′ G′′ G′′ G′′ )], (42)
− i,i i+1,i+1− i+1,i+1 i,i
atzerobiasis1inthepresenceofaMajoranafermion,while
the zero-energy Majorana bound state gives the differential whereG′′ = ImGR ,P istheprojectionoperatorontothe
i,i′ i,i′ e
conductance2e2/h. We havecalculatedthedifferentialcon- particlesubspace. We chooseanarbitraryiinthenormalre-
ductanceoftheNSjunction(Fig. 1),followingRefs.35–40. By gionduetocurrentconservation.
employingthe recursiveGreen’sfunctionmethod,we obtain The results for the differential conductance G at zero en-
the surface Green’s function41–43 of the semi-infinite Kitaev ergyareillustratedinFigs.8and9. We haveinvestigatedthe
7
FIG.8:(Coloronline)DifferentialconductanceoftheNSjunctionin FIG.9: (Coloronline) Differentialconductance oftheNSjunction
unitsof2e2/hwithrealdy whenN = 4,5. Thesystembelongsto inunitsof 2e2/hwithimaginarydy when N = 4,5. Thesystem
theclassBDI.Theenergyunitistx,andwehavetakendx =0.5and belongstotheclassD.Theenergyunitistx,andwehavetakendx =
dy =0.3ty.WhitelinesaretopologicalphaseboundariesinFig.2. 0.5anddy = 0.3ity. Whitelinesaretopologicalphaseboundaries
inFig.2.
cases with real d (class BDI) and with imaginary d (class
y y
D).In bothoftheclasses BDI andD, the figuresofthe con-
ductancestronglyresemblethephasediagrams(Fig. 2). The
brightregionswherethedifferentialconductanceisquantized
correspond to the topological regions. This means that the
localAndreevreflectionrate is 1 with each MZES. Itagrees
with the previousstudy33,44,45 that the Majorana fermion in-
ducestheresonantAndreevreflection.
We havecalculatedthedifferentialconductanceasafunc-
tionofthesuperconductingpairingphaseθandtheenergyof
injected electrons, which we show in Fig. 10. The behavior
reflects the behaviorof the energyspectrumgivenin Fig. 7. FIG.10: (Coloronline)DifferentialconductanceoftheNSjunction
Moreover,thedifferentialconductancepermodeisquantized inunitsof2e2/hasafunctionofthesuperconductingpairingphase
to2e2/hwhenθ = 0,π. ThisisbecauseoftheresonantAn- θandtheenergy. Theenergyunitistx,andwehavetakenµ = 0,
dreevreflectioninducedbyMajoranafermions. Furthermore, dx = 0.5, ty = 0.5, anddy = 0.25. TherearefourMZESinthe
the peak decays rapidly when the phase is switched on and caseofN =4.
therearenoMajoranafermions.
WehavealsocalculatedthecrossedAndreevreflectionrate
VII. CONCLUSIONANDDISCUSSION
in the NSN junction,wherethe two Majoranafermionscou-
ple.ThecouplingstrengthbetweenMajoranafermionsattwo
edges is on the order of exp( L/ξ), where ξ is the super- MotivatedbytheInSbnanowiresuperconductorsystem,we
−
conductingcoherencelength. When ξ L, since there are have investigated Kitaev-ladder topological superconductors
≪
nooverlapsbetweentheMZESlocalizedattherightandleft based on the effective Hamiltonian (1). Especially, we have
edges, thecrossedAndreevreflectionrate isquitesmall. On revealedthetopologicalphasetransitionbychangingthesu-
theotherhand,whenξ >L,therearesomeoverlapsbetween perconductingpairing phase difference between the x and y
theMZESlocalizedattherightandleftedges,andthecrossed directions(i.e.,thephaseofd /d ). Thisistheproblemnot
y x
Andreevreflectionrateremainsfinite. addressedinthepreviousstudies12–21onthep+ip-wavepair-
8
ingsystem. Wehavefoundthatthepairingphaseplaysacru- Noteaddedinproof. We becameawareofthenicepapers
cial role to determine the topological class. If the phase is onasimilarsystem50–52
0 or π, the system belongs to the class BDI with n MZES
(n = 0,1,2,...); otherwise it belongs to the class D with n
MZES(n = 0,1). Thephasegradientofthe bulksupercon-
ductor may be controlled by forming the SQUID configura-
Acknowledgements
tion. The proximity-inducedpairing in the wires also gets a
phasegradient,yieldinganimaginaryparttod ,andtheclass
y
changestoD.Thistopologicalclasschangecanbeobserved WeareverygratefultoY.Tanakaformanyhelpfuldiscus-
by differential conductance measurement. We note that our sions on the subject. This work was supported in part by
systemismappedtotheKitaev-spin-laddersystem47,48inthe Grants-in-Aid for Scientific Research from the Ministry of
caseofd = 0. Itmaybepossiblethatourmodelisrealized Education, Science, Sports and Culture No. 24224009 and
y
byusingultra-coldatomicsystems49. No. 25400317.
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