Exactly solvable spin chain models corresponding to BDI class of topological superconductors

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by  S. A. Jafari

ExactlysolvablespinchainmodelscorrespondingtoBDIclassoftopologicalsuperconductors
S. A. Jafari1,2,3,∗ and Farhad Shahbazi4,3,†
1Department of Physics, Sharif University of Technology, Tehran 11155-9161, Iran
2Center of excellence for Complex Systems and Condensed Matter (CSCM),
Sharif University of Technology, Tehran 1458889694, Iran
3SchoolofPhysics, InstituteforResearchinFundamentalSciences, Tehran19395-5531, Iran
4Department of Physics, Isfahan University of Technology, Isfahan 84156, Iran
WepresentanexactlysolvableextensionofthequantumXYchainwithlongerrangemulti-spininteractions.
TopologicalphasetransitionsofthemodelareclassifiedintermsofthenumberofMajoranazeromodes,n
M
which are in turn related to an integer winding number, n . The present class of exactly solvable models
W
belongtotheBDIclassintheAltland-Zirnbauerclassificationoftopologicalsuperconductors. Weshowthat
6 timereversal(TR)symmetryofthespinvariablestranslatesintoaslidingparticle-hole(PH)transformationin
1 thelanguageofJordan-Wigner(JW)fermions–aPHtransformationfollowedbyaπshiftinthewavevector
0 (πPH).PresenceofπPHsymmetryrestrictsthen (n )ofTRsymmetricextensionsofXYtoodd(even)
W M
2 integers.TheπPHoperatormayserveinfurtherdetailedclassificationoftopologicalsuperconductorsinhigher
r dimensionsaswell.
p
A PACSnumbers: 75.10.Pq,74.20.Rp,03.65.Vf

7
The spectrum of Quantum XY model is exhausted by the where φ is the phase string defined as φ = π(cid:80) c†c
j j (cid:96)<j (cid:96) (cid:96)
] emergent JW fermions1,2. The anisotropy of the exchange convertstheaboveHamiltonianto,
l
e coupling generates p-wave superconducting pairing between (cid:88) (cid:88)
r- spinless JW fermions leading to unpaired Majorana fermion H =2 (Jsc†jcj+s+λscjcj+s+h.c.), (4)
t (MF)atendsofanopenchain3. AddingfurtherneighborXY s=1,r j
s
. couplings in general spoils the exact solvability because the
t wherelongerrangeexchangeandanisotropyparameterJs,λs
a JWtransformationincorporatesappropriate(non-local)phase
give rise to hopping and pairing between s’th neighbors, re-
m strings in order to fulfill anti-commutation algebra2. In the spectively. Note the important role played by σz phases is
- contextoftheIsinginatransversefield(ITF)modelitwasre- to cancel the unwanted JW phases which renders the nXY
d
centlyshownthataddingappropriatelyengineeredthree-spin
n Hamiltoniantothequadraticform(4).
interactionscanstillleaveitexactlysolvable4. GiventhatITF
o Foreven(odd)r thegeneralizedterminvolvesn = r+1
c and XY model are related by a duality transformation5,6 we spinswhichwillbeodd(even)undertheTR.Letusnowfigure
[ expect similar extensions to work for the XY model. In this
outhowdoestheJWdictionarytranslatetheTRoperationof

3 letterweclassifygeneralizationsoftheXYmodelwitharbi- spinvariables. TheTRchangesthesignofspinoperators(cid:126)σj.
v traryn-spininteractionsintermsofaπPHsymmetrythatisa Sign reversal of σz with (cid:96) < j implies that under TR the
6 PHtransformationfollowedbyasignalternationinonesub- non-localphasestri(cid:96)ngistransformedaseiφj →(−1)j−1eiφj.
5 lattice. We show that in presence of πPH corresponding to ThereforetheTRofspinsforJWfermionstranslatesto,
4 everyMFtherewillbeapartnerMFwhichwillcorrespond-
2 inglyrestrictthepossiblewindingnumbers. Letusstartwith c →(−1)j−1c† ⇔c →−c† (5)
0 j j k π−k
theXYHamiltonian,
. whichisnothingbutthePHtransformationfollowedaπshift
1 (cid:88) (cid:88)
0 HXY = (J1+λ1)σjxσjx+1+ (J1−λ1)σjyσjy+1, (1) inthek-space–orslidingPH–thatwillbedenotedbyπPHin
6 j j thispaper.TheπPHcanfurtherresolvethetopologicalclassi-
1 ficationoftheAltland-Zirnbauer(AZ)classificationoftopo-
towhichweaddan-spininteraction,
: logicalsuperconductors7 thatarebasedonPH,TR,andsub-
v
i (cid:88) (cid:32)r(cid:89)−1 (cid:33) lattice symmetries8–11. In the following we discuss in detail
X HnXY =HXY+ (Jr+ηaλr)σja σjz+k σja+r, (2) twoprototypicalcasescorrespondingtolongerrangeinterac-
r j,a k=1 tionwithranger =2,3involvingn=3,4spins,respectively.
a
where a = x,y and η = −η = 1. Here r = n−1 de- Thenwepresentgeneralargumentsforarbitraryranddiscuss
x y
notes the range of n-spin interaction. J is the longer range aneven-odddichotomyrelatedtotheπPHtransformation.
r
exchangeandλ denotesthelongerrangeXYanisotropy.For 3XY Model. In this case the k-space representation of the
r
thisHamiltonian(nXYmodel)thequantityq =(cid:81)N σz isa JWHamiltonianis,
(cid:96)=1 (cid:96)
constantofmotion. Twopossibleq = ±1valuescorrespond H =(cid:88)c†(cid:126)h ·(cid:126)τ c , c† =(c†,c ) (6)
tonumberparityofJWfermionsandhencetheabovegeneral- k k k k k −k
k
izationisexpectedtogiveasuperconductingsystem. Indeed
theJWtransformation2, whereτa, a=x,y,zstandsforPaulimatricesintheNambu
spaceand(cid:126)h(k)=ε zˆ+∆ yˆwith
σz =1−2c†c , σx =eiφj(c +c†), σy =−ieiφj(c −c†), k k
j j j j j j j j j
(3) ε =J cosk+J cos2k, ∆ =−λ sink−λ sin2k. (7)
k 1 2 k 1 2
2
FIG.2. (Coloronline)Wavefunctionandwindingpatternforrepre-
sentativepointsinregionsvariousphasesofFig.1.Theredandblue
curvesrepresentthecoherencefactorsu andv asafunctionofk
k k
andthegreencurvecorrespondstotheφ /π. Blackarrowsareunit
FIG.1. (Coloronline)Thephasediagramof3XYmodelinparam- k
vectorsconstructedfromAndersonpseudovector(∆ ,ε )atevery
eter space. The phase boundary curves correspond to gap closing k k
kinthe1BZ.Thecoordinate(λ /λ ,J /J )isshowninthetopleft
separatingtopologicallydistinctphasescharacterizedbyawinding 2 1 2 1
ofeachpanel.Theresultingn isusedtolabelregionsofFig.1.
numbern asindicatedinthefigure. W
W
the other phases with n (cid:54)= 0 are separated from the triv-
W
Note that pairing with (hopping to) a neighbor at distance
ially gapped phase by a gap closing. The panel (b) of Fig. 2
r = 2 has added a λ sin2k (J cos2k) term to the Ander-
2 2 corresponds to λ = J = 0 where λ (cid:54)= 0 is adiabatically
sonpseudovector(cid:126)hrepresentationoftheHamiltonianmatrix. connected to the2Ising l2imit λ = 1. T1his is why the phase
1
This general feature holds for any r. The eigenvalues and boundary(9)isdeterminedbytheratioλ /λ .
2 1
eigenvectorsof(cid:126)h(k)aregivenby, Let us show that for any r the nXY model falls into BDI
class8 which in turn allows for winding number classifica-
E =±(cid:113)ε2 +∆2, |ψ−(cid:105)=(cid:18)iuk(cid:19),|ψ+(cid:105)=(cid:18)vk(cid:19), (8) tion. First let us check the TR symmetry. For spinless
k k k k vk k iuk fermions TR operator T is simply a complex conjugation.
Using the Nambu space representation, Eq. (6), we have
where the coherence factors are parameterized in terms of a T(cid:126)h(k)T =(cid:126)h(−k)whichrepresentstheTRsymmetryofthe
phase φ = tan−1(∆ /(ε +E )) as u = sin(φ /2) and
k k k k k k JWHamiltonian. NowdefiningtheoperatorC = τ K asthe
x
vk = cos(φk/2). TheJWHamiltonian(6)isdiagonalizedin PH transformation, one finds C(cid:126)h(k)C = −(cid:126)h(k) that checks
termsofBogolonsγ† =−iu c† +v c .
k k k k −k the PH symmetry. Finally for the chiral symmetry we have,
Theboundariesofthephasediagramofthe3XYmodelcan (cid:126)h(k)CT = −CT(cid:126)h(k). Let us emphasize that for every r,
be analytically calculated by investigating the gap closing of
only sin(rk) functions appear in the pairing term and hence
the spectrum (8) that happens when both ε and ∆ vanish
k k the above properties that rest on odd parity of ∆ apply to
whichgivesfollowingequationsforthephaseboundaries, k
nXY model. Since for the present spinless JW fermions one
(cid:12) (cid:12) has C2 = T2 = +1, the nXY belongs to BDI class in the
J2 =±1, or J2 = (λ2/λ1) for (cid:12)(cid:12)λ2(cid:12)(cid:12)≥1/2, (9) AZ classification of topological superconductors and hence
J J 1−2(λ /λ )2 (cid:12)λ (cid:12)
1 1 2 1 1 allowsforinteger(windingnumber)classificationofthetopo-
logical phases. However both range of possible integers and
which has been plotted in Fig. 1. It is remarkable that the
whethertheyareevenoroddisdeterminedbyrwhichinturn
phaseboundaryisgivenintermsofratiosJ /J andλ /λ .
2 1 2 1 isconnectedtothepresenceorabsenceoftheslidingPHsym-
Thisproperty isalso generalandthe phasediagram isdeter-
metry,πPH.Tosetthestagesfordiscussionofarbitraryr,let
minedonlyintermsoftheratiosJ /J andλ /λ . InFig.1
r 1 r 1 usconsiderthenextprototypicalcaseof4-spininteractions.
each region is labeled by a winding number n . This topo-
W 4XY Model. The four-spin term preserves the TR symme-
logical invariant corresponds to the number of times the unit
try of the spin model. The components of Anderson pseu-
circleiscoveredbythevector(∆ ,ε )askvariesacrossthe
k k dovector are given by, ε = J cos(k) + J cos(3k) and
firstBrillouinzone(1BZ)12. Thesevectorsarerepresentedby k 1 2
∆ = λ sin(k) + λ sin(3k). The phase boundaries will
blackarrowsinFig.2whichcorrespondtogreencurverepre- k 1 2
be given by setting ε = ∆ = 0. Therefore the gap
senting φ /π. The winding pattern of arrows and the global k k
k closing curves of this model are the lines λ /λ = 1 and
variation profile of φ does not change as long as the phase 2 1
k J /J =−1,andthecurve
boundariesofFig.1arenotcrossed. Thismeansthatn isa 2 1
W
topologicalinvariant13. Forthe3XYmodelfivepossibleval- λ J J λ /λ
ues n = 0,±1,±2 can be extracted from analysis similar 2 =1, or 2 =−1, or 2 = 2 1 , (10)
W λ J J 1+2λ /λ
1 1 1 2 1
toFig.2. Theresultingn valuesareusedtolabelregionsof
W
Fig.1. Thephasewithn = 0isadiabaticallyconnectedto for λ /λ ≥ 1 and λ /λ ≤ −1/3. These curves partition
W 2 1 2 1
the trivially gapped phase that can be reached by an applied the the parameter space (λ /λ ,J /J ) into seven regions
2 1 2 1
field h → ±∞ that couples to σz without gap closing. This representedintherightpanelofFig.3eachcharacterizedby
makesthen = 0regiontopologicallytrivial. Thisiswhile n = ±1,±3. Theleftpanelofthisfigurerepresentssame
W W
3
ToproceedfurtherletusfirstelucidatethemeaningofπPH
inthelanguageofMFs:IfwerepresenttheJW”electron”and
”hole” operators as c† = a +ib and h = a(cid:48) −ib(cid:48) and if
j j j j j j
wesearchforMFswithvanishingb,b(cid:48)component,thenevery
zeromodesolution(A ,0,A ,0,...)ismappedbyπPHtoa
1 2
partnerMF(A(cid:48),0,A(cid:48),0,...)withA(cid:48) =−(−1)jA .
1 2 j j
For even values of r where odd number of spin variables
are added to the XY model, consider any point in the phase
diagram (see left panel of Fig. 3) with given number n
M
of MFs. Obviously the generalized r + 1-spin interaction
breaks TR symmetry. In this case the overall minus arising
FIG. 3. (Color online) Color map of the number of MFs, n in from TR transformation can be absorbed by the transforma-
M
theparameterspacefortwoextensions3XY(left)and4XY(right) tion (λ ,J ) → −(λ ,J ). This means that for every MF
r r r r
correspondingtospinclustersofranger = 2,3,respectively. The atpoint(λ ,J ,λ ,J )ofthephasediagram,itspartnerMF
1 1 r r
4XYmodelhasπPHsymmetryinJWrepresentationwhichrestricts corresponds to point (λ ,J ,−λ ,−J ). This explains the
1 1 r r
nM toevenvaluesandnW tooddvaluesonly.Obviouslytheorigin inversion symmetry in the left panel of Fig. 3. This can also
inbothcasescorrespondtor=1whichisnotcontinouslyconnected
beseenfromEq.(12)thatmapstoitselfundersimultaneous
totopologicallydistinctcasesofr=2(left)norr=3(right).
changeofx→−xand(λ ,J )→−(λ ,J ).Foroddvalues
r r r r
of r, there are even number of spins giving a TR symmetric
termandhencetheTRoperationdoesnotproduceanyminus
setofdatafor3XYmodel. Inbothr = 2,3cases|n | ≤ r
W
signinthen-spinterm. ThereforecorrespondingtoeveryMF
whileforoddronlyoddvaluesofn arepicked. Toexplain
W
atanypoint(λ ,J ,λ ,J ),thepartnerMFalsoexistsatthe
themeaningofthecolorcodeinthisfigure,letusdiscussthe 1 1 r r
samepointintheparameterspace. Thiscanalsobeseendi-
numbern ofMFsforageneralr.
M
rectlyfromEq.(12): AlthoughinngeneralEq.(12)admits2r
Majoranaendmodes. Tofurtherunderstandtheproperties
solutionssuchthatthenumbern ofthemsatisfying|x|<1
of nXY model, let us now consider an open nXY chain and M
is0 ≤ n ≤ 2r. However,foroddr thisequationbecomes
discuss the Majorana zero modes of the chain. Presence of M
MFsrequiresequalspin(orspinlesspairing)12,14 whichengi- anequationofdegreerintermsofX =x2whichimpliesthat
neeredinone15 ortwo16 dimensions. InthecaseofXYmod- thesolutionsalwayscomeinpairs±xgivingpartnerMFsas
elsthespinlesspairingemerges3. LetusnowseehowdoMFs Aj ∼ (±x)j. This explains why in the right panel of Fig. 3
appearinnXYmodel. IntermsofMFsa = c +c†, b = only colors corresponding to even nM appear. The presence
j j j j of πPH for odd r implies that corresponding to every MF at
i(cj −c†j)thenXYmodelbecomes, thechainend,itspartnerobtainedbysignalternationinone-
sublatticeisalsoacceptablesolutionandhencen isalways
(cid:88) (cid:88) M
H =i (Js+λs)ajbj+s+(λs−Js)bjaj+s. (11) even. NowletusdiscusshowdoesπPHrestrictnW.
s=1,r j Consider an arbitrary point in the phase diagram corre-
We search for MFs of type a localized near the origin spondingtopairingamplitudes{λs}. SincetheresultingJW
Hamiltonian(4)isTRsymmetric,theactionofTRissimply
for the nXY model, i.e. zero-energy states of the form
(A ,0,A ,...,A ,0)withA ∼xj whichgives, i→−iwhichcanbeabsorbedby{λs}→{−λs}. Thelater
1 2 N j
onotherhandamountstochangingthesingofthehorizontal
(J +λ)+(J +λ)xr−1−(λ −J)xr+1−(λ −J)x2r=0. (12) component∆ oftheAndersonpseudovector. Thereforeitis
r r 1 1 1 1 r r k
theimplicationofTRsymmetryofJWHamiltonianthatcor-
If we searcher for solutions of type (0,B1,...,0,BN), we respondingtoeverywindingnumbernW, thereisawinding
wouldobtainasimilarequationbutwithx→x−1. Tohavea number−n irrespectiveofwhetherr isevenorodd. Now
W
normalizableMajoranazeromodeoftypeaweneedsolutions let us argue that the possible values of n are bounded by
W
thatsatisfy|x|<1. Fig.3representsacolormapofthenum- therangerofinteraction. Everytime∆ vanishes,thepseu-
k
ber nM of MFs of type a localized in one end for r = 2,3. dovectorpointsverticallyeithertothesouthornorthpole.For
Thefirstthingtonoteisthatthephaseboundariesinthe3XY the nXY model the maximum number of the zeros of ∆ in
k
modelobtainedfromtheMFcountinganalysispreciselycoin- the1BZproducedbycombinationfirstandr’thharmonicsof
cideswiththatinFig.1. Thisisalsotrueforthe4XYmodel, sinfunctionisrwhichimplies|n |≤r.
W
andtheboundariesgivenbyEq.(10)preciselycoincidewith
Toseewhenthemaximumnumberofzerosin∆ arereal-
k
thatintherightpanelofFig.3. Thecolorcodeineachpanel
ized,itisenoughtoconsiderthelimitJ ∼ rλ (cid:29) λ ∼ J
r r 1 1
ofthisfigureindicatesthenumberofMFsoftypeaboundto
where there are 2r +1 zeros for ∆ in the 1BZ which give
k
left end. For each panel we have explicitly indicated n . It
W maximal winding |n | = r. In this limit the secular equa-
W
canobservedthatinbothcases, tion (12) reduces to x−2r = (r − 1)/(r + 1) the absolute
value of which is always less than unity for every r > 0
n =n −r (13)
W M
which realizes maximum number of MFs equal to 2r. This
wherer istherangeofinteraction. Letuspresentaheuristic meansthatthemaximumvaluesofn andn happeninthe
W M
argumentthattheaboveformulaholdsforanyr. same limit. Now the point with minimum n is the TR of
W
4
the maximal n . The TR for JW Hamiltonian is equivalent latedbyA ↔−(−1)jA .
W j j
to{λ } → {−λ }whichisaπ/2rotationaroundthez-axis It can be noticed that the Z index defined by ν =
s s 2
forspinvariablesandtheoperationa↔bforMFswhiches- sign(ε )sign(ε )12, gives ν = −1 when r is odd. How-
0 π
sentiallyexchangesx ↔ x−1 andhencemappingeverystate ever,whenriseven,ν = sign(|J /J |−1). Forevenr,the
r 1
withn MFstoastatewith2r−n MFs. Thereforeboth ν = +1(−1) corresponds to even (odd) values of n . The
M M W
upperandlowerboundsof2r+1possibleintegers|n |≤r physicalinterpretationofν isasfollows: Whenν =+1(−1)
W
and0 ≤ n ≤ 2r describethesamephysicalstate. Finally, theidentityofBogolonsdoesnotchange(changes)fromhole-
M
since both n and n are unique topological labels of the like to particle-like when k spans the range [0,π]. The pres-
M W
same state, the mapping between the two sets must be one- enceofπPHsymmetryforoddrguaranteesthattheaboveZ
2
to-one. Sincetheendsoftwochainsofintegersmaptoeach index takes only one value −1 which means that the charge
other,weheuristicallyexpectEq.(13)toholdforanyr. character of Bogolons at k = 0 and k = π are opposite.
BreakingπPHallowsforboth±1values.
Now let us discuss why for odd values of r = 2p+1 the
Summary. To summarize, we have presented an exactly
n is always odd. The n changes by half between each
W W solvable extension of the quantum XY model that involves
twoconsecutivezerosof∆ =λ sink+λ sin((2p+1)k).
k 1 r clusters of n = r +1 spins interacting at range r. The en-
Supposethatthisgapfunctionvanishesatsomepointk inthe
∗ suing JW representation is a topological superconductor in
1BZ.ByπPHsymmetry,relation(5)italsovanishesatπ−k .
∗ BDI class. We showed that the TR operation of original
The sign of the vertical component ε at k and π −k are
k ∗ ∗ spinvariablestranslatestoaslidingPHtransformationofJW
oppositeasthecosfunctionsappearinginverticalcomponent
fermions,πPH.ThepresenceofπPHimpliesthatcorrespond-
ε ofAndersonpseudovectorchangesignupongoingfromk
k ∗ ing to every MF wave function A , there is a partner MF
toπ−k whenrisodd. j
∗ whosewavefunctionis−(−1)jA whichinturnrestrictsthe
j
NowstartingfromtheXYmodel(p=0)andfocusingonly number of MFs to even values only. The πPH also implies
intherighthalfof1BZwithk > 0,atk = 0andk = π the thattherootsofpairingpotentialcomeinpairswhichrestricts
Andersonvectorspointtonorthandsouthpolesrespectively the Z winding numbers to odd integers only. The Bott pe-
(Fig.2-b)whichmeansthatintherighthalfof1BZonepicks riodicity11 implies that there should exist similar restriction
upahalf-integerwinding. Foreveryk ifthewindingvector in higher dimensions on topological invariants when πPH is
∗
points to some pole the one at π −k will point to opposite a symmetry. It will be interesting to study possible higher
∗
pole, corresponding to every pair of roots k ,π−k of ∆ , dimensional models with πPH symmetry in electronic sys-
∗ ∗ k
an integer winding is inserted to the right half of 1BZ. This tems16,18,19.Thenumbern ofMFsleavesauniquesignature
M
means that always half-integer windings are possible in the intunnelingexperimentsandhenceremainsdirectlyaccessi-
righthalfof1BZ.Thereforethewindingnumberpickedover bletoexperiments. Arrayofmagneticnano-particlesonasu-
the whole 1BZ is an odd number. That is why in the right perconductorisdescribedbyaneffectivetheorythatincludes
panelofFig.3weonlyhaveoddwindingnumbers. Thefact r = 2hoppingbetweenthespinlessfermions10,20 whichmay
that for odd r, only even number of MFs and only odd n s serveaspotentialplatformtomaterialize3XYmodel.
W
arepossibleisconsistentwithEq.(13). Thislineofreasoning Acknowledgement. We thank S. Moghimi Araghi and A.
impliesthatinthesimplecasep=0correspondingtoXY(or T. Rezakhani useful discussions. We are grateful to C. W. J.
Kitaev)chain,byπPHsymmetrythetopologicallynon-trivial Beenakker,AlexanderAltland,MehdiKargarianandAnthony
phasealwayshoststwoindependentMajoranaendmodesre- Leggettforinsightfulcommentsandcommunictations.
∗ [email protected] arxiv:1505.03535.
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