An exactly soluble model with tunable p-wave paired fermion ground states

0.15 MB·English

by  Yue Yu

Anexactly solublemodel withtunable p-wavepaired fermionground states
Yue Yu1 and Ziqiang Wang2
1InstituteofTheoreticalPhysics, ChineseAcademyofSciences, P.O.Box2735, Beijing100080, China
2Department of Physics, Boston College, Chestnut Hill, MA 02467
(Dated:January16,2009)
MotivatedbytheworkofKitaev,weconstructanexactlysolublespin-1 modelonhoneycomblatticewhose
2
groundstatesareidenticalto∆1xpx+∆1ypy+i(∆2xpx+∆2ypy)-wavepairedfermionsonsquarelattice,
with tunable paring order parameters. We derive a universal phase diagram for this general p-wave theory
whichcontainsagappedAphaseandatopologicallynon-trivialBphase. Weshowthatthegaplesscondition
09 Gin-itnhveerBsiopnhassyemimsegtorivcergnaepdlebsys Ba gpehnaesrealnizeeadr tihneveprhsiaosne (bGou-inndvaerriseisonis) dsyemscmribeetrdybuynd1e+r1p-dxim↔ensi∆∆o11nxyalpyg.apTlehses
0 Majorana fermions in the asymptotic long wave length limit, i.e. the c = 1/2 conformal field theory. The
2 gapped B phase has G-inversion symmetry breaking and isthe weak pairing phase described by theMoore-
n ReadPfaffian. WeshowthatinthegappedBphase,vortexpairexcitationsareseparatedfromthegroundstate
a byafiniteenergygap.
J

6
1 I. INTRODUCTION the Kitaev modelis a specialcase ofa broaderclass of two-
dimensional spin-1 models whose ground states are equiva-
2
] lent to general paired fermion states in the p-wave channel.
l The low energy excitations of a topologically non-trivial
e Indeed, the vortex-free Kitaev Hamiltonian maps to an ex-
phase have remarkable properties. A well-known example
- actBCSfermionpairingmodelwithi(p +p )-waveattrac-
r is the quasihole excitation of the Laughlin state in the frac- x y
t tions on a square lattice9. Our generalized model includes
s tional quantum Hall effect (FQHE), which carries fractional
t. charge and anyon statistics. The most intriguing possibil- both the px + ipy wave paired states and the original Ki-
a taev model as special limits. It is an exactly soluble model
ity of a topologicalphase of matter is the nonabelianFQHE
m with minimalthree andfour-spininteractions. We show that
proposed by Moore and Read1 for even-denominator filling
- factors, e.g. ν = 52. The quasiparticle excitations, vor- the vortex-free ground states of this model are described by
d 2 ∆ p +∆ p +i(∆ p +∆ p )-wave pairedfermion
n tices in the Moore-Read Pfaffian wave function, have non- 1x x 1y y 2x x 2y y
stateswithtunablepairingorderparameters∆ onasquare
o abelianstatistics1whichplaysafundamentalroleintopologi- ab
lattice. Wefindthatthestructureofthephasediagramisde-
c calquantumcomputation3,4. ThekeyingredientintheMoore-
[ ReadPfaffianstateisthatthetopologicallynontrivialpartof termined by the geometry of the underlying Fermi surface.
3 the wave function is asymptomatically the same as the pair It contains both topologically trivial (A) and nontrivial (B)
phases. The A phase is always gapped and corresponds to
v wavefunctioninap +ip -wavefermionpairedstate6inthe
x y the strong pairing phase. The B phase can be either gapped
1 weakpairingphase7. Theexistenceoftheexoticnon-abelian
or gapless even if T-symmetry is broken. We find that gap-
3 statisticsisthuslikelyagenericpropertyofthemoretangible
6 less excitations in the B phase is protected by a generalized
time-reversalsymmetry(T-symmetry)breakingp-wave pair-
0 inversion(G-inversion)symmetryunderp ↔ ∆1yp andthe
8. ingRsetcaetenst.ly, Kitaev constructed a spin-1 model with link- emergence of a gapped B phase is thusxtied t∆o1Gx-iynversion
0 2 symmetry breaking. For instance, the p +ip wave paired
dependent Ising couplings on the honeycomb lattice4. Ki- x y
7 state is gapped while p + ip -wave paired state is gapless
y y
0 taev showed that the modelis equivalentto a bilinear Majo- althoughtheybothbreakthe T-symmetry. Thecriticalstates
: rana fermionmodeland is thusexactly soluble. A topologi-
v of the A-B phase transition remains gapless whether or not
calnon-trivialgaplessphase(theBphase)wasdiscovered.(A
i T- and G-inversion symmetries are broken, indicative of its
X Jordan-Wignertransformationtoamodelwithtwo-Majorana
topological nature. Indeed, if all ∆ are tuned to zero, the
r fermionsforthismodelhasbeenproposedin5). In thepres- topologicalA-Bphasetransitionisfraobmabandinsulatortoa
a
ence of a T-symmetry breaking term, the B phase becomes
free Fermigas. The Fermi surface shrinksto a pointzero at
gapped and exhibits vortex excitations obeying nonabelian
criticality.
statistics. The modelalso hasa topologicallytrivial, gapped
Aphase. Thetwophasesareseparatedbyatopologicalphase We show that the gapped B phase is a weak pairing state
transitionviaagaplesscriticalstate. Thesepropertiesstrongly whiletheG-inversionsymmetricgroundstates areextended.
resemblethe weak and strongpairingphasesand the critical Thegaplessphasewasnotwell-understoodbefore. Weshow
stateinthepx+ipy-wavepairedstatesofspinlessfermions7. thattheeffectivetheorynearthephaseboundarycorresponds
TheinterconnectionsamongtheKitaevmodel,thep-wave to 1+1-dimensionalmassless Majorana fermions in the long
pairedfermions,andtheMoore-ReadPfaffiananditsexcita- wave length limit, i.e., a c = 1/2 conformal field theory or
tionshavenotbeenwellunderstoodpreviously. Inparticular, the2-dimensionalIsingmodel.Thevortexexcitationsareim-
it is important to understand the universal properties among portantin thefamilyof Kitaevmodelssince thevortexexci-
these systems, analogous to finding the universality class in tationsmayobeyanyonstatistics4. Thevortexexcitationen-
statistical mechanics models. In this paper, we show that ergieshavebeennumericallyestimatedintheA phase10 and
2
theBphase11. Westudythevortexexcitationsinthegapped
Bphaseinthecontinuumlimitandshowthatthevortexpair y 14 x 15 y 16 x 17 y18 x
excitations cost a finite energy. This is consistent with the
z z z
results of numerical calculations11 and suggests that vortex 12x 13 y 2 x 3 y 4 x19 y20
excitationsmayhavewell-definedstatistics.
z z z z
11 y10x 1 y 6 x 5 y22 x 21
II. THEGENERALIZATIONOFKITAEVMODEL z z z
x 9 y 8 x 7 y24 x 23 y
We extend the Kitaev model on the honeycomblattice by
introducingminimalthree-andfour-spintermsintheHamil-
x y x y x y x y x y x y
tonian,
z z z z z z z
y x y x y x y x y x y x
H = −J σxσx−J σyσy −J σzσz
x i j y i j z i j
X X X
x−links y−links z−links FIG. 1: (Color online) Upper panel: The honeycomb lattices and
− κ σzσy σx links. Lowerpanel: Thevortexexcitations. Theemptycirclesde-
xXb b b+ez b+ez+ex noteWP =1andthefilledcirclesdenotevorticeswithWP =−1.
− κ σxσy σz
x w w+ex w+ex+ez
X
w andthusσˆa =ib c. TheHamiltoniannowreads
− κ σzσx σy a
yX b b+ez b+ez+ey
b H = i J uac c −i Kx c c
− κy σwyσwx+eyσwz+ey+ez Xa a−Xlinks a ij i j Xb b,b+ez b b+ez+ex
X
w
− i Kx c c
− λxXσbzσby+ezσby+ez+exσbz+ez+ex+ez Xw w+ex−ex,w−ez−ex w+ex−ex w−ez−ex
b
− i Λx c c
− λy σbzσbx+ezσbx+ez+eyσbz+ez+ey+ez, (1) Xb b,b+2ez+ex b b+2ez+ex
X
b
− i Λx c c
where σx,y,z are Pauli matrices, x-,y-,z-links are shown in X w,w−2ez−ex w w−2ez−ex
w
Fig. 1(upper panel), ′w′ and ′b′ label the white and black
+ y−partners (2)
sites of lattice, and e ,e ,e are the positive unit vectors,
x y z
which are defined as, e.g., e = e ,e = e ,e = e .
12 z 23 x 61 y where Kx = κ uz ux , Λx =
Jx,y,z, κx,y and λx,y are tunable real parameters. The orig- λ uz b,ub+xez x b,ub+zez b+ez+ex,b+ez ebt,cb+a2nedz+ueax =
inal Kitaev model has κα = λα = 0, α = x,y. Adding a ibxabba,bo+neza-lbi+nekzs,.b+Ietzc+aenxbeb+sehzo+wenx,bt+haetz+theex+Heazmiltoniancijom-
T-symmetry breaking external magnetic field correspondsto i j
muteswithua andthustheeigenvaluesofua =±1because
κ = κ = κ 6= 0 and a κ -term4,5. It is importantto note ij ij
x y z
(ua)2 = 1. Since the four spin terms we introduced are
thatthegeneralizedHamiltonianmaintainstheZ2gaugesym- ij
related to the hopping between the ’b’ and ’w’ sites, Lieb’s
metryactedbyagroupelement,e.g.,
theorem8 is still applicable. The third spin terms are ’b’ to
WP =σ1xσ2yσ3zσ4xσ5yσ6z ’b’ and ’w’ to ’w’ and Lieb’s theorem is not directly ap-
plicable. However, according to Kitaev4, one can still take
with [H,W ] = 0. In fact, one can construct Z gauge
P 2 ua = −ua = 1. TheHamiltonianforthegroundstatefree
invariant spin models with higher multi-spin terms, e.g., bw wb
oftheZ vortices(W =1forallP)isgivenby
σzσy σyσyσx,σzσy σyσyσzσy and σzσy σyσyσyσz , and 2 P
9 10 1 2 3 9 10 1 2 3 4 9 10 1 2 3 16
so on. One can also add the ’z’-partners of κ and λ
x,y x,y
termssothatthemodelbecomesmoresymmetric. However, H = iJ˜ (c c −c c )
0 x s,b s−ex,w s,w s−ex,b
addingthesetermornotwillnotaffectourresultinthispaper Xs
aswewillexplainlater. + iλ˜ (c c +c c )
WenowwritedowntheMajoranafermionrepresentationof xX s,b s−ex,w s,w s−ex,b
s
thisspinmodel.Letbx,y,zandcbethefourkindsofMajorana κx
fermions with b2x,y,z = 1 and c2 = 1. The spin operator is + i 2 (cs,bcs+ex,b+cs,wcs−ex,w)
givenby Xs
i 1 + ypartners+iJz cs,bcs,w (3)
σˆa = 2(bac− 2ǫabcbbbc). Xs
RestrictingtothephysicalHilbertspace,oneneedstorequire4 wheresrepresentsthepositionofaz-link,λ˜ = Jα+λα and
α 2
D =b b b c=1 J˜ = Jα−λα.
x y z α 2
3
where E = ξ2 +(∆ )2+(∆ )2 is the dispersion,
p p 1,p 2,p
q
A ∆p = ∆1,p + i∆2,p, and (up,vp) are the coherence fac-
Jy=0, Jx=JzA(0,) (0B,0) (,0)AJx=0, Jy=Jz vtopr/suwpi=th−|u(pE|p2 −=ξp21)/(1∆∗p+. Eξpp),|vp|2 = 12(1 − Eξpp) and
Jz=0, Jx=Jy
IV. PHASEDIAGRAMINTERMSOFTHEp-WAVE
FIG.2: Leftpanel: Theeffectivesquarelattice. Rightpanel: Phase STATES
diagraminthespacespannedby(J˜x,J˜y,Jz).Thisisa(1,1,1)cross
sectionwithallJ’spositive. We now turn to the properties of this p-wave paired state
oflinkfermions. Itisinstructivetoconsiderthefreefermion
dispersion.Theconditionξ =0definesatopologicaltransi-
p
III. MAPPINGTOAp-WAVEPAIREDSTATE tionbetweenabandinsulatorandametalwithaFermisurface
intheabsenceofpairing.Thesolutionisgivenby|cosp∗|=
x
The next step is to map the Majorana fermion H to a |cosp∗|=1,p∗ =(0,0),(0,±π),(±π,0),(±π,±π),where
0 y
fermionmodel.Definingfermionsonthez-linksby9 J ±J˜ ±J˜ = 0. Withoutloss ofthe generality,one con-
z x y
d =(c +ic )/2, d† =(c −ic )/2, sidersonlyJ˜x,y > 0andJz > 0. Then,ξp∗ = 0corresponds
s s,b s,w s s,b s,w totheinnertriangleofthe(1,1,1)-crosssectionin(J˜ ,J˜,J )
x y z
space(seeFig.2(rightpanel)).Noticethatthep-wavepairing
H becomes
0 gapfunctions∆ vanishatp∗andtherefore,thistriangleis
a,p
H = J (d†d −1/2)+J˜ (d†d −d d† ) the gaplesscriticalboundaryseparatingthe A and B phases.
0 zX s s x s s+ex s s+ex Outsidethetriangle,ξp > 0. ThustheAphaseisgapped. In
s thelimitp →p∗,thepaircorrelationg ≡ v /u isanalytic
+ λ˜x (d†s+exd†s−ds+exds) (4) nearp∗,implyingtightlyboundpairsinppositpionaplspaceand
X
s hencetheA-phaseasthestrong-pairingphase(Seebelowfor
+ iκ (d d +d†d† )+ypartners. detaileddiscussions)7. Theglobalstructureofthephasedia-
xX s s+ex s s+ex gramisinvariantinthegeneralized(J˜ ,J˜,J )spacebecause
s x y z
our minimal three- and four-spin extension does not change
Thisisaquadraticmodelofspinlessfermionsdsonthesquare thetopologyoftheunderlyingFermisurface.
lattice(Fig.2, leftpanel)withgeneralp-wavepairing. Ifwe The nature of the B phase is much more intriguing. In-
includethe’z’partnerofthethreeandfourspintermsineq. side the triangle, ξ , ∆ and ∆ can be zero individ-
p 1,p 2,p
(1),wehaveadditionalcorrespondingtermsineq. (4)which ually. The gapless condition (E = 0) requires all three
p
arethenextnearestneighbortermsinthesquarelattice.These to be zero at a common p∗. This can only be achieved if
termswillnotqualitativelyaffectourresult.ReturningtoMa- (i) one of the ∆ = 0 or (ii) ∆ ∝ ∆ . If either
a,p 1,p 2,p
joranafermions,alinkfermionisasuperpositionoftwoMa- (i) or (ii) is true, ξ and ∆ can vanish simultaneously, i.e.
p p
jorana fermions in a link. Therefore, the paring of the link E = 0atp∗,andthepairedstateisgapless. Otherwise,the
p
fermionsreflectsthe’pairing’ofMajoranafermions. B phase is gapped. Note that contrary to conventional wis-
AfteraFouriertransformation,eq. (4)becomes dom, T-symmetry breaking alone does not guarantee a gap
openingintheBphase.Thesymmetryreasonbehindthegap-
∆
H = ξ d†d + 1,p(d†d† +d d ) lessconditionoftheBphasebecomesclearinthecontinuum
0 Xp p p p 2 p −p p −p limitwhereEp = 0impliesthatthevortex-freeHamiltonian
mustbeinvariant,up to a constant, underthe transformation
∆
+ i 22,p(d†pd†−p−dpd−p) (5) px ↔ ηpy and J˜x ↔ η−2J˜y with η = ∆∆aa,,xy with a = 1 or
2 and for nonzero ∆. We refer to this as a generalized in-
wherethedispersionandthepairingfunctionsare version (G-inversion) symmetry since it reduces to the usual
mirror reflection when η = 1. This (projective) symmetry
ξp =Jz −J˜xcospx−J˜ycospy, protectsthe gapless nature of fermionicexcitations and may
∆ =∆ sinp +∆ sinp , a=1,2 beassociated with theunderlyingquantumorder12. Kitaev’s
a,p ax x ay y
originalmodelhas∆ = 0, andis thusG-inversioninvari-
1,i
with ∆ = κ and ∆ = λ˜ . We have thus antandgapless. Themagneticfieldperturbation4breaksthis
1,x(y) x(y) 2,x(y) x(y)
shownthatthe groundstate ofthe extendedKitaev modelin G-inversionsymmetryand the fermionicexcitation becomes
Eq.(1)areequivalenttogeneralp-wavepairedfermionstates. gapped. Aspecialcase withG-inversionsymmetrybreaking
ThequasiparticleexcitationsaregovernedbytheBdG equa- is∆p ∝sinpx+isinpy,i.e. thepx+ipy-wavepairedstate
tions discussedbyReadandGreeninthecontinuumlimit7.
The N-fermion ground state wave func-
E u =ξ u −∆∗v , E v =−ξ v −∆ u (6) tion in the general p-wave paired state can
p p p p p p p p p p p p
4
be written down as a Pfaffian for N even: is topologically trivial and has zero spectral Chern number,
Ψ(r ,...,r ) =1/(2N/2(N/2)!) sgnP N/2g(r − whilethegappedBphaseischaracterizedbytheChernnum-
1 1 P i=1 P2i−1
rP2i) where g(r) is the “pair coPrrelation”,Qi.e. the Fourier ber±14. ThisfactwasalreadydiscussedbyReadandGreen
transform of g = v /u in the BCS wave function inthecontextofp +ip pairedstate. HerewefollowRead
p p p x y
|Ωi = |u |1/2exp(1 g d†d† )|0i. The wave and Green7 to study the topologicalinvariantin a generalp-
p p 2 p p p −p
function eQxhibits very differPent behaviors in the long wave wavestate.
lengthlimitindifferentparameters13. IntheAphase,ξp > 0 In continuum limit, p = (px,py) lives in an Euclidean
as p → 0 , thus gp ∝ ∆p. The analyticity of gp leads to space R2. However, the constraint |up|2 + |vp|2 = 1 pa-
g(r)∝e−µr asinthestrongpairingphaseofapurepx+ipy rameterizesa sphere S2. As |p| → ∞, ξp → Ep such that
state7. In the gapped B phase with G-inversion symmetry vp → 0. Therefore, we can compactify R2 into an S2 by
breaking, ξp < 0 as p → 0. Defining p′i = ∆aipi with adding∞toR2wherevp →0. Thesphere|up|2+|vp|2 =1
a = 1,2 and i = x,y, it followsthat gp ∝ p′+1ip′ , leading can also be parameterized by a pseudospin vector np =
x y (∆ ,−∆ ,ξ )/E because|n | = 1. (u ,v ) thusde-
tog(r) = 1 withx′ = ∆−1x andthusaweak-pairing 1,p 2,p p p p p p
x′1+ix′2 a ai i scribesamappingfromS2(p∈R2)toS2(spinor|np|=1).
phase. Identifying z′ = x′1 + ix′2, we see that the ground The winding number of the mapping is a topologicalinvari-
state of the gapped B phase corresponds exactly to the ant.The north pole is u = 1,v = 0 at |p| = ∞ and
p p
Moore-ReadPfaffian. Itis easy to show that, u andv∗ obey the south pole is u = 0,v = 1 at p=0. In the n
p p p
the same BdG equation in this general p-wave paired state, parametrization,n = (0,0,ξp) = (0,0,1)at|p| = ∞and
such that the anti-particle of the quasiparticle ψ = (u,v) is 0 E
itself, i.e.,itisMajoranafermionobeyingDiracequationsin (0,0,ξEp) = (0,0,−1)atp = 0, correspondingtoeitherthe
2+1-dimensions7. northpoleorsouthpole.
We now discuss the nature of the gapless B phase in the In the strong pairing phase, we know that up → 1 and
generalmodelwithG-inversionsymmetry.Inthiscase,Ep = vp → 0 as p → 0 (or equivalently, ξp > 0). This means
0 at p = ±p∗ which are the solutions of ξ = 0 and, say, thatfor arbitraryp, (up,vp) mapsthe p-sphereto the upper
p
∆ = ∆ = 0. At p∗, the fermion dispersions are gen- hemisphereandthewindingnumberiszero.Thatis,thetopo-
p 1,p
erally given by 2D Dirac cones. However, by a continuous logicalnumberν =0inthestrongpairingphase.
variation of the parameters, one can realize a dimensional In the weak paring phase, up → 0 and vp → 1 as p →
reduction near the phase boundary where the effective the- 0. This means that the winding number is nonzero (at least
wrapping once). For our case, the winding number can be
ory is in fact a 1+1 dimensional conformal field theory in
directlycalculatedandisgivenby
the long wave length limit. Let us consider parameters that
are close to the critical line with |sinp∗| ≪ |cosp∗| where
a a 1
gq = sgn[qx∆1xcosp∗x + qy∆1ycosp∗y] ≡ sgn(qx′) with ν = 4π Z dpxdpynp· ∂pxnp×∂pynp =1 (9)
q=p−p∗. DoingtheFouriertransform,wefind (cid:0) (cid:1)
DefiningP(p)= 1(1+n ·~σ),whichistheFouriercompo-
g(r) = dq′dq′eiqx′x′+iqy′ysgn(q′) nentof the project2ionoperpatorto the negativespectralspace
Z x y x
oftheHamiltonian,thiswindingnumbercanbeidentifiedas
= δ(y′) dq′ qx′ sinq′x′ ∼ δ(y′). (7) thespectralChernnumberdefinedbyKitaev4
Z x|q′| x x′
x
1
ν = Tr[P (∂ P ∂ P −∂ P ∂ P )]dp dp(10)
The δ(y)-functionindicates that the pairingin the gapless B 2πiZ − px − py − py − px − x y
phasehasaone-dimensionalcharacterandthegroundstateis
aone-dimensionalMoore-ReadPfaffian. TheBdGequations where P− = I − P is a projective operator. This spectral
ChernnumbervanishesinthestrongpairingAphasebuttakes
reduceto
anintegervalueintheweakparingBphase. Thus,thephase
i∂tu=−i∆1x(1+iη)∂x′v, i∂tv =i∆1x(1−iη)∂x′u, (8) transitionfromAtoBisatopologicalphasetransition.
with η = ∆1y. Thus, the gapless Bogoliubov quasiparti-
clesareone-∆d1ixmensionalMajoranafermions. Thelongwave VI. VORTEXEXCITATIONS
lengtheffectivetheoryforthegaplessBphasenearthephase
boundaryis thereforethe massless Majorana fermion theory We have discussed the universal behaviors of the ground
in1+1-dimensionalspace-time,i.e.ac=1/2conformalfield state. We now turn to discuss the Z vortex excitation in
2
theoryorequivalentlyatwo-dimensionalIsingmodel. the spin model which corresponds to setting W = −1 for
P
agivenplaquette. TheHamiltonianintheMajoranafermion
representationisbilinearandtheenergiesofthevorticescan
V. TOPOLOGICALINVARIANTINAANDBPHASES beestimatedbothin theAphase10 andtheBphase11. How-
ever,itremaineddifficulttoobtainanalyticalsolutionsofthe
Wenotethatthereisnospontaneousbreakingofacontin- wave functions with two well-separated vortices. We have
uous symmetry associated with the phase transition from A shownthatthegroundstatesectorisequivalenttothep +ip
x y
toBphases. KitaevhasshownthattheAphaseinhismodel pairing theoryfor fermionson the square lattice. Therefore,
5
thePfaffianstateisthegroundstatewavefunctioninthecon- checkthatsince[H , g′(K 6= 0)d† d† ] = 0,the
0 K,k k K+k K−k
tinuum limit in the weak pairing phase. Our strategy is to K6=0sectordoesnotPplayanontrivialroleincalculatingthe
evaluatetheenergyofthetrialwavefunctioncontainingvor- energyE ofsuchavortexpair. Thelatterisgivenby
v
ticesabovethePfaffianstateinthecontinuumlimit. Fortwo
E =hw ,w |H|w ,w i=hw ,w |H |w ,w i
well separated half-vortices located at w and w shown in v 1 2 1 2 1 2 0 1 2
1 2
Fig.1(lowerpanel),theMoore-Readtrialwavefunctionhas = E |u δg |2hw w |d d†|w w i
k k k 1 2 k k 1 2
beenwell-studied1,7andisgivenby X
k
Ψ(z1,...zN;w1,w2)∝Pf(g′(zi,zj;w1,w2)), (11) =XEk|ukδgk|2/(1+|gk′0|2), (15)
k
(z −w )(z −w )+(w ↔w )
g′(z ,z ;w ,w )∝ 1 1 2 2 1 2 .
1 2 1 2 z1−z2 wheregk′0 = gk′ + k1(w1wB2K¯2 − (ww11+ww22K¯)C)|K→0 andδgk =
g′0 −g . Physically, the factor hw w |d d†|w w i = 1−
Thesecondquantizedstate correspondingtothis wavefunc- k k 1 2 k k 1 2
|g′0|2/(1+|g′0|2) = 1/(1+|g′0|2) in Eq.(15)is the quasi-
tionreads k k k
holedistributionwhenthetwovorticesarelocatedatw and
1
|w ,w i∝exp{1 g′(z ,z ;w ,w )d† d† } (12) w2. Thus, Ev indeed corresponds to the energy cost to ex-
1 2 2 1 2 1 2 r1 r2 cite the vortex pair. We have evaluated the vortex pair en-
X
r1,r2
ergy E in different limits. First, if w and w were sent
v 1 2
PerformingaFouriertransformation,wehave to infinity before K → 0, then g′0 → g and we recover
k k
the ground state. Second, if K → 0 while w and w re-
1 2
|w1,w2i∝exp{21Xgk′(K)d†K+kd†K−k}, mtenaidnsfitoni|tke,|2thiennbEokth|utkhδegskm|2a/ll(1an+d|lgak′r0g|e2)k∼limEkit|su.kT|2h,ewehxiccih-
K,k
tation energyis thus high butfinite due to the shortdistance
wherek=k1−k2andK=k1+k2aretherelativeandthe cut-off. (Recall that ξk < 0 in the B phase). It is indepen-
totalmomentaofthepairsandg′(K)istheFouriertransform dent of w and w and as a result the vortices are decon-
k 1 2
ofg′(r ,r ). Onecanshowthat, fined. The third case is when the vortices are far from the
1 2
originsuchthat|Kw |and|Kw |arefinite. Intheshortdis-
1 2
g′(K)∼(1 − A )δ(K) tance, large k limit, Ek|ukδgk|2/(1+|gk′0|2) ∼ |k|−4. On
k k w1w2|k|2k the other hand, in the long wavelength limit with |k| → 0,
1 B (w1+w2)C |uk| ∼ |k|, |gk′0| → |k|−3 and Ek → constant such that
+k(w1w2|K|2K¯2 − w1w2|K|2K¯) Ek|ukδgk|2/(1+|gk′0|2) ∼ |k|2 → 0. As a result, the vor-
texpairenergyE isfreeofinfrareddivergencesandisonly
1 v
=g′δ(K)+ g˜(K) (13) weakly dependent on w − w . Therefore, the vortices are
k k 1 2
also deconfined. Finally, since E increases with the paring
k
where A, B and C are positive constants and g˜(K) is inde- parametersκandλinEq.(4),Ev isexpectedtoincreasewith
pendentofk. Thus, increasingparinggapparameters.Theseresultsareconsistent
with the finite size numerical calculations of the vortex pair
|w ,w i∝exp{1 g′d†d† + (1/k)g′(K)d† d† } energy in the lattice model11. Our analytical results suggest
1 2 2 k k −k K+k K−k thatthevortexpairdescribedbyEq.(11),whilecostingahigh
X X
k K,k
energyinthebulk,correspondstolowenergyexcitationsnear
Such a vortex pair is shown in Fig. 1 where the red z-links theedgeofthesystem.
haveu =−1andallothershaveu =1.Thecorrespond- We note an importantdifferencebetween thisp-wave the-
bw bw
ingHamiltoniancanbewrittenasH = H +δH,whereH ory and a conventional p-wave superfluid: Instead of spon-
0 0
isthevortex-freeHamiltonianandδH isthevortexpart. The taneously breaking the U(1) symmetry in an usual p-wave
latterisexpressedasasumofthepairingandchemicalpoten- superfluid, only the discrete Z2 symmetry is broken and the
tialtermsaccordingtoEq.(5)overtheredz-linksextending U(1)symmetryis absentin the presentmodel. The vortices
intheξ-direction(i.e.,thedirectionwithx = y)betweenthe studiedherearethusZ2vorticesinsteadofU(1)vortices. As
vortices. ItisstraightforwardtoshowthatδH hasthefollow- aconsequence,inthegappedBphase,thevorticesareinthe
ingexpectationvalueinthevortexstate, deconfinementphase15 instead of being logarithmicallycon-
finedinthep-wavesuperfluid.Thisfactcanbeeasilyseenbe-
hw ,w |δH|w ,w i (14) cause∆ arerealandthereisnoU(1)phasefactorwhose
1 2 1 2 ax(y)
i(eiw1(pξ+p′ξ)−eiw2(pξ+p′ξ)) gradientgivesrisetoavectorfieldofthevortex.Ouranalysis
∝ f(p ,p′)=0, ofthevortexpairenergyalsoshowsthisdifference.
p +p′ ξ ξ
pXξ,p′ξ ξ ξ The finiteness of Ev and the vanishing of hδHi = 0 im-
ply that the vortex excitations are separated either from the
where f(p ,p′) is an analytical function of p + p′. This groundstateorotherexcitations.Thisisconsistentwithanal-
ξ ξ ξ ξ
meansthattherearenoacontinuumspectrumabovethevor- ysisofRead andGreenontheU(1)vortexexcitationsin the
tex pairs and then the vortex pairs are also separated from p-wave paired state7,14. To determine how close the Moore-
other higherenergy excitations. On the other hand, one can Read vortex state is to the exact vortex excitations in this
6
modelrequiresanumericalcalculationoftheoverlappingbe- on a square lattice with general pairing parameters. Based
tweentheexacteigenstatesandtheMoore-Readvortexwave on the general p-wave paired states, we analyzed the phase
functions. Amoreimportantquestionistherealizationofthe diagramofthesystemandthepropertiesoftopologicallydif-
Read-Moorefour-vortexstate whichhasa two-folddegener- ferentphases. We foundthat our phase diagram is universal
acywiththevorticesobeyingnon-abelianstatistics1.Weleave and includes both the Kitaev modeland the Pfaffian state in
thesestudiestofutureworks. itsuniversalityclass.
The authors thank the KITPC for warm hospitality where
VII. CONCLUSIONS
this work was initiated and finalized during the Program on
Quantum Phases of Matter. The authors are grateful to Z.
We have constructed an exactly soluble spin model with Nussinov, N. Read, X. Wan, X.-G. Wen, K. Yang, J. W. Ye,
two-, three-andfour-spincouplingsonahoneycomblattice. andY.S.Wuforusefuldiscussions.Thisworkwassupported
Thegroundstate sector ofthismodelonthe honeycomblat- inpartbyNNSFofChina,theMinistryofScienceandTech-
tice is mapped to a p-wave paired state of the link fermions nologyof China, a fundfromCAS and the U.S. DOE Grant
No. DE-FG02-99ER45747.
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