Gauge symmetry in Kitaev-type spin models and index theorems on odd manifolds

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by  Yue Yu

Gauge symmetry in Kitaev-type spin models and index theorems on odd manifolds
Yue Yu1
1Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China
(Dated: February 1, 2008)
Weconstructanexactlysolublespin-1 modelonahoneycomblattice,whichisageneralizationof
2
Kitaevmodel. Thetopologicalphasesofthesystemareanalyzedbystudyofthegroundstatesector
of this model, the vortex-free states. Basically, there are two phases, A phase and B phase. The
behaviorsofbothAandBphasesmaybestudiedbymappingthegroundstatesectorintoageneral
p-wavepairedstatesofspinlessfermionswithtunablepairingparametersonasquarelattice. Inthis
8 p-wavepairedstatetheory,theAphaseisshowntobethestrongpairedphase,aninsulatingphase.
0 TheBphasemaybeeithergapped orgapless determinedbythegeneralized inversion symmetryis
0 brokenornot. ThegappedBistheweakpairingphasedescribedbyeithertheMoore-ReadPfaffian
2 stateofthespinlessfermionsoranti-Pfaffianstateofholesdependingonthesignofthenextnearest
neighborhoppingamplitude. AphasetransitionbetweenPfaffianandanti-Pfaffianstatesarefound
n
inthegapped Bphase. Furthermore,weshowthatthereisahiddenSU(2)gaugesymmetryinour
a
J model. In the gapped B phase, the ground state has a non-trivial topological number, thespectral
firstChern numberorthechiral centralcharge, whichreflectsthechiralanomaly oftheedgestate.
2
We proved that the topological number is identified to the reduced eta-invariant and this anomaly
2
may be cancelled by a bulk Wess-Zumino term of SO(3) group through an index theorem in 2+1
] dimensions.
l
e
PACSnumbers: 75.10.Jm,03.67.Pp,71.10.Pm
-
r
t
s
. I. INTRODUCTION statistics. The B phase is gapless at special points of
t
a Brilliouin zone. When the B phase is gapped by a per-
m turbation, it is topologically non-trivial and has an odd-
The conceptofthe topologicalorderrecentlyiswidely
- integer spectral Chern number. (We call the gapless B
d interesting the condensed matter physicists because it phase the B1 phase and gapped one the B2 phase.) Ki-
n maydescribe the different’phases’withoutbreakingany
taev showed that if the spectral Chern number is odd,
o global continuous symmetry of the system [1]. However,
there must be unpairedMajoranafermions andthen the
c unlike the conventional order related to the symmetry
[ vortex excitations obey non-abelian statistics. Consis-
of the system in Landau’s phase transition theory, the tent with the non-abelian statistics, the fusion rules of
3 topological order of quantum states is not well defined
the superselection sectors of Kitaev model are the same
v yet. For example, in the quantum Hall effects, the topo-
as those of the Ising model. However, the source of the
9 logical property of the quantum stats may be reflected
non-abelianphysicshasnotbeenclearlyrevealedyet. On
2
by the filling factor of the Landau level which may be
8 the other hand, the first Chern number can only relate
thought as a topological index, the first Chern number
3 toanabeliangroupandtherefore,anoddspectralChern
. in magnetic Brilliouin zone [2, 3]. Nevertheless, only the number leads to a non-abelian physics but an even one
4
firstChernnumbercannotfullyscorethetopologicalor-
0 did not is topologically hard to be understood.
der of the quantum Hall states. In a given filling factor,
7 Although Kitaev model has a very special spin cou-
thequasiparticlesmayobeyeitherabelianornon-abelian
0 pling,itsveryattractivepropertiescausedabunchofre-
: statistics. On the other hand, the edge state may par- centstudies[8,9,10,11,12,13,14,15,16,17,18,19,20].
v
tially image the topological properties of the bulk state
i It is convenient to understand Kitaev model if one can
X [4]. In quantum Hall system, it was seen that the edge
map this model to a familiar model. In fact, Kitaev
state may be described by a conformal field theory [5].
r model may be mapped into a special p-wavepairedBCS
a Thus, according to the bulk-edge correspondence due to
state if only the vortex-free sector of the model is con-
the gauge invariance, it shows that the bulk state is de-
sidered[11]. We recentlygeneralizedKitaevmodeltoan
termined by a Chern-Simons topological field theory [6].
exactly soluble model whose vortex-free part is equiva-
However, the bridge between the microscopic theory of
lent to ∆ p +∆ p +i(∆ p +∆ p )-wave paired
1x x 1y y 2x x 2y y
the two-dimensional electron gas and the Chern-Simons
fermionstateswithtunablepairingorderparameters∆
ab
theory was not spanned.
on a square lattice. [12]. The phase diagram of our
Kitaev recently constructed an exactly soluble spin model has the same shape as that of Kitaev model, i.e,
model in a honeycomb lattice [7]. Using a Majorana theboundaryoftheA-Bphasesarecorrespondingtothe
fermionrepresentation,hefoundthequantumstatespace points p = (0,0),(0, π) and ( π,0) in the first Bril-
± ±
is characterized by two different topological phases even liouin zone. The A phase is gapped and may be iden-
there is not any global symmetry breaking. The A tified as the strong paring phase of the p-wave paired
phase is agappedphase whichhasa zerospectralChern state [21]. The B phase can be either gapped or gapless
number and the vortex excitations obey abelian anyonic even if T-symmetry is broken. We find that gapless ex-
2
citations in the B phase, i.e., the B1 phase, is protected termandareducedeta-invariantη¯isaninteger[25]. We
byageneralizedinversion(G-inversion)symmetryunder show that difference between the WZ term and a part
p ∆1yp andtheemergenceofagappedB(B2)phase of the eta-invariant gives an ambiguity of the WZ term.
isxt↔hus∆1txiedyto G-inversion symmetry breaking. For in- Another part of this eta-invariantis identical to the chi-
stance, the p +ip wave paired state is gapped while ral central charge, a half of the spectral Chern number.
x y
p +ip -wave paired state is gapless although they both Thus, an odd Chern number corresponds to a πi ambi-
y y
break the T-symmetry. The critical states of the A-B guity while an even Chern number to a 2πi ambiguity.
phase transition remains gapless whether or not T- and Accordingtothebulk-edgecorrespondence,theformeris
G-inversionsymmetriesarebroken,indicativeofitstopo- consistent with k = 2 while the latter is consistent with
logical nature. Indeed, if all ∆ are tuned to zero, the k =1.
ab
topologicalA-Bphasetransitionisfromabandinsulator The restof this workwas organizedasfollows. InSec.
toa freeFermigas. The Fermisurfaceshrinkstoapoint II, we recallKitaev model and show the SU(2) gaugein-
zero at criticality. variance. In Sec. III, we will describe the generalized
model. In Sec. IV, we give the phase diagram of the
In this paper, we further generalize the model pro-
system. In Sec. V, we consider the continuous limit of
posed by the present author and Wang in ref. [12] to
our model and show that the low energy effective theory
a model whose square lattice mapping includes a next
is the Majorana fermions coupled to a SO(3) gauge field
nearest hopping of the spinless fermions. In this case,
in a pure gauge. In Sec. VI, we apply the index theorem
the A phase is still a strong pairing phase as before.
on odd manifold to our model. In Sec. VII, we present
However, the B2 phase has more fruitful structure. The
a understanding to the edge state from the index theo-
particle-holesymmetryisbrokenevenifthechemicalpo-
rem point of view. The section VIII is our conclusions.
tentialandthe pairingparametersvanish. Near the long
We arrange three appendices. Appendix A is to address
wave length limit (p = (0,0) critical line), the effec-
∗
the mathematic expression of the index theorem on odd
tive chemical potential has the different sign from that
manifoldbecausemostofphysicistsarenotfamiliarwith
of the nearest neighbor hopping amplitude. Near other
it. In Appendix B, we give an introduction to the rep-
two critical lines (0,π) and (π,0), when the next near-
resentation to the spin-1/2 in the conventional fermion
est neighbor amplitude is positive, the effective chemical
and Majorana fermion. And in Appendix C, for com-
potential is also positive. When the next nearest neigh-
pleteness, we recall the vortex excitations in our model
bor amplitude is negative, the effective chemical poten-
although it was studied in our previous work [12].
tial is also negative. A positive chemical potential cor-
responds to a closed Fermi surface of the particles and
then a Pfaffianofthe particles while a negativechemical
II. KITAEV MODEL
potential to a closed Fermi surface of holes and then an
anti-Pfaffian of the holes of the spinless fermion. There-
We first recall some basic results of Kitaev model,
fore,aPfaffian/anti-Pfaffianphasetransitionhappens in
which is a spin system on a honeycomb lattice [7]. The
the B2 phase. This Pfaffian/anti-Pfaffian phase transi-
Hamiltonian is given by
tionhasbeenseeninthecontextoftheν =5/2fractional
quantum Hall effect [22, 23]. The model we present here
H = J σxσx J σyσy J σzσz,
is exactly the same as a toy model on square lattice to ki − x i j − y i j − z i j
study the Pfaffianandanti-Pfaffianphysics[22]. The B1 x−Xlinks y−Xlinks z−Xlinks
andB2phaseswhenthenextnearestneighborhoppingis where σa are Pauli matrices and ’x-,y-,z-links’ are three
absent are corresponding to the particle-hole symmetry different links starting from a site in even sublattice [7].
is conserved or spontaneously broken. This model is exactly solvable if one uses a Majorana
The another topic of this paper is trying to reveal the fermion representation for spin. Kitaev has shown that
mathematical connotation behind the topological order. his Hamiltonian has a Z2 gauge symmetry acting by a
WeemphasizethatthereisahiddenSU(2)gaugesymme- group element, e.g., for (123456) being a typical plaque
tryinthismodelifthemodelisrepresentedbyMajorana
W =σxσyσzσxσyσz
fermion operators. This non-abelian gauge symmetry is P 1 2 3 4 5 6
the source of the non-abelian physics of the model. The with[H ,W ]=0. Infact,wecanshowthatthismodel
ki P
non-abelian degrees of freedom in the A phase are con- has an SU(2) gauge symmetry in the Majorana fermion
fined while in the B2 phase, the non-abelian degrees of representation. Letb andcbefourkindsofMajorana
x,y,z
freedom are deconfined. There is a Wess-Zumino(ZW) fermions with b2 =b2 =b2 =c2 =1 and define
x y z
term for SU(2)/Z group whose lever k may character
2
the confinement-deconfinement phases. A level k WZ (χcd)= 1 bx−iby bz −ic . (1)
term corresponds to a level k SU(2)/Z2 Chern-Simons 2(cid:18) bz +ic −bx−iby (cid:19)
topological field theory. It was known that k =1 theory
One observes SU(2) gauge invariant operators
can only have abelian anyonwhile k =2 theory includes
non-abelian anyons [24]. A recently proved index theo- 1 i 1
rem in 2+1 dimensions shows that the sum of this WZ σˆa = Tr[χ†χ(σa)T]= (bac ǫabcbbbc) (2)
2 2 − 2
3
with respect to the local gauge transformation χcd ( f (p))σy + ∆(p)σz with ∆( p) = ∆(p), f =
1 1
U[26cc]′.χcI′tdaisndeatshyento(χc†h)ecdck→th(aχt†)σˆcca′/(U2−m1a)cy′dsfeorrveUa∈sSsUp(i→2n)- 2(−fJ +=4fJco+si21fpx).cosA√2s3spuymaenψd f(2−p)=to4Jbceo−sth21pexssoinlu√t2i3opnys
1 2 ±
1/2 operators. Replacing σa by σˆa, Kitaev model has of Schrodinger equation H (p,∆)ψ (p)= E(p)ψ (p)
0 ± ±
a hidden SU(2) gauge symmetry which is trivial in the ±
withE(p)= f 2+ ∆2. Afternormalization,wehave
spin operator representation. The constraint D = 1 is | | | |
also gauge invariant because D = b b b c = iσˆxσˆyσˆz. L·~σψ± =±ψp± with
x y z
Under this constraint,σˆa takes the form ib c−after using
a 1
b b b c = 1. The SU(2) symmetry of σa = ib c can be L= ( f (p), f (p),∆(p)). (7)
x y z a 2 1
√3JE(p) − −
directly checked
σa =ibac=ib′ac′, (3) Ewxitphlicpitlya,nndeapr pthe=dpua∗l =vec−to32rsp1of+n23pa2nmdodn(p,1,itp2is)
1 2 1 2
where L = 1 (δp ,δp ,∆(p∗)) (δpˆ ,δqˆ ,∆ˆ) with Eˆ(p) =
Eˆ(p) y x √3J ≡ y x
bb′x′y =−αα12 αα12 −ββ12 −−ββ12  bbxy  (4) (pδpˆ(yδ,px)δ2pˆ+x,(δ∆pˆy)).2A+c∆co2r/d3inJg2.toNKeaitrapev,=on−epc∗a,nitdiesfinLe=a
b β β α α b − −
 c′z   −β 1 β2 α1 α2  cz  spectrum Chern number by using the vector field L. We
 ′   2 1 − 2 1   will be back to this issue when studying the index theo-
rem.
with α2+α2+β2+β2 =1 and b b b c=b b b c =1.
1 2 1 2 x y z ′x ′y ′z ′
UsingJordan-Wignertransformation,avarietyofKitaev
model on a brick-wall lattice has been exactly solved [8]
and a real space ground state wave function is explicitly
shown [11]. This variety should correspond to another
gauge fixed theory. A
After some algebras, Kitaev transferred the Hamilto- Jy=0, Jx=Jz (0,0) Jx=0, Jy=Jz
nian to a free Majorana fermion one [8]
B
(0,) (,0)
A A
1
H = H(p)µνc p,µcp,ν, (5) Jz=0, Jx=Jy
2p;µX,ν=b,w −
where H(−p) = −H(p) and cq,µ are the Fourier com- FIG. 1: Phase diagram in (Jx,Jy,Jz)-space.
ponents of a Majorana fermion operators and µ = b or This is a (1,1,1)-cross section in all positive region.
w refers to the even or odd position in a z-link [7]. The
groundstateisvortex-freeandthecorrespondingHamil-
tonian H (p) is given by
0
0 if(p)
H0(p)=(cid:18) if∗(p) 0 (cid:19), (6) III. GENERALIZED EXACTLY SOLUBLE
− MODEL
withf(p)=2(Jxeip·n1+Jyeip·n2+Jz). Herewestillfol-
low Kitaev and choose the basis ofthe translationgroup
n = ( 1/2,√3/2). In the next section, we will see
1,2
±
that deforming the angle between the x-link and y-link y 14 x 15 y 16 x 17 y18 x
to a rectangle will be much convenient. The eigenen-
z z z
ergy may be obtained by diagonalizing the Hamiltonian, 12x 13 y 2 x 3 y 4 x19 y20
which is E (p) = f(p). The phase diagram of the
0 ±| | z z z z
model has been figured out in Fig. 1. Kitaev calls
11 y10x 1 y 6 x 5 y22 x 21
the gapped phase as A-phase and the gapless phase B
phase. The A phase is topologically trivial and gapped. z z z
Itisthe strong-couplinglimit ofSU(2)likethe antiferro- x 9 y 8 x 7 y24 x 23 y
magnetic Heisenberg model [26] and can be explained as
the strong paring phase in the p wave sence [21]. After
perturbed by an external magentic field, the B phase is
FIG. 2: (Color online): Thehoneycomb lattices and links.
gapped and has a non-zero spectral Chern number and
then is topologically non-trivial citeki. Without lose of
generality, we consider J = J = J = J. The effective We now consider the Hamiltonian which is generaliza-
x y z
Hamiltonian is then given by H (p,∆) = ( f (p))σx + tion of the Kitaev model in honeyconmb lattice to the
0 2
−
4
following one this spin model and then the Hamiltonian reads
H = J σxσx J σyσy J σzσz
− x i j − y i j − z i j
xXlinks y Xlinks z Xlinks
− − −
κ σzσy σx
− x b b+ez b+ez+ex
Xb
κ σxσy σz
− x w w+ex w+ex+ez
Xw
κ σzσx σy
− y b b+ez b+ez+ey
Xb
κ σyσx σz
− y w w+ey w+ey+ez
Xw
− λxXb σbzσby+ezσby+ez+exσbz+ez+ex+ez H = iXa aXlinksJauaijcicj −iXb Kbx,b+ezcbcb+ez+ex
λ σzσx σx σz (8) −
− yXb b b+ez b+ez+ey b+ez+ey+ez − iXw Kwx+ex−ex,w−ez−excw+ex−excw−ez−ex
− BbXb σbzσby+ezσbz+ez+exσby+ez+ex−ey − iXb Λxb,b+2ez+excbcb+2ez+ex
− BwXw σwxσwz+exσwx+ex−eyσwz+ex−ey−ez − iXw Λxw,w−2ez−excwcw−2ez−ex
B σy σxσy σx + y partners
− wXb b−ey b b+ez b+ez−ex + i− β c c
− BbXb σbz−ey−ezσbx−eyσbxσby+ezσby+ez+exσbz+ez+ex+ez + iXb αb,b+ez+ex−ey bcb+cez+ex−ey
where w and b labels the white and black sites in lat- Xw w,w+ex−ey−ez w w+ex−ey−ez
′ ′ ′ ′
tice and e ,e ,e are the positive unit vectors, which + i α c c
x y z w,b+ey+ex w b+ey+ex
are defined as, e.g., e12 = ez,e23 = ex,e61 = ey (See Xw
Fig. 2). J , κ , λ and B are real parameters.
x,y,z x,y x,y b,w + i β c c (9)
This is a generalization of Kitaev model with the three- b,b+ez+ey+ex+ez b b+ez+ex+ey+ez
Xb
spin,four-spinandsix-spinterms. Itis easyto checkthis
generalized Hamiltonian still has a Z gauge symmetry
2
acting by a group element, e.g., W = σxσyσzσxσyσz
P 1 2 3 4 5 6
with [H,W ] = 0. In fact, one can add more Z gauge
P 2
invariant multi-spin terms, e.g.,
σzσy σyσyσx,
9 10 1 2 3
σzσy σyσyσzσy,
9 10 1 2 3 4
σzσy σyσyσyσz ,
9 10 1 2 3 16
and so on. The site indices are shown in Fig. 2. For our
purpose, however, we restrict on (8).
where Kx = κ uz ux , Λx =
λ uz bu,bx+ez x b,ub+zez b+ez+ex,b+ez b,b+2ez+exetc
anxdb,bu+aez b=+ez,bi+baezb+aexonb+eaz+-leinx,kbs+.ez+exβ+ez =
B uz ij ux i j uy b,b+ez+ex−ey and
αb b,b+ez b+ez,b+ez+ex=b+ez+exB,b+uezx+ex−eyuy
uzw,w+ex−ey−ez etc. Itw cwa,nw+ebxe ws+heoxw,wn+ext−haeyt
thwe+eHx−aemy,iwlt+oenxi−aeny−cezommutes with ua and thus the
ij
eigenvalues of u = 1. Since the four spin and six-spin
ij
FIG. 3: The effective squarelattice. terms we introduced±are related to the hopping between
the ’b’ and ’w’ sites, Lieb’s theorem [27] is still applied.
We now use the Majorana fermion representation for Following Kitaev, we take u = u = 1 and the
bw wb
−
5
vortex-free Hamiltonian is given by Or it is
H = iJ˜ (c c c c ) H0 = (−td†idj +∆ijd†id†ij +h.c.)−µ (d†idi−1/2)
0 xXs sb s−ex,w− s,w s−ex,b Xhiji Xi
+ iλ˜xXs (cs,bcs−ex,w+cs,wcs−ex,b) −t′hXhijii(d†idj +h.c.)+δXi,±(d†idi±ex −d†idi±ey), (13)
κx
+ i 2 Xs (cs,bcs+ex,b+cs,wcs−ex,w) wThheerpeatri=ng−paJ˜xra+2mJ˜ye,tetr′s=ar−eBde,fiµne=d b−yJ∆z and δ ==Jλ˜˜x−2J˜y+.
+ y partners+iJ c c iκ . Thelastequalityineq.(13)isthetoi,yi±mexo,ydelHaxm,yil-
z sb sw x,y
Xs tonian describing Pfaffian/anti-Pfaffianstates [22]. Note
that the pairing free Hamiltonian is particle-hole sym-
+ iB c c +iB c c
bXs s,b s+ex−ey,w wXs s,w s+ex−ey,b metry if ∆ij = µ = t′ = 0. The t′-term breaks the
particle-hole symmetry. The δ-term breaks the π/2 ro-
+ iB c c +iB c c
w s,w s+ey+ex,b b s,b s+ex+ey,w tational symmetry. After Fourier transformation ds =
Xs Xs 1 eipsd , we have
= iJ˜xXs (csbcs−ex,w−cs,wcs−ex,b) √LxLy Pp · p ∆1p
+ iλ˜xXs (cs,bcs−ex,w+cs,wcs−ex,b) H0 = Xp ξpd†pdp+ 2 (d†pd†−p+dpd−p)
κx ∆2p
+ i 2 Xs (cs,bcs+ex,b+cs,wcs−ex,w) + i 2 (d†pd†−p−dpd−p) (14)
where the dispersion relation is
+ y partners+iJ c c
z sb sw
Xs ξ =J J˜ cosp J˜ cosp +2Bcosp cosp (15)
p z x x y y x y
− −
+ iB (c c c c ) (10)
− s,b s+ex−ey,w− s,w s+ex−ey,b and the pairing functions are
Xs
+ iB+Xs (cs,bcs+ex−ey,w+cs,wcs+ex−ey,b) ∆∆1p2 ==∆∆1xssiinnppx++∆∆1yssiinnppy, (16)
p 2x x 2y y
+ iB− (cs,bcs+ex+ey,w−cs,wcs+ex+ey,b)
Xs with ∆ = κ and ∆ = λ˜ . After Bogoli-
1,x(y) x(y) 2,x(y) x(y)
+ iB+ (c c +c c ) ubov transformation,
s,b s+ex+ey,w s,w s+ex+ey,b
Xs
αp =updp−vpd†−p,
where s represents the position of a z-link, λ˜α = Jα+2λα α†p =u∗pd†p−vp∗d−p (17)
and J˜α = Jα−2λα. B± = Bb±2Bw. the Hamiltonian can be diagonalized
Tosimplify thepairing,onetakesB+ =0anddenotes
B ≡B−. Defining a fermion on z-links by [11, 12] H0 = Epα†pαp+const. (18)
Xp
d =(c +ic )/2, d =(c ic )/2, (11)
s s,b s,w †s s,b− s,w and the Bogoliubov quasiparticles have the dispersion
the vortex-free Hamiltonian becomes an effective model
E = ξ2 +(∆1)2+(∆2)2. (19)
p p p p
of spinless fermions on a square lattice (Fig.3) q
The Bogoliubov-de Gennes equations are given by
H0 = Jz (d†sds−1/2) E u =ξ u ∆ v , E v = ξ v ∆ u (20)
Xs p p p p− ∗p p p p − p p− p p
+ BXs (d†sds±ex±ey −dsd†s±ex±ey) with
+ J˜x(d†sds+ex −dsd†s+ex) vp/up =1−(Epξ−ξp)/∆∗p,
+ λ˜xXs (d†s+exd†s−ds+exds) |up|2 = 2(1+ Epp),
1 ξ
+ iκxXs (dsds+ex +d†sd†s+ex)+y partners.(12) |vp|2 = 2(1− Epp). (21)
6
IV. PHASE DIAGRAM is closed. This is the d-hole paired phase. Thus, there is
a Pfaffian/anti-Pfaffiantransition as µ is across zero.
eff
We now study the phase diagram in parameter space. There areother twocriticallines (0,π) and(π,0) near
The phase diagram when t = 0 has been discussed whichtherearealsogaplessexcitations. Thecriticalcon-
′
in our previous work [12], which has the same shape dition is given by
as that in original Kitaev model (with (J ,J ,J ) in
x y z J J˜ J˜ 2B = µ δ+2t
Fig.1 substituted by (J˜x,J˜y,Jz)) but the structures z∓ x± y− − ∓ ′
of the B phase are more fruitful. After including =2t′ µeff =0. (25)
−
the t-term, the phase boundary is still in (p ,p ) =
′ x y Inside of the B phase but near the critical lines,
(0,0),(0, π),( π,0),( π, π) as we know before. For
± ± ± ±
thepresentmodel,itisJ˜ J˜ J˜ =0withJ˜ =J +2B
z± x± y z z µeff 2t′ &0. (26)
for (0,0),( π, π) and J 2B for (0, π),( π,0). −
z
± ± − ± ±
In (J˜ ,J˜,J˜) space, the phase diagram are of the To satisfy this condition, µ has to be the same sign
x y z eff
same shape as that of original Kitaev model (Fig. 1, as that of t. Therefore, if t > 0, the system is in the
′ ′
(J ,J ,J ) is replaced by (J˜ ,J˜,J˜) ). The A phase Pfaffian state and the system is in anti-Pfaffian state if
x y z x y z
is a strong pairing phase. The nature of the B phase t′ <0.
is much more intriguing. Inside the B phase, ξp, ∆1,p Weseethatwhent′ <0,thesignoftheeffectivechem-
and ∆ can be zero individually. The gapless condi- ical near the critical line (0,0) has the opposite depen-
2,p
tion (Ep = 0) requires all three to be zero at a common denceonthesignoft′totheeffectivechemicalnearother
p∗. This can only be achieved if (i) one of the ∆a,p =0 two critical lines. For t′ > 0, the sign of the effective
or (ii) ∆ ∆ . If either (i) or (ii) is true, ξ and chemicalnearthecriticalline(0,0)maychangefromthe
1,p 2,p p
∝
∆ canvanishsimultaneously,i.e. E =0atp ,andthe oppositetothesameasthatnearothertwocriticallines.
p p ∗
pairedstateisgapless. Otherwise,theBphaseisgapped. Hence, if t′ = 0, there must be a Pfaffian/anti-Pfaffian
6
Notethatcontrarytoconventionalwisdom,T-symmetry transition inside of the gapped B phase.
breaking alone does not guarantee a gap opening in the At t′ = 0, the Pfaffian and anti-Pfaffian states are
B phase. The symmetry reason behind the gapless con- degenerate. As we have seen [12], in the gapless B
dition of the B phase becomes clear in the continuum phase, there are two gapless Majorana excitations at
limit where E = 0 implies that the vortex-free Hamil- nodal points while in the gapped B phase, the particle-
p
tonian must be invariant, up to a constant, under the hole symmetry is spontaneously broken, i.e., the ground
transformationp ηp andJ˜ η 2J˜ withη = ∆a,y state is either Pfaffianor anti-Pfaffian. All abovediscus-
x ↔ y x ↔ − y ∆a,x sions are consistent with those in ref. [22].
with a = 1 or 2 and for nonzero ∆. We refer to this
as a generalized inversion (G-inversion) symmetry since
it reduces to the usual mirror reflection when η = 1.
V. CONTINUOUS LIMIT, DIRAC EQUATIONS
This (projective) symmetry protects the gapless nature
AND SO(3) GAUGE THEORY
of fermionic excitations and may be associated with the
underlying quantum order [28]. Kitaev’s original model
Infact,thegroundstatewavefunctionforageneralp-
has ∆ =0, and is thus G-inversion invariant and gap-
1,i
wavepairedstatecanalsobecalculatedinthecontinuous
less. The magnetic field perturbation [7] breaks this G-
limit. The BCS wave function is given by
inversionsymmetryandthefermionicexcitationbecomes
gapped.
1
(0,T0)h.eIcnotnhtiinsucuamse,limit takes place near the critical line |Ωi=Yp |up|1/2exp(2Xp gpd†pd†−p)|0i, (27)
J˜z J˜x J˜y 2t′ = µeff 2t′ =0. (22) where gp = vp/up. For even fermion number N, the
− − − − −
Pfaffian ground state wave function reads
with µ =J J˜ J˜. Slight inside of the B phase,
eff z x y
− − N/2
µeff &−2t′. (23) Ψ(r1,...,r1)∝ sgnP g(rP2i−1 −rP2i) (28)
XP Yi=1
Ift <0,µ ispositiveandthesystemisinthePfaffian
′ eff
phase. If t′ >0, withgp istheFouriertransformofg(r). FortheAphase,
g ∆ and the analyticity of g leads to g(r) e µr
p p p −
µ .2t (24) ∝ ∝
eff ′ as the same calculationin a pure p +ip strong pairing
| | x y
state. IntheBphase,iftheG-chiralsymmetryisbroken,
may be either positive or negative. µ >0 means that
eff
defining p = ∆ p with a = 1,2 and b = x,y, g
thhoeleeFleecrtmroinsuFrfearcmeiissuorpfeanceedi.sTclhoisseidst(h∆eijd-=pa0r)ticwlehipleaitrhede p′+1ip′ and′athenga(br)b= x′+1ix′ withx′a =∆−ab1xb. Thpis∝is
x y 1 2
phase. Ontheotherhand,µ <0meansthattheelec- a Pfaffian state with z = x +ix and is corresponding
eff ′ ′1 ′2
tronFermisurfaceisopenedwhiletheholeFermisurface to a weak paired gapped fermion state [21].
7
In the long wave length limit (small p limit), we ap- wavelengtheffective theory forthe gaplessB phasenear
proximate ξ m=J˜ J˜ J˜ and define ψ(t,s)= the phase boundary is therefore the massless Majorana
p z x y
≈− − −
(u(t,s),v(t,s)) with (u,v) the Fourier transformation of fermion theory in 1+1-dimensional space-time, i.e. a
(u ,v ). TheBdGequationsinthegappedBphasereads c = 1/2 conformal field theory or equivalently a two-
p p
dimensional Ising model.
eµaγ ∂ ψ+imψ=0 (29)
a µ
with the vielbein eµt = δ and eij = ∆ . γ = σz
µt ij 0 VI. TOPOLOGICAL INVARIANTS AND INDEX
and γ = σzσi. Furthermore, it may be generalized to
i THEOREM
a curve space with spin connection term added [21]. In
general, the equation of u is not compatible to that of
A. Spectral Chern Number and η-invariant
v and the fermions are Dirac fermions. However, it is
easy to show that for this p-wave paired state, u and
We note that there is no spontaneous breaking of a
v obey the same equation, i.e., the anti-particle of the
∗
continuous symmetry in the phase transition from A to
quasiparticle is itself. Then the fermions are Majorana
B phases. Kitaev has shown that the A phase in his
ones.
model is topological trivial and has zero spectral Chern
We now turn to local gauge transformation. As we
numberwhilethegappedBphasehasthisChernnumber
have pointed out, taking u =1 means the whole SU(2)
ij
1 [7]. This fact was also already realized by Read and
gauge symmetry is fixed. If making an SU(2) gauge
±
Green in the p +ip paired state. Here we follow Read
transformation, we will run out the vortex-free state. x y
and Green [21] to study this topological invariant for a
Keeping in the vortex-free state, the gauge transforma-
general p-wave state.
tionis confined inanSU(2)/Z SO(3)one. Therefore,
2
∼ In continuous limit, p = (p ,p ) is in an Euclidean
making an SO(3) transformation, x y
spaceR2. However,thereisaconstraint u 2+ v 2 =1,
p p
ψ′(r,t)=U(r,t)ψ(r,t), (30) whichparameterizesasphereS2. As p | | ,ξ|p | Ep.
| |→∞ →
Then, v 0 as p . Therefore, we can compact
p
Dirac equations read R2 as an S→2 by ad|d|in→g ∞in which v 0 to R2. The
p
∞ →
eµaγaDµψ′+imψ′ =0, (31) snphe=re(|∆up(1|)2,+∆|v(p2)|2,ξ=)/1Ecanbeacalsuosebenpara=m1et.er(iuzed,vby)
p p p p p p p p
where the covariantderivative Dµ =∂µ+Aµ with Aµ = describes a m−apping from S2 (p |R2|) to S2 (spinor
Uga−u1g∂eµwUitwhitahvUan∈isShOin(g3)s.trTenhgisthgaFug=edpAot+enAtial Ais =a p0u.re t|nopp|olo=gic1a).linTvhareiawnitn.Tdihnegnnourtmhbpeorl∈eofistuhe =ma1p,vpin=g 0isaat
∧ p p
We now discuss the nature of the gapless B phase p = andsouth pole u =0,v =1 at p=0. For n
p p p
in the general model with G-inversion symmetry. In |pa|ram∞etrization,n =(0,0,ξp)at p = and(0,0,ξp)
this case, E = 0 at p = p which are the solu- 0 E | | ∞ E
p ∗ at p=0, either the north pole or south pole.
±
tions of ξ = 0 and, say, ∆ = ∆ = 0. At p ,
p p 1,p ∗ For strong pairing phase, we know that u 1 and
p
the fermion dispersions are generally given by 2D Dirac →
v 0 as p 0 (or equivalently, ξ > 0). This
p p
cones. However, by a continuous variation of the pa- → →
meansthatforarbitraryp,(u ,v )mapsp-spheretothe
p p
rameters, one can realize a dimensional reduction near
up-hemishpere and then winding number is zero. That
the phase boundary where the effective theory is in fact
is, the topological number is zero in the strong pairing
a 1+1 dimensional conformal field theory in the long
phase.
wave length limit. Let us consider parameters that are
For the weak paring phase, u 0 and v 1 as
p p
close to the critical line with sinp cosp where → →
| ∗a| ≪ | ∗a| p 0. This means that the winding number is not zero
g = sgn[q ∆ cosp + q ∆ cosp ] sgn(q ) with →
q x 1x ∗x y 1y ∗y ≡ x′ (atleastwrappingonce). Forourcase,itmaybedirectly
q=p p . Doing the Fourier transform, we find
∗ calculated and the winding number is given by
−
g(r) = Z dqx′dqy′eiqx′x′+iqy′ysgn(qx′) ν = 41π Z dpxdpy∂pxnp×∂pynp·np =1 (34)
q δ(y )
= δ(y′)Z dqx′ |qxx′′|sinqx′x′ ∼ x′′ . (32) pDoenfiennintgofPt(hpe)=pro21j(e1ct+onpper·a~σt)o,rwthoicthheisntehgeaFtiovuerisepreccotrma-l
The δ(y)-function indicates that the pairing in the gap- space of the Hamiltonian, this winding number is iden-
less B phase has a one-dimensional character and the tified as the spectral Chern number defined by Kitaev
ground state is a one-dimensional Moore-Read Pfaffian. [7]
The BdG equations reduces to
1
ν = Tr(P (∂ P ∂ P ∂ P ∂ P )]dp dp (.35)
i∂tu=−i∆1x(1+iη)∂x′v, i∂tv =i∆1x(1−iη)∂x′u(,33) 2πiZ − px − py − py − px − x y
with η = ∆1y. Thus, the gapless Bogoliubov quasipar- There is a topological invariant called Atiyah-Padoti-
∆1x
ticles are one-dimensional Majorana fermions. The long Singereta-invraiant[33]whichreflectsthe asymmetryof
8
the spectrum of the Dirac operator where the first term is equal to Γ/2πi with Γ the WZ
term. The Maslov triple index τ (B,P,g) is an integer
µ
1
η¯(S2)= lim dλρ(λ)sgn(λ)λ−s, (36) [39]. Wedonothaveaphysicalexplanationofτµ(B,P,g)
2s 0Z | | yet and it possibly relates to the central charge of the
→
theory[40]. η¯isthereducedeta-invariant. Thefirstterm
whereρ(λ)isthespectraldensity. Transformingthevari-
in (40) determines the bulk state topological properties
ablefromλtop,themeasureoftheintegrationfrom∆
p
andthelattertwotermsreflecttheboundarytopological
to q includes a Jacobian determinant [29],
properties. That the index Ind T τ is an integer
g µ
−
1 1 determines the bulk-boundary correspondence.
dλ...= d∆(1)d∆(1) ...
Z 4π Z p p Ep The WZ term Γ is defined for the fundamental rep-
resentation of SU(2) but g is restricted to a subgroup
1 1
= d2pJ(∆ ,p) ... SU(2)/Z SO(3). The reduced eta-invariant is given
p 2
4π Z Ep by ∼
1
= d2pTr[J(n ~σ,q)...]
16πiZ p· η¯(B,g)=η¯(S+2,gτ=1)+η¯(S2,gτ=0)
−
NthoeteHtahmaitlt−on21inanp,·o~σneishtahse signature matrix sgn(H(p))of =[+η¯([Sη¯+2(S,g2τ,=g1)−)η¯+(Sη¯+2(S,g2τ,=g0)] )]
+ τ=0 τ=0
−
η¯(S2) = 1 lim dλρ(λ)sgn(λ)λ−s ≡∆η¯(S+2)+η¯(S2,gτ=0), (41)
2s 0Z | |
→ because η¯(Y) = 0 for Y may contract to a cycle S1.
1 1
= ZS2dpTr[2sgn(H(p))ρ(2sgn(H(p)))],(37) Iηn(D(gSe2n,egral,))].η¯(S±2B,egc1a,0u)se =o12f[dima(kernDon(S-z±2e,rgo1,0))gap+,
1,0
dim(ke±rD(S2)) = 0. Therefore, η¯(S2,g) =
where
1∆η(S2) +± 1η(D(S2,g )). The discrete eigen-
1 i ∂sgn(H(q))∂sgn(H(q)) 2 + 2 τ=0
ρ( sgn(H(q))) = states of the Dirac operator do not contribute to η¯
2 16π(cid:18) dqx dqy because there is no asymmetry ofthe spectrum for these
∂sgn(H(q))∂sgn(H(q)) states. η¯(S2,gτ=0) = η¯(S2) is just the eta-invariant
. (38) calculated in the previous subsection.
− dq dq (cid:19)
y x
Now, the index theorem reads
Defining the spectral projector I P = P (q) = 1(1
sgnH(q)), the eta-invariant is exa−ctly equ−al to one2 ha−lf Ind Tg =Γ/2πi−ν/2−∆η¯(S+2)+τµ(B,P,g). (42)
of the spectral Chern number defined by Kitaev [7]
The integrity of Ind T τ (B,P,g) requires
g µ
−
1
η¯(S2)= Tr[P (q)dP (q) dP (q)]. (39)
4πiZ − − ∧ − ∆Γ/2πi≡Γ/2πi−∆η¯(S+2)=ν/2 mod(Z). (43)
Kitaevidentifiesonehalfofν asthechiralcentralcharge DaiandZhanghavethought∆η¯(S2)asanintrinsicform
+
c . Our result shows that this chiral central charge is
ofthe WZ term[25]andthen ∆Γis infactanambiguity
ju−st the eta-invariant. Physically, it is easy to be under- ofthe WZ term. It is seenthat,if ν is even, one requires
stood because both c and η¯ reflect the anomaly of the ∆Γ = 2πi integer. If ν is odd , ∆Γ/2πi is required
spectrum of the syste−m. to be a hal×f integer. In our model, it is known that for
an SO(3) group, ∆Γ = πi integer, which is consistent
×
with ν = 1. In general, the index theorem (40) gives a
B. Index Theorem
constraint to the WZ term. An odd ν requires an even
level k WZ term in the effective action and the minimal
In the continuous limit, if the space is compacted as one is k = 2. This is consistent with the non-abelian
S2,the 2+1space-timeisaballX withaboundaryB = anyonic statistics of the vortex excitations. For an even
Sbo+2t(tτom=h1a)l∪veYs o∪fSa−2s(pτh=ere0)anwdheYreiSs±2a acryelinthdeert.opNaonwd, aνb,etlhiaenmainnyimonailcvsatlauteisotficks.is one, it is consistentwith the
we can apply the index theorem (A2) to this spectral
problemoftheDiracoperatorD inX withboundaryB
µ
[25]. The general form of thye index theorem in an odd-
VII. EDGE EXCITATIONS AND BULK-EDGE
dimensional manifold is briefly reviewed in Appendix A.
CORRESPONDENCE
In 2+1-dimensions, the index theorem for the Toeplitz
operator reads
In the previous discussion, the two-dimensional space
1 is taken to be a tours without boundary. If we consider
Ind Tg = 24π2 Z Tr[(gdg−1)3]−η¯(B,g)+τµ(B,P,g()40) a two-dimensional space with edge instead of the torus,
X
9
Kitaev has shown that the gapless chiral edge excita- thattheoperatoractson. AtiyahandSinger(AS)proved
tions coexist with a non-zero spectral Chern number[7]. ageneralformoftheindextheoremforevendimensional
This is a general result if the bulk states are gapped. compact manifold [32]. It said that the analytical index
In fact, it was generally known that the eta-invariant of of an elliptical differential operator on the vector bundle
the Dirac operator can be related to the ground state is a global topological invariant which can be expressed
fermion charge [29, 30, 31]. By using the continuous bytheintegraloflocaltopologicalcharactersontheback-
equation ∂ ρ+ j = 0, the eat-invariant is related to groundmanifold. Ageneralizationofthetheoremtoeven
t
∇·
the edge current integrated along the one-dimensional manifoldswithboundary,so-calledAtiyah-Patodi-Singer
edge S, i.e.,[29] (APS) theorem [33], e.g., for a Dirac operator D on a
manifold M with boundary ∂M, reads
η =2 ds j , (44)
ZS · IndD = Aˆ(M)+ 1(h[∂M]+η[∂M])+ω(∂M),(A1)
wherethefactor’2’isdifferentfrom’1’in(104)ofKitaev Z 2
M
in [7] because of 1 factor in (5). Thus, a non-zero eta-
2
invariant corresponds to a non-vanishing net edge cur- where Aˆ is the Hirzebruch Aˆ-class of M, h is the di-
rentandthen the gaplesschiraledge excitations. Hence, mensions of the zero modes of the boundary Dirac op-
the index theorem already explained the bulk-edge cor- erator D and η is the APS eta-invariant defined by
∂M
respondence. η(∂M)=lim sgn(λ)λ s where λ is the eigen-
valueofD∂Ms.→ω0(P∂Mλ6=)0isaChe|rn|−-Simonstermcausedby
the non-product boundary metric and physically giving
VIII. CONCLUSIONS anomaly in quantum field theory [34]. There are many
applications of the AS and APS theorem in physics, e.g,
We studied a generalized Kitaev model whose vortex- see Refs.[29, 30, 31]. Recently, index theorems were ap-
free sector can be mapped to a p-wave paired state with pliedtochiralp-wavesuperconductors[35]andgraphene
the next nearest neighbor hopping. The phase diagram [36].
is figured out. The property of the gapped B phase was There are partners of AS and APS theorems on odd-
very interesting for a Pfaffian/anti-Pfaffian phase tran- dimensionalmanifoldsX (d=odd)[25,37,38]. Again,we
sition was found in this phase. According to the gauge consider Dirac operator D. L = λLλ is the spectrum
invariance of the spin-1/2 theory in the fermion repre- space of D where λ is the subspPace with eigenvalue λ.
L
sentation, we found the low-lying effective theory of the P isaprojectoperatordefinedbyPL=L+ = λ 0Lλ.
model is described by Majorana fermion coupled to a Let trivially take its value on CN, i.e., a staPte ≥ψ
L ∈ L
gauge field. The existence of this non-dynamic gauge is extended to an N vector which transforms under a
field enabled us to understand the mathematic connota- group GL(N,C). For Toeplitz operator T = PgP on
g
tionbehindthesetopologicalorders. Theedgeconformal odd-dimensionalmanifolds X withboundaryB inwhich
anomalycan be cancelledby the bulk WZ term in terms the groupelement g is not an identity, an index theorem
of the recently proved index theorem on odd manifold. is given by [25]
1
Ind T = Aˆ(RX)Tr[exp( R )]ch(g)
IX. ACKNOWLEDGEMENT g −(2πi)d+21 ZX − L
η¯[B,g]+τ (B,P,g), (A2)
µ
The author wouldlike to thank Z. Nussinov, N. Read, −
X.Wan,Z.H.Wang,X.-G.Wen,T.Xiang,Y.S.Wu,K. whereg GL(N,C)(orasubgrouplikeSU(N)); RX and
Yang,J.W.YeandM.Yuforusefuldiscussions. Theau- R aret∈hecurvaturesofthebackgroundmanifoldX and
L
thorisgratefultoWeiping Zhangforhimtoexplainhow ; η¯[B,g] is a reduced eta-invariant and τ , which is an
µ
to understand the index theorem (A2). Especially, the Linteger, is the Maslov triple index [39]. We do not have
authorgreatlyappreciatesZiqiangWangforoursuccess- aphysicalexplanationofτ yetanditpossiblyrelatesto
µ
fulcooperationinref. [12]. Partofthisworkisthegener- the central charge of the theory [40]. ch(g) is odd Chern
alizationofourco-work. Thisworkwassupportedinpart character defined by
by the national natural science foundation of China, the
national program for basic research of MOST of China d−1
2 n!
and a fund from CAS. ch(g)= Tr[(g 1dg)2n+1]. (A3)
−
(2n+1)!
nX=0
APPENDIX A: INDEX THEOREMS: Thefirsttermin(A2)determinesthebulkstatetopologi-
MATHEMATICAL PREPARATION calpropertiesandthelattertwotermsreflectthebound-
ary topological properties. That the index Ind T τ
g µ
−
Index theorems relate the analytical index of a differ- is an integer determines the bulk-boundary correspon-
entialoperatortothetopologicalindexofavectorbundle dence.
10
APPENDIX B: SPIN AND MAJORANA That is,
FERMIONS
bx =c† +c , by =i(c† c ),
Inthisappendix,weintroducetheMajoranarepresen- ↑ ↑ ↑− ↑
bz =c† +c , c=i(c† c ).
tationofspin-1/2operators. ConsiderthePaulimatrices ↓ ↓ ↓− ↓
It is easy to check that
0 1 0 i 1 0
σx = ,σy = − ,σz = , (B1)
(cid:18)1 0(cid:19) (cid:18) i 0 (cid:19) (cid:18)0 1(cid:19)
− b2 =1, c2 =1
a
with σxσy = iσz and so on. The spin-1/2 matrices are b b = b b ,cb = b c (B6)
a b b a a a
− −
one-half of the Pauli matrices: S= ~σ. Casimir of SU(2)
2
group requires S S = S(S +1) = 3/4 , i.e., ~σ ~σ = 3. Therefore,usingthe Majoranafermions,onecanexpress
· ·
Using the conventional fermion operators c and c , the the spin operators by
spin-1/2 operators can be expressed by ↑ ↓
i
σˆx = (b c b b ,
σˆx =c†c +c†c 2 x − y z
↑ ↓ ↓ ↑ i
σˆy = i(c†c c†c ) σˆy = (byc bzbx),
− ↑ ↓− ↓ ↑ 2 −
σˆz =c†↑c↑−c†↓c↓. (B2) σˆz = 2i(bzc−bxby). (B7)
oTnheathaiss σσˆˆxaσˆ=y =c†sσiσˆsasz′.csH′.owSeinvceer,{itc,ics†e}a=sy1t,ocs2ee=tch†a2t= 0, Correspondingly, the constraint reads
i
(σˆa)2 =n +n 2n n Tx = (b c+b b )=0,
↑ ↓− ↑ ↓ 2 x y z
. Toinsurethe Casimiroperatorconstraint,onerequires Ty = i(b c+b b )=0,
y z x
2
n +n =1, n n =0 i
↑ ↓ ↑ ↓ Tz = (b c+b b )=0. (B8)
z x y
2
These are equivalent to an SU(2) constraint
Compactly,itisD =b b b c=1. Inanexplicitgaugein-
Tx =c†c† +c c =0, variant form, D = iσˆxxσˆyyzσˆz =1. Using this constraint,
↑ ↓ ↓ ↑ −
Ty =i(c†c† c c )=0, the spin operators are simplified as
↑ ↓− ↓ ↑
Tz =n +n 1=0 (B3) σˆa =ib c. (B9)
↑ ↓− a
with TxTy = iTz and so on. It is well-known taht
σˆa/2,Ta/2 form anSO(4) SU(2) SU(2)/Z Lie
2
{ } ∼ × APPENDIX C: ISOLATED VORTEX
algebra.
EXCITATIONS
Thefermionexpressionbringsextradegreesoffreedom
and then there is an SU(2) gauge invariant of the spin
operators. Defining The isolated vortex excitations above the Pfaffian
ground state has been discussed in [12]. For self-
c c containing of the paper, we repeat the paragraph which
(χαβ)=(cid:18)c↑† c↓† (cid:19), (B4) discuss the vortex excitations. The Z2 vortex excitation
−
↓ ↑ inthespinmodelwhichcorrespondstosettingW = 1
P
−
the spin operator may be rewritten as for a given plaquette. Although the Hamiltonian in the
fermion representationis bilinear, it is difficult to obtain
1
σˆa = Tr[χ χ(σa)T] (B5) analytical solutions of the wave function with vortex ex-
†
2
citations[7]. Ourstrategyistoevaluatetheenergyofthe
Making a gauge transformation χ g χ and Moore-Read [5] trial wave function with two well sepa-
αβ αγ γβ
σtˆhaenisχg†αaβug→e iχnv†αaγrgiγ†aβntf.or g ∈SU(2) and g†→=g−1, one has r.aItnedthhealgfa-vpopretdiceBsplohcaasteedwiatthwp1xa+nidpyw-2washvoewpnaiirninFg,ig. 4
Majorana fermions are related to the conventional
Ψ(z ,...z ;w ,w ) Pf(g (z ,z ;w ,w )),
fermion through 1 N 1 2 ∝ ′ i j 1 2
(z w )(z w )+(w w )
1 1 2 2 1 2
1 1 g′(z1,z2;w1,w2) − − ↔ .
c↑ = 2(bx−iby), c↓ = 2(bz −ic) 1∝ z1−z2
c†↑ = 12(bx+iby), c†↓ = 21(bz +ic) |w1,w2i∝exp{2rX1,r2g′(z1,z2;w1,w2)d†r1d†r2} (C1)