Projected BCS states and spin Hamiltonians for the SO(n)_1 Wess-Zumino-Witten model

0.29 MB·English

by  Hong-Hao Tu

Projected BCS states and spin Hamiltonians for the SO(n) WZW model
1
Hong-Hao Tu
Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany
(Dated: December 11, 2012)
We propose a class of projected BCS wave functions and derive their parent spin Hamiltonians.
ThesewavefunctionscanbeformulatedasinfiniteMatrixProductStatesconstructedbychiralcor-
relatorsofMajoranafermions. In1D,thespinHamiltonianscanbeviewedasSO(n)generalizations
ofHaldane-Shastrymodels. Wenumerically computethespin-spincorrelation functionsandR´enyi
entropiesfor n=5 and 6. Together with theresults for n=3 and 4, we conclude that these states
2 arecritical and theirlow-energy effectivetheory istheSO(n) Wess-Zumino-Witten model. In2D,
1
1
we show that the projected BCS states are chiral spin liquids, which support non-Abelian anyons
0
for odd nand Abelian anyonsfor even n.
2

c PACSnumbers: 75.10.Pq,11.25.Hf,03.65.Fd
e
D
Introduction.– Efficient description of quantum many- mericalresultswithfieldtheorypredictionsfromSO(n)
1
0
bodysystemsisachallengingprobleminmodernphysics, criticality. Togetherwiththeknownresultsforn=3and
1
as the dimension of the Hilbert space scales exponen- 4, we expect that for general n these states are critical

] tially with the number of particles. For strongly inter- and belong to the SO(n)1 WZW universality class. We
l
e acting many-body systems, much of our understanding also show that the projected BCS states with modified
- of their properties comes from physically motivated trial Cooper pair wave functions provide a good description
r
st wave functions and/or exact solutions of specific mod- for Ising ordered and disordered phases close to SO(n)1
. els. A great success of the trial wave function approach criticality. In2D,theprojectedBCSstatesarechiralspin
t
a is the celebrated Laughlin wave function for the frac- liquids with p+ip pairingsymmetry. We find that these
m tional quantum Hall effect at 1/m (with m odd) filling topological states support non-Abelian Ising anyons for
- [1]. Toward exact results, Bethe’s solution of the spin- odd n and Abelian anyons for even n, respectively.
d
1/2Heisenbergchain[2]andintegrabilityofthespin-1/2 Projected BCS wave function.– Constructing the pro-
n
Haldane-Shastry model [3] provide invaluable insight for jectedBCSwavefunctionsreliesonaslave-particlerepre-
o
c critical spin-1/2 chains. sentationoftheSO(n)algebra. Letusstartfroma1Dpe-
[ riodicchainwithevenN sites,wherethenvectorsineach
Thejustificationoftrialwavefunctionsisusuallyadif-
site are represented by using singly occupied fermions,
2 ficulttask. Forexample,therelevanceofAnderson’sres-
v onating valence bond (RVB) state [4] for the mechanism |nai = c†a|0i (a = 1,...,n). In terms of fermions, the
81 ofhigh-Tc superconductivity is still a controversialissue. SO(n) generators are written as Lab = i(c†acb − c†bca),
A useful technique for justifying trial wave functions is where1 a<b n. Toremoveunphysicalstatesinthis
4 ≤ ≤
1 to study their parent Hamiltonians for which the trial fermionicrepresentation,asingle-occupancyconstraintis
0. wave functions are exact ground states. For Laughlin required, na=1c†j,acj,a = 1 ∀j =1,...,N, which defines
1 wave function, the parent Hamiltonian which consists of a GutzwilPler projector PG. Then, the projected BCS
2 certainHaldanepseudopotentials[5]differsfromphysical wave function of our interest is defined by
1 Coulomb interactions but their difference can be viewed
Xiv: aspsina-1pAerKtuLrTbasttiaotnea[6n]d. itSsimpailraerntsiHtuaamtiioltnonaiarinse[s7],fowrhtihche |Ψi=PGexpXi<j zi−1 zj Xa=n1c†i,ac†j,a|0i, (1)
contains an extra biquadratic term apart from Heisen-  
ar berg interactions. Since the spin-1 AKLT model can be where na=1c†i,ac†j,a creates an SO(n) singlet between
adiabatically connected to the spin-1 Heisenberg chain sitesiaPndj. Notethat Ψ isacoherentsuperpositionof
without closing the gap, it is widely believed that the valence-bondsinglets of|aribitraryrange(See Fig. 1)and
AKLT state qualitatively captures the physics of spin-1 hence can be viewed as an RVB state [4]. If we choose
Heisenberg chain. z = exp(i2πj), the amplitude of the Cooper pair wave
j N
In this work, we propose a class of projected BCS function 1/zi zj is the inverse of the chord distance
| − |
statesandderivetheirparentHamiltonians. Thesestates betweenthesites. Underthischoice, Ψ isbothrealand
| i
canalsoberepresentedasinfiniteMatrixProductStates translationally invariant, which is the uniform case that
(MPS) [8] constructed from chiral correlators of Majo- we will consider in the following.
rana fermions. In 1D, the spin Hamiltonians are SO(n) Before discussing the properties of Ψ for general n,
| i
generalizations of Haldane-Shastry models. We numer- we establish the relation between (1) and some known
ically calculate the spin-spin correlation functions and results. Forn=3,afterswitchingtothestandardspin-1
the R´enyientropiesforn=5and6andcomparethe nu- basis n1 = 1 ( 1 1 ), n2 = i ( 1 + 1 ),
| i √2 |− i−| i | i √2 |− i | i
2
which the Majorana fields act is an infinite-dimensional
Hilbert space. This allows the infinite MPS (3) to de-
scribe the expected SO(n) criticality with unbounded
1
increase of the entanglement entropy.
+ ThekeybenefitoftheinfiniteMPSformulationisthat
a parentHamiltonian can be derived,such that (3) is its
exact groundstate. As shown in [9], the presence of null
.,*- ,+. vectors in conformal field theories (CFT) leads to a set
ofoperatorswhichannihilatetheinfiniteMPS.Following
this approach,we have derived such operators for (3)
*
w 3
FIG. 1: (color online). Schematic of a valence bond config- Λab = ij[2Lab Lab(L~ L~ )+(L~ L~ )Lab],
i 3 j − n 1 i i· j i· j i
uration in the projected BCS state (1). The valence bonds jX(=i) −
(blue)areSO(n)singlesformedbytwoSO(n)vectors. Inthe 6
uniform case zj =exp(i2Nπj), the periodic chain is viewed as wherew (z +z )/(z z )andL~ L~ LabLab.
a unit circle and |zi−zj| is the chord distance between two ij ≡ i j i− j i· j ≡ a<b i j
sites. Since Λaib|Ψi = 0 ∀i,a,b and iLaib|Ψi =P0 ∀a,b, we
candefineaparentHamiltonianPH = i,a<b(Λaib)†Λaib+
n3 = 0 , we find that the projected BCS state (1) 2(N3−2) a<b( iLaib)2 +E0 whose gPround state is the
|is eiquiva|leint to the spin-1 Haldane-Shastry state, which infinitePMPS(P3)withenergyE0. ChoosingE0 =−92(n−
1)N(N2 4), the explicit form of H is given by
has been considered in Ref. [9–11]. It was shown [9, 10] −
that this state is criticalandits low-energyeffective the- n+2 n 4
ory is an SU(2)2 (or equivalently SO(3)1) WZW model. H = − wi2j[ 3 (L~i·L~j)+ 3(n− 1)(L~i·L~j)2]
In a recent work [12], a projected BCS wave function Xi=j −
6
similar to (1) is used as a variational ansatz to describe n−4 w w (L~ L~ )(L~ L~ ). (4)
ij ik i j i k
thephasesinspin-1bilinear-biquadraticchain,including −3(n 1) · ·
− i=Xj=k
theTakhtajan-Babujianmodel[13]whichalsobelongsto 6 6
SU(2)2WZWuniversalityclass[14]. Forn=4,afterrep- Generically, the Hamiltonian (5) is a long-ranged SO(n)
resenting the four SO(4) vectors by two spin-1/2 states, bilinear-biquadratic model with three-spin interactions.
wefindthattheprojectedBCSstate(1)canberewritten For n = 4, as we expected, the Hamiltonian only has
as a product of two decoupled spin-1/2 Haldane-Shastry inverse-square Heisenberg exchange interactions, which
states. Animmediate consequenceofthis decomposition can be decomposed into two spin-1/2 Haldane-Shastry
isthattheSO(4)stateiscriticalandrepresentsthefixed Hamiltonians due to SO(4) SU(2) SU(2).
point of the SO(4)1 WZW model. Jastrow versus Pfaffian.–≃It is ×well-known that the
Infinite MPS and parent Hamiltonian.– From the SO(n) Lie algebra has a sharp difference between even
known results for n = 3 and 4, one may speculate that and odd n [17]. As we shall see, this leads to distinct
for general n the projected BCS state (1) belongs to the forms of the wave function (1) in Cartan basis for even
SO(n)1 WZW universality class. Let us further uncover and odd n: The former has a pure Jastrow form, while
this relationship by formulating (1) as an infinite MPS. the latter includes a Pfaffian factor. To see this differ-
The SO(n)1 WZW model has a primary field with con- ence, let us consider SO(2l) and SO(2l+1) (l: integer)
formalweighthv =1/2in the vector representation[15], and choose mutually commuting Cartan generators as
which can be naturally interpreted as Majorana fermion L12,L34,...L2l 1,2l. For SO(2l), a convenient choice of
−
fieldsχa(z)(a=1,...,n). ThisMajoranarepresentation the Cartan basis is defined by 0,...,m = 1,...,0 =
α
of the primary field allows us to rewrite the projected (n2α in2α 1 )/√2 (α = 1|,...,l), wher±e m is ithe
− α
BCS state (1) as the following infinite MPS: e|igenvia±lue| of L2iα 1,2α. For the vectors 0,...,m =
− α
|
1,...,0 , we label their positions in the spin chain by
n ± i
|Ψi= Ψ(a1,...,aN)|na1,...,naNi, (2) x(1α) <···<x(Nαα). Inthisbasis,the wavefunction(1)for
a1,..X.,aN=1 even n=2l takes the form
wherethe coefficients arethe chiralcorrelatorsofN Ma- l
Ψ( m )=ρ (z z )mα,imα,j, (5)
jorana fields [16] m i j
{ } −
αY=1Yi<j
Ψ(a ,...,a )= χa1(z )χa2(z ) χaN(z ) . (3)
1 N 1 2 N
h ··· i where ρ = sgn(x(1)...x(1)...x(l)...x(l)) (sgn: signa-
m 1 N1 1 Nl
Unlike usual MPS with finite matrix dimensions, the ture of a permutation) if m = 0 α and ρ = 0
i α,i ∀ m
state(3)isaninfiniteMPS[8]sinceitsancillaryspaceon otherwise. P
3

For SO(2l +1), apart from the vectors 0,...,mα = >] −2 (a)
f±L|nua1n2bl,c+e.t1l.iioi.n,n,g0w(it1,hh)tiechfihroerrpiesoodesaxidtniisnnotinsh=sialanb2tylea+ddxd(11b0i)tyiiso<awnlalr·li·tC·vtaee<nrcttaaxonsr(N|00|)g0,e,tn.h.ee.r,aw0toiarv=se. x1212ln[(−1) <LLjj+x−−−−8645 Mffiitt Cηη (==N 11=−..22240250) −3 −2 −1 0
ln[sin(πx/N)]
l
1
Ψ({m})=ρmPf0(zi−zj)αY=1Yi<j(zi−zj)mα,imα,j, (6) 1212LL>]jj+x−−42 (b)
whiemreαρ,im==0s∀gαn(xan(10d).ρ.m.x=(N00)0,xo(1t1h)e.r.w.xis(Ne11,).a.n.dx(1tlh)e..P.xfa(Nlffi)l)ainf xln[(−1) < −−−865 Mffiitt Cηη (==N 11=−..2454020) −3 −2 −1 0
Pfactor Pf0(zi−1zj) is restricted to the positions for the ln[sin(πx/N)]
extra vector 0,...,0 .
| i FIG. 2: (color online). Spin-spin correlation function
Numerical results.– The power-law decaying correla- (logarithmic scale) ln[(−1)xhL12L12 i] as a function of
j j+x
tion functions and the universal scaling of entanglement
ln[sin(πx/N)] in the projected BCS state (1) for N = 200
entropy [18] are characteristic behaviors of conformal and (a) n = 5 and (b) n = 6. The solid lines (red) are fits
critical points in 1D. Even though these quantities are oftheform ln[(−1)xhL12L12 i]=ηln[sin(πx/N)]+A,where
j j+x
difficult to be computed analytically for (1), the Jastrow η and A arefittingparameters. Thedotted lines are also fits
and Pfaffian forms (6) and (7) of the wave functions are of this formula but with the field theory prediction η = n/4
(Ref. [20]).
very suitable for determining them numerically via the
Metropolis Monte Carlo (MC) method [19]. Below we

focus on the projected BCS state (1) with n = 5 and (a)
3
6 and provide numerical evidence that they belong to
SO(5)1 and SO(6)1 WZW models, respectively. (2)SL2 MC (N=200)
1 fit c = 2.31
For critical spin chains in SO(n)1 WZW universality fit c = 2.5
class, field theory predicts that for n < 8 the spin-spin −1 .2 −1 −0.8 −0.6 −0.4 −0.2 0
correlation function behaves as LabLab ( 1)x/xη ln[sin(πL/N)]/4
h j j+xi ∼ −
with η =n/4 [20]. For the projected BCS state (1) with
n=5and6,wehavecomputedthetwo-pointspincorre- 4 (b)
Nlato=r 2hL001j2.LT1j+2hxei.crFitiigca.l2exsphoownesntthsethnautmbeersictafiltrewsuitlhtsofuorr (2)SL23 MC (N=200)
1 fit c = 2.76
numerical data are η = 1.22 for SO(5) and η = 1.42 for fit c = 3

SO(6) (solid lines in Fig. 2), which agree very well with −1.2 −1 −0.8 −0.6 −0.4 −0.2 0
ln[sin(πL/N)]/4
the field theory predictions (dotted lines).
The entanglement entropy that is easily accessible via FIG. 3: (color online). R´enyi entropy S(2) as a function of
L
MC method is the R´enyi entropy S(2) = lnTrρ2 (See ln[sin(πL/N)]/4 in the projected BCS state (1) for N =200
Ref. [8, 21–23]), where ρ is the redLuced d−ensity mLatrix and (a) n = 5 and (b) n = 6. The solid lines (red) are fits
ofthestateinasubsystemLoflengthL. ForSO(n)1WZW of the form SL(2) = cln[sin(πL/N)]/4+c′2, where c and c′2
are fitting parameters. The dotted lines are also fits of this
modelwithc=n/2weexpectS(2) =cln[sin(πL/N)]/4+
L formula but the central charge c is fixed to c = n/2 of the
c′2 [18], where c′2 is a constant. For n=5 and 6, we plot SO(n)1 WZW model.
S(2) as a function of ln[sin(πL/N)]/4 for N = 200 in
L
Fig. 3. From our MC data, the estimates of the central
charge are c = 2.31 for SO(5) and c = 2.76 for SO(6) is whether there exist a modified version of (1) and its
(solid lines in Fig. 3), which are close to the predicted parent Hamiltonian that represent the fixed point of the
c=n/2 (dotted lines) but show some deviations. SO(n) WZW model.
1
The origin of the small deviations of the numericalre- Away from SO(n) criticality.– After showing that
1
sultsandtheSO(n) predictionsmaybeduetothepres- the projected BCS state (1) captures the physics of the
1
ence of marginally irrelevant terms in the SO(n) WZW SO(n) WZW model, it is natural to ask whether simi-
1 1
model for (1) and its parent Hamiltonian (5), unlike the larprojectedwavefunctionsarerelevantforgappedspin
SU(n) Haldane-Shastry models [24, 25] (including the chains away from (but close to) SO(n) criticality. Let
1
spin-1/2 Haldane-Shastry model for n = 2) being the us restrictourselvesto SO(n)symmetric models forsim-
fixed points of the SU(n) WZW model. For n = 3, the plicity. According to the well-known result by Witten
1
presence ofmarginalterm inthe spin-1 Haldane-Shastry [26], the SO(n) WZW model is equivalent to n mass-
1
model has been confirmed numerically [9, 10]. If this lessMajoranafermions,i.e., nIsingmodelsatcriticality.
is also the case for n 5, an interesting open question Forthiscriticaltheory,theonlyrelevantperturbational-
≥
4
lowedby SO(n)symmetry is the mass termof Majorana onic quasiparticle excitations in these 2D states, which
fermions. Thus, the low-energy effective theory has the have intriguing properties depending on the parity of n.
following Hamiltonian density: A more detailed analysis of these 2D states will be pre-
sented elsewhere.
n n
iv Interestingly, the quasiparticles built upon the SO(n)
= (ξν∂ ξν ξν∂ ξν) im ξνξν, (7)
H −2 R x R− L x L − R L states support non-Abelian statistics for odd n. Let us
Xν=1 Xν=1
adapttheCFTapproachofcreatingquasiholeexcitations
whereξν areright(left)movingMajoranafermions,v in FQH states [33] to our spin system. For odd n, the
R(L)
andmaretheirvelocityandmass. Herewehaveassumed SO(n)1 WZW model has three primary fields: identity
four-fermioninteractionsare weakandcanbe neglected, field I (singlet), vector field v (Majorana fermions), and
sincetheyaremarginalandonlyrenormalizethemassof spinor field s. Following the CFT approach, creating
Majorana fermions at low-energy limit [27]. quasiparticles in the SO(n) state is achieved by adding
The SO(n) criticality corresponds to m = 0. The spinor fields s in the chiral correlator (3). Then, the
1
two gapped phases adjacent to the SO(n) criticality statisticsofquasiparticlesareencodedinthefusionrules
1
are i) Ising ordered phase (m < 0) and ii) Ising disor- of the primary fields. In fact, the spinor fields have a
dered phase (m > 0). For these two phases, we note nontrivialfusionrules s=I+v,togetherwiths v=s
× ×
that they can be well described by modified projected andv v =I. Thesefusionrules resemblethosein Ising
×
BCS states. Actually, these two gapped phases and an CFT(σ σ =I+ε,σ ε=σ, andε ε=I), whichare
× × ×
SO(n) critical point (Reshetikhin model [28]) are real- responsiblefor the non-Abelianstatistics ofIsing anyons
1
izedintheSO(n)bilinear-biquadraticchain[20,29].The [37]. Indeed, the Majorana free field representation of
ideal example that belongs to the Ising ordered phase SO(n)1WZWmodelallowsustoidentifythespinorfields
is the SO(n) AKLT model [7, 30, 31], whose ground s with conformal weight hs = n/16 as a product of n
state can be represented as a projected BCS state, by Ising σ fields (hσ = 1/16). Thus, we conclude that the
replacing g = 1/(z z ) in (1) with g = 1 [12]. For SO(n) states support non-Abelian Ising anyons for odd
ij i j ij
−
the Ising disordered phase, the ground state of the spin n. Note that the case with n=3 recovers the physics of
chain is dimerized [20] and hence the valence bonds are the Moore-Readstates [33, 38], while for oddn 5 they
≥
short-ranged. In this case, a proper Cooper pair wave are natural generalizations of the Moore-Read states.
function for the projected BCS state can be chosen as Now we show that the SO(n) states only support
g exp( z z /ξ),whereξ isthe lengthscaleofthe Abelian anyons for even n. This subtle difference roots
ij i j
∼ −| − |
valence bonds. In the extreme case, a Cooper pair wave in the fusion rules of the SO(n) primary fields. In con-
1
function that is nonvanishing only between neighbor- trasttooddncase,theSO(n) WZWmodelwithevenn
1
ing sites yields Majumdar-Ghosh-like state, which cor- has two spinor primary fields s ands [15], apartfrom
+
−
responds to perfect dimerization. These results imply the usual identity and vector fields. The fusion rules of
that both Ising ordered and disordered phases close to spinorandvectorfieldsares v =s ands v =s .
+ +
× − −×
SO(n) criticality are well described by projected BCS Dependingontheparityofn/2,thefusionrulesinvolving
1
stateswithproperlychoseng . Indeed,forn=3,itwas twospinorfieldsares s =s s =I, s s =v
ij +× + −× − +× −
shown [12] that the projected BCS states with Cooper for even n/2 and s s = s s = v, s s = I
+ + +
× −× − × −
pair wave functions generated from Kitaev’s Majorana for oddn/2. However,due to the absence of multiplicity
chains [32] are good variational wave functions for the in the fusion outcome, only Abelian anyons can exist in
Haldane (Ising ordered) and the dimerized (Ising disor- the SO(n) states with even n.
dered) phases. Conclusion and perspective.– To conclude, we have
2Dchiralspinliquids.–Afterestablishingtherelevance proposed a class of projected BCS states and derived
of projected BCS states (1) for SO(n) criticality in 1D, their parent spin Hamiltonians. These wave functions
1
we move on and discuss their properties in a 2D square also have an infinite MPS form generated by chiral cor-
lattice, wherethe z’sin(3)arenowcomplex coordinates relatorsofMajoranafermions. In1D,theycanbeviewed
of the lattice sites. In an analogy with fractional quan- as SO(n) generalizationsof Haldane-Shastrymodels and
tum Hall (FQH) states constructed by conformal blocks capture the physics of SO(n) WZW model. These re-
1
of their gapless edge CFTs [33, 34], the chiral correla- sults indicate that modified projected BCS states are
tor (3) from SO(n) WZW model (n massless Majorana good variational ansatz for describing Ising ordered and
1
fermions) yields chiral spin liquids, which break time re- disordered phases close to SO(n) criticality. In 2D, the
1
versalsymmetryandarespincounterpartsofFQHstates SO(n) states are chiral spin liquid states, which support
[35]. From the projected BCS form (1), the Cooper pair non-Abelian Ising anyons for odd n and Abelian anyons
wavefunction1/(z z )nowcorrespondstothetopolog- forevenn. Anopenquestionthatdeservesfurtherinves-
i j
−
icalphaseofp+ipsuperconductors[36]supportingchiral tigation is whether these 2D chiral spin liquids are rel-
gapless Majorana edge modes, which justifies the above evant for physical models and materials [39]. Moreover,
bulk-edge correspondence. Below we focus on the any- our 2D toy models may also shed light on an important
5
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6
Supplemental material
Equivalence of the projected BCS state and the infinite MPS
In this Section, we prove the equivalence of the projected BCS state and the infinite MPS.
Projected BCS state
Let us first expand the projected BCS state
n
1
|Ψi = PGexp z z c†i,ac†j,a|0i
Xi<j i− j Xa=1
 
n
1
= PG (cid:18)1+ z z c†i,ac†j,a(cid:19)|0i
aY=1Yi<j i− j
n N
1
= PGaY=1Na=0X(Na even)x(a)<X<x(a)Pfa(zi−zj)c†x1(a),a···c†xN(aa),a|0i
 1 ··· Na 
N n
1
= P Pf ( )
G a
··· z z
N1,N2,...,NXn=0(Na even)x(1)<X<x(1)x(2)<X<x(2) x(n)<X<x(n)aY=1 i− j
1 ··· N1 2 ··· N2 1 ··· Na
(c† c† )(c† c† ) (c† c† )0
× x(1),1··· x(1),1 x(2),2··· x(2),2 ··· x(n),n··· x(n),n | i
1 N1 1 N2 1 Nn
where N (a = 1,...,n) is the number of c fermions (i.e. na vector in the configuration), x(a) < < x(a) are
a †a | i 1 ··· Na
the positions of c fermions in the lattice, and the Pfaffian factor Pf ( 1 ) is restricted to the positions for the c
fermions. †a a zi−zj †a
The next step is to implement the Gutzwiller projection. Note that the Gutzwiller projector P requires single
G
occupancy. Thus, the positions of fermions (x(1) < < x(1), x(2) < < x(2), ... , x(n) < < x(n)) must be
all different from each other, so that each site1has ex·a·c·tly oNn1e fer2mion.··A·s a rNes2ult, we h1ave ·n·· N N=a N. After
a=1 a
implementing the Gutzwiller projector, we obtain P
N n
1
Ψ = Pf ( )
a
| i z z
N1,N2,...,Nn=0(N1+NX2+···+Nn=N andNa even)allallowedXx1(a)<···<xN(aa)aY=1 i− j
(c† c† )(c† c† ) (c† c† )0
× x(1),1··· x(1),1 x(2),2··· x(2),2 ··· x(n),n··· x(n),n | i
1 N1 1 N2 1 Nn
The final step is to rearrange the positions of fermionic operators so that they can be identified as a spin state.
This rearrangementonly results in a sign, depending on the positions of fermions
N n
1
Ψ = Pf ( )
a
| i z z
N1,N2,...,Nn=0(N1+NX2+···+Nn=N andNa even)allallowedXx1(a)<···<xN(aa)aY=1 i− j
sgn(x(1),...,x(1),x(2),...,x(2),...,x(n),...,x(n))x(1),...,x(1),x(2),...,x(2),...,x(n),...,x(n)
× 1 N1 1 N2 1 Nn | 1 N1 1 N2 1 Nni
where x(1),...,x(1),x(2),...,x(2),...,x(1),...,x(1) isaspinconfigurationlabeledbythepositionsofthevector na
| 1 N1 1 N2 1 Nni | i
and sgn(x(1),...,x(1),x(2),...,x(2),...,x(n),...,x(n)) is the signature of permutation due to the sign factor coming
1 N1 1 N2 1 Nn
from fermionic anticommutation relations.
Thus, the projected BCS wave function can be written as
n
1
Ψ( x(1) , x(2) ,... x(n) )=sgn(x(1),...,x(1),x(2),...,x(2),...,x(n),...,x(n)) Pf ( ) (1)
{ } { } { } 1 N1 1 N2 1 Nn aY=1 a zi−zj
7
where x(a) is the set of positions satisfying x(a) < <x(a) (N even and n N =N).
{ } 1 ··· Na a a=1 a
P
Infinite MPS
Let us now consider the infinite MPS
n
Ψ = Ψ(a ,...,a )na1,na2,...,naN
1 N
| i | i
a1,..X.,aN=1
where Ψ(a ,...,a ) are given by the chiral correlatorsof Majorana fermion fields χa (a=1,...,n)
1 N
Ψ(a ,...,a )= χa1(z )χa2(z ) χaN(z )
1 N 1 2 N
h ··· i
To evaluate Ψ(a ,...,a ), we use the two-point correlator of Majorana fermions
1 N
δ
χa(z)χb(w) = ab
h i z w
−
The multipoint correlators of Majorana fermions are obtained by Wick’s theorem
hχa(z1)χa(z2)···χa(zNa)i=(cid:26)Pfa(z0i−1zj) NNaaeovdedn
Therefore, to obtain a nonvanishing Ψ(a ,...,a ), we must have even N. Additionally, N , the number of vectors
1 N a
na in the spin configuration, must also be even for all a=1,...,n.
| i
To comparewith the projectedBCSwavefunction, let us evaluate the superpositioncoefficientof the infinite MPS
for a spin configuration, which has N vector na at positions x(a) < < x(a) (N even and n N = N).
a | i 1 ··· Na a a=1 a
Taking into account the anticommuting nature of Majorana fermion fields, we first pick up the vePctors n1 in the
| i
spin configuration and rewrite the infinite MPS as
Ψ( x(1) , x(2) ,... x(n) ) = sgn(x(1),...,x(2),y ,...,y ) χa=1(z )χa=1(z ) χa=1(z )
{ } { } { } 1 N1 1 N−L1 h x(11) x(21) ··· x(N11) i
χb(z )χb(z ) χb(z ) (b=1)
×h y1 y2 ··· yN−N1 i 6
1
= sgn(x(1),...,x(2),y ,...,y )Pf ( )
1 N1 1 N−L1 a=1 zi zj
−
χb(z )χb(z ) χb(z )
×h y1 y2 ··· yN−N1 i
where the positions y < < y correspond to the vectors nb with b = 1. The above steps can be repeated
1 ··· N−N1 | i 6
from b=2 to n. In the end, we obtain
n
1
Ψ( x(1) , x(2) ,... x(n) )=sgn(x(1),...,x(1),x(2),...,x(2),...,x(n),...,x(n)) Pf ( ) (2)
{ } { } { } 1 N1 1 N2 1 Nn aY=1 a zi−zj
Comparing with Eq. (1), we conclude that the infinite MPS and the projected BCS state are equivalent.
Derivation of the parent Hamiltonian
In this Section, we derive the parent Hamiltonian for the infinite MPS.
Brief summary of the SO(n) WZW model
1
For infinite MPS associated to WZW models, the derivation of the parent Hamiltonian relies on the existence of
nullvectorsintherepresentationspacesofKac-Moodyalgebra[9]. ForSO(n) WZWmodel, theKac-Moodyalgebra
1
is defined by
[Jab,Jcd]=ifab,cd,efJef +nδ δ n,m Z (3)
n m n+m ab,cd n+m,0 ∈
8
where repeated indices are summed over and the SO(n) structure constant fab,cd,ef is given by
fab,cd,ef =δ δ δ +δ δ δ δ δ δ δ δ δ
ad be cf bc ae df ac be df bd ae cf
− −
For odd n (n 3), the SO(n) WZW model has three primary fields respectively in singlet (denoted by I), vector
1
≥
(v) and spinor representation (s), whose conformal weights are h = 0, h = 1/2 and h = n/16, respectively. For
I v s
evenn(n 4),apartfromtheprimaryfields insingletandvectorrepresentations(h =0andh =1/2),the SO(n)
I v 1
≥
WZW model has two primary fields in spinor representations (denoted by s and s ), whose conformal weights are
+
−
h = h = n/16. The SO(n) WZW model has central charge c = n/2 and can be constructed by combining n
s+ s− 1
Ising models (c=n 1).
× 2
For both odd and even n, the primary fields in the vector representation have conformal weight h =1/2 and are
v
naturally interpreted as Majorana fermions, which are the key ingredients for us to construct the infinite MPS.
For each Majorana fermion χa (a=1,...,n), a primary state χa can be defined by
| i
χa =χa(0)0
| i | i
where 0 is the vacuum of the WZW model and satisfies Jab 0 =0. When acting on the Kac-Moody currents, the
| i n>0| i
primary states satisfy
n
Jab χc = (Lab) χd
0 | i − cd| i
Xd=1
Jab χc = 0 (n>0) (4)
n | i
where Lab are given by
(Lab) =i(δ δ δ δ )
cd ac bd bc ad
−
Note that Lab form a closed SO(n) algebra
[Lab,Lcd] = i(δ Lbc+δ Lad δ Lbd δ Lac)
ad bc ac bd
− −
= ifab,cd,efLef
Null vectors and parent Hamiltonian
To derive the parent Hamiltonian, one has to find the null vectors in the SO(n) Kac-Moody algebra. The null
1
vectors are descendant states satisfying
Jab φ =0 (n>0)
n | i
For our purpose, we look for null vectors with the following form:
φd = Wd Jab χc
| i aX<b,c abc −1| i
where Wd are the coefficients that have to be determined. They satisfy the orthonormal condition
abc
(Wadb′c)∗Wadbc =δd′d
aX<b,c
Ingeneral,thetensorWd correspondstoaClebsch-Gordandecomposition. FortheSU(2) WZWmodel,theSU(2)
abc k
Clebsch-Gordancoefficientsareknown[9]. However,wearenotawareofaclosedformfortheSO(n)Clebsch-Gordan
coefficients. To overcome this difficulty, let us consider the norm of φd
| i
hφd|φdi = a′X<b′,c′aX<b,c(Wad′b′c′)∗Wadbchχc′|J1a′b′J−ab1|χci
= Wad′b′c′Ma′b′c′,abcWadbc
a′X<b′,c′aX<b,c
= (Wd) MWd
†
9
where Wd is viewed as a column vector and M is a matrix defined by
Ma′b′c′,abc =hχc′|J1a′b′J−ab1|χci
If φd is a null state, φd φd = (Wd) MWd = 0. Since M comes from the norm of two descendent states, it is
†
| i h | i
a positive-semidefinite matrix satisfying (Wd) MWd 0. Therefore, identifying the orthonormal vectors Wd that
†
≥
belong to the kernel of M gives us all null states φd . For our SO(n) WZW model, we can write down the explicit
1
| i
form of M
Ma′b′c′,abc == hχχcc′′|J[J1aa′b′b′′J,−aJb1a|bχ]cχic
h | 1 −1 | i
= hχc′|i fa′b′,ab,efJ0ef +δa′b′,ab|χci
Xef
 
= χc′  i fa′b′,ab,ef (Lef)cg χg +δa′b′,ab χc 
h | − | i | i
Xef Xg
 
= i fa′b′,ab,ef(Lef)cc′ +δa′b′,abδc′,c
−
Xef
where we used Kac-Moody algebra (3) and the properties of the primary state (4). In this way, the null vectors for
the SO(n) WZW model are obtained.
1
Let us mention that the above approach is a systematic way of finding null vectors and can be easily generalized
to other WZW models. The role of the positive-semidefinite matrix M is similar to the Gram matrix for defining the
Kac determinant [15] in conformal field theory.
After obtaining Wd for all the null vectors, we define the following K tensor [9]
abc
Kaa′bb,′c,c′ = (Wad′b′c′)∗Wadbc (a<b and a′ <b′)
Xd
Let us write these K tensors as n×n matrices, Kaa′bb,′c,c′ =(Kaa′bb′)c,c′. Then, Kaa′bb′ have the following compact form
Kaa′bb′ = 32δab,a′b′ − 6(nn+21)ifab,a′b′,cdLcd+ 6(nn−41)(LabLa′b′ +La′b′Lab)
− −
Following Ref. [9], we define an operator Λab (1 a<b n)
i ≤ ≤
N
Λab = w (K(i))abLcd
i ij cd j
j=X1(=i) Xc<d
6
N
2 n+2 n 4
= w Lab ifab,cd,efLefLcd+ − (LabLcd+LcdLab)Lcd
ij(cid:20)3 j − 6(n 1) i j 6(n 1) i i i i j (cid:21)
j=X1(=i) − −
6
N
2 1 1
= w Lab Lab(L~ L~ )+ (L~ L~ )Lab
ij(cid:20)3 j − n 1 i i· j 3 i· j i (cid:21)
j=X1(=i) −
6
where w (z +z )/(z z ) and L~ L~ LabLab. These operators annihilate the infinite MPS, Λab Ψ =0
ij ≡ i j i− j i· j ≡ a<b i j i | i
i,a,b. Moreover,theinfiniteMPSisanSO(nP)singletandtherefore Lab Ψ =0 a,b. Then,anSO(n)symmetric
∀ i i | i ∀
parent Hamiltonian can be defined by P
H = (Λaib)†Λaib+J ( Laib)2+E0 (J ≥0)
iX,a<b Xa<b Xi
whose ground state is the infinite MPS with energy E .
0
10
In 1D, we use z = exp(i2πj) to ensure translational invariance. Choosing J = 2(N 2)/3 and E = 2(n
j N − 0 − −
1)N(N2 4)/9, we arrive at the following explicit form of H:
−
n+2 n 4 n 4
H = w2[ (L~ L~ )+ − (L~ L~ )2] − w w (L~ L~ )(L~ L~ ). (5)
− ij 3 i· j 3(n 1) i· j − 3(n 1) ij ik i· j i· k
Xi=j − − i=Xj=k
6 6 6
To obtain the above form, the following identities are quite useful
Lab(L~ L~ )Lab = L~ L~ (i=j)
i i· j i i· j 6
Xa<b
Lab(L~ L~ )2Lab = 2(n 1) (n 2)(L~ L~ ) (L~ L~ )2 (i=j)
i i· j i − − − i· j − i· j 6
Xa<b
Lab(L~ L~ )(L~ L~ )Lab = 2(L~ L~ ) (L~ L~ )(L~ L~ ) (i=j =k)
i i· j i· k i j · k − i· k i· j 6 6
Xa<b
Jastrow and Pfaffian wave functions in Cartan basis
In this Section, we derive the explicit Jastrow and Pfaffian forms of the wave functions in Cartan basis.
Cartan basis
Let us first define the Cartan basis.
The SO(n) algebra is defined by
[Lab,Lcd]=i(δ Lbc+δ Lad δ Lbd δ Lac)
ad bc ac bd
− −
For n = 2l and 2l + 1, we can choose at most l (rank of the algebra) mutually commuting generators as
L12,L34,...,L2l 1,2l. In the vector basis, the SO(n) generators are defined by Lab = i(na nb nb na )
−
| ih | − | ih |
(1 a<b n). Diagonalizing the Cartan generators gives us the following Cartan basis:
≤ ≤
1
1,0,... = (n2 +in1 )
| i √2 | i | i
1
1,0,... = (n2 in1 )
|− i √2 | i− | i
1
0,1,0,... = (n4 +in3 )
| i √2 | i | i
1
0, 1,0,... = (n4 in3 )
| − i √2 | i− | i
.
.
.
1
0,0,...,1 = (n2l +in2l−1 )
| i √2 | i | i
1
0,0,..., 1 = (n2l in2l−1 )
| − i √2 | i− | i
For SO(2l), the above basis is already complete. For SO(2l+1), we have an additional vector n2l+1 , which is
| i
annihilated by all Cartan generators. Thus, we have the following extra vector for SO(2l+1):
0,0,...,0 = n2l+1
| i | i
Thus, the Cartan basis for SO(2l) and SO(2l+1) can be compactly written as
1
0,...,mα = 1,...,0 = (n2α in2α−1 ) (α=1,...,l)
| ± i √2 | i± | i
0,0,...,0 = n2l+1
| i | i
Note that m is the eigenvalue of the Cartan generator L2α 1,2α.
α −