Bethe Ansatz and Symmetry in Superintegrable Chiral Potts Model and Root-of-unity Six-vertex Model

0.18 MB·English

by  Shi-shyr Roan

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1 Bethe Ansatz and Symmetry in Superintegrable Chiral
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Potts Model and Root-of-unity Six-vertex Model

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m Shi-shyr Roan2
- Institute of Mathematics
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a Academia Sinica
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s Taipei , Taiwan
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t (email: [email protected] )
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Abstract

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v We examine the Onsager algebra symmetry of τ(j)-matrices in the superintegrable chiral
3 Potts model. The comparison of Onsager algebra symmetry of the chiral Potts model with
4 the sl -loop algebra symmetry of six-vertex model at roots of unity is made from the aspect of
2
5
functional relations using the Q-operator and fusion matrices. The discussion of Bethe ansatz
1
for both models is conducted in a uniform manner through the evaluation parameters of their
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5 symmetry algebras.
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a 2003 PACS: 05.50.+q, 02.20.Uw, 75.10Jm
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2000 MSC: 14H70, 17B67, 82B23
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d Key words: Chiral Potts model, Six-vertex model, Onsager Algebra, sl2-loop algebra
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r 1 Introduction
a
The symmetry of quantum spin chains and the related lattice models has recently attracted cer-
tain attention due to their close connection with diverse areas of physics as well as mathematics.
However, up to thepresent stage, only limited knowledge is available about thesymmetry of lattice
vertex models, and few exact results are obtained in this area. Even the sl -loop algebra symmetry
2
1Talk presented at the 23rd International Conference “Differential Geometric Methods in Theoretical Physics
(DGMTP)” at NankaiInstitute, Tianjin, China, Aug. 20-26, 2005.
2ThisworkissupportedinpartbyNationalScienceCouncilofTaiwan underGrantNoNSC94-2115-M-001-013.
1
of thesix-vertex modelat roots of unity, foundin [11], has not beenfully understoodtill now, given
that much accomplishment has been made on the study of evaluation parameters related to the
symmetry algebra representation. Some conjectures supported by numerical evidence remain to
be answered, (see [12] [14] and references therein). Though the understanding of the symmetry of
eight-vertex model in [15] [16] is still rudimentary in the present stage, the conjectural functional-
relation-analogy discovered in the study on the eight-vertex model and the N-state chiral Potts
model (CMP)3 in [7] did lead to exact results about the Onsager algebra symmetry of τ(j)-models
in the superintegrable CPM [20] [21]. In the study of CPM, the lack difference-property for the
rapidities is considered as the characteristic nature which distinguishes CPM from other known
solvable lattice models. Nevertheless, progress made on the transfer matrix of CPM for the past
fifteen years, culminating in the recent Baxter’s proof of the order parameter [6], has provided the
sufficient knowledge for the understandingof the detailed nature about the symmetry of superinte-
grable CPM. By this, the study of CPM could suggest a promising method to help the symmetry
study about the six-vertex model at roots of unity as the limit case of eight-vertex model from the
approach of functional relations, a scheme proposed in [16]. In this paper, we examine the similar-
ity of the symmetry structure of two lattice models: the superintegrable CPM and the six-vertex
model at roots of unity. The symmetry of superintegrable CPM is described by the Onsager alge-
bra, obtained in [20][21], with a shortexplanation in Sec. 2. Thesix-vertex modelat roots of unity
possesses the sl -loop algebra symmetry by the works in [11] [12] [14]. We present a discussion of
2
symmetry of six-vertex model, parallel to the theory of CPM, through Bethe roots and evaluation
parameters in Sec. 3, and give some concluding remarks in Sec. 4.
Notations. In this paper, Z, C denote the ring of integers, complex numbers respectively, N
is a positive integer 2, ZN = Z/NZ, and i= √ 1, ω = e2Nπi. Denote by X,Z the Weyl operators
≥ − n
of the vector space CN, defined by X n = n+1 , Z n = ωn n for n Z , and CN is the
N
| i | i | i | i ∈ ⊗
n-tensor product of CN.
2 The N-state Chiral Potts Model
2.1 Rapidity and functional Relation of chiral Potts model
In the study of CPM as a descendent of the six-vertex model, Bazhanov and Stroganov obtained
the following 3-parameter family of Yang-Baxter solutions for the inhomogeneous R-matrix of six-
vertex model,
tω 1 0 0 0
−
0 t 1 ω 1 0
R(t)=  − −  ,
0 t(ω 1) ω(t 1) 0
 − − 
 0 0 0 tω 1 
 − 
 
with the CN-operator entries parametrized by a four-vector ratio p = [a,b,c,d] [8, 20]:
b2 td2X (bc ωadX)Z
b2Gp(t) = t(bc −adX)Z−1 tc−2+ωa2X ! , t ∈ C , (1)
− − −
which satisfy the following Yang-Baxter equation:
R(t/t′)(G (t) 1)(1 G (t′))= (1 G (t′))(G (t) 1)R(t/t′).
p p p p
aux aux aux aux
O O O O
3In this paper, the discussion of the chiral Potts model will be confined only to the full homogeneous lattice by
takingp=p′ in [7].
2
Hence the same relation holds for the monodromy matrix of size L, L G (t) with G (t) =
ℓ=1 p,ℓ p,ℓ
(2)
G (t) at the site ℓ. Therefore, the τ -matrix,
p p N
L
τ(2)(t):= tr ( G (ωt)) for t C, (2)
p aux p,ℓ
∈
ℓ=1
O
L
formacommutingfamilyof( CN)-operators. TheZ -operatorsX,Z of CN atthesitej giverise
N
⊗
L
to the Weyl operators XjZj of CN with the relations: ZiXj = ωδi,jXjZi, [Zi,Zj]= [Xi,Xj] = 0
⊗
L
and ZN = XN = 1. Then the spin-shift operator of CN, X := L X , defines the Z -charge
j j ⊗ j=1 j N
(2)
Q, and commutes with τp (t). Q
The rapidities of N-state CMP are elements in the genus (N3 2N2 + 1) curve W in the
−
projective 3-space P3, defined by the equivalent sets of equations:
kaN +k′cN =dN, aN +k′bN = kdN, kxN = 1 k′µ−N,
W : − (3)
( kbN +k′dN = cN ⇐⇒ ( k′aN +bN = kcN ⇐⇒ ( kyN = 1 k′µN
−
where [a,b,c,d] P3, (x,y,µ) = (a,b,d) C3, k′ is a parameter with k2 = 1 k′2 = 0,1. By
∈ d c c ∈ − 6
eliminating µN in the last set of above equations , and using the variables t := xy,λ := µN, one
arrives the hyperelliptic curve of genus N 1, tN = (1−k′λ)(1−k′λ−1), as a N2-unramified quotient
− k2
of (3). The rapidities possess a large symmetry group, in which the following two will be needed
in our later discussion,
U : [a,b,c,d] [ωa,b,c,d], (x,y,µ) (ωx,y,µ), (t,λ) (ωt,λ);
7→ ⇐⇒ 7→ ⇐⇒ 7→ (4)
C :[a,b,c,d] [b,a,d,c], (x,y,µ) (y,x,µ−1), (t,λ) (t,λ−1).
7→ ⇐⇒ 7→ ⇐⇒ 7→
The Boltzmann weights W ,W of the CPM, depending on two rapidities p,q W, are
p,q p,q
∈
defined by
Wp,q(n) = n dpbq−apcqωj, Wp,q(n) = n ωapdq−dpaqωj,
Wp,q(0) j=1 bpdq−cpaqωj Wp,q(0) j=1 cpbq−bpcqωj
Q Q
whichsatisfytheN-periodicitypropertyfornbytheconstraint(3)ofrapidities. TheCPMtransfer
L
matrix of size L with periodic boundary condition, L+1= 1, is the ( CN)-operator defined by
⊗
L
T (p;q)σ1′,...,σL′ = W (σ σ′)W (σ σ′ ) , σ ,σ′ Z . (5)
cp σ1,...,σL p,q l − l p,q l − l+1 l l ∈ N
l=1
Y
For a fixed p ∈ W, {Tcp(p;q)}q∈W form a commuting family of operators by the following well-
known star-triangle relation of Boltzmann weights:
N−1
W (σ′ n)W (σ n)W (n σ′′) = R W (σ σ′)W (σ′ σ′′)W (σ σ′′),
qr pr pq pqr pq pr qr
− − − − − −
n=0
X
1
where R = fpqfqr with f = detN(Wpq(i−j)) N. Then T (p;q) commutes with X and the
pqr fpr pq (cid:18) Nn=−01Wpq(n) (cid:19) cp
spatial translation operator S (which defines the total momentum P Z ). Denote T (p;q) :=
R Q ∈ L cp
T (p;q)S . The transfer matrix T (p;q) can be derived from τ(2)(t ) with p W as the auxiliary
cp R cp p q
∈ b
3
“Q”-operator, as discussed in [2] on the TQ-relation of theeight-vertex model. Onearrives τ(2)T -
cp
relation ((4.20) in [7])4 using the automorphism U in (4):
τ(2)(t )T (p;Uq) = ϕ (q)T (p(;q)+ϕ (Uq)XT (p;U2q) (6)
p q cp p cp p cp
τ(2)(t )= ϕ (q)T (p;q)+ϕ (Uq)XT (p;U2q) T (p;Uq)−1 (7)
p q p cp p cp cp
⇐⇒
(cid:18) (cid:19)
where ϕ (q) := ((yp−ωxq)(tp−tq))L, ϕ (q) := (ωµ2p(xp−xq)(tp−tq))L. By (7) and the commutativity of
p yp2(xp−xq) p yp2(yp−ωxq)
T (p; ), [τ(2)(t ),T (p;q′)] = 0 for p,q,q′ W. The fusion operators τ(j)(t) for 0 j N are
cp p q cp
∗ ∈ ≤ ≤
determined by the following T T -relation for 0 j N, ((3.46) for (l,k) = (j,0) in [7]) with
cp cp
≤ ≤
(0) (1)
τ := 0, τ := I,
p p
b
z(t )z(ωt ) z(ωj−1t )
T (p;q)T (p;CUjq)= r h τ(j)(t )+ q q ··· q Xjτ(N−j)(ωjt ) (8)
cp cp p,q j;p,q p q α (λ ) p q
(cid:18) p q (cid:19)
b
L L L
where z(t) = ωµ2p(xpyp−t)2 , α (λ ) = k′(1−λpλq)2 and r = N(xp−xq)(yp−yq)(tNp −tNq ) ,
yp4 p q λq(1−k′λp)2 p,q (xNp −xNq )(ypN−yqN)(tp−tq)
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
L
h = j−1 yp2(xp−ωmxq) . By (8), one can derive the fusion relations of τ(j)s ((4.27) of
j;p,q m=1 (yp−ωmxq)(tp−ωmtq)
(cid:18) (cid:19)
[7] ): Q
τ(j)(t)τ(2)(ωj−1t) = z(ωj−1t)Xτ(j−1)(t)+τ(j+1)(t), 1 j N,
p p p p
≤ ≤ (9)
(N+1) (N−1)
τ (t) := z(t)Xτ (ωt)+u(t)I
p p
where u(t) := α (λ) + α (λ−1). Note that with τ(2)(t) in (2) for p P3, the validity of fusion
p p p
∈
relation (9) provides a characterization of the rapidity constraint (3) for p W, (Theorem 1 in
∈
[20]). Using (7) and (9), one can express τ(j) in terms of T (p;q), hence the τ(j)T -relations for
p cp cp
1 j N +1, ((4.34) in [7]):
≤ ≤
τ(j)(q) = T (p;q)T (p;Ujq) j−1 ϕ (q)ϕ (Uq) ϕ (Um−1q)
p cp cp m=0 p p ··· p ×
(cid:18) (10)
ϕ (Um+1q) ϕ (UPj−1q)T (p;Umq)−1T (p;Um+1q)−1Xj−m−1 .
p p cp cp
···
(cid:19)
By substituting (10) in (8), one obtains a single functional equation of T (p;q) ((4.40) of [7]):
cp
T (p;q) = N−1C (q)T (p;q)T (p;Umq)−1T (p;Um+1q)−1X−m−1, (11)
cp m=0 m;p cp cp cp
where C (qb) = ϕ (q) Pϕ (Um−1q)ϕ (Um+1q) ϕ (UN−1q)(Nyp2N−2(yp−yq)(yp−xq))L.
m;p p ··· p p ··· p (ypN−yqN)(ypN−xNq )
2.2 Bethe Ansatz and Onsager algebra symmetry in superintegrable chiral
Potts model
1
For CPM in the superintegrable case, i.e., the rapidity p given by µp = 1, xp = yp = η2, where
η := (11−+kk′′)N1 , simplification occurs for the functional relations. We shall omit the label p appeared
in all operators for the superintegrable case, simply write τ(2)(t), T (q), etc. As q tends to p, to
cp
the first order of small ε, one has ((1.11) in [1]),
T (q) = 1[1+ε(N 1)L]+εH(k′)
cp
−
4τp(2)(q),Tcp(p;q) in this paperare theoperators τk(2=)0,q, Tq in [7] respectively.
4
where H(k′) = H + k′H is the Z -quantum chain in [17, 18], with the expression: H =
0 1 N 0
2 L N−1 Xℓn , H = 2 L N−1 ZℓnZℓ−+n1, which satisfy the Dolan-Grady relation [13],
− ℓ=1 n=1 1−ω−n 1 − ℓ=1 n=1 1−ω−n
hence generate the Onsager algebra representation where only the spin-1 subrepresentation oc-
P P P P 2
1
curs as irreducible factors [1, 3, 5, 10, 19]. Through the gauge transform by M = dia.[1,η2], the
monodromy matrix G(t) in (1) becomes
1 tX (1 ωX)Z
G(t)= − − , t = η−1t,
t(1 X)Z−1 t+ωX
!
− − −
e
e e e
which is again a Yang-Baxter soluteion (1) for a = be= c = d = 1. Hence τ(2)(t) = τ(2)(t), where
τ(2)(t) is the trace of the L-size monodromy matrix associated to G(ωt). Write τ(j)(t) = τ(j)(t),
the fusion relation (9) has the form: e e
e e e e e
τ(j)(t)τ(2)(ωj−1t) = (1 ωj−1t)2L τ(j−1)(t) ωLX +τ(j+1)(t), 1 j N ,
− ≤ ≤ (12)
τ(N+1)(t) = (1 t)2L τ(N−1)(ωt)ωLX +2(1 tN)L .
− −
e ee e e e e e e
By examining ecommuteators of Hk weith tehe entriees of the monodreomy matrix constructed from
G(t), one can show [H ,τ(2)(t)] = 0 for k = 0,1. It follows the Onsager algebra symmetry of τ(j)-
k
model (Theorem 1 of [21]). However, the understanding of the detailed nature of Onsager algebra
symemetry in the superintegrable CPM still requires the full knowledge about eigenvalues of CPM
transfer matrix, which was solved by the Bethe-ansatz method in [1, 4, 5] as follows.
For parameters v ,...,v with ( v )N = 0,1 and v v−1 = 1,ω for i = j, consider the rational
1 mp − i 6 i j 6 6
function
N−1 (1 tN)L(ωjt)−Pa−Pb mp
(t)= ω−Pb − , F(t):= (1+ωv t) (13)
i
P (1 ωjt)LF(ωjt)F(ωj+1t)
jX=0 − e e iY=1
e e e
where Pa,Pb are integers satisfying 0 re(:= Pa+e Pb) Ne 1, Pb Pa Q+L (mod N). (t)
≤ ≤ − − ≡ P
is invariant under t ωt, hence depending only on tN. The criterion of (t) as a t-polynomial is
7→ P
the following constraint for v s, ((4.4) in [1], (6.22) in [4]): e
j
e e e e e
v +ω−1 mp v ω−1v
( i )L = ω−r i− l, i = 1,...,m . (14)
v +ω−2 − v ωv p
i i l
lY=1 −
Here the non-negative integer m satisfies the relation LP m (Q 2P m ) (mod N). The
p b p b p
total momentum P is given by eiP = ω−Pb mp 1+ωvi . T≡he above−relati−on is indeed the Bethe
i=1 1+ω2vi
equation of τ(2)-model (Theorem 3 of [21]). Then (t) is a simple tN-polynomial of degree m =
Q E
P
[(N−1)L−r−2mp] with negative real roots, (Theorem 2 of [21]). Let s ,...s be the tN-zeros of
N 1 mE
P(t), and defineG(λ) = mj=E1 λ+1±(2λλ−1)wj wherewj :=e (sj−sjη−−12N)12.eThen PP((1t)) = G(λ)G(eλ−1). One
has the following expression of T (q)-eigenvalues ((1.11) in [1] and (21) in [5]):
Q cp e
e
Tcp(q) = NL((ηη−−2N21xxNqq −−11))LL(η−21xq)Pa(η−21yq)Pbµ−qPµFF((te1q))G(λq) (= e−iPTcp(q)), (15)
b
which gives the energy value of H(k′) ((2.23) of [1]):
mE
E = 2P +Nm (N 1)L+k′ (N 1)L 2P Nm +2(P P ) +N(1 k′) w .
µ E µ E b a j
− − − − − − − ±
(cid:18) (cid:19) j=1
X
5
Therefore the τ(2)-degenerate states associated to the Bethe roots v s form an irreducible Onsager-
i
algebra-representation space of dimension 2mE, which we associate the following normalized CPM
transfer matrix5:
Q(q) = Tcp(q)(1−η−2NxNq )L (=: e−iPQ(q)).
NL(1 η−21xq)L(η−21xq)Pa(η−21yq)Pbµ−qPµ
−
b
The Q-eigenvalues and the functional equation (11) now become
Q(q) = F(tq) (λ ), Q(Cq)= eiPF(tq) (λ−1), (16)
F(1)G q F(1)G q
Q(Cq) = ω−Pb Ne−1 (1−tNq )Lω−bmrQ(q)Q(Umqe)−1Q(Um+1q)−1 . (17)
NL m=0 (1−ωmtq)L
P e
The relations, (6) b(10) (8), now become
e
τ(2)(tq)Q(Uq) = ω−Pa(1 tq)LQ(q)+ωPb(1 ωtq)LQ(U2q), (18)
− −
τ(j)(tq)= ωe(j−1)ePb mj−=10 jk(−=1−10(ω1m−tωqk)tLqe)LQ(q)Q(Umq)−1Q(Uejq)Q(Um+1q)−1ω−r , (19)
(cid:18)Q (cid:19)
NLetrQe(q)Q(CUjq)=Pω−jPb(1−tNq )L τ(je)(t )+ ω−jPa(1−tNq )L τ(N−j)(ωjt ), 0 j N. (20)
q jm−=10(1−ωmtq)Le q Nm−=j1(1−ωmtq)L q ≤ ≤
e e
By (16), ethe relabtion (18) yiQeld ((6.18) in [4e]): e Q e e
e e
τ(2)(t )F(ωt ) = ω−Pa(1 t )LF(t )+ωPb(1 ωt )LF(ω2t ). (21)
q q q q q q
− −
Using (19) and (20), one obtains the τ(j)-polynomial expression (Theorem 3 (ii) of [21]):
e e e e e e e
τ(j)(t)= ω(j−1)Pb kj−=10(1−ωkt)eL mj−=10 (1−Fω(tm)Ft)(LωFjt()ωωm−tm)F(P(aω+mP+b)1t), 2≤ j ≤ N,
trF(t)F(ωjt) (t)Q= ω−jPb(1−tN)LPτ(j)(t)+ eω−jPa(e1−tN)L τ(N−j)(ωjt), 0 j N. (22)
e e P j−1 (1−ωmet)L N−1(1−ωmt)L ≤ ≤
m=0 me=j e e
e e
e e e e Q e e Q e e
3 Six-vertex Model at Reoots of Unity e
3.1 Bethe equation of the six-vertex model
L
The transfer matrix of the six-vertex model of an even size L is the ( C2)-operator constructed
⊗
from the Yang-Baxter solution6
L =  z12q(q−2σZ q−−1z)−σ21+q,σ2Z, z12q(qσ2Z−q−z1−2)1σq−−2σZ  , σZ,σ± : Pauli matrix,
− −
 
for the R-matrix
z−21q z21q−1 0 0 0
 −0 z−21 z12 q q−1 0 
R(z) =  0 q −q−1 z−2−1 z12 0 ,
 0 −0 0− z−21q z21q−1 
 − 
 
5The Q-operator herediffers from theQcp in [21] by a scale factor: (η−21xq)Pa(η−21yq)Pbµ−qPµQ(q)=Qcp(q).
6Here we use theconvention in [14], Eq. (1.3): a=isinh1(v−iγ), b=−isinh1(v+iγ), c=−isinhiγ, with the
2 2
variables z = −ev,q = −eiγ. Note that the q here differs from the q in [14] by minus sign for its connection with
Uq(sl2) as Eq. (2.3) in [12], where −e−z is equal to z12 here.
c
6
as the trace of monodromy matrix: T(z) = tr ( L L (z)) for z C. The logarithmic z d -
aux ℓ=1 ℓ ∈ dz
derivative of T(z) at z = q−1 gives rise to the XXZ chain with the periodic boundary condition:
N
L
1 1
H = (σ1σ1 +σ2σ2 + σ3σ3 ), = (q+q−1),
XXZ −2 ℓ ℓ+1 ℓ ℓ+1 △ ℓ ℓ+1 △ 2
ℓ=1
X
which has been a well-studied Hamiltonian initiated by Bethe in 1931 [9]. A major result of the
study is that the ground state energy for the value SZ(= 1 σZ) is determined by an appropriate
2 ℓ ℓ
solution of the following Bethe equation for the variable v = z−1:
P−
v +q−1 m v q−2v L
( i )L = q−L+2m i− l, m = SZ . (23)
v +q − v q2v 2 −| |
i i l
lY=1 −
TheBethe-equation techniquewasfurtherextendedtothemethodofBaxter’sTQ-relation ineight-
vertex model; when applying to the six-vertex model, there exists a non-degenerated commuting
family of Q-operators with the following relation (see, Chapter 9 of [2]):
T(z)Q(s) = q−2|SZ|(1 qz)LQ(U−1s)+(1 q−1z)LQ(Us). (24)
− −
Heresisasuitablemulti-valued complexcoordinateofz, andU isas-automorphismwhichinduces
the z-transform: z q2z. Note that there are many such Q-operators, however all give the same
7→
Bethe equation (23) through Eq.(24).
3.2 Evaluation polynomial and fusion relation of six-vertex model at roots of
unity
For the root of unity case with q2N = 1, i.e. q2 = ω, the six-vertex model possesses the sl -loop
2
algebra symmetry [11]. The Bethe state corresponding to the Bethe roots is the “highest weight”
vector of an irreduciblerepresentation of thesl -loop algebra, with the evaluation parameters char-
2
acterized by the Drinfeld polynomial [12, 14]. By studyingthecreation sl -loop current operator in
2
the ABCD-algebra, the Drinfeld polynomial for a Bethe root v m of (23) is given by Eq. (3.9)
{ i}i=1
in [14]. Denote t = qz, and define the integer r by r L m (mod N), 0 r N 1. The
≡ 2 − ≤ ≤ −
Drinfeld polynomial is indeed the tN-polynomial associated the following polynomial P(t)7,
e
P(t) = N−1(1e−ωjt)L(ωjt)−r , F(t) := m (1+q−1v t), e (25)
i
F(ωjt)F(ωj+1t)
j=0 i=1
X e e Y
e e e
which has a similar form as (13). Indeeed withe F(t) in (13) or (25), let H(t) = 1−tN,1 t in
1−t −
CPM, six-vertex model respectively. The function Pe(t) defined by P(t) = jNe=−01 HF((ωωjjett))FL((ωωjj+t)1−ter) is
e
invariant under t ωt. The condition on roots of F(t) so that P(t) is a pPolynomialeis proevided
7→ e e
by Bethe equation (14), (23) respectively. Define the T(2)-operator by T(2)(t) = zL2Te(z) inethe
six-vertex modele, and eω−Pb(1−tN)Lτ(2)(ω−1t) in CPM. Theen Eqs. (18e), (24) are combined into one
(1−ω−1t)L(1−t)L e
T(2)Q-relation:
e e e
e e
T(2)(t)Q(q) = ω−rH(t)LQ(U−1q)+H(ω−1t)LQ(Uq), UN = 1. (26)
7In [14], the evaluation function is given by (3.9) there: Y(v) = N−1a(v + 2(j + 1)iγ), where a(v) =
e e e j=0
mi=1sinh21si(nvhiL−v21)(svi−nhiγ21)(vi−v+2iγ) in (2.47) of [14]. In terms of variablesPz,t and Bethe roots vis here, a(v) =
2Q2m−L( mi=1vi)(q−1z)−2L+m mi=1(1(+1−viqz−)1(1z+)Lω−1viz), which implies Y(v)=22me−L( mi=1vi)t−2L+m+rP(t).
Q Q
Q e e
7
TheT(j)-operators are defined recursively through the following fusion relation for j 1 by setting
≥
T(0) = 0, T(1) = H(ω−1t)L,
T(j)(t)T(2)(ωj−1t) = ω−rH(ωj−1t)LT(j−1)(t)+H(ωj−2t)LT(j+1)(t). (27)
e
By (26), the induction-argument yields the T(j)Q-relation for j 0:
e e e e ≥ e e
j−1
T(j)(t) = Q(U−1q)Q(Uj−1q) (ω−krH(ωk−1t)LQ(Uk−1q)−1Q(Ukq)−1). (28)
k=0
X
By this, one obtaiens the boundary condition of the fusion erelation:
T(N+1)(t)= ω−rT(N−1)(ωt)+2H(ω−1t)L. (29)
In CPM case, with the identification T(j)(t) = ω−(j−1)Pb(1−tN)Lτ(j)(ω−1t), Eqs. (27)-(29) are the
e e j−2 (1−ωekt)L
k=−1
same as Eqs. (12), (19). While in six-vertexemodel, theQfusioen releation aend T(j)Q-relation hold for
any Q-operator satisfying T(2)Q-relation (26). e
For a polynomial F(t) with Bethe roots v s, by (28) the corresponding T(2)-eigenvalue is deter-
i
mined by the relation
e
T(2)(t)F(t)= ω−rH(t)LF(ω−1t)+H(ω−1t)LF(ωt). (30)
Using Eq.(27) to express T(j+1) in T(j) and T(j−1), by induction argument, one obtains the form
e e e e e e
of T(j)-eigenvalues from Eq.(30):
j−1 H(ωk−1t)Lω−kr
T(j)(t) =F(ω−1t)F(ωj−1t) , j 1, (31)
F(ωk−1t)F(ωkt) ≥
k=0
X e
which implies e e e
e e
trF(ω−1t)F(ωj−1t)P(t)= T(j)(t)+ω−jrT(N−j)(ωjt), 0 j N. (32)
≤ ≤
Eqs.(30)-(32) in CPM case are the same as Eqs. (21), (22). Note that Eq.(32) is a consequence of
e e e e e e
theQQ-relation (20) inCPMcase, which encodesthedetailed natureof Onsager algebra symmetry
inthederivationofEq.(16). However inthecaseofroots-of-unitysix-vertexmodel,theQQ-relation
hasyetbeenfound,eventhoughthesl -loopalgebrasymmetrytogetherwithevaluationparameters
2
hasalreadybeenknown[11,12,14]. BasedontheunderstandingintheCPMcase, wenowdescribe
a similar, but speculated, structure about the QQ-relation in six-vertex case as follows. Consider
the curve W and its symmetries U,C:
W : w2 = tN, U : (w,t) (w,ωt), C :(w,t) ( w,t).
7→ 7→ −
1
FgiovresodridseNto, tthhee aabboovvee scyumrvmeeeitsriepsarbaymUetr=iezeϕd−b2y[N2s], e=C =t2,ϕaNn.dTethhee pauoltyonmoeomrpiahlisPm(t)ϕin: s(257→) aqrse
expected8 to have the simple tN-roots s1,...,sM with P(e0) 6= 0. Define G(w) = Mj=1(√sj −w),
e
G(w)G(−w) P(t)
then = . In the eigen-space of T(z) corresponding to F(t) determiQned by a Bethe
G(0)2 P(0) e
root, The Q-operator has the following Q-eigenvalues:
e
e
G(w) G( w)
Q(q)= F(t) , Q(Cq)= F(t) − for q = (w,t).
G(0) G(0)
The above conditions reveal tehe sl -loop symmetry oef six-vertex model, as thee role of Eq.(16) for
2
the Onsager algebra symmetry in CPM. Hence such a Q-matrix, if exits, must possess certain
constraints in order to incorporate the symmetry of six-vertex model as discussed in [2] Secs.
9.1-9.5.
8The statement is truein thecase r=0, which has been justified in [12].
8
4 Discussion
In this paper we have examined the symmetry structure of the superintegrable CPM and the six-
vertex model at roots of unity by the method of functional relations. For the superintegrable
CPM, exact results about the Onsager algebra symmetry of the τ(j)-models are obtained using
the explicit form of eigenvalues of the CPM-transfer matrix. Based on common features related to
evaluation parameters of the symmetry algebra representation, we discussed the Bethe ansatz of
both theories in a unified manner. By this, in the six-vertex model at roots of unity, we obtained
the fusion relation of T(j)-matrices, T(j)Q-relation from the TQ-relation, and further indicate the
specialnatureofQ-operatorinaccordwiththerequiredsl -loopalgebrasymmetryofthesix-vertex
2
model.
Acknowledgements
The author thanks the conference organizers for the invitation, and the local committee at Nankai
University for the hospitality and hard work. I also wish to thank Professor F. Hirzebruch for the
warm hospitality in November, 2005 during the author’s one-month stay at Max-Planck-Institut
fu¨r Mathematik in Bonn, where this report was completed.
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