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Correlation Functions of the XXZ model for ? < ?1

M ichioJimboa , Kei Mikib , Tetsuji Miwac and Atsushi Nakayashikid

arXiv:hep-th/9205055v2 19 May 1992

a

Department of Mathematics,Faculty of Science, Kyoto University, Kyoto 606, Japan

b

Department of Mathematical Science, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan

c

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan

d

The Graduate School of Science and Technology, Kobe University, Rokkodai, Kobe 657, Japan

Dedicated to Professor Chen Ning Yang on the occasion of his seventieth birthday

Abstract. A new approach to the correlation functions is presented for the XXZ model in the anti-ferroelectric regime. The method is based on the recent realization of the quantum a?ne symmetry using vertex operators. With the aid of a boson representation for the latter, an integral formula is found for correlation functions of arbitrary local operators. As a special case it reproduces the spontaneous staggered polarization obtained earlier by Baxter.

2

M. Jimbo et al.

§0. Introduction In this article we consider the one-dimensional in?nite spin-chain HXXZ = ? 1 2

∞ y y x x z z σk+1 σk + σk+1 σk + ?σk+1 σk k=?∞

(0.1)

known classically as the XXZ model. We shall limit ourselves strictly to the antiferroelectric regime ? < ?1. Our aim is to ?nd an exact expression for the spin correγ γn γ y x z lation functions σi11 · · · σin , where σi = σi , σi , σi and · = vac| · |vac denotes the ground state average. Here the ground state |vac means one of the two ground states in the anti-ferroelectric regime. In Baxter’s paper [1] they are denoted by |± . The present paper is based entirely on the framework of the recent work [2], which discribes the realization of the quantum a?ne symmetry for (0.1) using the q-deformed vertex operators. Let us recall below the contents of [2] which are relevant to the subsequent discussions. The Hamiltonian (0.1) is an operator acting on the in?nite tensor product V ?∞ = · · · V ?V ?V ? · · · of the two dimensional space V = C2 . The space V ?∞ admits also an action of the quantum a?ne algebra U = Uq (sl2 ) via the iterated coproduct. A na¨ ?ve computation shows that the algebra U provides an exact symmetry for (0.1). Namely ′ if ? = (q + q ?1 )/2, then [HXXZ , U ′ ] = 0 where U ′ = Uq (sl2 ) denotes the subalgebra ‘without the grading operator d’ [2], while d plays the role of the boost operator. However the actions of (0.1) and U are both de?ned only formally, and the issue is how to extract the theory free from the di?culties of divergence. The basic idea in [2] is to replace the formal object V ?∞ by the level 0 U -module Fλ,? = V (λ)?V (?)?a ? Hom(V (?), V (λ)), (0.2)

where V (λ) (λ = Λ0 , Λ1 ) denotes the level 1 highest weight U -module, and V (λ)?a signi?es its dual (the super?x indicates that the U -module structure is given via the antipode a). The choice λ = ? (resp. λ = ?) is responsible for the even (resp. odd) particle sector. V (Λ0 ) (resp. V (Λ1 )) means that we are working in the boundary condition z z z z σ2k = 1, σ2k+1 = ?1 (resp. σ2k = ?1, σ2k+1 = 1) for k >> 0, and V (Λ0 )?a (resp. ?a V (Λ1 ) ) means the same thing for k << 0. The tensor product should be completed in the q-adic sense to allow for in?nite sums (see [2] for a precise treatment). To make contact with the na¨ picture of V ?∞ , one utilizes the embedding of V (λ) into the half ?ve in?nite tensor product · · · V ?V ?V [3]. This is supplied by iterating the vertex operators Φ?V : V (λ)?→V (?) ? V. λ (0.3)

(It is conjectured [2] that there is a unique normalization of (0.3) which makes the in?nite iteration convergent.) Similarly V (?)?a embeds to the other half in?nite tensor product V ?V ?V ? · · ·, giving altogether the embedding Fλ,? ?→ · · · V ? V ? V ? · · · . (0.4)

Eq. (0.4) provides the principle of interpreting the notions de?ned in the picture V ?∞ by pulling them back to Fλ,? . The translation operator T is the ?rst such example. (The Hamiltonian (0.1) itself is de?ned in terms of T and the grading operator d; see [2].) In γ this paper we follow the same principle to formulate the local operators σk as acting on Fλ,? .

Correlation Functions of the XXZ model for ? < ?1

3

In the language of (0.2) the ground state (vacuum) vector is given by the identity element of Fλ,λ = Hom(V (λ), V (λ)). The inner product of two vectors f, g ∈ Fλ,λ is given as the trace f |g = trV (λ) (q ?2ρ f g) trV (λ) (q ?2ρ ) (0.5)

where ρ = Λ0 + Λ1 (we have changed the normalization from [2], see Sect. 1 below). Thus γ γn the correlation functions vac|σi11 · · · σin |vac can be expressed as the trace of products of the vertex operators (0.3). To perform the evaluation of these traces, we invoke the bosonization method. The realization of the q-deformed currents on level one modules was done in [4] using (ordinary) bosons. In the same spirit we derive the formulas for the vertex operators (0.3) in terms of bosons. This leads to an explicit formula for the correlators in terms of certain integrals of meromorphic functions. We verify that in the simplest case of the z one-point function σk this formula reproduces the known result for the spontaneous staggered polarization due to Baxter [1]. Jacques Perk noted a strong similarity between our formula and the formula for the Ising model spin-spin correlation functions given in [5,6]. The bosonization of the vertex operators also enables us to compute the n-point functions discussed in Sect. 6.8 of [2]. They are the matrix elements (not the trace) of the product of the vertex operators with respect to the highest weight vectors. The conjectural formula (6.39) of [2] is thus proved. In this paper, we do not go into details on this matter. The plan of the paper is as follows. In Sect. 1 we formulate the local operators and their correlators using the scheme mentioned above. In our algebraic formulation the algebra U , intertwiners, etc. are a priori de?ned over the base ?eld F = Q(q) (q an indeterminate, see the remark at the end of Sect. 2); we shall see however that the resulting formulas are meaningful for complex values of q with |q| < 1. In Sect. 2 we outline the bosonization of the vertex operators. The basic ingredients are the Drinfeld realization of the algebra U ′ and the result of [4], which we recall brie?y. Unlike the classical case (q = 1) only one of the components Φ? (z) of (0.3) has the exponential form, while the other one does not but is given as the q-commutator of Φ? (z) with a generator f1 of U . Sect. 3 is devoted to the formula for the correlators.

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M. Jimbo et al.

§1. Vacuum expectation values of local operators Let us recall some notations of [2]. We denote by V = F v+ ⊕ F v? the 2-dimensional vector space on which the Pauli matrices σ x , σ y , σ z act. We consider V as the 2-dimensional ′ Uq sl(2) -module. The XXZ-Hamiltonian formally acts on the in?nite tensor product V ?∞ of V . We label the components of the tensor product by integers k ∈ Z from right (k → ?∞) to left (+∞ ← k). In our mathematical scheme, we replace the semi-in?nite tensor product of the components k ≥ 1 by the irreducible highest weight Uq sl(2) module V (Λi ) (i = 0, 1) Let us de?ne the action of the local operators on V (Λi ). Let L ∈ End(V ? · · · ? V ). The operator L naturally acts on the tensor product of the components n ≥ k ≥ 1 of V ?∞ . We want to interprete this action as one in End V (Λi ) . For this purpose, we use the vertex operators [2]: they are the intertwiners ? Φ?V (z) : V (λ) → V (?) ? Vz , λ ? Φ? (z) : V (λ) ? Vz → V (?),

λV n

? ? ? Φ?V (z)(v) = Φ?V (z)(v) ? v+ + Φ?V (z)(v) ? v? (1.1) λ λ+ λ? ? ? Φ? (z)(v ? v± ) = Φ? ± (z)(v),

λV λV

where λ = Λ0 , ? = Λ1 or λ = Λ1 , ? = Λ0 , and Vz = V ? F [z, z ?1] signi?es the Uq sl(2) module associated to V with the spectral parameter z (see [2], Sect. 6). Precisely speaking, we consider the vertex operators as the generating functions of their Fourier components in terms of the spectral parameter z. They are normalized as ? ? ΦΛ1 V (z)(|uΛ0 ) = |uΛ1 ? v? + · · · , ΦΛ0 V (z)(|uΛ1 ) = |uΛ0 ? v+ + · · · , Λ1 Λ0 ? ? ? ? ΦΛ1 + (z) = ΦΛ1 V (z/q 2 ), ΦΛ1 ? (z) = ?q ?1 ΦΛ1 V (z/q 2 ),

Λ0 V Λ0 ?

(1.2)

? ? ΦΛ0 V + (z) = ?q ΦΛ0 V (z/q 2 ), Λ1 Λ1 ?

Λ0 V ? ΦΛ0 V ? (z) Λ1

Λ0 +

? = ΦΛ0 V (z/q 2 ), Λ1 +

(q 2 ; q 4 )∞ ? λ ? Φ (z)Φ?V (z) = idV (λ) . λ (q 4 ; q 4 )∞ ?V Here |uΛi denotes the highest weight vector of V (Λi ). We have used the standard ∞ notation (z; p)∞ = j=0 (1 ? zpj ). Set Λn = Λn?2 for simplicity of notation. Given an L as above, we de?ne the (i) operator L = ρzn ,...,z1 (L) ∈ End V (Λi ) by L= (q 2 ; q 4 )n ? Λi ∞ ?Λ ΦΛi+1 V (z1 ) ? · · · ? ΦΛi+n?1 (zn ) i+n V (q 4 ; q 4 )n ∞ ?Λ V ?Λ V ? idV (Λ ) ? L ? Φ i+n (zn ) ? · · · ? Φ i+1 (z1 ).

i+n

Λi+n?1

Λi

In this paper, we use the following convention (di?erent from that of [2]) for the invariant bilinear form on P = ZΛ0 ⊕ ZΛ1 ⊕ Zδ: (Λ0 , Λ0 ) = 0, (Λ0 , α1 ) = 0, (Λ0 , δ) = 1, (α1 , α1 ) = 2, (α1 , δ) = 0, (δ, δ) = 0. Note that Λ1 = Λ0 + α1 /2, δ = α0 + α1 . We set ρ = Λ0 + Λ1 as usual, and also α = α1 for simplicity. We identify P ? = Zh0 ⊕ Zh1 ⊕ Zd as a subset of P via ( , ). We have ρ = 2d + α1 /2.

Correlation Functions of the XXZ model for ? < ?1 The vacuum expectation value L L

(i) zn ,...,z1 (i) zn ,...,z1

5

of the local operator L is given by

(i)

=

trV (Λi ) q ?2ρ ρzn ,...,z1 (L) . trV (Λi ) q ?2ρ

(1.3)

This is a consequence of the formula for the invariant inner product in the space V (Λi ) ? V (Λi )?a [2]. For the XXZ model correlator we specialize the spectral parameters to z1 = · · · = zn . We expect that the formula unspecialized would give the equal-row vertical arrow correlator for the inhomogeneous six-vertex model with zk being the trigonometric spectral parameter of the k-th vertical line. Here we use the usual language of vertex models on the 2-dimensional square lattice, in which the ?uctuation variables are described as arrows sitting on vertical or horizontal edges. Note the following selection (i) rule, which is speci?c to the six vertex model: L zn ,...,z1 = 0 for L = Eε′ εn ? · · · ? Eε′ ε1 n 1 ′ (Eij is a matrix unit) such that ε1 + · · · + εn = ε1 + · · · + ε′ . n §2. Bosonization

′ Let us ?rst recall Drinfeld’s realization of the quantum a?ne algebra U ′ = Uq sl(2) [7]. It is an associative algebra generated by the letters {x± | k ∈ Z}, {al | l ∈ Z=0 }, γ ±1/2 k and K, satisfying the following de?ning relations.

γ ±1/2 ∈ the center of the algebra, γ k ? γ ?k 1 , [ak , K] = 0, [ak , al ] = δk+l,0 [2k] k q ? q ?1 Kx± K ?1 = q ±2 x± , k k 1 ± [ak , xl ] = ± [2k]γ ?|k|/2 x± , k+l k ± ± ±2 ± ± ±2 ± ± xk+1 xl ? q xl xk+1 = q xk xl+1 ? x± x± , l+1 k 1 + ? (k?l)/2 (l?k)/2 [xk , xl ] = (γ ψk+l ? γ ?k+l ), q ? q ?1 where [n] = (q n ? q ?n )/(q ? q ?1 ) and {ψr , ?s | r ∈ Z≥0 , s ∈ Z≤0 } are related to {al | l ∈ Z=0 } by

∞ ∞

ψk z ?k = K exp (q ? q ?1 )

k=0 ∞ k=1

ak z ?k ,

∞ ?1

??k z = K

k=0

k

?1

exp ?(q ? q

)

k=1

a?k z k .

The standard Chevalley generators {ei , fi , ti } are given by the identi?cation

?1 t0 = γK ?1 , t1 = K, e1 = x+ , f1 = x? , e0 t1 = x? , t1 f0 = x+ . 0 0 1 ?1

We use the coproduct ?(ei ) = ei ? 1 + ti ? ei , ?(fi ) = fi ? t?1 + 1 ? fi , i ?(ti ) = ti ? ti .

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M. Jimbo et al.

Next recall the construction of the level one irreducible integrable highest weight representations of U ′ in terms of bosons [4]. Let P = Z α , Q = Zα be the weight/root 2 lattice of sl(2), and let F [P ], F [Q] be their group algebras. The basis elements of F [P ] 1 are written multiplicatively as enα (n ∈ 2 Z). As an F [Q]-module, F [P ] = F [P ]0 ⊕ F [P ]1 where F [P ]i = F [Q]eiα/2 . Introduce a U ′ -module structure on the space W = F [a?1 , a?2 , · · ·] ? F [P ] in the following way. First let {ak }, eβ and ?α act on W as ak = the left multiplication by ak ? 1 for k < 0, = [ak , ·] ? 1 for k > 0, eβ1 (f ? eβ2 ) = f ? eβ1 +β2 , ?α (f ? eβ ) = (α, β)f ? eβ . We let also K = 1 ? q ?α , γ = q ? id.

The actions of the generators {x± } are given through the generating functions X ± (z) = n ± ?n?1 as follows. n∈Z xn z X + (z) = exp a?n ?n/2 n an ?n/2 ?n α ?α e z , q z exp ? q z [n] [n] n=1 n=1 a?n n/2 n q z exp [n] n=1

∞ ∞ ∞

X ? (z) = exp ?

an n/2 ?n ?α ??α . q z e z [n] n=1

∞

With these actions W becomes a U ′ -module. The submodules F [a?1 , a?2 , · · ·] ? F [P ]i are isomorphic to the irreducible highest weight modules V (Λi ) with the highest weight vectors uΛ0 = 1 ? 1 and uΛ1 = 1 ? eα/2 . The grading operator d is introduced by

r

?d(an1 1 · · · anr r ? eβ ) = ?i ?i

j=1

nj i j +

(β, β) (Λi , Λi ) n1 (a?i1 · · · anr r ? eβ ). ? ?i 2 2

We remark that the trace of p?d X + (z)X ? (z) on V (Λ0 ) is calculated in [8]. We shall describe the vertex operators in terms of the representation constructed above, and then compute a similar trace for them (see Sect. 3). ? ? Let the components Φ± (z) = Φ?V (z) be de?ned as in (1.1). From the condition λ± ? that it intertwines the action of x0 we ?nd ? ? Φ+ (z) = [Φ? (z), x? ]q , 0 (2.1)

where [X, Y ]q = XY ? qY X. The intertwining relations with the Chevalley generators ? determine Φ± (z) uniquely, but it is not easy to get the expression of the vertex operators from them only. We better use the relations with Drinfeld’s generators an . We lack the formulas for the coproduct of them, in general. However we have the following partial information which will su?ce for our purpose [9].

Correlation Functions of the XXZ model for ? < ?1 Proposition 2.1. For k ≥ 0 and l > 0 we have

k?1

7

?(x+ ) = x+ ? γ k + γ 2k K ? x+ + k k k

i=0

γ (k+3i)/2 ψk?i ? γ k?i x+ i

l?1

2 mod N? ? N+ ,

?(x+ ) ?l

=

x+ ?l

?γ

?l

+K

?1

?

x+ ?l

+

i=1

γ (l?i)/2 ??l+i ? γ ?l+i x+ ?i

2 mod N? ? N+ ,

?(al ) = al ? γ

l/2

+γ

3l/2

?(a?l ) = a?l ? γ

?3l/2

+γ

? al ?l/2

mod N? ? N+ , ? a?l mod N? ? N+ .

2 Here N± and N± are left F [γ ± , ψr , ?s |r, ?s ∈ Z≥0 ]-modules generated by {x± |k ∈ Z} m and {x± x± |m, n ∈ Z} respectively. m n

By using Proposition 2.1 and noting that N± v± = 0, N+ v? ? F [z, z ?1]v+ , we get the exact relations [k] k ? z Φ? (z) k > 0, k [k] ? ? [a?k , Φ? (z)] = q ?5k/2 z ?k Φ? (z) k > 0, k ? ? ? [Φ? (z), X + (w)] = 0, t1 Φ? (z)t?1 = q Φ? (z). 1 ? [ak , Φ? (z)] = q 7k/2 ? These conditions along with the normalization (1.2) determine the form of Φ? (z) completely. Explicitly we have, on V (Λi ) (i = 0, 1), ? Φ? (z) = exp a?n 7n/2 n an ?5n/2 ?n α/2 e (?q 3 z)(?α +i)/2 . (2.2) q z exp ? q z [2n] [2n] n=1 n=1

∞ ∞

It can be checked directly that the operators given by (2.1, 2.2) enjoy the correct intertwining properties. Let us make a comment on the base ?eld F = Q(q). In the above expression of the bosonization of the vertex operators, the square root of q appears. But, it can be absorbed in the boson oscillators if we change the de?nition of their commutation relations. Note also that the combination (?α + i)/2 in the power of ?q 3 z produces only an integer power acting on V (Λi ). In any event, the appearance of q 1/2 is only super?cial, and our theory is free from the ambiguity in the choice of the square root.

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M. Jimbo et al.

§3. Calculation of correlators As explained in Sect.1, for the evaluation of the correlators it is necessary to calculate the trace of the product of the vertex operators. Using the boson realization described in Sect.2, we shall treat the trace of the form tr(x?d y α O) where x, y are complex parameters with |x| < 1. Since q ?2ρ = q ?4d?α , the choice x = q 4 , y = q ?1 will be relevant to the correlators. The calculation is simpli?ed by the technique of Clavelli and Shapiro ([10], Appendix C). Their prescription is as follows. Introduce a copy of bosons {bn } satisfying [am , bn ] = 0 and the same commutation relations as the an . Let b?n an + b?n (n > 0), an + n (n < 0). an = ? n 1?x x ?1 ? For a linear operator O = O({an }) on the Fock space Fa = F [a?1 , a?2 , · · ·], let O = O({?n }) be the operator on Fa ? Fb (Fb = F [b?1 , b?2 , · · ·]) obtained by substituting an a ? for an . We have then ? 0|O|0 ? trFa x?d O = ∞ , n n=1 (1 ? x )

n2 ? ? where d = ? n>0 [n][2n] a?n an and 0|O|0 denotes the usual expectation value with respect to the Fock vacuum |0 = 1 ? 1 ∈ Fa ? Fb , 0|0 = 1. In order to apply their method to our case, we need the following. Set

f (z) = (zx; x)∞ (q x/z; x)∞ ,

2

g(z) =

(zq 2 xn ; q 4 )∞ . (zq 4 xn ; q 4 )∞ n=1

∞

? Let us denote the operators X ? (z) and Φ± (z) with an substituted for an by J(z) and ? φ± (z). Then the explicit expressions of J(z) and φ? (z) (after the normal ordering) are J(z) = f (1) exp ?

∞

q n/2 n (z a?n ? z ?n b?n ) [n] n=1

∞

× exp

q n/2 z ?n an ? (xz)n bn ?α ??α , e z [n] 1 ? xn n=1 q 7n/2 z n a?n ? q ?5n/2 z ?n b?n [2n] n=1

∞

φ? (z) = g(1) exp × exp ?

q ?5n/2 z ?n an ? q 7n/2 (xz)n bn α/2 e (?q 3 z)(?α +i)/2 on V (Λi ). [2n](1 ? xn ) n=1 1?

∞

They satisfy the following relations, q 2 ξ2 ξ2 ξ1 ξ2 1? f f , ξ1 ξ1 ξ1 ξ2 z2 z1 , g φ? (z1 )φ? (z2 ) =: φ? (z1 )φ? (z2 ) : g z2 z1 x 1 J(ξ)φ? (z) =: J(ξ)φ? (z) : , (1 ? q 2 w?1 )f (w) ?q ?1 w , φ? (z)J(ξ) =: J(ξ)φ? (z) : (1 ? w)f (w) w dξ φ+ (z) = (q ? q ?1 ) : J(ξ)φ? (z) : . (1 ? w)(1 ? q 2 w?1 )f (w) q2 <|w|<1 2πi J(ξ1 )J(ξ2 ) =: J(ξ1 )J(ξ2 ) :

Correlation Functions of the XXZ model for ? < ?1

9

Here w = ξ/q 2 z and : · · · : denotes the normal ordering with respect to the bosons an and bn (We do not normal order the operators enα and ?α ). In the above equations, all ∞ factors of the form 1/(1 ? z) should be understood as n=0 z n . This fact speci?es the contour for the expression φ+ . Using the formulas above, (1.3) is evaluated as follows. Set

ε ,···,ε Pε′n,···,ε′1 (zn , · · · , z1 | x, y | i) =

n 1

×

trV (Λi )

(q 2 ; q 4 )n ∞ (q 4 ; q 4 )n ∞ ?Λ ? ?Λ V ? Λi+1 V x?d y α ΦΛi V ε′ (z1 ) · · · ΦΛi+n?1ε′ (zn )ΦΛi+n εn (zn ) · · · ΦΛi ε1 (z1 ) Λi+1 i+n V n i+n?1

1

trV (Λi ) (x?d y α )

. (3.1)

We have L

(i) zn ,...,z1 ε ,···,ε = Pε′n,···,ε′1 (zn , · · · , z1 | q 4 , q ?1 | i).

n 1

for L = Eε′ εn ? · · · ? Eε′ ε1 (see the remark at the end of Sect. 1). n 1 Introduce the following notations A = {a1 , · · · , as } = {j | ε′ = ?1}, j (s + t = n,

2

B = {b1 , · · · , bt } = {j | εj = +1},

a i < aj ,

bi < bj

2 ?1

for i < j), ; x)∞ (xz ?1 ; x)∞ .

h(z) = (q z; x)∞ (xz; x)∞ (q z

We prepare the integration variables ξa (a ∈ A), ζb (b ∈ B) and set ηj = ξaj (1 ≤ j ≤ s), ? ? = ζbn+1?j (s < j ≤ n), η = j ηj and z = j zj . Then we have

ε ,···,ε Pε′n,···,ε′1 (zn , · · · , z1 | x, y | i)

n 1

= (?1)t q ×

a∈A

a+

b∈B

b?n(n+1)/2 a∈A Ca 2

dξa 2πi(ξa ? za )

b∈B

Cb

dζb 2πi(ζb ? zb )

a∈A a<j≤n

zj ? q ξa zj ? ξa

2

b∈B b<j≤n

ζb ? q zj ζb ? zj

j<k

ηk ? ηj ηk ? q 2 ηj (3.2)

×

h(1)

n

j<k

h(zj /zk )h(ηj /ηk ) h(ηj /zk )

j,k

2 z η 2m y 2m xm ?i/4 m∈Z+i/2 (?/?) . (x; x)∞ trV (Λi ) (x?d y α )

Note that the last factor of the above equation can be rewritten into z ? η ?

i (?(y z /?)2 x1+i ; x2 ) (?(?/y?)2 x1?i ; x2 ) ? η η z ∞ ∞ . 2 x1+i ; x2 ) (?y ?2 x1?i ; x2 ) (?y ∞ ∞

The contours of integration should be chosen as follows. Both Ca and Cb are anticlockwise, and the zi (1 ≤ i ≤ n) lie inside of Ca and outside of Cb . Other relevant poles with small multipliers q, x (e.g., q 2 zi ) are inside, and those with large multipliers are outside of the contours Ca and Cb . By specializing (3.2) to the case x = q 4 and y = q ?1 , we obtain the integral formula for the correlators. The case n = 1 gives the spontaneous + ? ?ε ε staggered polarization P0 = P+ (1 | 1)? P? (1 | 1) [1]. Note that Pε (1 | i) = P?ε (1 | 1 ? i). + + 2 2 2 2 2 2 By using (3.2) for P+ (1 | 1) and P+ (1 | 0) we get P0 = (q ; q )∞ /(?q ; q )∞ .

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M. Jimbo et al.

Acknowledgement. We wish to thank N. H. Jing for sending his preprint [8], and H. Ooguri and J. H. H. Perk for discussions. We are grateful to Brian Davies and Omar Foda for their collaborations in our previous works, on which is based the present paper. References [1] Baxter R J, Spontaneous staggered polarization of the F model, J. Stat. Phys., 9, (1973) 145–182. [2] Davies B, Foda O, Jimbo M, Miwa T and Nakayashiki A, Diagonalization of the XXZ Hamiltonian by vertex operators, RIMS preprint, 873, (1992). [3] Foda O and Miwa T, Corner transfer matrices and quantum a?ne algebras, Int. J. Mod. Phys. A, 7, supplement 1A (1992) 279–302. [4] Frenkel I B and Jing N H, Vertex representations of quantum a?ne algebras, Proc. Nat’l. Acad. Sci. USA, 85, (1988) 9373–9377. [5] Baxter R J, Solvable eight-vertex model on an arbitrary planar lattice, Phil. Trans. Roy. Soc., A289, (1978) 315–346. [6] Au-Yang H and Perk J H H, Critical Correlations in a Z-invariant Inhomogeneous Ising Model, Physica, 144A, (1987) 44–104. [7] Drinfeld V G, A new realization of Yangians and quantum a?ne algebras, Soviet Math. Doklady, 36, (1988) 212–216. [8] Jing N H, On a trace of q analog vertex operators, in Quantum Groups, Spring Workshop on Quantum Groups Argonne National Laboratory 16 April–11 May 1990, eds. T. Curtright, D. Fairlie and C. Zachos. World Scienti?c, Singapore 1992 [9] Chari V and Pressley A, Quantum a?ne algebras, Commun. Math. Phys., 142, (1991) 261–283. [10] Clavelli L and Shapiro J A, Pomeron Factorization in General Dual Models, Nucl. Phys. B, 57, (1973) 490–535.

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