A Vertex Operator Approach for Form Factors of Belavin's $(\mathbb{Z}/n\mathbb{Z})$-Symmetric Model and Its Application

0.24 MB·English

by  Yas-Hiro Quano

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7(2011), 008, 16 pages
A Vertex Operator Approach for Form Factors
Z Z
of Belavin’s ( /n )-Symmetric Model
and Its Application⋆
Yas-Hiro QUANO
Department of Clinical Engineering, Suzuka University of Medical Science,
Kishioka-cho, Suzuka 510-0293, Japan
E-mail: [email protected]
1
1
Received October 22, 2010, in final form January 07, 2011; Published online January 15, 2011
0
2 doi:10.3842/SIGMA.2011.008

n
a Abstract. A vertex operator approach for form factors of Belavin’s (Z/nZ)-symmetric
J
5 model is constructed on the basis of bosonization of vertex operators in the A(n1−)1 model
and vertex-face transformation. As simple application for n = 2, we obtain expressions for
1
z x
2m-point form factors related to the σ and σ operators in the eight-vertex model.

]
h Key words: vertex operator approach; form factors; Belavin’s (Z/nZ)-symmetric model;
p integral formulae
-
h
2010 Mathematics Subject Classification: 37K10;81R12
t
a
m
[
1 Introduction

5
v
2 In [1] and [2] we derived the integral formulae for correlation functions and form factors, respec-
8 tively, of Belavin’s (Z/nZ)-symmetric model [3, 4] on the basis of vertex operator approach [5].
3
Belavin’s(Z/nZ)-symmetricmodelisann-stategeneralizationofBaxter’seight-vertexmodel[6],
4
. which has (Z/2Z)-symmetries. As for the eight-vertex model, the integral formulae for correla-
0
1 tion functions and form factors were derived by Lashkevich and Pugai [7] and by Lashkevich [8],
0 respectively.
1
It was found in [7] that the correlation functions of the eight-vertex model can be obtained
:
v
by using the free field realization of the vertex operators in the eight-vertex SOS model [9], with
i
X insertion of the nonlocal operator Λ, called ‘the tail operator’. The vertex operator approach for
r higher spin generalization of the eight-vertex model was presented in [10]. The vertex operator
a
approach for higher rank generalization was presented in [1]. The expression of the spontaneous
polarization of the (Z/nZ)-symmetric model [11] was also reproduced in [1], on the basis of
vertex operatorapproach. Concerningformfactors, thebosonization schemefortheeight-vertex
model was constructed in [8]. The higher rank generalization of [8] was presented in [2]. It was
shown in [12, 13] that the elliptic algebra U (sl ) relevant to the (Z/nZ)-symmetric model
q,p N
provides the Drinfeld realization of the face type elliptic quantum group (sl ) tensored by
q,λ N
B
a Heisenberg algebra. b
The present paper is organized as follows. In Section 2 we review the basbic definitions of
the (Z/nZ)-symmetric model [3], the corresponding dual face model A(1) model [14], and the
n 1
vertex-face correspondence. In Section 3 we summarize the vertex operat−or algebras relevant to
the (Z/nZ)-symmetric model and the A(1) model [1, 2]. In Section 4 we construct the free field
n 1
representations of the tail operators, in t−erms of those of the basic operators for the type I [15]
⋆ThispaperisacontributiontotheProceedingsoftheInternationalWorkshop“RecentAdvancesinQuantum
Integrable Systems”. The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2010.html
2 Y.-H. Quano
(1)
and the type II [16] vertex operators in the A model. Note that in the present paper we
n 1
use a different convention from the one used in [−1, 2]. In Section 5 we calculate 2m-point form
factors of the σz-operator and σx-operator in the eight-vertex model, as simple application for
n = 2. InSection 6wegive someconcludingremarks. Usefuloperator productexpansion(OPE)
formulae and commutation relations for basic bosons are given in Appendix A.
2 Basic definitions
The present section aims to formulate the problem, thereby fixing the notation.
2.1 Theta functions
The Jacobi theta function with two pseudo-periods 1 and τ (Imτ > 0) are defined as follows:
a
ϑ (v;τ) := exp π√ 1(m+a) [(m+a)τ +2(v+b)] ,
b −
(cid:20) (cid:21) m Z
X∈ (cid:8) (cid:9)
for a,b R. Let n Z>2 and r R>1, and also fix the parameter x such that 0 < x < 1. We
∈ ∈ ∈
will use the abbreviations,
[v] := xvr2−vΘx2r(x2v), [v]′ := [v]r r 1, [v]1 := [v]r 1,
| 7→ − | 7→
v := xvr2−vΘx2r( x2v), v ′ := v r r 1, v 1 := v r 1,
{ } − { } { }| 7→ − { } { }| 7→
where
Θ (z) = (z;q) qz 1;q (q;q) = qm(m 1)/2( z)m,
q − −
∞ ∞ ∞ m Z −
(cid:0) (cid:1) X∈
(z;q ,...,q ) = 1 zqi1 qim .
1 m ∞ − 1 ··· m
i1,..Y.,im>0(cid:0) (cid:1)
Note that
1/2 v π√ 1 ǫr ǫr
ϑ , − = exp [v],
1/2 r ǫr π − 4
(cid:20) − (cid:21)(cid:18) (cid:19) r (cid:16) (cid:17)
0 v π√ 1 ǫr ǫr
ϑ , − = exp v ,
1/2 r ǫr π − 4 { }
(cid:20) (cid:21)(cid:18) (cid:19) r
(cid:16) (cid:17)
where x = e ǫ (ǫ > 0).
−
For later conveniences we also introduce the following symbols:
r−1n−j gj(z−1) x2n+2r−j−1z xj+1z
rj(v) = z r n g (z) , gj(z) = {x2n j+1z x}2{r+j 1z}, (2.1)
j − −
{ }{ }
r n−j gj∗(z−1) x2n+2r−j−1z ′ xj−1z ′
rj∗(v) = zr−1 n g (z) , gj∗(z) = {x2n j 1z x}2{r+j 1z} , (2.2)
j∗ { − − }′{ − }′
j(n−j) ρj(z−1) (x2j+1z;x2,x2n) (x2n−2j+1z;x2,x2n)
χj(v) = (−z)− n ρ (z) , ρj(z) = (xz;x2,x2n)∞(x2n+1z;x2,x2n) ∞, (2.3)
j
∞ ∞
where z =x2v, 1 6 j 6 n and
z = (z;x2r,x2n) , z = (z;x2r 2,x2n) .
′ −
{ } ∞ { } ∞
Form Factors of Belavin’s (Z/nZ)-Symmetric Model and Its Application 3
In particular we denote χ(v) = χ (v). These factors will appear in the commutation relations
1
among the type I and type II vertex operators.
TheintegralkernelforthetypeIandthetypeIIvertexoperatorswillbegivenastheproducts
of the following elliptic functions:
[v+ 1 w] [v 1]
f(v,w) = 2 − , h(v) = − ,
[v 1] [v+1]
− 2
[v 1 +w] [v+1]
f (v,w) = − 2 ′, h (v) = ′.
∗ [v+ 1] ∗ [v 1]
2 ′ − ′
2.2 Belavin’s (Z/nZ)-symmetric model
Let V = Cn and εµ 06µ6n 1 be the standard orthonormal basis with the inner product
ε ,ε = δ . Bela{vin}’s (Z/n−Z)-symmetric model [3] is a vertex model on a two-dimensional
µ ν µν
h i
square lattice such that the state variables take the values of (Z/nZ)-spin. The model is
L
(Z/nZ)-symmetric in a sense that the R-matrix satisfies the following conditions:
(i) R(v)ik = 0, unless i+k = j +l, mod n,
jl
(ii) R(v)i+pk+p = R(v)ik, i,j,k,l,p Z/nZ.
j+pl+p jl ∀ ∈
Thedefinitionof theR-matrix intheprincipalregimecan befoundin[2]. ThepresentR-matrix
has three parameters v, ǫ and r, which lie in the following region:
ǫ >0, r > 1, 0< v < 1.
(1)
2.3 The A model
n−1
The dual face model of the (Z/nZ)-symmetric model is called the A(1) model. This is a face
n 1
model on a two-dimensional squarelattice , the dual lattice of , such−that the state variables
∗
L L
take the values of the dual space of Cartan subalgebra h of A(1) :
∗ n 1
−
n 1
−
h = Cω ,
∗ µ
µ=0
M
where
µ 1 n 1
− 1 −
ω := ε¯ , ε¯ = ε ε .
µ ν µ µ µ
− n
ν=0 µ=0
X X
The weight lattice P and the root lattice Q of A(1) are usually defined. For a h , we set
n 1 ∈ ∗
−
n 1
−
a = a¯ a¯ , a¯ = a+ρ,ε = a+ρ,ε¯ , ρ = ω .
µν µ ν µ µ µ µ
− h i h i
µ=1
X
An ordered pair (a,b) h 2 is called admissible if b = a+ε¯ , for a certain µ(0 6 µ 6 n 1).
∗ µ
∈ −
c d
For (a,b,c,d) h 4, let W v be the Boltzmann weight of the A(1) model for the state
∈ ∗ b a n 1
(cid:20) (cid:12) (cid:21) −
c d (cid:12)
configuration round a fa(cid:12)ce. Here the four states a, b, c and d are ordered clockwise
b a (cid:12)
(cid:20) (cid:21)
4 Y.-H. Quano
c d
from the SE corner. In this model W v = 0 unless the four pairs (a,b),(a,d),(b,c)
b a
(cid:20) (cid:12) (cid:21)
and (d,c) are admissible. Non-zero Boltzmann(cid:12)weights are parametrized in terms of the elliptic
(cid:12)
theta function of the spectral parameter v. Th(cid:12)e explicit expressions of W can be found in [2].
We consider the so-called Regime III in the model, i.e., 0 < v < 1.
2.4 Vertex-face correspondence
Let t(v)a be the intertwining vectors in Cn, whose elements are expressed in terms of theta
a ε¯µ
functions−. As for the definitions see [2]. Then t(v)a ’s relate the R-matrix of the (Z/nZ)-
a ε¯µ
−
(1)
symmetric model in the principal regime and Boltzmann weights W of the A model in the
n 1
regime III −
c d
R(v v )t(v )d t(v )c = t(v )c t(v )bW v v . (2.4)
1− 2 1 a⊗ 2 d 1 b⊗ 2 a b a 1− 2
b (cid:20) (cid:12) (cid:21)
X (cid:12)
(cid:12)
(cid:12)
Let us introduce the dual intertwining vectors satisfying
n 1 n 1
− t∗µ(v)aa′tµ(v)aa′′ = δaa′′′, − tµ(v)aa ε¯νt∗µ′(v)aa−ε¯ν = δµµ′. (2.5)
−
µ=0 ν=0
X X
From (2.4) and (2.5), we have
c d
t (v )b t (v )aR(v v ) = W v v t (v )a t (v )d.
∗ 1 c⊗ ∗ 2 b 1− 2 b a 1− 2 ∗ 1 d ⊗ ∗ 2 c
d (cid:20) (cid:12) (cid:21)
X (cid:12)
(cid:12)
(cid:12)
For fixed r > 1, let
c d c d
S(v) = R(v) , W v = W v ,
− |r7→r−1 ′ b a − b a (cid:12)
(cid:20) (cid:12) (cid:21) (cid:20) (cid:12) (cid:21) (cid:12)r r 1
(cid:12) (cid:12) (cid:12) 7→ −
(cid:12) (cid:12)
(cid:12)
(cid:12) (cid:12)
and t (v)b is the dual intertwining vector of t(v)a. Here, (cid:12)
′∗ a ′ b
t(v)a := f (v)t(v;ǫ,r 1)a,
′ b ′ − b
with
v2 (r+n 2)v (n 1)(3r+n 5)
− − −
x−n(r 1)− n(r 1) − 6n(r 1)
f (v) = − − −
′
n (x2r 2;x2r 2)
− −
− ∞
(x2z 1;x2n,x2r 2) (x2r+2n 2z;x2n,x2r 2)
−p − − −
∞ ∞, (2.6)
× (z 1;x2n,x2r 2) (x2r+2n 4z;x2n,x2r 2)
− − − −
∞ ∞
and z = x2v. Then we have
c d
t (v )b t (v )aS(v v )= W v v t (v )a t (v )d.
′∗ 1 c⊗ ′∗ 2 b 1− 2 ′ b a 1− 2 ′∗ 1 d ⊗ ′∗ 2 c
d (cid:20) (cid:12) (cid:21)
X (cid:12)
(cid:12)
(cid:12)
Form Factors of Belavin’s (Z/nZ)-Symmetric Model and Its Application 5
3 Vertex operator algebra
3.1 Vertex operators for the (Z/nZ)-symmetric model
Let (i) be the C-vector space spanned by the half-infinite pure tensor vectors of the forms
H
ε ε ε with µ Z/nZ, µ = i+1 j (mod n) for j 0.
µ1 ⊗ µ2 ⊗ µ3 ⊗··· j ∈ j − ≫
The type I vertex operator Φµ(v) can be defined as a half-infinite transfer matrix. The opera-
tor Φµ(v) is an intertwiner from (i) to (i+1), satisfying the following commutation relation:
H H
Φµ(v )Φν(v )= R(v v )µν Φν′(v )Φµ′(v ).
1 2 1− 2 µ′ν′ 2 1
µ′,ν′
X
When we consider an operator related to ‘creation-annihilation’ process, we need another
type of vertex operators, the type II vertex operators that satisfy the following commutation
relations:
Ψ (v )Ψ (v )= Ψ (v )Ψ (v )S(v v )µ′ν′,
∗ν 2 ∗µ 1 ∗µ′ 1 ∗ν′ 2 1 − 2 µν
µ′,ν′
X
Φµ(v )Ψ (v )= χ(v v )Ψ (v )Φµ(v ).
1 ∗ν 2 1− 2 ∗ν 2 1
Let
ρ(i) = x2nHCTM : (i) (i),
H → H
where H is the CTM Hamiltonian defined as follows:
CTM
1 ∞
H (µ ,µ ,µ ,...) = jH (µ ,µ ),
CTM 1 2 3 v j j+1
n
j=1
X
µ ν 1 if 06 ν < µ 6 n 1,
Hv(µ,ν) = n−1+−µ ν if 06 µ 6 ν 6 n−1. (3.1)
(cid:26) − − −
Then we have the homogeneity relations
Φµ(v)ρ(i) = ρ(i+1)Φµ(v n), Ψ (v)ρ(i) = ρ(i+1)Ψ (v n).
− ∗µ ∗µ −
(1)
3.2 Vertex operators for the A model
n−1
For k = a+ρ,l = ξ+ρand06 i 6 n 1,let (i) bethespaceofadmissiblepaths(a ,a ,a ,...)
− Hl,k 0 1 2
such that
a = a, a a ε¯ ,ε¯ ,...,ε¯ for j =0,1,2,3,...,
0 j j+1 0 1 n 1
− ∈ { − }
a = ξ+ω for j 0.
j i+1 j
− ≫
The type I vertex operator Φ(v)a+ε¯µ can be defined as a half-infinite transfer matrix. The
a
operator Φ(v)a+ε¯µ is an intertwiner from (i) to (i+1) , satisfying the following commutation
a Hl,k Hl,k+ε¯µ
relation:
c d
Φ(v )cΦ(v )b = W v v Φ(v )cΦ(v )d.
1 b 2 a b a 1− 2 2 d 1 a
d (cid:20) (cid:12) (cid:21)
X (cid:12)
(cid:12)
The free field realization of Φ(v )b w(cid:12)as constructed in [15]. See Section 4.2.
2 a
6 Y.-H. Quano
The type II vertex operators should satisfy the following commutation relations:
ξ ξ
Ψ (v )ξcΨ (v )ξd = Ψ (v )ξcΨ (v )ξbW c d v v ,
∗ 2 ξd ∗ 1 ξa Xξb ∗ 1 ξb ∗ 2 ξa ′(cid:20) ξb ξa (cid:12)(cid:12) 1− 2(cid:21)
Φ(v1)aa′Ψ∗(v2)ξξ′ = χ(v1−v2)Ψ∗(v2)ξξ′Φ(v1)aa′. (cid:12)(cid:12)
Let
ρ(l,ik) = Gax2nHl(,ik), Ga = [aµν],
06µ<ν6n 1
Y −
(i) (1)
where H is the CTM Hamiltonian of A model in regime III is given as follows:
l,k n 1
−
(i) 1 ∞
H (a ,a ,a ,...)= jH (a ,a ,a ),
l,k 0 1 2 n f j−1 j j+1
j=1
X
H (a+ε¯ +ε¯ ,a+ε¯ ,a) = H (ν,µ),
f µ ν µ v
and H (ν,µ) is the same one as (3.1). Then we have the homogeneity relations
v
(i) (i+1)
ρ ρ
Φ(v)a′ a+ρ,l = a′+ρ,lΦ(v n)a′, Ψ (v)ξ′ρ(i) = ρ(i+1) Ψ (v n)ξ′.
a Ga Ga′ − a ∗ ξ k,ξ+ρ k,ξ′+ρ ∗ − ξ
ξ′
The free field realization of Ψ (v) was constructed in [16]. See Section 4.3.
∗ ξ
3.3 Tail operators and commutation relations
In [1] we introduced the intertwining operators between (i) and (i) (k = l+ω (mod Q)):
H Hl,k i
T(u)ξa0 = ∞ tµj( u)aj : (i) (i),
− aj+1 H → Hl,k
j=0
Y
T(u) = ∞ t ( u)aj+1 : (i) (i),
ξa0 ∗µj − aj Hl,k → H
j=0
Y
which satisfy
ρ(i) = (x2(rx−22r;;xx22rr−)2)∞ (n−1)(n−2)/2 G1 T(u)aξρ(l,ik)T(u)aξ. (3.2)
(cid:18) ∞ (cid:19) ′ξ k≡Xl+ωi
(modQ)
In order to obtain the form factors of the (Z/nZ)-symmetric model, we need the free field
representations ofthetailoperatorwhichisoffdiagonalwithrespecttotheboundaryconditions:
Λ(u)ξ′a′ =T(u)ξ′a′T(u) : (i) (i) , (3.3)
ξa ξa Hl,k → Hl′k′
where k = a+ρ, l = ξ+ρ, k = a +ρ, and l = ξ +ρ. Let
′ ′ ′ ′
n 1
L a′0 a′1 u := − t ( u)a1tµ( u)a′0.
(cid:20) a0 a1 (cid:12) (cid:21) µ=0 ∗µ − a0 − a′1
(cid:12) X
(cid:12)
(cid:12)
Form Factors of Belavin’s (Z/nZ)-Symmetric Model and Its Application 7
Then we have
Λ(u)ξ′a′0 = ∞ L a′j a′j+1 u .
ξa0 j=0 (cid:20) aj aj+1 (cid:12) (cid:21)
Y (cid:12)
(cid:12)
From the invertibility of the int(cid:12)ertwining vector and its dual vector, we have
ξ′a ξ′
Λ(u ) = δ . (3.4)
0 ξa ξ
Note that the tail operator (3.3) satisfies the following intertwining relations [1, 2]:
Λ(u)ξ′cΦ(v)b = L c d u v Φ(v)cΛ(u)ξ′d, (3.5)
ξb a b a − d ξa
d (cid:20) (cid:12) (cid:21)
X (cid:12)
Ψ (v)ξcΛ(u)ξda′ = L ξc(cid:12)(cid:12) ξd u+∆u v Λ(u)ξca′Ψ (v)ξb, (3.6)
∗ ξd ξaa Xξb ′(cid:20) ξb ξa (cid:12)(cid:12) − (cid:21) ξba ∗ ξa
(cid:12)
where (cid:12)
ξ ξ ξ ξ
L c d u = L c d u .
′
ξ ξ ξ ξ
(cid:20) b a (cid:12) (cid:21) (cid:20) b a (cid:12) (cid:21)(cid:12)r r 1
(cid:12) (cid:12) (cid:12) 7→ −
We should find a r(cid:12)(cid:12)epresentation of Λ(cid:12)(cid:12)(u)ξ(cid:12)(cid:12)′a′ and fix the constant ∆u that solves (3.5) and (3.6).
ξa
4 Free filed realization
4.1 Bosons
In[17,18]thebosonsBj (1 6 j 6 n 1,m Z 0 )relevanttoellipticalgebrawereintroduced.
m
− ∈ \{ }
For α,β h we denote the zero mode operators by P , Q . Concerning commutation relations
∗ α β
∈
among these operators see [17, 18, 2].
We will deal with the bosonic Fock spaces , (l,k h ) generated by Bj (m > 0) over
the vacuum vectors l,k : Fl,k ∈ ∗ −m
| i
Fl,k = C[{Bj1,Bj2,...}16j6n]|l,ki,
− −
where
l,k = exp √ 1(β Q +β Q ) 0,0 ,
1 k 2 l
| i − | i
and (cid:0) (cid:1)
1
t2 β t 1= (t β )(t β ), β = , β < β .
0 1 2 0 1 2
− − − − r(r 1)
−
p
4.2 Type I vertex operators
Let us define the basic operators for j = 1,...,n 1
−
j j+1
U−αj(v) = zr−r1 : exp−β1 √−1Qαj +Pαj logz) + Bm−mBm (xjz)−m :,
m=0
(cid:0) (cid:1) X6
 
j
Uωj(v) = zr2−r1j(nn−j) :expβ1 √−1Qωj +Pωj logz) − m1 x(j−2k+1)mBmkz−m :,
m=0 k=1
(cid:0) (cid:1) X6 X
 
8 Y.-H. Quano
where β = r 1 and z = x2v as usual. The normal product operation places P ’s to the
1 − −r α
right of Qβ’s, qas well as Bm’s (m > 0) to the right of B m’s. For some useful OPE formulae
−
and commutation relations, see Appendix A.
In what follows we set
π = r(r 1)P , π = π π = rL (r 1)K .
µ − ε¯µ µν µ− ν µν − − µν
The operatoprs K , L and π act on as scalars ε ε ,k , ε ε ,l and ε ε ,rl
µν µν µν l,k µ ν µ ν µ ν
F h − i h − i h − −
(r 1)k , respectively. In what follows we often use the symbols
− i
G = [K ], G = [L ].
K µν ′L µν ′
06µ<ν6n 1 06µ<ν6n 1
Y − Y −
For 0 6 µ 6 n 1 the type I vertex operator Φ(v)a+ε¯µ can be expressed in terms of U (v)
− a ωj
and U (v) on the bosonic Fock space . The explicit expression of Φ(v)a+ε¯µ can found
−αj Fl,a+ρ a
in [15].
4.3 Type II vertex operators
Let us define the basic operators for j = 1,...,n 1
−
j j+1
V−αj(v) = (−z)r−r1 : exp−β2 √−1Qαj +Pαj log(−z) − Am−mAm (xjz)−m:,
m=0
(cid:0) (cid:1) X6
r j(n−j) 
Vωj(v) = (−z)2(r−1) n
j
1
:exp β √ 1Q +P log( z) + x(j 2k+1)mAk z m :,
×  2 − ωj ωj − m − m − 
m=0 k=1
(cid:0) (cid:1) X6 X
 
where β = r and z = x2v, and Aj = [rm]x Bj . For some useful OPE formulae and
2 r 1 m [(r 1)m]x m
− −
commutationqrelations, see Appendix A.
For 0 6 µ 6 n 1 the type II vertex operator Ψ (v)ξ+ε¯µ can be expressed in terms of V (v)
− ∗ ξ ωj
and V (v) on the bosonic Fock space . The explicit expression of Ψ (v)ξ+ε¯µ can found
−αj Fξ+ρ,k ∗ ξ
in [16].
4.4 Free field realization of tail operators
In order to construct free field realization of the tail operators, we also need another type of
basic operators:
1
W−αj(v) = ((−1)rz)r(r−1)
j j+1
O O
× : exp−β0 √−1Qαj +Pαj log(−1)rz) − m−m m (xjz)−m :,
m=0
(cid:0) (cid:1) X6
 
where β = β + β = 1 , ( 1)r := exp(π√ 1r) and Oj = [m]x Bj . Concerning
0 1 2 √r(r 1) − − m [(r 1)m]x m
− −
useful OPE formulae and commutation relations, see Appendix A.
We cite the results on the free field realization of tail operators. In [1] we obtained the free
ξa′
field representation of Λ(u) satisfying (3.5) for ξ = ξ:
ξa ′
ν
dz
Λ(u)ξξaa−−εε¯¯νµ = GKI j=µ+12π√−j1zjU−αµ+1(vµ+1)···U−αν(vν)
Y
Form Factors of Belavin’s (Z/nZ)-Symmetric Model and Its Application 9
ν 1
−
× (−1)Ljν−Kjνf(vj+1−vj,πjν)G−K1, (4.1)
j=µ
Y
where v = u and µ < ν. In [2] we obtained the free field representation of Λ(v)ξ+ε¯n−1a+ε¯n−1
µ ξ+ε¯µa+ε¯n−2
satisfying (3.6) as follows:
( 1)n µ[a ] [ξ 1]
Λ(u)ξξ++εε¯¯µn−a+1aε¯+n−ε¯n2−1 = (x−1 −x)(xn2−r;2xn2−r1)3 µn−[11]− ′GKG′L−1
− ′
− ∞
n 2
− dzj W u r 1 V (v ) V (v )
×IC′j=µ+12π√−1zj −αn−1 − −2 −αn−2 n−2 ··· −αµ+1 µ+1
Y (cid:0) (cid:1)
n 2
−
× (−1)Lµj−Kµjf∗(vj −vj+1,πµj)G−K1G′L, (4.2)
j=µ+1
Y
for 0 6 µ 6 n 2 with ∆u = n 1 and v = u. Concerning other types of tail operators
ξa′ − − −2 n−1
Λ(u) , the expressions of the free field representation can be found in [1, 2].
ξa
4.5 Free field realization of CTM Hamiltonian
Let
n 1 j n 1
H = ∞ [rm]x − x(2k 2j 1)mBk (Bj Bj+1)+ 1 − P P
F [(r 1)m]x − − −m m− m 2 ωj αj
m=1 − j=1k=1 j=1
X XX X
n 1 j n 1
= ∞ [rm]x − x(2j 2k 1)m(Bj Bj+1)Bk + 1 − P P (4.3)
[(r 1)m]x − − −m− −m m 2 ωj αj
m=1 − j=1k=1 j=1
X XX X
be the CTM Hamiltonian on the Fock space [19]. Then we have the homogeneity relation
l,k
F
φµ(z)qHF = qHFφµ q−1z ,
and the trace formula (cid:0) (cid:1)
xnβ1k+β2l2
tr x2nHFG = | | G .
Fl,k a (x2n;x2n)n−1 a
∞
(cid:0) (cid:1)
Let ρ(l,ik) = Gax2nHF. Then the relation (3.2) holds. We thus indentify HF with free field
(i) (1)
representations of H , the CTM Hamiltonian of A model in regime III.
l,k n 1
−
5 Form factors for n = 2
In this section we would like to find explicit expressions of form factors for n = 2 case, i.e., the
eight-vertex model form factors. Here, we adopt the convention that the components 0 and 1
for n = 2 are denoted by + and . Form factors of the eight-vertex model are defined as matrix
−
elements of some local operators. For simplicity, we choose σz as a local operator:
σz = E(1) E(1),
++− −−
where E(j) is the matrix unit on the j-th site. The free field representation of σz is given by
µµ′
σz = εΦ (u)Φε(u).
∗ε
ε=
X±
c
10 Y.-H. Quano
Here, Φ (u) is the dual type I vertex operator, whose free filed representation can be found
∗ε
in [1, 2].
The corresponding form factors with 2m ‘charged’ particles are given by
1
F(i)(σz;u ,...,u ) = Tr Ψ (u ) Ψ (u )σzρ(i) , (5.1)
m 1 2m ν1···ν2m χ(i) H(i) ∗ν1 1 ··· ∗ν2m 2m
(cid:16) (cid:17)
where c
(x4;x4)
χ(i) = Tr (i)ρ(i) = (x2;x2)∞.
H
∞
In this section we denote the spectral parameters by zj = x2uj, and denote integral variables by
w = x2va.
a
By using the vertex-face transformation, we can rewrite (5.1) as follows:
1
F(i)(σz;u ,...,u ) = t u u + 1 l1 t u u + 1 l2m
m 1 2m ν1···ν2m χ(i) ′ν∗1 1− 0 2 l ··· ′∗ν2m 2m− 0 2 l2m−1
l1,X...,l2m (cid:0) (cid:1) (cid:0) (cid:1)
ε t (u u )k tε(u u )k1
× ∗ε − 0 k1 − 0 k2
k≡l+Xi(mod2)εX=± k1X=k±1k2=Xk1±1
Tr Ψ (u )l Ψ (u )l2m−1Φ (u)k Φ(u)k1Λ(u )l2mk2[k]x4HF ,
× (i) ∗ 1 l1··· ∗ 2m l2m ∗ k1 k2 0 lk [l]
Hl,k(cid:18) ′ (cid:19)
where H is the CTM Hamiltonian defined by (4.3).
F
Let
1
F(i)(σz;u ,...,u ) = ε t (u u )k tε(u u )k1
m 1 2m ll1···l2m χ(i) ∗ε − 0 k1 − 0 k2
k≡l+Xi(mod2)εX=± k1X=k±1k2=Xk1±1
Tr Ψ (u )l Ψ (u )l2m−1Φ (u)k Φ(u)k1Λ(u )l2mk2[k]x4HF . (5.2)
× (i) ∗ 1 l1··· ∗ 2m l2m ∗ k1 k2 0 lk [l]
Hl,k(cid:18) ′ (cid:19)
Then we have
F(i)(σz;u ,...,u ) = F(i)(σz;u ,...,u )
m 1 2m ll1···l2m m 1 2m ν1···ν2m
ν1,X...,ν2m
tν1 u u + 1 l tν2m u u + 1 l2m−1.
× ′ 1− 0 2 l1··· ′ 2m− 0 2 l2m
(cid:0) (cid:1) (cid:0) (cid:1)
For simplicity, let l =l j for 1 6j 6 2m. Thenfrom therelation (3.4), Λ(u )l2mk2 vanishes
j − 0 lk
unless k = k 2. Thus, the sum over k and k on (5.2) reduces to only one non-vanishing
2 1 2
−
term. Furthermore, we note the formula
0 u u 1+k
εt∗ε(u−u0)kk−1tε(u−u0)kk−−21 = (−1)1−i{ }[{u −u00][k− 1] }.
ε= − −
X±
Here, we use k l i (mod 2). The sum with respect to k for the trace over the zero-modes
− ≡
parts can be calculated as follows:
2m m 1
u u0 1+k ( zj)2(rr−l1)−k2(x−1z)−l+(r−r1)k − ( wa)−rr−l1+k
{ − − } − −
k l+i(mod2) j=1 a=1
≡ X Y Y
( 1)rx−r+1z0 −r−l1+kr xrr−l21−2kl+(r−r1)k2
× −
(cid:0) (cid:1)