- Email: [email protected]
annihilation operators and form factors of the XXZ model
14 March 1994 PHYSICS LETTERS A
ELSEVIER
Physics Letters A 186 (1994) 217-224
Creation/annihilation operators and form factors of the X X Z model Kei Miki Department ofMathematicalScience, Facultyof EngineeringScience, Osaka University, Toyonaka, Osaka560, Japan Received 23 September 1993; accepted for publication 21 December 1993 Communicated by A.P. Fordy
Abstract
Two kinds of creation/annihilation operators for the XXZ model in the antiferroelectric regime are constructed. They presumably have a well defined low temperature limit. As an application, the axioms of form factors are discussed.
1. Introduction
In Ref. [ 1 ], the X X Z model in the antiferroelectric regime is diagonalized in the thermodynamic limit. This model is defined by the Hamiltonian
Hxxz=-½
~ k=
(a~+ 1o'~ + o'~+ i o'~ + z~o'~+1o'~),
zt= ½( q + q - 1 ) ,
0
,
--oo
acting on the infinite tensor product ... ® V® V® ... o f V = C 2. This Hamiltonian commutes with Uq(sl2). The key idea o f Ref. [ 1 ] was the identification o f the left and right semi-infinite tensor products o f Vwith the level 1 highest weight representation V(Ai) and the level - 1 lowest weight representation V(Ai)*a o f Uq(sl 2 ), respectively. Here i = 0 or 1, depending on the boundary conditions. By this, the model was cast into the framework of representation theory. The transfer matrix and creation/annihilation operators (CAOs) were constructed in terms o f type I and type II vertex operators (VOs) (in the terminology ofRef. [ 1 ] ), respectively. Unfortunately, contrary to the type I case, type II VOs do not have a well defined low temperature limit x--,0. As a result, the CAOs are defined only as an analytic continuation. This fact seems to make unclear whether the axioms of form factors [ 2 ] hold in this model. On this problem, some explicit calculations were carried out in Ref. [ 3 ]. In this Letter, we shall construct two kinds o f CAOs that presumably have a well defined low temperature limit. This is a generalization o f the construction o f the CAOs for the Ising model [ 4 ]. Two kinds o f CAOs (right and left CAO) and their c o m m u t a t i o n relations are first introduced in Ref. [5 ] in the context o f the G e l f a n d - L e v i t a n - M a r c h e n k o equation. As an application, we shall discuss the axioms o f form factors along the lines o f Appendix B o f Ref. [2]. In this approach, the existence of two kinds o f CAOs plays an important role. This Letter is organized as follows. In Section 2, the results o f Ref. [ 1 ] are summarized. In Section 3, we introduce new operators that seem to have a low temperature limit. In Section 4, using these operators, we construct CAOs. In Section 5, the axioms o f form factors are discussed. 0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved
SSD10375-9601 (93)E1030-J
218
K. M i k i / Physics Letters A 186 (1994) 2 1 7 - 2 2 4
2. Notations
In order to fix notations, we briefly summarize the well known results [ 1 ] in a slightly different convention. Unless otherwise mentioned, we follow the notation of Ref. [ 1 ] except that we use the inner product on weight space normalized by (ai, oti) = 2 [6].
2.1. Finite dimensional representation Let us denote a two-dimensional representation in the principal picture by V~= Cv+ ®Cv_. For later purposes, we take as bases of Vc® Vxc and Vx~® Vo
w±~=~v+®v±,
Wo=V+®V_-V_®V+,
w~=½(v+®v_+v_®v+)
U±l=V±®V±/~,
Uo=½(v+®v_-v_®v+),
and
u~=v+®v_ +v_®v+,
respectively. Let us further denote a three-dimensional representation spanned by Wk ( k = 0, -+ 1 ) by Wo These representations are isomorphic to their dual representations by the following correspondences, ttta+
V~+_,¢~-V¢,
v*~--~v~,
. a -+
W~+_,¢~-W¢,
W~--~W_k.
(2.1)
2.2. S matrix Setting ( z ) ~ = I-[~°=o ( 1 - z x 4 " ) , we define an S matrix and its matrix elements by
s(o=-~-'
(x~)°°(x~/~)~ R(O,
S(Ov~®vB= E s~,P~,(Ov~,®vp,,
( X 2 / ~ 2 ) oo ( X 4 ¢ 2 ) ~
-~(O:v+_®v±~v±®v+_,
ot'ff
1 -x 2
1_~2
v±®v~~~v~®v±
l_x2QXV±®V~.
This S matrix enjoys the unitarity condition and the crossing symmetry,
Slz(~)S2~((-1)=id,
S~,~ce(()=SP_-~%;(1/x~) .
2.3. Vertex operators and L operators By an intertwiner of the type OVwv, for example, we shall mean an intertwiner of Uq(slz) such that O~vV: V ( A I - i ) V~ V(AI-i) W ~ U® V. In this notation, type I and type II VOs are intertwiners of the type ~-V(Ai) , ~ V(ai)V~ and V(AI_,) ~ ~v~,_,), respectively. Let us denote them and their components as follows, VeV(AD, V A1 - i ) q'v{a,)
V~ = ~ V ( 0 =
~v(A-,o~_~v(A~-),vr= , q)v(O= ~-v(a),~(~,) v_,
E ~(O®v,. Ot
,
Z ~(O®v*, Ot
"~v(a'-i>~-~v'-'r(a'-'))=~(O= Y~v*®~u.(O, "*" V~V(AD ~, - V(AD ot
q~v,v(a,_,) v(A,) = ~,v(o = y~ v,~® ~'~(~) ot
We normalize them by the conditions
Here va, is a highest weight vector with weight Au v], its dual vector, and the upper and lower signs correspond to i = 0 and 1, respectively. Denoting any component of the VOs by 0 (~) and using the same convention on the double sign, the VOs satisfy the following equalities,
K. Miki /PhysicsLettersA 186 (1994)217-224 zDO(Oz-O= O(z~),
D= - p + ½i
• v(Oo~V(O=g-t×id, q~(-O=
on V(Ai),
p=Ao +Am,
~V(Oo~v(O=g-t×idv(~,)®v,
g=(x2)oo/(x4)~,
~,~(-O=(-1)('~:t)/2P,~(~)
(- 1)(a+')/z~'~(O,
219
on V(Ai) .
(2.2)
In addition, due to (2.1), the components of the VOs have the property o~(0
= ~-'~(Ux),
~u'~(O = ~ u _ ~ ( U x ) .
For other operators appearing later, we shall change the position of indices in a similar manner without mentioning it. Finally, by the product of type I and type II VOs, we introduce L operators [ 7 ] as follows, ( + ) v(a~) v~-Lv~v(.,) =L(+)~,v*(O= y. v*®L(+)~P(O®vp=g~%(O~V~(x~),
(7)V(Ai)V~_r(-) Lv~v(+) --
v, V 2 ( O =
E
v*®L(-),~a(O®va=g~(~)~v,(O,
ot,~O
where ~'= _~/ql/2.
2.4. X X Z model As explained in Section 1, the left (right) semi-infinite tensor product of Vis identified with the level 1 highest (level - 1 lowest) weight representation. In this Letter, we shall regard a right module as a left module via an anti-automorphism b of Uq(sl2) such that
b ( y ) = x P a ( y ) x -p,
a:
antipode,
yeUq(sl2)
and denote this fact by the superscript b. This convention corresponds to that of Ref. [4 ]. Then roughly speaking, the above identification is summarized as V ( A i ) , . ~ , . . . ® V I ® V I = ®Cvot,
V(Ai).b,.~ V~.b ® V 1*b ® . . . . ® C v ~* ,
where VOg~''''®V~t2®Vot
I,
Vog--l~Otl®Vo£2®....
~ = ( a l , 0 t 2 .... ),
o t j = ( - - 1 ) j+i
(j>>l).
By this identification, setting Yg= V(Ao) ® V(A1 ), the space of states 3r is regarded as 3r= Horn (a~, ~ ) _ ~ ® ~ . In particular, in our convention, the vacuum state corresponds to I v a c ) ~ = x ° idv(a,). Now we introduce the map r which sends the state f e ~ to the state rfe ~r obtained from f b y reversing the order of the infinite tensor product and negating all spins. This map relates two kinds of CAOs introduced in Section 4. This is defined as follows. Let us define a map q: V(A~)-o V(A~)* by
q(yvn,) = v.],(boto) (y),
y~ Ua(sl2),
where to is an automorphism of Ua(sl 2 ) such that to (e~) = f , to ~ ) = e. to (t~) = t 7 ~. In terms of the semi-infinite tensor product of V, q satisfies the relation q(v~,) = v 1. Then r: Hom(V(A~), V(Aj) ) ~ H o m ( V ( A j ) , V(A,) ) is defined by
( n ( v ) , f u ) = (~l(u), rfv),
feHom(V(A~), V(Aj)),
ue V(A~),
ve V(As),
since (v*, fv~ ) = (v~, rfv. ). In later sections, we need the following formulae r ~ , ~ ( O = ~,~((-1),
r~,~(() = ~ , ~ ( ( - t ) ,
T2=id.
Next let us introduce a symmetric inner product ( , ) on ~ and a bra-ket notation by
(f,g)=Yr~e(rfg),
fge~,
(Ivac>~, O l v a c ) i ) = j ( v a c l O l v a c ) i ,
O e H o m ( ~ , 3r) .
K. M i k i / P h y s i c s Letters A 186 (1994) 2 1 7 - 2 2 4
220
We further define an anti-automorphism * and an automorphism - in the space of operators (9 = H o m (:~, ~ ) by
( f , , O g ) = ( O * f g ) = ( r f O-rg),
0~(9,
fg~.~.
In later sections, we shall need the VOs on Yf* since we express Oe (9 as an element of Horn ( ~ , a~) ® Horn ( af*, of*). They are obtained from those on of by the transposition ': Horn (V(Ai), V(A/) )--, Horn ( V(Aj)*, V(A~)*) defined by
(v*, gu) = ('gv*, u>,
geHom(V(A~), V(A~) ),
u~ V(A~),
v*e V(Afl*.
Set
~.(~) = '~-~(~), Then we find, ~(*)'~
V~V(AI-i) *b ~
•
Z ~)ol ot
®~.((),
fib v(~,_,).~ V( Ai )*bV¢
= Z ~"(O®v. Ol
Since this operation transforms type I (II) VOs on a¢~into type II (I) VOs on ~¢'*,we have to change the position of the indices of the latter accordingly.
3. New operators We shall introduce new operators that seem to have a well defined low temperature limit. These shall be used for the construction of the CAOs in the next section. Let us define j k((), and ~u(+ ) (() and Z k ( ( ) by Or=+ k=O,+ 1
~e(.-) ((1 = ~.(~)
-
~(.+)(~),
Or=+
g@V'(()qbv2(x~)=
~
zk(()®wk+id®wo •
k=0, + 1
Note that J v w(() = Y~,,kV~®J~ k(~)®Wk is an intertwiner of the type JV~d~/))wr since (x+x-1)p~2R(x): Uk~Wk ( k = 0, + 1 ), u~w,0. Here PI2 is the permutation operator. On the other hand, ~ - + ) ( ( ) = Z~ v* ® ~(,-+) (() and Zw(() = Yk zk(()®Wk are not intertwiners. However, we shall change the position of their indices as if they are. Hereafter the summation over paired indices should be understood. Setting (o= ( J ( j and me 2 Z - - [or-- ( -- 1 ) i ] / 2
it can be easily shown that these operators satisfy
J'~k(()JBk(() =g'-'5~,
~e(.+)(O = - z k ( ( ) & k(¢),
j,k(()j~,l(() =g,-,sk,
e(-)"(() = J % ( ( ) r ~ ( ( ) ,
[ ~ , ( ( l ) , Zk(~'2)] =--gg'5~i)((12)Jo, k((2)
1 --X 2
gg'= -- 1 +x 2'
on V(Ai) .
(3.1) (3.2) (3.3)
Noting (2.2) and the equality
L (+)v, v2(~),~ r,(~) + ~ V2(x~)L(-)v, v3(~) = ~v, (~)®u,, we can further show
(3.4)
K. Miki / Physics Letters A 186 (1994) 217-224 r~v~_+)(¢) = ~v(~:)a ( ( - , ) ,
rj, k(l) = ( _ 1 ) ' + k J a k ( ¢ - - ' ) ,
221
r27~(~')= ( _ 1 )~Zk(~,),
where ~' = - ~- ~/q 1/2, For later use, we define operators acting on ~¢g*with the similar understanding about the indices by
ffotk(l)=tJ--°t_k(l),
~¢( + )°t ( l ) = t ~/(~x) ( l ) ,
,~k(¢)=t~--k(¢) .
On the low temperature property of these operators we have the following conjecture. For Ill = 1, (i) (v*~, J~k(~)V#), (V~,, 7t~+)(l)Va) and (v*~, 27k(~')V#) have finite limits at x = 0 . (ii) ( v*~,J , k(~) Va ) ~ 0 at x = 0 for a finite number of at and # for a fixed/~ and at, respectively. Note the fact ( v * ~ , O ( l ) v a ) = c o n s t × l ~ for some integer n at x = 0 , and the relations (v*~,Ov~)= ( v~, rOy,,) = ( ' O r .*, v a ).
4. Creation and annihilation operators In this section, we shall define O * ( l ) , O ~ ( l ) and O* ( l ) from O~(1) ~ (9 by the rule
o ~ ( o = o ~ ( ¢ - ' ) *,
Oa(l)=o-'~(l-')
-,
O*(l)=o-s(l)
*-
(a, a = + ) .
Besides we shall write down only the formulae from which the others are obtained by * and We introduce two kinds ofCAOs ~o'~(l), ~0" ( l ) and Oa(l), ~ * ( l ) ( Ill = 1 ) by
~o'~(¢)=g(~(x~)L<-)'~y(l))®(cba(x~)~Y(~))eHom(Yt~,
~e)®Hom(Yg*, ~ * )
and the above rule. Explicitly ~0" ( l ) is given by
~ ( ¢ ) =g(L ~*) J ( l ) ~ ( ~ ) )® (4b(x~') 4~,(~) ). We further define the ~ operator and the transfer matrix z ( l ) by -~Pa&(ll ~) =g'Jak(l)®J&k(~) =*~¢t--&(~l l / X ) =*~P--ct&(l/xl l ) = .~e_~ - d ( l / X I ~ / X ) ,
T(() = g ~ ( ( ) ® ~ " ( ( ) • Hereafter, for simplicity, we shall denote the operators of the form O@id and i d @ 0 by o and 6, using the corresponding lowercase letter. In terms of the operators defined in Section 2, ~0~(l) is rewritten as
q)~,(~) =j,~k(l) o (ak(~) _ 6k(~) )
(4.1)
=~u(-)~(~) -~'(--'-"3a(Ux)Le'~((ll/x) =~u(-)~(~) + Le%(ll O~-'7~(I).
(4.2)
From the above, we can see that the CAOs have a low temperature limit if the conjecture in Section 2 is correct. Noting
~((l()[vac>i=J'~alvac)l_i,
~a(llC)~S(lll)--O~,
i(vacl~"a(lll)=l_i(vaclJ",~,
Le,~a (ll l) Le~a(ll l) = ~ ,
(4.3) (4.4)
we can further show from (4.2), ~o~(l) Ivac)~=0,
~(vacl to~(~) =~ (vac I ~u"(~),
C a ( l ) = £~ a ( l l ~)~o'~( l ) •
(4.5) (4.6)
In addition, thanks to (3.4), the CAOs satisfy the following relation,
~c'( l) =~'~( ~) +~o*_,~(l / x ) .
(4.7)
K. Miki / Physics Letters A 186 (1994) 217-224
222
This relation plays an important role in showing the axioms of form factors in the next section. Finally the c o m m u t a t i o n relations a m o n g the CAOs, the 50 operator and the transfer matrix are given by q'~ ((,)tp~((2) = S~;(~21 )rp~,(~2)~'(~, ) +ga~OJ( >(~12),
(4.8)
~ (*¢ , ) ~ ( ¢*~ )
(4.9)
* * = S"# ~,,,,(¢,~)~#,(¢2)~,,,(¢1),
[q~"((, ), ¢*(¢2)1 =g~U)'~((,2)50~,~ (¢2 [ ( 2 ) ,
(4.10)
[~*(¢, ), ¢*(¢2) ] = 0 ,
(4.1 1 )
~a ,~(¢, i ¢, )tp~(¢2 ) = S~,,~,(~,2)¢0#,(~2)~,'~(~ ~,a , I¢, ),
(4.12)
(-x~)~( T(~2)~0~(~1)=~21
on
(--X~21)oo(
x 3r~ '
---Z37Y Z---~0~t~ )z((2) , -x (~12)~
(4.13)
S~,~,((~2)50#'~((21 (2)~='~(¢~ I ~ , ) = 50~ a,((~ 1¢1 )50~,((2 ] ( 2 ) S ~ J ' ( ( , 2 ) ,
(4.14)
[50" a ( ( , I(,), z((2)] = 0 ,
(4.15)
V(A~)®V(A:)*o. The
first three relations are derived from (4.6), ( 4 . 7 ) , (4.1 1 ), ( 4 . 1 2 ) and
[ ~'a(~, ), ¢*(~2) ] =gg(')~(¢,2) 50'~ ,(~2 [ ¢2) •
(4.16)
The last equality ( 4 . 1 6 ) , in turn, follows from (3.3). Note that the 5 ° operator defined here has the property similar to T + (0) in Ref. [8].
5. Form factors
In this section, except for (4.7) and the c o m m u t a t i o n relations between ~/"(~) ( ~ a ( ( ) ) and the operator O ( z ) ~ ~ under consideration, we shall need only the relations among the CAOs and the 50 operator, and their
actions on the v a c u u m states. Noting (4.3), (4.6) and (4.1 1 ), set ) a t ....... = ~0cq( ( 1 ) . . . ~ 0 " ~ . ( ( , ) [ v a c ) i = ~0a,((,)...~0,xl ( ( l ) I v a c ) i + n
,
a"'"a' J<~n ..... ~ I =i
...~
ot 1,.,.,Otn
"*',
) ''"
Using the a u t o m o r p h i s m - , we have FOi>(~m, ..., ~, [ g i l l , "', ~n) ~...... fl'a ........ = F ~ + m i + n ) ( ~ l -' , .", ~m' [2--l l ~ n l , "'', ~I-' )--P'"'"--/~" ........ --c~, ,
(5.1)
where O ( z ) = O ( z - I ) - . Now we discuss the axioms of form factors along the lines of Appendix B of Ref. [ 2 ]. As a concrete example, we take O ( z ) = ak(~) and omit the subscript O of form factors. Note that ( - 2 ) X O ( q - l / z ) in the case k = 0 is the order operator. Let us rewrite F (i') ((~ Iz I ~z, ..., ~2,) - " ' . , ......~, as follows. Using (4.5), replace ~0by q/. After taking the c o m m u t a t o r of q/and O, express q/in terms of ~oand (0" by using (4.7), and move (o to the right to the vacuum. Then we obtain the first equality of
K. Miki / PhysicsLettersA 186 (1994)217-224
223
F < " ) ( ~ Izl~2 .... , ~ 2 . ) - ~ , ~ ...... ~.
=g(i°(z[(1/X, ( 2 ,
(2n)al . ...... . ~ .. + i ( v.a c l
.
. [ ~ / ~ - -. ~ 1 ( ( I ). , O ( z ) ]
[(2,
(2n)Ot2i ...... 2n
+ g E ~O,-° ((It)G~t(z[ (2, ..., (271)o~1;ot2. . . . . . 2n l
=F('-il-i)(z[~2
..... (2.,x(l),~, ...... , . . ~ , + l _ i ( v a c l
[ q T - ~ ( ( , ) , O ( z ) ] l ( 2 .... , ~2. > ~¢2,...,Ot2n i
+ g ]~ fi,~,('-;) (~u) G}
(5.2)
1
Here
G~t(zt(2 .... ,(2.)~,;,~2 ...... 2. 1 Otl = S Or2,..., Otl--`~'~`~.~`~-'(~z-1~(DFw~(z~(2~z-1~+1~(~")`~.~.:`~-'~`~+~.~`~2~
G}2 (z[(2 ..... (2.)~,.~2 ...... 2. O/I
O~1+1,...,Ot2n
' i~#] , . . . , ~ n ( ( 1 S,,,,...,~,,
.... ,
(,. I~)=Op#,,
=S
for m = O ,
¢~1,..., Ofm-- 1 ~pt
#'
O~m~
I~l,...,~,_~(~l,-.-, ~,~-i [~)S#,,~-(~m/~),
for m>~ 1 ,
a n d S ~ ..... ~#~,[#, ~l'""~" (~[ ~1, ..., (,~) is similarly defined. The second equality follows f r o m (5.1). This relation can be simplified as follows. Set f ~ ° ( z l ( ~ ..... (2.)~, ...... 2 . = F " ° ( z l ( 1
.... , G.),,,,....,~2.+g~ d(,~l,-°((ll)G}°(z[~2,..., ~2.)~1;~2.....,~2. t
--gg' d(al,- 0 ( ( l / x z ) i (vac IJ~, k(z) 1(2, -.., (2. >~2 i ...... 2.,
f(2i)(z[(,, . (2.),~,,...,,~2.=VU')(zl(,, . . . . . .
, (2.)~,
2 n - g 2 d~2. (i) ((2. ~-1 )G} ~- o (z[ (~, ..., (2n-I)ot2n;a, ...... 2 , - 1
......
l
+gg'd(.~).((E./XZ)~-i(vaclj,~2.k( z) l~ .... , ~=.-1 ~ ¢Xl,...,ot2n1-i 1 G: ° (z[
~2, ..., ~2.)~,;.,2 ...... 2. = G ~ ( z l ~2, ..., (2n)o!l;Ot2....,ot2n - - G } l - i ) ( z [ ~2 . . . . . ~ 2 n ) o t , ; ~ 2 ...... 2 n ,
d~ ') ( ( ) =
Z
m e 2 l _ [ o ~ _ ( _ l )i]/2 xm'~- X - m
Then noting [V,((,),ak(~z)]=_gg,6(d)(~lz)j,~
k((:),
[Oa(~l), a a ( ~ ) ] =0,
on V ( A i ) ® V ( A : ) *b ,
(5.2) is rewritten as f t i) (zl (i/x, if: .... , ~:.),~,.....a2. = f ~ l - i ) ( z [ ~2 .... , ~2., x~, ),~2.....a2.,,~, • The above equation holds as a L a u r e n t series in (1. Setting p = ½[oq + ( - 1 );], this implies as an analytic function o f (~,
224
K. Miki / Physics Letters A 186 (1994) 217-224
F ( " ) ( z l ( l ..... ~2,) ........ 2 , = F ( l - i l - i ) ( z l ~ 2 ..... ~2,, X 2~1)~2 ...... 2....
g(x~.)P -
--
1 --x2~2l G}i)(zl ~2, ..., ~1,)-,;~2 ...... 2.,
gg'((a/z)P 1 - (~x/z) 2
t(vaclj~,k(z)l~2,
"'"
~2. ~ ¢~2,...,Ot2n
a t ~l =
~dx,
a t ~l = z .
The first two equalities correspond to the axioms of form factors [2].
Acknowledgement T h e a u t h o r w o u l d like to t h a n k P r o f e s s o r E. D a t e for useful c o m m e n t s . T h i s w o r k is partially s u p p o r t e d by a G r a n t - i n - A i d for Scientific R e s e a r c h o n P r i o r i t y Areas, the M i n i s t r y o f E d u c a t i o n , Science a n d Culture, J a p a n .
References
[ 1 ] B. Davies, O. Foda, M. Jimbo, T. Miwa and A. Nakayashiki, Commun. Math. Phys. 151 ( 1993 ) 89. [ 2 ] F. Smirnov, Form factors in completely integrable models of quantum field theory (World Scientific, Singapore, 1992). [ 3 ] S. Pakuliak, Annihilation poles for form factors in XXZ model, RIMS preprint 933 ( 1993 ). [ 4 ] O. Foda, M. Jimbo, T. Miwa, K. Miki and A. Nakayashiki, Vertex operators in solvable lattice models, RIMS preprint 922 (1993). [ 5 ] I.M. Khamitov, Theor. Math. Phys. 62 ( 1985 ) 217; 63 ( 1985 ) 486; J. Sov. Math. 40 ( 1988 ) 115. [6] M. Jimbo, T. Miwa, K. Miki and A. Nakayashiki, Phys. Lett. A 168 (1992) 256. [7] N. Yu. Reshetikhin and M.A. Semenov-Tian-Shansky, Lett. Math. Phys. 19 (1990) 133. [8] D. Bernard and A. LeClair, Nucl. Phys. B 399 (1993) 709.
ELSEVIER
Physics Letters A 186 (1994) 217-224
Creation/annihilation operators and form factors of the X X Z model Kei Miki Department ofMathematicalScience, Facultyof EngineeringScience, Osaka University, Toyonaka, Osaka560, Japan Received 23 September 1993; accepted for publication 21 December 1993 Communicated by A.P. Fordy
Abstract
Two kinds of creation/annihilation operators for the XXZ model in the antiferroelectric regime are constructed. They presumably have a well defined low temperature limit. As an application, the axioms of form factors are discussed.
1. Introduction
In Ref. [ 1 ], the X X Z model in the antiferroelectric regime is diagonalized in the thermodynamic limit. This model is defined by the Hamiltonian
Hxxz=-½
~ k=
(a~+ 1o'~ + o'~+ i o'~ + z~o'~+1o'~),
zt= ½( q + q - 1 ) ,
0
,
--oo
acting on the infinite tensor product ... ® V® V® ... o f V = C 2. This Hamiltonian commutes with Uq(sl2). The key idea o f Ref. [ 1 ] was the identification o f the left and right semi-infinite tensor products o f Vwith the level 1 highest weight representation V(Ai) and the level - 1 lowest weight representation V(Ai)*a o f Uq(sl 2 ), respectively. Here i = 0 or 1, depending on the boundary conditions. By this, the model was cast into the framework of representation theory. The transfer matrix and creation/annihilation operators (CAOs) were constructed in terms o f type I and type II vertex operators (VOs) (in the terminology ofRef. [ 1 ] ), respectively. Unfortunately, contrary to the type I case, type II VOs do not have a well defined low temperature limit x--,0. As a result, the CAOs are defined only as an analytic continuation. This fact seems to make unclear whether the axioms of form factors [ 2 ] hold in this model. On this problem, some explicit calculations were carried out in Ref. [ 3 ]. In this Letter, we shall construct two kinds o f CAOs that presumably have a well defined low temperature limit. This is a generalization o f the construction o f the CAOs for the Ising model [ 4 ]. Two kinds o f CAOs (right and left CAO) and their c o m m u t a t i o n relations are first introduced in Ref. [5 ] in the context o f the G e l f a n d - L e v i t a n - M a r c h e n k o equation. As an application, we shall discuss the axioms o f form factors along the lines o f Appendix B o f Ref. [2]. In this approach, the existence of two kinds o f CAOs plays an important role. This Letter is organized as follows. In Section 2, the results o f Ref. [ 1 ] are summarized. In Section 3, we introduce new operators that seem to have a low temperature limit. In Section 4, using these operators, we construct CAOs. In Section 5, the axioms o f form factors are discussed. 0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved
SSD10375-9601 (93)E1030-J
218
K. M i k i / Physics Letters A 186 (1994) 2 1 7 - 2 2 4
2. Notations
In order to fix notations, we briefly summarize the well known results [ 1 ] in a slightly different convention. Unless otherwise mentioned, we follow the notation of Ref. [ 1 ] except that we use the inner product on weight space normalized by (ai, oti) = 2 [6].
2.1. Finite dimensional representation Let us denote a two-dimensional representation in the principal picture by V~= Cv+ ®Cv_. For later purposes, we take as bases of Vc® Vxc and Vx~® Vo
w±~=~v+®v±,
Wo=V+®V_-V_®V+,
w~=½(v+®v_+v_®v+)
U±l=V±®V±/~,
Uo=½(v+®v_-v_®v+),
and
u~=v+®v_ +v_®v+,
respectively. Let us further denote a three-dimensional representation spanned by Wk ( k = 0, -+ 1 ) by Wo These representations are isomorphic to their dual representations by the following correspondences, ttta+
V~+_,¢~-V¢,
v*~--~v~,
. a -+
W~+_,¢~-W¢,
W~--~W_k.
(2.1)
2.2. S matrix Setting ( z ) ~ = I-[~°=o ( 1 - z x 4 " ) , we define an S matrix and its matrix elements by
s(o=-~-'
(x~)°°(x~/~)~ R(O,
S(Ov~®vB= E s~,P~,(Ov~,®vp,,
( X 2 / ~ 2 ) oo ( X 4 ¢ 2 ) ~
-~(O:v+_®v±~v±®v+_,
ot'ff
1 -x 2
1_~2
v±®v~~~v~®v±
l_x2QXV±®V~.
This S matrix enjoys the unitarity condition and the crossing symmetry,
Slz(~)S2~((-1)=id,
S~,~ce(()=SP_-~%;(1/x~) .
2.3. Vertex operators and L operators By an intertwiner of the type OVwv, for example, we shall mean an intertwiner of Uq(slz) such that O~vV: V ( A I - i ) V~ V(AI-i) W ~ U® V. In this notation, type I and type II VOs are intertwiners of the type ~-V(Ai) , ~ V(ai)V~ and V(AI_,) ~ ~v~,_,), respectively. Let us denote them and their components as follows, VeV(AD, V A1 - i ) q'v{a,)
V~ = ~ V ( 0 =
~v(A-,o~_~v(A~-),vr= , q)v(O= ~-v(a),~(~,) v_,
E ~(O®v,. Ot
,
Z ~(O®v*, Ot
"~v(a'-i>~-~v'-'r(a'-'))=~(O= Y~v*®~u.(O, "*" V~V(AD ~, - V(AD ot
q~v,v(a,_,) v(A,) = ~,v(o = y~ v,~® ~'~(~) ot
We normalize them by the conditions
Here va, is a highest weight vector with weight Au v], its dual vector, and the upper and lower signs correspond to i = 0 and 1, respectively. Denoting any component of the VOs by 0 (~) and using the same convention on the double sign, the VOs satisfy the following equalities,
K. Miki /PhysicsLettersA 186 (1994)217-224 zDO(Oz-O= O(z~),
D= - p + ½i
• v(Oo~V(O=g-t×id, q~(-O=
on V(Ai),
p=Ao +Am,
~V(Oo~v(O=g-t×idv(~,)®v,
g=(x2)oo/(x4)~,
~,~(-O=(-1)('~:t)/2P,~(~)
(- 1)(a+')/z~'~(O,
219
on V(Ai) .
(2.2)
In addition, due to (2.1), the components of the VOs have the property o~(0
= ~-'~(Ux),
~u'~(O = ~ u _ ~ ( U x ) .
For other operators appearing later, we shall change the position of indices in a similar manner without mentioning it. Finally, by the product of type I and type II VOs, we introduce L operators [ 7 ] as follows, ( + ) v(a~) v~-Lv~v(.,) =L(+)~,v*(O= y. v*®L(+)~P(O®vp=g~%(O~V~(x~),
(7)V(Ai)V~_r(-) Lv~v(+) --
v, V 2 ( O =
E
v*®L(-),~a(O®va=g~(~)~v,(O,
ot,~O
where ~'= _~/ql/2.
2.4. X X Z model As explained in Section 1, the left (right) semi-infinite tensor product of Vis identified with the level 1 highest (level - 1 lowest) weight representation. In this Letter, we shall regard a right module as a left module via an anti-automorphism b of Uq(sl2) such that
b ( y ) = x P a ( y ) x -p,
a:
antipode,
yeUq(sl2)
and denote this fact by the superscript b. This convention corresponds to that of Ref. [4 ]. Then roughly speaking, the above identification is summarized as V ( A i ) , . ~ , . . . ® V I ® V I = ®Cvot,
V(Ai).b,.~ V~.b ® V 1*b ® . . . . ® C v ~* ,
where VOg~''''®V~t2®Vot
I,
Vog--l~Otl®Vo£2®....
~ = ( a l , 0 t 2 .... ),
o t j = ( - - 1 ) j+i
(j>>l).
By this identification, setting Yg= V(Ao) ® V(A1 ), the space of states 3r is regarded as 3r= Horn (a~, ~ ) _ ~ ® ~ . In particular, in our convention, the vacuum state corresponds to I v a c ) ~ = x ° idv(a,). Now we introduce the map r which sends the state f e ~ to the state rfe ~r obtained from f b y reversing the order of the infinite tensor product and negating all spins. This map relates two kinds of CAOs introduced in Section 4. This is defined as follows. Let us define a map q: V(A~)-o V(A~)* by
q(yvn,) = v.],(boto) (y),
y~ Ua(sl2),
where to is an automorphism of Ua(sl 2 ) such that to (e~) = f , to ~ ) = e. to (t~) = t 7 ~. In terms of the semi-infinite tensor product of V, q satisfies the relation q(v~,) = v 1. Then r: Hom(V(A~), V(Aj) ) ~ H o m ( V ( A j ) , V(A,) ) is defined by
( n ( v ) , f u ) = (~l(u), rfv),
feHom(V(A~), V(Aj)),
ue V(A~),
ve V(As),
since (v*, fv~ ) = (v~, rfv. ). In later sections, we need the following formulae r ~ , ~ ( O = ~,~((-1),
r~,~(() = ~ , ~ ( ( - t ) ,
T2=id.
Next let us introduce a symmetric inner product ( , ) on ~ and a bra-ket notation by
(f,g)=Yr~e(rfg),
fge~,
(Ivac>~, O l v a c ) i ) = j ( v a c l O l v a c ) i ,
O e H o m ( ~ , 3r) .
K. M i k i / P h y s i c s Letters A 186 (1994) 2 1 7 - 2 2 4
220
We further define an anti-automorphism * and an automorphism - in the space of operators (9 = H o m (:~, ~ ) by
( f , , O g ) = ( O * f g ) = ( r f O-rg),
0~(9,
fg~.~.
In later sections, we shall need the VOs on Yf* since we express Oe (9 as an element of Horn ( ~ , a~) ® Horn ( af*, of*). They are obtained from those on of by the transposition ': Horn (V(Ai), V(A/) )--, Horn ( V(Aj)*, V(A~)*) defined by
(v*, gu) = ('gv*, u>,
geHom(V(A~), V(A~) ),
u~ V(A~),
v*e V(Afl*.
Set
~.(~) = '~-~(~), Then we find, ~(*)'~
V~V(AI-i) *b ~
•
Z ~)ol ot
®~.((),
fib v(~,_,).~ V( Ai )*bV¢
= Z ~"(O®v. Ol
Since this operation transforms type I (II) VOs on a¢~into type II (I) VOs on ~¢'*,we have to change the position of the indices of the latter accordingly.
3. New operators We shall introduce new operators that seem to have a well defined low temperature limit. These shall be used for the construction of the CAOs in the next section. Let us define j k((), and ~u(+ ) (() and Z k ( ( ) by Or=+ k=O,+ 1
~e(.-) ((1 = ~.(~)
-
~(.+)(~),
Or=+
g@V'(()qbv2(x~)=
~
zk(()®wk+id®wo •
k=0, + 1
Note that J v w(() = Y~,,kV~®J~ k(~)®Wk is an intertwiner of the type JV~d~/))wr since (x+x-1)p~2R(x): Uk~Wk ( k = 0, + 1 ), u~w,0. Here PI2 is the permutation operator. On the other hand, ~ - + ) ( ( ) = Z~ v* ® ~(,-+) (() and Zw(() = Yk zk(()®Wk are not intertwiners. However, we shall change the position of their indices as if they are. Hereafter the summation over paired indices should be understood. Setting (o= ( J ( j and me 2 Z - - [or-- ( -- 1 ) i ] / 2
it can be easily shown that these operators satisfy
J'~k(()JBk(() =g'-'5~,
~e(.+)(O = - z k ( ( ) & k(¢),
j,k(()j~,l(() =g,-,sk,
e(-)"(() = J % ( ( ) r ~ ( ( ) ,
[ ~ , ( ( l ) , Zk(~'2)] =--gg'5~i)((12)Jo, k((2)
1 --X 2
gg'= -- 1 +x 2'
on V(Ai) .
(3.1) (3.2) (3.3)
Noting (2.2) and the equality
L (+)v, v2(~),~ r,(~) + ~ V2(x~)L(-)v, v3(~) = ~v, (~)®u,, we can further show
(3.4)
K. Miki / Physics Letters A 186 (1994) 217-224 r~v~_+)(¢) = ~v(~:)a ( ( - , ) ,
rj, k(l) = ( _ 1 ) ' + k J a k ( ¢ - - ' ) ,
221
r27~(~')= ( _ 1 )~Zk(~,),
where ~' = - ~- ~/q 1/2, For later use, we define operators acting on ~¢g*with the similar understanding about the indices by
ffotk(l)=tJ--°t_k(l),
~¢( + )°t ( l ) = t ~/(~x) ( l ) ,
,~k(¢)=t~--k(¢) .
On the low temperature property of these operators we have the following conjecture. For Ill = 1, (i) (v*~, J~k(~)V#), (V~,, 7t~+)(l)Va) and (v*~, 27k(~')V#) have finite limits at x = 0 . (ii) ( v*~,J , k(~) Va ) ~ 0 at x = 0 for a finite number of at and # for a fixed/~ and at, respectively. Note the fact ( v * ~ , O ( l ) v a ) = c o n s t × l ~ for some integer n at x = 0 , and the relations (v*~,Ov~)= ( v~, rOy,,) = ( ' O r .*, v a ).
4. Creation and annihilation operators In this section, we shall define O * ( l ) , O ~ ( l ) and O* ( l ) from O~(1) ~ (9 by the rule
o ~ ( o = o ~ ( ¢ - ' ) *,
Oa(l)=o-'~(l-')
-,
O*(l)=o-s(l)
*-
(a, a = + ) .
Besides we shall write down only the formulae from which the others are obtained by * and We introduce two kinds ofCAOs ~o'~(l), ~0" ( l ) and Oa(l), ~ * ( l ) ( Ill = 1 ) by
~o'~(¢)=g(~(x~)L<-)'~y(l))®(cba(x~)~Y(~))eHom(Yt~,
~e)®Hom(Yg*, ~ * )
and the above rule. Explicitly ~0" ( l ) is given by
~ ( ¢ ) =g(L ~*) J ( l ) ~ ( ~ ) )® (4b(x~') 4~,(~) ). We further define the ~ operator and the transfer matrix z ( l ) by -~Pa&(ll ~) =g'Jak(l)®J&k(~) =*~¢t--&(~l l / X ) =*~P--ct&(l/xl l ) = .~e_~ - d ( l / X I ~ / X ) ,
T(() = g ~ ( ( ) ® ~ " ( ( ) • Hereafter, for simplicity, we shall denote the operators of the form O@id and i d @ 0 by o and 6, using the corresponding lowercase letter. In terms of the operators defined in Section 2, ~0~(l) is rewritten as
q)~,(~) =j,~k(l) o (ak(~) _ 6k(~) )
(4.1)
=~u(-)~(~) -~'(--'-"3a(Ux)Le'~((ll/x) =~u(-)~(~) + Le%(ll O~-'7~(I).
(4.2)
From the above, we can see that the CAOs have a low temperature limit if the conjecture in Section 2 is correct. Noting
~((l()[vac>i=J'~alvac)l_i,
~a(llC)~S(lll)--O~,
i(vacl~"a(lll)=l_i(vaclJ",~,
Le,~a (ll l) Le~a(ll l) = ~ ,
(4.3) (4.4)
we can further show from (4.2), ~o~(l) Ivac)~=0,
~(vacl to~(~) =~ (vac I ~u"(~),
C a ( l ) = £~ a ( l l ~)~o'~( l ) •
(4.5) (4.6)
In addition, thanks to (3.4), the CAOs satisfy the following relation,
~c'( l) =~'~( ~) +~o*_,~(l / x ) .
(4.7)
K. Miki / Physics Letters A 186 (1994) 217-224
222
This relation plays an important role in showing the axioms of form factors in the next section. Finally the c o m m u t a t i o n relations a m o n g the CAOs, the 50 operator and the transfer matrix are given by q'~ ((,)tp~((2) = S~;(~21 )rp~,(~2)~'(~, ) +ga~OJ( >(~12),
(4.8)
~ (*¢ , ) ~ ( ¢*~ )
(4.9)
* * = S"# ~,,,,(¢,~)~#,(¢2)~,,,(¢1),
[q~"((, ), ¢*(¢2)1 =g~U)'~((,2)50~,~ (¢2 [ ( 2 ) ,
(4.10)
[~*(¢, ), ¢*(¢2) ] = 0 ,
(4.1 1 )
~a ,~(¢, i ¢, )tp~(¢2 ) = S~,,~,(~,2)¢0#,(~2)~,'~(~ ~,a , I¢, ),
(4.12)
(-x~)~( T(~2)~0~(~1)=~21
on
(--X~21)oo(
x 3r~ '
---Z37Y Z---~0~t~ )z((2) , -x (~12)~
(4.13)
S~,~,((~2)50#'~((21 (2)~='~(¢~ I ~ , ) = 50~ a,((~ 1¢1 )50~,((2 ] ( 2 ) S ~ J ' ( ( , 2 ) ,
(4.14)
[50" a ( ( , I(,), z((2)] = 0 ,
(4.15)
V(A~)®V(A:)*o. The
first three relations are derived from (4.6), ( 4 . 7 ) , (4.1 1 ), ( 4 . 1 2 ) and
[ ~'a(~, ), ¢*(~2) ] =gg(')~(¢,2) 50'~ ,(~2 [ ¢2) •
(4.16)
The last equality ( 4 . 1 6 ) , in turn, follows from (3.3). Note that the 5 ° operator defined here has the property similar to T + (0) in Ref. [8].
5. Form factors
In this section, except for (4.7) and the c o m m u t a t i o n relations between ~/"(~) ( ~ a ( ( ) ) and the operator O ( z ) ~ ~ under consideration, we shall need only the relations among the CAOs and the 50 operator, and their
actions on the v a c u u m states. Noting (4.3), (4.6) and (4.1 1 ), set ) a t ....... = ~0cq( ( 1 ) . . . ~ 0 " ~ . ( ( , ) [ v a c ) i = ~0a,((,)...~0,xl ( ( l ) I v a c ) i + n
,
a"'"a' J<~n ..... ~ I =i
...~
ot 1,.,.,Otn
"*',
) ''"
Using the a u t o m o r p h i s m - , we have FOi>(~m, ..., ~, [ g i l l , "', ~n) ~...... fl'a ........ = F ~ + m i + n ) ( ~ l -' , .", ~m' [2--l l ~ n l , "'', ~I-' )--P'"'"--/~" ........ --c~, ,
(5.1)
where O ( z ) = O ( z - I ) - . Now we discuss the axioms of form factors along the lines of Appendix B of Ref. [ 2 ]. As a concrete example, we take O ( z ) = ak(~) and omit the subscript O of form factors. Note that ( - 2 ) X O ( q - l / z ) in the case k = 0 is the order operator. Let us rewrite F (i') ((~ Iz I ~z, ..., ~2,) - " ' . , ......~, as follows. Using (4.5), replace ~0by q/. After taking the c o m m u t a t o r of q/and O, express q/in terms of ~oand (0" by using (4.7), and move (o to the right to the vacuum. Then we obtain the first equality of
K. Miki / PhysicsLettersA 186 (1994)217-224
223
F < " ) ( ~ Izl~2 .... , ~ 2 . ) - ~ , ~ ...... ~.
=g(i°(z[(1/X, ( 2 ,
(2n)al . ...... . ~ .. + i ( v.a c l
.
. [ ~ / ~ - -. ~ 1 ( ( I ). , O ( z ) ]
[(2,
(2n)Ot2i ...... 2n
+ g E ~O,-° ((It)G~t(z[ (2, ..., (271)o~1;ot2. . . . . . 2n l
=F('-il-i)(z[~2
..... (2.,x(l),~, ...... , . . ~ , + l _ i ( v a c l
[ q T - ~ ( ( , ) , O ( z ) ] l ( 2 .... , ~2. > ~¢2,...,Ot2n i
+ g ]~ fi,~,('-;) (~u) G}
(5.2)
1
Here
G~t(zt(2 .... ,(2.)~,;,~2 ...... 2. 1 Otl = S Or2,..., Otl--`~'~`~.~`~-'(~z-1~(DFw~(z~(2~z-1~+1~(~")`~.~.:`~-'~`~+~.~`~2~
G}2 (z[(2 ..... (2.)~,.~2 ...... 2. O/I
O~1+1,...,Ot2n
' i~#] , . . . , ~ n ( ( 1 S,,,,...,~,,
.... ,
(,. I~)=Op#,,
=S
for m = O ,
¢~1,..., Ofm-- 1 ~pt
#'
O~m~
I~l,...,~,_~(~l,-.-, ~,~-i [~)S#,,~-(~m/~),
for m>~ 1 ,
a n d S ~ ..... ~#~,[#, ~l'""~" (~[ ~1, ..., (,~) is similarly defined. The second equality follows f r o m (5.1). This relation can be simplified as follows. Set f ~ ° ( z l ( ~ ..... (2.)~, ...... 2 . = F " ° ( z l ( 1
.... , G.),,,,....,~2.+g~ d(,~l,-°((ll)G}°(z[~2,..., ~2.)~1;~2.....,~2. t
--gg' d(al,- 0 ( ( l / x z ) i (vac IJ~, k(z) 1(2, -.., (2. >~2 i ...... 2.,
f(2i)(z[(,, . (2.),~,,...,,~2.=VU')(zl(,, . . . . . .
, (2.)~,
2 n - g 2 d~2. (i) ((2. ~-1 )G} ~- o (z[ (~, ..., (2n-I)ot2n;a, ...... 2 , - 1
......
l
+gg'd(.~).((E./XZ)~-i(vaclj,~2.k( z) l~ .... , ~=.-1 ~ ¢Xl,...,ot2n1-i 1 G: ° (z[
~2, ..., ~2.)~,;.,2 ...... 2. = G ~ ( z l ~2, ..., (2n)o!l;Ot2....,ot2n - - G } l - i ) ( z [ ~2 . . . . . ~ 2 n ) o t , ; ~ 2 ...... 2 n ,
d~ ') ( ( ) =
Z
m e 2 l _ [ o ~ _ ( _ l )i]/2 xm'~- X - m
Then noting [V,((,),ak(~z)]=_gg,6(d)(~lz)j,~
k((:),
[Oa(~l), a a ( ~ ) ] =0,
on V ( A i ) ® V ( A : ) *b ,
(5.2) is rewritten as f t i) (zl (i/x, if: .... , ~:.),~,.....a2. = f ~ l - i ) ( z [ ~2 .... , ~2., x~, ),~2.....a2.,,~, • The above equation holds as a L a u r e n t series in (1. Setting p = ½[oq + ( - 1 );], this implies as an analytic function o f (~,
224
K. Miki / Physics Letters A 186 (1994) 217-224
F ( " ) ( z l ( l ..... ~2,) ........ 2 , = F ( l - i l - i ) ( z l ~ 2 ..... ~2,, X 2~1)~2 ...... 2....
g(x~.)P -
--
1 --x2~2l G}i)(zl ~2, ..., ~1,)-,;~2 ...... 2.,
gg'((a/z)P 1 - (~x/z) 2
t(vaclj~,k(z)l~2,
"'"
~2. ~ ¢~2,...,Ot2n
a t ~l =
~dx,
a t ~l = z .
The first two equalities correspond to the axioms of form factors [2].
Acknowledgement T h e a u t h o r w o u l d like to t h a n k P r o f e s s o r E. D a t e for useful c o m m e n t s . T h i s w o r k is partially s u p p o r t e d by a G r a n t - i n - A i d for Scientific R e s e a r c h o n P r i o r i t y Areas, the M i n i s t r y o f E d u c a t i o n , Science a n d Culture, J a p a n .
References
[ 1 ] B. Davies, O. Foda, M. Jimbo, T. Miwa and A. Nakayashiki, Commun. Math. Phys. 151 ( 1993 ) 89. [ 2 ] F. Smirnov, Form factors in completely integrable models of quantum field theory (World Scientific, Singapore, 1992). [ 3 ] S. Pakuliak, Annihilation poles for form factors in XXZ model, RIMS preprint 933 ( 1993 ). [ 4 ] O. Foda, M. Jimbo, T. Miwa, K. Miki and A. Nakayashiki, Vertex operators in solvable lattice models, RIMS preprint 922 (1993). [ 5 ] I.M. Khamitov, Theor. Math. Phys. 62 ( 1985 ) 217; 63 ( 1985 ) 486; J. Sov. Math. 40 ( 1988 ) 115. [6] M. Jimbo, T. Miwa, K. Miki and A. Nakayashiki, Phys. Lett. A 168 (1992) 256. [7] N. Yu. Reshetikhin and M.A. Semenov-Tian-Shansky, Lett. Math. Phys. 19 (1990) 133. [8] D. Bernard and A. LeClair, Nucl. Phys. B 399 (1993) 709.