Nuclear Physics B330 (1990) 523-556
North-Holland
COMMON STRUCTURES BETWEEN FINITE SYSTEMS AND
CONFORMAL FIELD THEORIES THROUGH
QUANTUM GROUPS
V. PASQUIER and H. SALEUR
Service de Physique Th3orique* de Saclay, F-91191 Gif-sur-Yvette Cedex, France
Received 15 March 1989
We discuss in this paper algebraic structures that are common to finite integrable lattice
systems and conformal field theories. The concept of quantum group plays a major role in our
study, and a detailed theory of representations of Uq[SU(n + 1)] for q a root of unity is given. We
obtain in particular a discrete analog of the Feigin-Fuchs construction, with corresponding
concepts of null vectors or unitarity. The modular transformation S-matrix is also obtained from
finite lattice considerations.
Introduction
T h e recent developments of integrable systems and conformal field theories [1]
have exhibited numerous similarities. In the c o n t i n u u m limit, there is now a rather
well-established correspondence between the two [2]. The aim of this paper is to
unravel some c o m m o n algebraic structures which already appear at the finite level.
T h e key ingredient in conformal theories is the study of representations of the
Virasoro algebra, which has mainly been accomplished using the F e i g i n - F u c h s
construction [3]. We show here how the T e m p e r l e y - L i e b algebra [4] and the
q u a n t u m g r o u p [5] Uu[SU(2)] play a similar role for finite integrable systems.
In the first part of this paper, we study in detail the representation theory of
Uq[SU(2)] with special emphasis on the case q a root of unity.
In the second part we consider the X X Z [6] integrable hamiltonian with free or
periodic b o u n d a r y conditions. We discuss decomposition of the Hilbert space onto
representations of the T e m p e r l e y - L i e b algebra, and c o m p a r e the c o n t i n u u m limit of
the hamiltonians with the spectrum of L 0, L0. The net result is that a given
irreducible representation of the T e m p e r l e y - L i e b algebra can be associated to an
irreducible Virasoro representation in the case of an open chain, and to a direct sum
of Virasoro x Virasoro irreducible representations for the closed chain.
* Laboratoire de l'Institut de Recherche Fondamentale du Commissariat ~ l'Energie Atomique.
0550-3213/90/$03.50@Elsevier Science Publishers B.V.
(North-Holland)
V. Pasquiet; tt. Saleur / Finite systems and conformal fieM theories,
524
The third part contains a discussion of relations between these results and the
Feigin-Fuchs construction, using the recent formulation of the latter given by
Felder [7]. We also show how the S-matrix [8] of modular transformations for
characters can be recovered from finite lattice considerations.
Some of these developments are extended to Uq[SU(n + 1)] in the appendix.
The discussion of Uq[SU(2)] representations and of the open chain hamiltonian is
an extended version of our Les Houches paper [9], whereas the closed chain results
were motivated by the numerical analyses of Alcaraz et al. [10].
1. The algebra UJSU(2)!
Uq[SU(2)] is generated by S +, S , q -+sZ under the relations [5]
The algebra
q2S z _ q 2sZ
qS~S +q s ~ = q ± XS ± ,
[S +, S ] -
q_ q
1
(1,1)
'
which reduce to the usual relations among generators ( S z, S 5) of SU(2) in the limit
q --* 1. The following formulae for the coproduct, antipode and counit endorse Uq
with a Hopf algebra structure [11,12]
A(q± sz) =q± S~®q±
,90( q ± s ' ) = q -v-s ~ '
e(q± SZ) = 1 ,
sZ
A(S± )=qJ®S± +S+-®q
sZ,
(1.2)
5 " ( S 5) = _ q _+1S +,
(1.3)
e(S±) =0.
(1.4)
For a quantum spin chain of N spins s, the configuration space is of the form
(C 2,+ 1)U and one can easily find representations of the generators (1.1). In the case
s = ~ for instance, since ( q + l
q~l)/(q_q
1)= +1, the classical and quantum
commutation relations coincide for a single spin, and one finds, using the coproduct
(1.2)
q S = qOz/2 ® . . . ® qO~/2
S + = ~ S , + -. ~ q.° z /.2 ®.
i
® q,~/~ ® o i ± / 2 ® q ,z/~ ® . . . ® q ~/~,
(1.5)
i
where o z, o+ are the usual Pauli matrices (equal to twice the one-half spin
operators). There are N terms in each tensor product, and the i index in o~-+ means
that the Pauli matrix is in the ith position of the tensor product. For higher values
of s, one can first define Si + for a single spin by combining 2s quantum spin-{
representations (see below) and projecting onto the spin-s representation, and then
V. Pasquier, H. Saleur / Finite systems and conformal field theories
525
define S -+ for the whole chain via eq. (1.5). In the case s = 1 for instance, one finds
simply
+__
Si-
0
(q+q ~
0
0
0
0
0
Si
0
o
(q+q i
0
o0
i) "
(1.6)
~q+ q-1
The center of Uq is generated [12] by the q-analog of the Casimir element
32= S-S+q._
( qSZ+l/2 q-SZ-a/2) 2 -- ql/2 _ q-1/2 )2.
_q___q_ 1
q_q 1
(1.7)
The following commutation relation will be useful in the sequel:
[m]q!
[(s+)",(s-)"]=(s
) ....
,,
[ ~_i 2 -'njq! k=l
H
(m >~ n ) ,
q2SZ-m+k_q-2SZ+m
-q- _
- -q- -- 1
k
,
(1.8)
where [x]q = ( q ' - q - ' ) / ( q - q-l) and [ m ] q ! = [ m ] q [ m - l]q...[2]q[l]q and both
sides of eq. (1.8) act on states in K e r S ÷.
It is natural to keep here the ordinary scalar product, treating q as a formal
variable (i.e. "complex conjugation" leaves q unchanged). Then S + is the adjoint
of S - .
1.1. CASE W H E N q IS NOT A ROOT OF UNITY
The theory of representations of Uq[SU(2)] has been studied in detail [13] in the
case when q is not a root of unity, and is essentially equivalent to the U[SU(2)] case.
Accordingly, the configuration space (C2S+1) N of a spin chain can be split into a
direct sum of irreducible highest weight representations pj that are in one-to-one
correspondence with ordinary SU(2) representations, the basis vectors being simply
deformed by q, q x factors, pj's can be labeled by the value of the Casimir operator
(1.9)
To illustrate this we discuss the example s = ~, N = 2. The highest weight vector
with spin j = 1 is chosen to be ]a) = I1" 1"), and Pa follows by applying S - (1.5).
The orthogonal of O~ is then stable under the action of Uq[SU(2)], and constitutes a
representation P0- Since q is not a root of unity, there are no null-norm states and
Pl O /90 = ~ /
C 4 = Pl (~ /90"
V. Pasquier, H. Saleur / Finite systems and conformal fieM theories
526
la> = I t t >
It
S- la> =
qlt2 ItS> + q-it2 l i t >
Ib> = q-ln Itl> - qlt2 llt>
It
S-21a> =
\
Ill>
g
x
/
91
v
90
Fig. 1. Decomposition of the configuration space C 4 into P0 ~ Pl.
This is represented in fig. 1 where arrows denote the action of S -+. In Pl,
calculation of S -+: gives the matrices (1.6).
1.2. CASE W H E N q IS A ROOT OF UNITY: GENERALITIES
The case q being a root of unity (qP= +1) has attracted attention recently
[14,15]. Two essential new features happen then. First, the Casimir takes identical
values for highest weights of spins j, j ' related by one of the transformations.
j ' = j + np,
j'=p-
l - j + np,
(1.10)
where n ~ 7/. S 2 is no longer sufficient to label representations pj, the latter being
defined here by continuity from the irrational case, and some pj, pj, where j, j '
satisfy (1.10) can mix and get connected under the action of S -+.
The other important consequence of q being a root of unity is that S -+ become
nilpotent
(S+-)P=0.
(1.11)
Using eq. (1.8) it is easy to check that (S +-)P commutes with S + and q +sZ. To
show that it actually vanishes, we consider first the case of the configuration space
(C2) N, and evaluate the action of (S _+)mon a given vector ]a).
Due to the left and right strings of operators present in the representation (1.5),
Si+, S f (resp. Si-, Sf) ( i * j ) are not mutually local if q4= 1, and have the
monodromy properties
s,+sT= v - 2S; S?
s,-sg= q2SfSi-,
,
i <j,
i <j
(1.12)
V. Pasquier, H. Saleur / Finite systems and conformal field theories
527
(while S i + S f = Si-Sj÷ ). Acting on Ia ) with (S+) m say, one can flip a given set of m
spins down in m! ways, which gives due to (1.12) an overall [m] q! factor. The latter
vanishes for m = p since [ p ] q = 0. For configuration space (C2s+1) N (1.11) holds
true as well since S -+ are then defined via addition of spin-½ representations.
Relation (1.11) is responsible for the appearance of new highest (lowest) weights
inside the pj's. Note that, despite (1.11), states with values of S z differing by
multiples of p can still be connected using ( S +-)P/[ p]q! w h i c h is well defined [14].
1.3. STRUCTURE OF THE REPRESENTATIONS IF q IS A ROOT OF UNITY
We turn now to a more precise description of the representation theory in the
case of q a root of unity. We will mainly focus here on determining how Oj's can get
mixed. This is the most important question for the study of spin chains since all
states which are connected by the action of S -+ have the same energy for hamiltonians w i t h U q [ S U ( 2 ) ] symmetry. Our strategy consists in decomposing (C 2s+l)N in a
direct sum of representations 0j in the general irrational case, and then in determining the structure of the new highest weights vectors that appear when qP = +_1.
Let us first discuss an example. Consider (C2)3; this can be decomposed as
/93/2 -I-2Pal= in the following way. The highest weight of P3/2 is chosen to be
Ja ) = I T T T ), and other vectors in P3/2 are obtained by application of S . Now if
q3 = + , S la) is also a highest weight, hence has a zero norm. The orthogonal of
S - J0) is of dimension two, and contains already S - l a ) . Since the subspace S z = 1_2
is of dimension 3, there exists one v6cto,r Ib') which is not orthogonal to S - l a ) ,
hence { S - a l b ' ~ = {alS+b ') 4: O. This explains the exchange of up arrows between
s Z = 3 and S z = ~ spaces in fig. 2. The picture can now be completed by noticing
that ( S - ) 3 = 0 , hence the arrow going from S 2la) to S-31a) disappears,
while I$ $ $ ) can still be reached from IT T T) by applying S - 3 / [ 3 ] q ! . T h i s is
replaced by a new arrow connecting S - I b') a n d ( S - 3 / [ 3 ] q ! ) l a ) . We obtain finally
la> = I t t t >
q Ittl>
+ Itit>
+ q-
q Itil>
+ lltl>
+ q-1 I l l t >
Ib'>
Ic>
S-Ib'>
S- Ic>
I~ll>
Fig. 2. Structure of C 6 for q = e i~/3.
528
V. Pasquier, H. Saleur / Finite systems and conformalfield theories
- Ib') = [b') + S
[a) (fig. 2). The configuration space decomposes thus into a
big (type-I) representation which is a mixture of P3/2 and Pl/2, and a small type-II
representation 01/2. (03/2,Pl/2) is indecomposable, but is not irreducible since it
contains a sub-representation 03/2- 03/2 itself is not irreducible either, since it
contains { S - [ a ) , S Zla)}. Hence the Hermann-Weyl theorem does not apply any
more when q is a root of unity. Also it is clear that (03/2, P1/2) is not a highest
weight representation.
The analysis of the more general situations can be accomplished in the same way.
In the irrational case, (C2'+1) u decomposes into a sum of representations p. For
qP = -I- 1, and a given value of S 2 there are sets of indices like
S+S
(Jk>JI,--1 > "'" >Jl,
0~<Jl<~(P--1)},
(1.13)
which are related by the symmetries (1.10) (fig. 3) (if k is odd, Jk =Jl mod p, and if
k is even, Jk = P - 1 - J l mod p). If k = 1, Pl cannot get mixed, and it is still an
irreducible highest weight representation (type II). If k > 1, there occurs mixing of
the representations that we now analyze. We start with a highest weight ]a) at spin
Jk, and we build pj~ by acting with S (S-)P/(p)q!. In this way we obtain states
IJk, m ) as follows [14]
,
S-[jk, m)=[k-m+l]q]jk, m--1 ),
S+[jk,m)=[jk+m+l]q[jk, rn+l)
(1.14)
with the convention that
S+ [Jk, Jk) = 0,
S I k , - J k ) = 0.
(115)
PJk is not irreducible. It contains states that are annihilated by S + at spin j~_ 1, Jk 3
=Jk-1 - P , Jk-5 =Jk-1 - 2p - . . .
The state IJk, Jk-x) is annihilated by S + and (S+)P/[p]q!. Under the action of
S and ( S - ) P / [ p ] q ! it generates an irreducible sub-representation, all states of
which have a zero norm.
At spin Jk-1, there must exist a state Ib') such that S + [ b ' ) = IJk, Jk 1 -{- 1).
Acting with S-, ( S - ) P / [ p ] q ! w e generate from Ib') states [j~ 1, m ) as follows:
S [jk_t,m)=[k_t,--m+llqlJk
S+lJk
1, m ) = [Jk
,,m--l},
1,+m+l]q[Jk_l,m+l)+lJk--m--1
| [
[ ] J 1 --k m
] IJk, m + 1), (1.16)
q
V. Pasquier, H. Saleur / Finite ,s?'stems and conformal field theories
la>
J" - - r
I
I
I
I
la'>
Jk-1
Jk-2 = Jk-P
Ib'>
--T-- I
t
I
I
I
t
I
I
I
I
t
I
Jk-3 : Jk-l-P
Jk-t,
:
j,-2p
I-
t
t
I
I
I
I
I
t
I
I
I
I
Jk-5 = Jk-1-2P
-U
I
i
I
I
I
I
I
I
I
I
I
I
I
I
Fig. 3. Pairing of representations (P;s,, P;,, ~)"
529
530
V. Pasquier, H. Saleur / Finite systems and conformal field theories
with
S + l j ~ - , , Jk ,) = IJk, Jk-1 + 1),
(Jk+Jk 1--1)[Jk+Jk ']ql
S ]Jk
l'-Jk-D=\
2Jk 1
q[Jk--Jk lJ -Ijk-'jk-Dq'
(1.17)
As can be seen in formulas (1.16), S + connects Ojk and Ojk ~; 0jk , is a highest
weight representation generated from [b') if we factor out Ojk. States that are
annihilated by S + can be built at spin Jk-2 =Jk - P , Jk-4 =Jk - 2p, . . . by taking
appropriate linear combinations of IJk, m) and [Jk-1, rn). All states of (p/~, pj~ ~)
that belong to Ker S + are in Im(S +) p- 1.
The only states in (Oj,, Pj, ~) that are annihilated simultaneously by S + and
(S+)P/[p]q! are la) and l a ' ) - = ( S )J~-J~ ~la) as can be checked using formulas
(1.14) and (1.16). This ensures that there is no vector common to (PJk, PJ, ,) and its
orthogonal. Therefore we can repeat the same arguments in the orthogonal of
(pjk, pj~ ,). In this way we get pairs of representations that mix in larger structures
(type I) (pj~, PJk ,); (PJ~ 2' OA 3 ) ' ' ' ' Depending on the number of p's we end up
with a certain number of pj] that cannot mix and are still irreducible highest weight
representations (type II).
A special situation occurs if Jl = ~(P - 1) since p - 1 - J l =JP In this case the
representations pjk are still irreducible and do not pair. By convention we shall also
call them of type I.
This procedure thus allows us to completely decompose the configuration space
into a direct sum of indecomposable (under the action of S + and (S +)P/[p]q!)
representations.
It is useful to introduce the q-dimension of representations
dimq p/=
~
q 2sz= [ 2 j + 1]q,
(1.18)
states in Pt
which is invariant in the first of the transformations (1.10) and changes of sign m
the second one. Accordingly, we see that the representations mix in such a way that
their total q-dimension is zero. Also eq. (1.18) vanishes for the degenerate case
j = ½(np - 1). Thus the configuration space splits into:
type-I representations which have q-dimension zero, and are either mixed or of the
kind P(,,p-1)/2 (fig. 4),
type-II representations which have a nonzero q-dimension, and are still isomorphic
to U[SU(2)] ones.
1.4. TYPE-II REPRESENTATIONS, BRATTELI D I A G R A M
Type-II representations are described by their highest weight vector 1aj) such
that SZlaj) =jlaj), 0 <~j < ~p - 1. pj is really isolated if moreover laj) does not
belong to a larger O/, representation. As was mentioned in the above analysis of the
mixings, this would imply that lay) ~ Im(S+) p-a. Note that the highest weights of
V. Pasquier, H. Saleur / Finite systems and conformal field theories
(np-1)/2
[(n-2)p-1]/2
531
-7-
-T
I
I
I
I
I
I
IJ
[(n-4)p-1]/2
Fig. 4. Structure of a representation #~,,p 1)/2.
#/, (j'>~ {_p-1) all belong also to Im(S+) p-a. The highest weights of type-II
representations are thus completely characterized by the condition
KerS +
]aj) ~ I m ( S + ) P - 1
(1.19)
which is reminiscent of cohomology theory. Another, less natural, characterization
of these states is
K e r ( S - ) 2j+1
[aj) ~ i m ( S _ ) p _ l _ 2 j .
(1.20)
Their number /2~N) is immediately deduced from the pairing principle. If we
introduce F/N>, the number of pj representations in the U[SU(2)] case, it reads
~ ? ) N ) = ~j.(N) - - F ( N )
*p
1 j
_1_ ~ ' ( N )
~j+p
-- p ( N )
*p-1
j+p"]-'''"
In the case of (C2) N,
N
,) (
N
N / 2 + j + 1)"
(1.21)
532
V. Pasquier, H. Saleur / Finite systems and conformal field theories
0
2
0
3
0
0
1
3/2
0
Fig. 5. Bratteli diagram for U(SU(2)), Uq[SU(2)]
( q = e i~/4 ) for (C 2 ) u.
0
0
2
1
2
3
/.
Fig. 6. Bratteli diagram for (C3) u.
This has a simple geometrical interpretation in terms of paths on a Bratteli
diagram [16]. We first discuss the s-- ½ case. The U[SU(2)] diagram is shown in
fig. 5. The vertical scale is the number N of s = ~ spins that are tensorized to build
the configuration space, while points correspond to representations onto which this
space decomposes. Their spin is indicated, and multiplicities are given by the
number of paths ~!N) going tO the origin. For q P = +1 the similar diagram for
type-II representations is obtained by cutting the preceding one at j - - 5p
1 - 1,
keeping only the left part. Indeed representations that remain have spins 0 ~<j <
½p - 1, and the number of paths to the origin is easily seen to be given by eq. (1.21).
The higher-s case works out similarly (fig. 6).
1.5. QUESTIONS OF UNITARITY
It is natural to define scalar products in (C2S+l) u as in the SU(2) case. For q real,
the space splits into irreducible highest weight representations pj, and one finds,
V. Pasquier, H. Saleur / Finitesystems and eonformalfield theories
533
with the convention {ajlai ~ = 1,
((S
)"ajl(S-)mai)=~,,m[n]q!
fi
q2j-,,+k_q-2j+,,
k=l
k
q--q-~
'
(1.22)
which is strictly positive for n ~< 2j. Since S + is the adjoint of S , these representations are unitary. If q is complex eq. (1.22) can take negative values, and vanishes if
q is a root of unity. In this latter case, it is clear that none of the mixed
representations are unitary since they contain new highest weights which are null
states for eq. (1.22). The case of type-II representations is more interesting. The
various terms in eq. (1.22) are all of the form sin ~rp'x/p where x varies from 1 to
2 j inside Oj- sin ~rp'x/p is positive for x < p / p ' , and (1.19) is always positive if
j < p / 2 p ' . If p ' = 1 all type-II representations are unitary, as well as &p-~/2- If
p' > 1, some of the type II or &p 5)/2 representations are nonunitary. For instance,
(S ajlS aj)-
sin2~rjp'/p
sinTrp'/p <~0,
for
p <~j <~Min[ P ~p,
~
p
,p,
2. Spin chains and quantum groups
2.1 T H E SPIN -1
2.1.1.
XXZ
C H A I N WI TH I M A G I N A R Y S U R F A C E TERMS
We consider the chain of N spins s = ½ with hamiltonian [17]
H=
N-I
q+q-~
......
v
v
~ o io
i+~ +o/o/+1
+ - - o t2z O i + l
q_q
+ - - ( 021
-- OfV) '
(2.1)
i=1
where o are Pauli matrices.
H is obtained by taking the very anisotropic limit of the transfer matrix for the
six-vertex model with free boundary conditions and R-matrix [6]
1
R=
0
0
0
0
1 +q-Xx
-x
0
-x
1 +qx
0
=l+xe,
(2.2)
where x is an anisotropy parameter and e a Temperley-Lieb matrix [4]. One has
H=
N l(ql
~
i=1
q+ 2
)
2e i .
(2.3)
534
V. Pasquier, tt. Saleur / Finite systems and conformalfield theories
The surface terms in eq. (2.1) arise due to the weights of external vertices. We
restrict to the case q = e i'/"+ 1 a complex number of modulus one, when the chain is
massless. H ( q ) is then not hermitian, but, through relabeling of sites from N to 1, is
similar to H(q ~). Its eigenvalues are real, and the spectrum is invariant under spin
reflexion S z --* - S z. Note that, introducing the operator
(NI
U = exp iTr ~., jof
j= 1
,
(2.4)
]
one has
UH(q)U -l=-H(-q
1).
(2.5)
Hence eq. (2.1) has properties similar to the same hamiltonian where the o:o:
coupling has the opposite sign, which is more often considered [10]. Eq. (2.1)
commutes with the generators of Uq[SU(2)]
[H,S ±] =0
[H, Xz] = 0,
(2.6)
and the latter can be considered as the "symmetry group" of the chain. The
spectrum can then be classified according to representation theory of sect. 1. In the
case/~ irrational, the commutation of H with S -+ implies that all states in a given
representation OJ have the same energy. For a highest weight with spin j, the
corresponding energy appears thus 2 j + 1 times in the spectrum, in sectors of
charge - j ~< S z ~<j. Let us introduce the generating function of all scaled levels in
some S z >~0 sector
Z yN(E-Eo)/2,~O,+l)sin[,,/(~÷l)l= •
"~(~'=
all levels
yN(~ Eo),
(2.7)
all levels
where E 0 is the ground state energy (which occurs for S z = 0) and the prefactor in
the exponential ensures an isotropic continuum limit. The generating function
restricted to highest weights then reads
K(N)
j
=
Z
yN(E-Eo)_~j(N)
--
"
~(N)
--~j+l
•
(2.8a)
highest w e i g h t s
with spin j
F r o m what proceeds, it is clear that the coefficients of eq. (2.8a) in the expansion in
powers of y are positive integers. The number of terms in (2.8a) is Fj(u), the number
of 0i representations. Conversely one has
N/2
°~s(#)= E K~N,j=S z
(2.8b)
V. Pasquier, H. Saleur / Finite systems and conformal fieM theories
535
In the case/~ rational = p / p ' - 1, we define K) N) by continuity, following energies
of highest weights of the irrational case. Of course, K ( u ) ceases then to be the
generating function of highest weight levels. Some representations pj can mix;
consequently H can only be brought into triangular form inside the type-I representations, with all diagonal terms equal. One can then consider the generating function
restricted to type-II representations, the expression of which derives from (1.21)
_v(N)
x(N)_
J
_v(N)
*~p 1 - j
+ K(N)
*~j+p
_ /t-(N)
*~2p-1
j
+
...
(2.9)
Here also all coefficients in the expansion in powers of y are integers due to the
quantum symmetry.
2.1.2. To calculate eqs. (2.7)-(2.9) one can in principle use the Bethe ansatz
technique [17, 18]. We write an eigenstate as
(2.10)
la) = ~ _ , f ( x 1. . . . . Xn)[X 1. . . . . X n ) ,
where the x are locations of down spins on the chain, and the summation extends
on sets of increasing integers 1 <~ x I < x 2 < . • • < x~ <~ N ( n ~ N / 2 ) . The ansatz for
f is then
f ( x 1. . . . . x,,) = Y ~ e p A ( k p l , . . . . kp,,)ei(kp~x~+ ... +kp,,x,),
(2.11)
P
where the sum extends over all permutations and negations of the impulsions k, and
changes sign at each transformation (note that f vanishes if two k's are equal).
The coefficients A are given by [17]
A ( k 1. . . . . k , , ) = f i f l ( - k ~ )
/=1
1-[
l <~j<l~n
B(-k,,k,)e
ik,,
(2.12)
where
fl( k ) = (1 - q e - i k )e '(N+ ')* ,
(2.13)
B(kl, k2) = [ 1 - ( q + q-1)eik2+ei'*l+k2)]
×[1-(q+q
1)e-ik~ + ei(k2-kl)] .
(2.a4)
Note that if eik = q 1, f l ( - k ) = 0 and the amplitude (2.12) vanishes. Thus for
impulsions 7 = ~/(/~ + 1) such that e tv = q, there is no reflected wave, due to the
special boundary term. If
a(k)
= (1 - q - a e - , k )
(2.15)
536
V. Pasquier. H. Saleur / Finite systems and conformal field theories
and supposing that none of the A vanishes, compatibility conditions read
a(k,)fi(kj)
a(-kj)fl(-k])
=
e2/r% =
fi
t=,.t•j
j=l
. . . . . . .
B(k,,kO
n
(2.16)
or, taking logarithms
Nkj=vli+
~ [g'(kj,k,)+g'(kj,-k,)],
½
(2.17)
/=1,/:#j
where
eiq,(&, k2)
1 - ( q + q-1)ei& + e i(k,+k2)
(2.18)
1 - ( q + q-1)eik: + e i(kl+k2} "
The associated energy is
E=(N-1)(--
+ 4
2
j=l
cos kj -
2
"
(2.19)
In eq. (2.17), the l ' s are distinct integers I j ~ [1, N], and the ground state is
obtained by chosing ! ° = j , j = 1 .... , n.
The states built in this way can be shown, generalizing the proof of ref. [18] for
the SU(2) case, to be highest weights of the (quantum) S + operator: S + [a} = 0. In
establishing this property, the fact that none of the k's is equal to ~, is of course of
crucial importance. S - a} reads then, up to a normalization factor
S 1~)=
~2
f ( x ~ ..
n I - • • n,,
x,){
~2
1 ~.k--<
+
q'lxxa ""x,,)+
q 2{i-l)q~lx 1 . . . x i 1xxi
•Vt
1•
A <--¥1
...
x 1
q 2"q"Ix 1 ".. x , , x } }
" ' " X n } qX n < x <~ N
(2.20)
Similarly, the ( S - ) k I a } contain k impulsions "y, without reflected waves. The
total set of impulsions in eq. (2.20) can be seen as a solution of eq. (2.17) by
analyzing the set of allowed I 's. We shall not use this point of view in what follows
and consider states obtained in eq. (2.17) with distinct positive integers as the only
"Bethe states". The Bethe ansatz (together with use of Uq[SU(2)] symmetry) gives
all the eigenstates of eq. (2.1) provided that all highest weights of S ÷ are conversely
Bethe states. We shall suppose this holds true in the following, at least for /~
irrational.
V. Pasquier. H. Saleur / Finite sTstems and conformal field theories
537
2.1.3. In practice, it is possible [19, 20] to obtain the scaling behavior of energies
(2.19) in the asymptotic limit n, N--+ oo. The calculation is especially simple in the
case of the ground states in the S z = j sector for which { I } = {i0}, and one finds
lira
N~ ~
N/~ 0 =
1 (1
24
6
/* (/.t + 1)
lim N(/~j-/~'o) = (1 - 2 ~ j ) 2 - 1
,v+~
~-+1)
(2.21)
The central charge here is thus
c = 1 - 6 / # ( ~ + 1).
(2.22)
Although the usual X X Z chain has c = 1, the boundary terms introduced here lower
c to (2.22). It is interesting to notice that for arbitrary surface fields h ( o ( - ofv ), one
would also find c = 1 [23], in the calculation of eq. (2.21); the value h = (q - q - 1 ) / 2
plays a very singular role, associated to the Uq[SU(2)] symmetry. Introducing the
Kac formula
h~.~=
4 , ( , + 1)
(2.23)
we find that limN~ ~(/~"/ - - / ~ 0 ) = hi,l+2/The consideration of other sets of integers in eq. (2.17) gives a p r i o r i ( N ) states,
which is the total degeneracy of the S z sector. Due to the quantum symmetry
though, and since states inside a representation Oj are not Bethe states (at least for #
irrational), only Q N)= (N) _ (,, U ), ( j = ½N - n ) sets can actually correspond to
a solution of eq. (2.17). In this case one finds
lira ( / ~ - / ~ 0 ) = hl,1+2j + ).2 l i - li 'm.
i=1
N~oo
(2.24)
We did not find any argument stating when some set of I ' s indeed corresponds to
a solution. However, in the q = 1 case of the X X X chain, renormalization group
arguments suggest that the continuum limit is described by a free bosonic field [24].
The partition function on a rectangle N × T with periodic boundary conditions in
the time direction should read
• s~
=
z[Deplexp--cp(x+ T, v)=q~(x,y)
0q00cp-
(2.25)
538
V. Pasquier, H. Saleur / Finite systems and conformal field theories
where y = e -~T/L, ~ ( y ) = y 1 / 2 4 p ( y ) , p ( y ) = [-[~(1 - y " ) and g = 1. The generating
function which has to be taken of highest weight levels is then in this case
lim K ) N ) - y j2 _ y ( j + l ) 2
N~ ~
P (Y)
,
q = 1.
(2.26)
If we suppose now the set of integers in eq. (2.24) does not depend on /~ we get
yhL~+2j -- y h l , - l - 2 j
lira K~ N) = K 1 l + 2 j =
iv~ o~
"
(2.27)
P(y)
We have checked numerically the validity of eq. (2.27) by a finite size scaling study
of chains with N ~< 10. (2.27) is the character T r y L0 in an irreducible representations of the Virasoro algebra with c = 1 - 6//z(/~ + 1), /~ irrational, and highest
weight h =hL~+2/. In the case /~ rational, we find immediately the generating
function of highest weight energies for type-II representations
x(N)
= K)N) -- vlXp
( N ) 1 - j "~ **j+p
I~(N)
j
+
(2.28)
""
yhl,l+z(j+,,pl _ yhl, ~ 2~j+,p~
lim - (N)
N~
p- 1
0 ~<j < - -
(2.29)
n~Z
It coincides with the irreducible [21] character of Virasoro algebra for h = h1,1 + 2jOn the other hand we note that the levels excluded in eq. (2.28) correspond, due to
sect. 1, to highest weight states inside larger representations. The latter do not seem
to be associated to solutions of eq. (2.17) since some of their impulsions are equal to
7. It is therefore tempting to conjecture that highest weight of type-II representations are the only Bethe states with spin 0 4 j < ~ ( p - 1).
Of course one could also consider generating functions of the scaled energies of
states that belong to Ker S + and have the higher spin in their associated representation. For j = ( n p - 1)/2 one finds
(2.30)
X1, np = K1, np,
and for j = j o + k p ,
O~<jo< ~ ( p -
Xl,l+2/=
1), k>~ 1
Z
yhl,l+2~y+.p) __ yhl, I 2(j+npl
p(y)
,
(2.31)
ncZ
he[--k,-1]
which are also irreducible Virasoro characters, but outside the minimal Kac table.
V. Pasquier, H. Saleur / Finite systems and conformalfield theories
539
2.1.4. These results point out relations between the quantum group symmetry
for the lattice model, and the Virasoro symmetry in the continuum limit, with
corresponding concepts of null vectors, unitarity, etc. Also note that starting with a
non-hermitian hamiltonian and projecting on type-II representations, we get sectors
described in the continuum limit by characters Xl,1 +2j of the minimal table, with
unitarity. This is of course reminiscent of the Feigin-Fuchs construction. We shall
comment further on these points in the sequel.
The projection on type-II representations is easily done in calculating the partition function. Introducing a q 2sz term in the trace we get
z ( N ) = TryHq2SZ = Trtypenynq2SZ
=y-C/24
~.
(2j +
] ) q A. j (N) , (2.32)
o<~j<(p-1)/2
where y = e -~T/N. This is for N, Teven the partition function of the Q = (q + q-1)2
state critical Potts model on a rectangle ½T× ~N
1 with periodic (free) boundary
conditions in the time (space) direction. In the asymptotic limit one has
lim z ( N ) = y -c/24
N ~ oc
E
( 2 j + 1)qX1,1+2j(Y ) .
(2.33)
0 ~ j < ( p - 1)/2
(j integer)
Restriciting the Tr to type II highest weights with spin j gives Z) N)= y-c/24x(jN).
This is [22, 23] the partition function of the critical restricted solid on solid model of
Ap_ 1 type, with heights fixed to a = 1 (b = 2 j + 1) on the bottom (top) of the strip.
Such a result is expected since the number of states in X~u) is a number of paths on
the Bratteli diagram that corresponds exactly to the number of configurations of the
N-sites Ap x model with the above boundary conditions [22].
XXZ
2.2. THE SPIN-1
CHAIN WITH IMAGINARY SURFACE TERMS*
Integrable hamiltonians analogous to eq. (2.1) can be written for higher spins.
Then the commutation with Uq[SU(2)] allows one to write the R-matrix as a sum of
projectors onto various (quantum) spin representations, and coefficients are obtained by requiring the Yang-Baxter relations to be satisfied [12]. In the case s = 1
one finds [25]
H=
L-1
z z 21
~-, Si'Si+l _ ( S i ' S i + l ) 2 + ½ ( q - q - l~2~z~z
, ~ , ~ i + l - ½(q_ q -1 ) 2(SiSi+
)
i=l
_(q+q-l_2)[(cXsX
+½(q
_
Y v
+ sis,+
q - 1 ) 2 [ ( g i Z ) 2 + ( S i +Zl )
2
z z + ~,]
)sis,+l
]+½(qZ-q-Z)(sZ-sZ),
(2.34)
* The two following subsections deal with rather technical questions. We advise the nonspecialist
reader to turn directly to the periodic chain case (2.57).
V. Pasquier, H. Saleur / Finite systems and conformalfield theories
540
where the " b u l k " terms appeared in ref. [25], and surface terms are necessary as in
eq. (2.1) to ensure commutation with Uq[SU(2)]. In eq. (2.34), the operators
correspond to
,x=~_
l(i'Z)
o
,
,v=-v
1
' i)
o
-
,
,~=
i
(i ° !)
o
0
-
(2.3S)
while the generators of Uq[SU(2)] are obtained via (1.6) and the coproduct formula.
The same analysis of the spectrum symmetries as for the spin-½ XXZ chain can
immediately be carried out, and generating functions can be defined. The calculation of energies is of course more difficult here, due to the complexity of Bethe
ansatz equations. Heuristic arguments similar to those of ref. [26] can nevertheless
be used to conjecture the asymptotic limit of the spectrum.
We consider first the limit q --* 1. Here the model is known to renormalize on a
free superfield with bosonic coupling constant g = ½. According to the analysis of
[26] the fermionic degrees of freedom describe on the other hand an Ising model
with free boundary conditions. Using eq. (2.25) and
free
zlsing=•2(y)//•(yl/2)•(y2)
(2.36)
one finds
yS z2/:
~/2(y)
Zs~ = Nlira
~
~ s ~N)= ~ ( y l / 2 ) ~ ( y 2 )
T~(y) .
(2.37)
Therefore
lim K~N,=
P(Y)
( y j 2 / 2 - - y(j+l)2/2).
,¥~o¢
p(yl/2)p(y2)
(2.38)
This is an irreducible character of the c = 3 superconformal algebra in the NS
sector, with highest (L0) weight j2/2 (note that here j is always integer) [27].
In the case q = e i~/~+2, the central charge is known to become
(2.39)
c = ~ - 1 2 / / , ( / , + 2),
while the Kac formula in the NS sector reads
hr" =
[(/* + 2 ) r - / * s ] 2 8/,(/, + 2)
4
,
r - s even.
(2.40)
V. Pasquier, H. Saleur / Finite systems and conformal field theories
541
It is natural to conjecture then, as in eq. (2.27), that the limit of K) N) is then
P(y)
lim K~ x ) = KI.I+2j = e ( f / 2 ) p ( y 2 )
(yhl.,+2,_yhl, ~ ~,).
N ~ ao
(2.41)
In the case/~ = p / p ' - 2 rational, the mixing of representations occurs and generating functions of type-II highest weights can be introduced as in eq. (2.9). If
/~ = p - 2 is integer for instance one has
X~N)= K)N) _ v(N)
+ j~(N) + . . .
ZXp 1 j
*'j+p
(2.42)
lim - ( N ) _ .,(NS)
(2.43)
with
N___~ooXj
-- A.I,I+2j
is the character of the Neveu-Schwarz superconformal algebra in the irreducible
representation of highest (L0) weight h~, 1+2j.
2.3. THE XXZ CHAIN WITH REAL SURFACE TERMS
Another hamiltonian of interest is
N-1
H=
~ oi~o,+1 +
v 1 + Aoi z°i :+ 1 + p o ( + p ' o [
0 i v(I[+
(2.44)
i=1
where z~ = (q + q - 1 ) / 2 , and p and p' are real. It is hermitian, and does not present
the s Z ~ - S z symmetry. Because of its surface terms, H does not commute with
the quantum generators S ±, except in the SU(2) ( q = 1) case, and the only
symmetry of eq. (2.45) is U(1): [H, S z] = 0. Nevertheless, the spectrum of H (2.44)
turns out to be very similar to the one of (2.1) in the case
(p-A)(p'-A)
= 1
(2.45)
to which we restrict ourselves [28].
We consider first the Bethe ansatz solutions in the SZ>~ 0 sector. They are
obtained, starting from a reference state with all spins up, by lowering n down
spins. Eqs. (2.12) and (2.14) remain the same, while (2.13) and (2.15) have to be
replaced by
fl+(k ) = [ l + ( p - Z i ) e
ik]ei(N+l)k,
a+(k)=
ik].
[l+(p'--A)e
(2.46)
If e i k = y = A - - p ,
B ( - k ) vanishes and there is no reflected wave in this case.
Supposing none of the amplitudes A vanishes, compatibility equations read as
V. Pasquier, H. Saleur / Finite systems and conformal field theories
542
eq. (2.16), and the combination a+(k)fl+(k)/a+(-k)fl+(-k) is still equal to e 2iNk
due to (p - A ) ( p ' - A ) = 1. In what follows it can be convenient to parametrize
p = sinv tg
2
,
p' = -sin~, t g - - - ~
(2.47)
where A = cos 7, and iS is arbitrary.
The final equations read then as eq. (2.17), and the energy is
-
E - P+P'
2
+(N-1)A+4
j=l
coskj
q+q
2
"
(2.48)
Since the 8-dependent term in eq. (2.48) is the same for all values of n, we see that
the gaps for Bethe states with S z >/0 in eqs. (2.1) and (2.44) are exactly the same,
hence do not depend on 3 in eq. (2.47).
In the case of the hamiltonian (2.1) (with y _ q - i ) , other eigenstates were
obtained by acting with S - on Bethe states, getting in this way some of the
impulsions eik equal to q-1.
We can consider here the situation where one of the eik's (eik- say) is equal to
y = A - p . In this case, due to f l + ( - k n ) = 0 one gets only n - 1 compatibility
equations
a+(ki)fl (k,)
=e2iNkJ =
og+(-kj)~+(-ki)
fi
j=l
l=l,14=j O ( k j , k , )
'
n--1
'" . . . .
(2.49)
Putting the l -- n term on the left-hand side gives
a (k,)fl-(kj)
a-(-kj)fl-(-kj)
, 1 B(-k,,k,)
=,=,,I-],.j B(kj, k,)
,
j=l ..... n-l,
(2.50)
where
fi ( k ) = [1 - ( p + a ) e - i k l e
i(N+x)k,
a (k)=[1-(p'+k)e-ik].
(2.51)
Eq. (2.50) is the equation for Bethe states in the S Z < 0 sector, starting with a
reference state with all spins down and turning n - 1 spins up. The energies
associated to eqs. (2.49) and (2.50) are the same
E= ½(p+p') + ( N -
1)A + 4 ~ [ c o s k j - ½(q+ q - l ) ]
j=l
71 - - 1
=-½(p+p')+(N--1)A+4
Y" [coskj-½(q+q-1)].
j=l
(2.52)
V. Pasquier, H. Saleur / Finite systems and conformal field theories
543
All the discussion so far is based on Bethe ansatz solutions. In the preceding
discussion we assumed that, at least for/L irrational, eq. (2.17) has
solutions in the S z >1 0 sector. Let us assume now that eq. (2.49) has
(2.54a)
s ° l u t i ° n s in the S ' Z < O sect°r" W e expect then the missing (
eq. (2.53) to be exactly the
Xs z
U/2 + S 'Z levels (2.54) with S ' z = - S z -
1 ) levels
1, reproduced
by the folding procedure described above. With the hypothesis of completeness of
Bethe ansatz solutions, one should thus observe in the S Z > 0 sector all levels
appearing in the - S z - 1 sector, plus/'s(~ ) additional ones. The gaps of the former
(type I) can depend on 8, while due to (2.48), the gaps of the latter are independent
of 8 (type II).
We can now consider various generating functions of scaled levels. Analytic and
numerical calculations suggest here that
yhtA+2sz
lim ,~s~N~-
(2.54b)
whatever the sign of S z. Restricting to type-lI levels we find
(2.55)
type-n levels
in sector i > 0
is exactly the same as eq. (2.8a). Note here that although energies are the same in
both problems, wave functions are different, and type-II states do not seem to be
annihilated by any simple operator like S +.
Even though H does not commute with S +, additional coincidences appear in the
spectrum at rational values of #, and type-II levels, which are the same as the
highest weight levels of eq. (2.1) (defined for /~ irrational, and by continuity),
coincide accordingly. The combination (2.9) thus still makes sense for a finite value
of N here, and the Virasoro characters can be calculated in the same way.
544
V. Pasquier, H. Saleur /
Finite systems and conformal field theories
If we calculate the partition function Z (u) with trace restricted to type-II states
with spin 0 ~<j < ( p - 1)/2 we get
Z (N)=
Tr y n = y
,/24
type-II
states
x~N)
y"
(2.56)
0 ~<j < ( p - 1 ) / 2
without the multiplicity factor [2j + l]q of eq. (2.33).
2.4. T H E
XXZ
CHAIN WITH PERIODIC
BOUNDARY
CONDITIONS
It is also interesting to consider the chain [10]
N
q+q-1
1
x x
V V
H(e'~) = Z °i°i+, +°i°;+l
_[_
--°,°,+1
e i~p
e
z 2
2
i=1
i~
+ --aUOl+2 --OUO;+2
q+ q 1
2
aUOlZ
~-.
(2.57)
H is obtained by taking the very anisotropic limit of the transfer matrix for the
six-vertex model with twisted boundary conditions. In terms of the e i it reads
H=
N q+q-1
~
2
2e,,
(2.58)
i=1
where the ( N - 1) first e's are the same as eq. (2.2) and the last is modified
eN =
o
o
0
q-1
0
0
-e
_e~
'~
0
q
0
.
(2.59)
H is hermitian and equivalent to H(e -'~) through relabeling of spins. The spectrum
is not invariant under S z ~ - S z.
As shown already in eq. (2.6), all e~'s with 1 ~< i ~< N - 1 commute with Uq[SU(2)].
Due to the last e m term we find
[H(ei~), S z] = O,
S+H(e i ~ ) = H ( q 2 e i ~ ) S + + [ ( q - q - 1 ) S ~ - ( q + q
+[(q-l_q)S~_(q+q-1)s~oz][l_q
1)Sa+oz][1-q2(SZ+l)e
2(sz+l)ei~] ,
S - H ( e '~) = H( q?ei~)S -+ [(q l _ q ) S ~ _ ( q + q-1)S~o z ] [ 1 - q
+[(q-l-q)S~-(q+q
i~]
a(sz+l) ei~]
1)s~oZ][1-q-2(sZ+ale i~],
(2.60)
V. Pasquier, H. Saleur / Finite systems and conformal field theories
545
while similar relations can be written with higher powers of S +-. Although H(e i~)
does not commute with the quantum spin, it is possible to cancel the right-hand side
of eq. (2.60) by properly choosing the magnetization of states which are acted on.
Defining
Hk=H(qZk),
(2.61)
one finds
S + H k+l = Hks +-,
in sector S z = k,
(2.62a)
and more generally
( S ± ) " H k + - , = Hk(S+-) ° ,
in sector s z - - k ,
(2.62b)
which can be represented by the commutative diagram (fig. 7). Due to these
identities, some levels will be common to H 2 + , and H~ -+" where the lower index
denotes the charge sector (integer or half-integer depending on whether N is even or
odd).
Consider now some hamiltonian Hskz. It is possible to write the Bethe ansatz
equations in this case
k
N k j = 2~rlj + - ].t q- 1
N/2 - S z
+
Y'~
g~(kj, k,),
(2.63)
1=a,14:j
where I ' s are integers in ( - v1 N, ½N). A calculation similar to eq. (2.20) shows then
that for such Bethe states [a), ( s + ) k - S Z l a ) = 0 . The Hilbert space J~s Z thus
N k ) states(type I)obtained from )~k acting with ( S - ) (k-sz),
decomposes into ( m/2-
and(zv/?sZ)-(N/~_k)states(typelI)whichareannihilatedby(S+)(k-SZ).The
energies of the latter do not appear in the spectrum of H~S z . It is likely that all the
(S÷)"
'•k+n
[
Hk.n
k
ut,""~Jk
(S÷) .
~
~""J~k*n
Fig. 7. Commutative diagram associated to eq.
(2.62).
546
V. Pasquier, H. Saleur / Finite systems and conformal field theories
type-II states (for g irrational) are Bethe states. We can now define a generating
function
(N)'~"¢7 =
(y~)N(E
E
Eo)/2w(g+l)sin[v/(la+l)l(Y] (N/zp)(P) '
levels of n~z
(2.64)
\y ]
where E 0 is the ground state of some of the H k" hamiltonians, P is the impulsion,
and two variables y, ~ are necessary due to toroidal boundary conditions. Note that
the numerical factor in eq. (2.64) that ensures isotropic continuum limit is the same
as in (2.7). Restricting to type-II levels one gets
( N ) K ~ = (N)o~Sk~-(N)o~S~ .
(2.65)
The coefficients in the expansion in powers of y, .~ are all integers due to the above
"folding" mechanism which explains thus the observation of ref. [10].
In the rational case q P = +1, H k = H km°ap, and the following commutative
diagram can be drawn which involves only two hamiltonians (n = k - S z > O)
J~k p
(S+IP )"
Jt~sZ
(S+)">
"~k
(S+)P >"
JgaSZ+p
I
I
I
I
H s~
Ha
H sZ
Hk
J,
J,
$
$
(2.66)
One shows easily from the developments of sect. 1 that for type-I representations,
all states in Ker(S +)" are also in Im(S +) p- n. In the case of type-II representations,
this holds true for states with spin - n / 2 < SZ<~ ( p - 1 - n ) / 2 . Hence, for
1 <~k + S z < p - 1
(2.67)
the sequence in eq. (2.66) is exact except for the subspace oWs~. In this case it is
natural to consider K e r ( S + ) n / I m ( S + ) p - ' . The generating funtion of the corresponding scaled levels is
(N)D~z=(N)K~z (N) S z
+(N)Kk ~ _
•
-Kp n - k
S+p
"'" '
(2.68)
where the symmetry o ~ - o , ¢p ~ -q0 of the spectrum has been used.
In the continuum limit, free field mapping arguments [2] suggest that, writing
q = ei~r/~ + 1
N~oc
_
1
P(>')P(Y)
£
yD*k'+k+e(~+ll] 2 1/d-I*(/~+l)~[/*k' k e(p.+l)] 2
I/4/~(/*+I).
(2.69)
V. Pasquier, H. Saleur / Finite systems and conformalfield theories
547
In the c a s e / , + 1 = p one finds then [10]
lim
N~oo
p 2
(lV)Dsk.z+(N)DP_-s~= Dxkz+ DP-skz= Y'~ X r ,
k_SgXr,
k+S Z .
(2.70)
r=l
As in sect. 1 we can count the number of states in K e r ( S + ) " / I m ( S + ) p ~ for a
chain of length N and a sector of charge S z from formula (2.68). It corresponds to
the n u m b e r of paths of N steps on the Bratteli diagram (fig. 5) which go from spin
(n - 1 ) / 2 to (n - 1)/2 + S z. In particular for S z = 0, the paths connect identical
spins; hence the number of states, once summed over n, is the same as the number
of configurations of the Ap 1 model in periodic boundary conditions. Accordingly,
the partition functions [10] coincide
p
ZAp ~= ( y p )
1
,./24 y[ D~
(2.71)
n=l
In order to generalize (2.71) to other restricted solid on solid models [29], it is
convenient to consider representations of the Temperley-Lieb algebra on a closed
chain, with generators e i, 1 ~< i ~< N - 1 as in eq. (2.2) and e N defined by eq. (2.59),
with relations
eiei+_lei=ei,
e2i=(q+q
1)e i
(2.72)
and eN+ 1 = e 0. These representations R s z k are indexed by the total spin S z and
the angle % and since we restrict to values e i~ = q2k, the commutative diagram
(2.66) holds. So we obtain irreducible representations of eq. (2.72) by considering
the restrictions to Ker(S +) (k- s z ) / I m ( S +) p- k + s l with
1 <~ k + S z ~< p - 1 .
(2.73)
k + S z and k - S z are integers which we denote from now on by m and n. The
corresponding irreducible representation is p,,,.
The integrable restricted solid on solid models provide natural representations of
the T e m p e r l e y - L i e b algebra (2.72) which can be decomposed onto the pm,,'S.
This involves three steps. First it is convenient to introduce representations which
we
denote (~z) where k and S z are defined modulo p and are simply the direct
sum of the R j , k for S z' = S z mod p. Then from the commutative diagram we have
Here by the subtraction we mean for instance that any trace in p,~, is obtained by
taking the difference of traces.
V. Pasquier, H. Saleur / Finite systerns and conformal fieM theories
548
Second we consider the so called f-models [2], the Hilbert space of which for f
dividing p decomposes as
with k a multiple of p / f rood f and S z equal to 0 mod f. Then it is known how
the Hilbert space of restricted solid on solid models O~ADE decomposes onto the
[21
=
= l(ae12 - w 6 - w 4
+w2 -wl),
JtaE~ = { (it°30 -- a'g15 -- J~10 -- J~6 + J{'5 + J~3 + J~2 -- Jr1) ,
(2.76)
where subtractions are interpreted as in eq. (2.74).
Third, combining eqs. (2.74-2.76) we obtain the decomposition of J{'ADE in
terms of p.,.
"~ADE = (~'[mnPrnn,
(2.77)
where the y's are positive integers. It is noteworthy to observe that in the
continuum limit the decomposition (2.77) corresponds to the decomposition of the
partition function, where
p-2
Tr,,,, y n = (y37)-,724 E X,,,X . . . .
(2.78)
r=l
and y,,,, is the matrix of the modular invariant combination of characters [30].
3. Connections with conformal field theories
We pointed out in the last parts that several aspects in the study of quantum spin
chains and Uq[SU(2)] representations were reminiscent of the works initiated by
Feigin and Fuchs on the Virasoro algebra [31].
In the case of the open chain, the hamiltonian H x x z can be considered as the
discrete analog of the operator L 0. In the same way as L 0 is a member of the
Virasoro algebra, H x x z belongs to the Temperley-Lieb algebra. The latter algebra
appears to be the commutant of Uq[SU(2)], which is generated by S ÷, S-, q ±sZ.
Similarly, in the Feigin-Fuchs construction, the Virasoro algebra commutes with
the screening operators Q+, Q- and the U(1) charge Q3. In the case q = 1, to obtain
irreducible representations of Temperley-Lieb algebra we considered in a given
charge sector-j states belonging to Ker S ÷. This gives representations in one-to-one
V. Pasquier, H. Saleur / Finite systems and conformal field theories
549
correspondence with those of SU(2). In the continuum limit the spectrum of Hxx x
in an irreducible representation of Temperley and Lieb is the same as the one of L 0
in an irreducible representation of Virasoro with highest weight hj =j2.
In the case q = exp[i~r/(~ + )], /L ~ Q, the picture remains essentially the same.
The central charge of the Virasoro algebra becomes c = 1 - 6/t~(~ + 1), the boundary term in the XXZ chain which ensures Uq[SU(2)] invariance playing the role of
the charge at infinity in the Feigin-Fuchs construction. The spectrum of the
hamiltonian in an irreducible Temperley-Lieb representation is now the same as the
one of L o in an irreducible representation of Virasoro with h1,1+ 2 j = [ ( 1 - - 2jaj) 2 1]/4/~(/L + 1).
If q is a root of unity, the new feature is (S+) p = 0. A similar property is also
observed [7] in the Feigin-Fuchs construction for the screening operators Q-+. It is
then natural to restrict to type-II representations which are selected by considering
KerS+/Im(S+) p-1. In this case the Temperley-Lieb and Virasoro algebra representations are again in correspondence. These type-II representations are all unitary
only if/L is integer. Thus the theory of Uq[SU(2)] representations gives a discrete
analog of the concepts of null vectors or unitarity in conformal field theories.
The case of the closed chain is more intricate. We have classified irreducible
representations of the periodic Temperley-Lieb algebra (2.72) using again Uq[SU(2)].
These are characterized by two numbers m, n and are obtained from the commutative diagram (2.66) by restricting to Ker(S + ) " / I m ( S +)P-". In this case we deal with
all the quotients of this type (n was restricted to one in the open chain). In the
continuum limit, the spectrum of the hamiltonian is then in one-to-one correspondence with the one of L 0, L0 in the direct sum of representations of the product of
left and right Virasoro algebras
p
p,,,,, ~
2
(~) (r, m) ® (r, n ) .
(3.1)
r=l
It is also interesting to notice that the continuum limit S-matrix which encodes
modular transformations of affine characters can already be observed at the discrete
level. For simplicity we restrict to q--ei'~/P; also for the following reasoning it is
more convenient to use a lagrangian formalism. The hamiltonian Hxx z (2.57) is the
derivative with respect to the spectral parameter of the transfer matrix of the
six-vertex model [6]
d
H(e '~)= ~ulogT(u,e i~) ( u = 0 ) ,
(3.2)
where in the vertex model a Boltzmann weight e +i~ is associated to up (down)
arrow of a given vertical link at each column (fig. 8). The T(u, e i~) form, for a given
% a family of commuting transfer matrices. We consider now this vertex model on a
torus. The partition function in a sector of charge S z (modulo p), for e i~°= q2k, is
550
V. Pasquier, H. Saleur / Finite systems and conforrnal fieM theories
\
\
\
\
\
e '~
0
(0
/
/
/
/
e -i' )
I
/
Sz
Time ev0tuti0n
S z'
Fig. 8. Effect of the twisted boundary term on the
vertex model.
denoted by Z
Fig. 9. Boundary conditions corresponding to
Z s z " s z'.
( t, isea toseet t is ao erewri tenas
= Z _sZ .
(3.3)
S z"
where Z s z " s Z, is the partition function of the six-vertex model without cp-term, with
charges S z , S z" in the time (space) directions of propagation (fig. 9). The modular
properties of Z s ~ ' s~, are very simple; in particular under S-transformation one has
Zs~" s z' ~ Z
(3.4)
s ~', s ~ ,
/ L~
Therefore the modular properties of Z ( s z ) follow
k
q2kSZ'+2k,SZz(
1
k'
(3 5)
S ~', k'
where S z, k have the same parity and are defined modulo p. If we now consider the
action of the transfer matrix in the representation p,~, we obtain the partition
function from (2.74)
(3.6)
V. Pasquier, H. Saleur / Finite :(vstems and conformal field theories
T h e transformation of
Zmn under
551
S follows
s 1
Z .... - - , -2 p . , . ~,
sin
~rmm'
rrnn"
p sin p Z,,,,,,
(3.7)
as is known to hold true in the continuum limit, where
p-2
Zm.=
(3.8)
r=l
Appendix
THE ALGEBRA Uq[SU(n + 1)]
T h e algebra Uq[SU(n + 1)] is generated by E ", F ~, q +-H~/2 (0~ = 1 . . . . . n ) u n d e r
the following relations
q H°/2EBq- H~/2
=
qa~a/2EB '
qH"/2FB q H"/2 = q-~,.a/2F B,
[U% F ~] = a,~a(q H° - q - U " ) / q -- q - ' ,
[ E '~, E '~ ] = 0,
[ F '~, F '8 ] = 0,
E'~2E ~ - ( q + q - a) E'~E¢E'~ + EBE '~2 = O, I
F,~'-FB _ ( q + q - a ) F,~FBF~, + FBF,~~ = 0 j
for a~B = O,
(A.1)
for a,,¢ = - 1,
where a,/~ is the Cartan matrix. They reduce to the standard relations among
generators e'L f'*, h '~ of SU(n + 1) in the limit q--+ 1 (we use here the Chevalley
basis; in the SU(2) case this would correspond to e = S ÷, f = S , h = 2 S z in the
"physicist's" notations of sect. 1). U q [ S U ( r / + 1)] can be endorsed with a H o p f
algebra structure; in particular the coproduct reads
k (q -+g~/2) = q +-H"/2 ® q
+_ H ~ / 2 ,
A ( E a) = qlt~/2 @ E ~ + E ~ ® q - H ~ / 2 '
A(F~)=qtt°/2®F~+
r~®q
,"/2.
(a.2)
In the fundamental representation, the relations (A.1) are identical to those for
552
V. Pasquier, H. Saleur / Finite systems and conformal field theories
U[SU(n + 1)]. Therefore, tensorizing N of these representations one has
qH~/2 = q1,~/2 ® • . . ® qt'~/2,
E '~ = ~ E f
= E q h~/2 ® . . . ® qh./2 ® e~ ® q-h"~2 ® . . . ® q-h"/2,
i
i
r,~= ~F,~=
i
y ' q h " / 2 ® . . . ® q h ° / 2 ® f i ~ ® q - h " / 2 ® . . . ®q-h"~2.
(a.3)
i
In the Uq[SU(3)] case, one finds the quadratic and cubic q-deformed Casimir
operators as
(q+q-l)C
2 [resp,(q-q-1)C3]
q2+(4H~ + 2H2)/3 q_ q - 2 - ( 4 H l + 2H2)/3 .4_ q-2(nl-H2)/3 _ 3
(q-q
+ qa+(~q+2H2)/3FiE1 + q-1
1) 2
(H~+2H2)/3F2E2
_ q,H,-142)/3(F1F 2 _ q F z F 1 ) ( E I E 2 -
q-IE2E1 )
+ ( - ) same thing where q and q-1 are exchanged.
(A.4)
On highest weights [aj, j2)
(q+q-1)C
2 (resp.(q-q-1)C3)
q 2+(4ja+2h)/3 q- q - 2 - ( 2 j l + 4 j 2 ) / 3 "4- q -2(jl
--_
(q-q-l)
J2)/3 -- 3
+(_)(q~q-1).
2
(a.5)
In the limit q --+ 1 we recover
62 ~"Jl + J2 + ~ ( j2 + j22 + Jl J2 ),
C3= _{(Jl -J2)[ 1 +Jl +J2 + {(2Jl2 + 2J22 + 5jlJ2)] •
(A.6)
Introducing numbers X~ such that
jo~q-l:~k
a
1--Xa,
~ ha=0,
or=0
(A.7)
553
V. Pasquier, H. Saleur / Finite systems and conformal field theories
eq. (A.5) reads
(q + q-1)C2[resp" ( q - q-1)C3] =
qXo x~-X2+q Xo-Xl+X2+q-X,,+~l-x2_3
( q _ q_t)2
+(_)
(q ~ q - t ) .
(A.8)
In the SU(n + 1) case, for highest weights
E"la~jo}) = O,
H"la~jo}) = A l a ( j . } ) ,
the values of the n q-deformed Casimir operators are obtained, generalizing eq.
(A.8), by expanding the expression
qZ:_o~oX~exp u
~,~
{e.=_+l}
(A.9)
a=0
in powers of u. They are invariant under the Weyl group action which permutes the
~'s.
In the case when q is not a root of unity, representations p{jo} are in one-to-one
correspondence with U[SU(n + 1)] ones. If q is a root of unity, a different structure
emerges. We represent each p (defined here by continuity) by a point J = Z] = l j , A~
(fig. 10)
in the weight diagram. R is the half sum of positive roots R = ~,~=IR,~
~ "
and o denote elements of the Weyl group with parity e(o). It is then easy to check
that the Casimir operators (A.9) are invariant under the transformations
J~J'=o(J+R)-R+p
n~R~ .
1
Fig. 10. Weight diagram for SU(3).
(A.10)
v. Pasquier,H. Saleur / Finitesystemsand conformalfield theories
554
Restricting to tensor products of fundamental representations with the quantum
operators defined by (A.3), one has the nilpotency property
(E~)P=(F~)P=O.
(A.11)
We now give some conjectures about representation theory. We define the q-dimension of p{j,~} by
dimqO(jo}= y" qZH~= ~ qR.tt.
states
(A.12)
states
It can be explicitly calculated via Weyl character formula
dimq Ofj.} =
Ewe(O)q"(S+n) R
Ewe(O)qO(n). n
(A.13)
For Uq[SU(3)]
qjl+j2+2 _
dim
q lOjl ' J2 =
q-(j~+h+2)
_
qj~+l + q-(A+l) _ q j2+1 + q-(J2+l)
(q__q-1)(q+q
(1 + j l ) ( 1 +j2) (1 + Ja__+J2]
2 }'
if
1_2)
q ---, 1
(A.14)
dimq p(j~} goes to e(o)dimq p{jo} in the transformation (A.10). It is then natural to
suppose that for q P = A-l, representations split into type-I representations
with q-dimension 0 that are either mixed or such that (A.12) vanishes, and
type-II representations with nonzero q-dimension. The nonzero O'S for which
eq. (A.12) vanishes are those for which E ~ = l j ~ = p - n and those deduced
by symmetries (A.10). Type-II representations are thus inside the fundamental
domain of length p - n
(fig. 12). Tensorizing N fundamental representations
(say 1,0 . . . . . 0) (fig. 11), the degeneracy of the space characterized by eigenvalues
Fig. 11. The three fundamental representations of
SU(3).
Fig. 12. Fundamental domain for Uq[SU(3)], q =
eiCr/5.
1/. Pasquier, H. Saleur /
H~ = ~"
1 - 2~', E ] = 0 ~ "
= N,
Finite systems and conformal field theories
555
is the multinomial coefficient
(A.15)
The number of representations p(jo} is then
/~{(N ) -
J.}-
In the case qP=
should then read
± 1,
E
o~W
E(O)(o(Y+n)-n
N
)"
(1.16)
the formula giving the number of type-II representations
N
t2'N)=
E s(o) E o(J+R)-R+p £ n,~R,~
{J.}
o~ W
{n~}
(A.17)
a=l
It counts the number of paths on the properly factorized Bratteli diagram.
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