On $τ^{(2)}$-model in Chiral Potts Model and Cyclic Representation of Quantum Group $U_q(sl_2)$

0.13 MB·English

by  Shi-shyr Roan

9
0
0
2
On τ(2)-model in Chiral Potts Model and Cyclic
n
a Representation of Quantum Group U (sl )
J q 2

2
2

] Shi-shyr Roan
h
c Institute of Mathematics
e
m Academia Sinica
- Taipei , Taiwan
t
a
(email: [email protected] )
t
s
.
t
a
m
-
d
n
o Abstract
c
[ We identify the precise relationship between the five-parameter τ(2)-family in the N-state

4 chiral Potts model and XXZ chains with Uq(sl2)-cyclic representation. By studying the Yang-
v
Baxter relation of the six-vertex model, we discover an one-parameter family of L-operators in
6
1 terms of the quantum group Uq(sl2). When N is odd, the N-state τ(2)-model can be regarded
2 as the XXZ chain of U (sl ) cyclic representations with qN = 1. The symmetry algebra of the
q 2
0
τ(2)-model is described by the quantum affine algebra U (sl ) via the canonical representation.
. q 2
6
In generalfor anarbitraryN, we showthat the XXZ chainwith a U (sl )-cyclic representation
0 q 2
8 for q2N =1 is equivalent to two copies of the same N-statbe τ(2)-model.
0
:
v
i 2008 PACS: 05.50.+q, 02.20.Uw, 75.10Jm
X
r 2000 MSC: 17B37, 17B80, 82B20
a
Key words: τ(2)-model, XXZ chain, Quantum group U (sl )
q 2
1 Introduction
In the study of N-state chiral Potts model (CPM), Bazhanov and Stroganov [12] discovered a five-
parameter family of L-operators of τ(2)-model in the six-vertex model with a particular field (see
1
[9] page 3), i.e. the solution of Yang-Baxter (YB) equation for theasymmetric six-vertex R-matrix,
(see (2.1) and (2.2) in this paper). The chiral Potts transfer matrix can be constructed as the Bax-
ter’s Q-operator of the τ(2)-model [11, 29]. By the functional relations between the fusion matrices
of τ(2)-model and the chiral Potts transfer matrix, one can compute the eigenvalue spectrum of the
(homogeneous) superintegrableCPM[1,6,7], wheretheτ(2)-degeneracy occurswith the symmetry
algebra described by Onsager algebra [24]. Furthermore, one can discuss the eigenvalue spectrum
and calculate the order parameter in CPM through functional-relation method [4, 5, 10, 22, 30].
Though much progress has been made on issues related to the eigenvalues, we still lack enough
information at present about the eigenvectors in CPM. Consequently, some important problems
whichrequiretheknowledgeofeigenvectors, suchasthecalculation ofcorrelation functions,remain
unsolvedin thetheory. Ontheother hand,the(”zero-field”) six-vertex model([8]Sect. 3) withthe
symmetric R-matrix has been a well-studied theory with profound knowledge known in literature
about their eigenvalues and eigenvectors. The understandingof the structure was further extended
to XXZ models of higher spin [20], or more generally the model associated to a representation of
the quantum group U (sl ). When q is a Nth root of unity, it was shown in [15, 16, 17, 23, 25, 27]
q 2
that the degeneracy of XXZ models occurs with the extra sl -loop-algebra symmetry induced from
2
the structure of the quantum affine algebra U (sl ). For odd N, as a special member of a one-
q 2
parameter U (sl )-cyclic representation, the spin-(N −1)/2 XXZ chain can be identified with the
q 2
b
superintegrable N-state τ(2)-model [28]. Hence the superintegrable τ(2)-model also carries the sl -
2
loop-algebrasymmetryforthedegeneracy(ofcertainsectors), compatiblewiththeOnsager-algebra
symmetry inherited from the chiral Potts transfer matrix [28]. The interrelation of these two sym-
metries can lead to some useful information about eigenvectors of the superintegrable model (see,
e.g. [2, 3]). Theaim of this paperis to extend the equivalent relationship between the N-state τ(2)-
model and XXZ model with U (sl )-cyclic representation to the whole five-parameter τ(2)-family
q 2
foundin[12]foranyN. ThroughtheYBrelationinthesix-vertexmodel,wefindacharacterization
of quantum group U (sl ) through a one-parameter L-operators of the YB solution (see (2.5), (2.6)
q 2
in the paper). When q is a root of unity, the algebra U (sl ) possesses a three-parameter family
q 2
of cyclic representation. Together with the rescaling of the spectral parameter, XXZ chains with
U (sl )-cyclic representation also carry a five-parameter L-operators. By extending the argument
q 2
in [28], we show the identification of τ(2)-models and the XXZ chains with U (sl )-cyclic represen-
q 2
tation. Note that such a precise connection between τ(2)-family in CPM and that XXZ-family with
symmetricR-matrix seems not to have appeared in theliterature before, to thebestof theauthor’s
knowledge, even in odd N case except a one-parameter family of τ(2)-models in [28]. We hope an
identification of these models will help to provide some useful clues to understand the eigenvectors
of CPM in certain special cases.
This paper is organized as follows. In section 2, we recall the definition of τ(2)-model, quantum
group U (sl ), and the quantum affine algebra U (sl ). In section 3, we first describe the three-
q 2 q 2
parameter family of cyclic representations of U (sl ) for a root of unity q. The case of qN = 1 for
q 2
b
odd N is discussed in subsection 3.1 where the τ(2)-model is identified with the XXZ chain with
cyclic representation of U (sl ), hence the quantum space carries a canonical representation of the
q 2
2
quantumaffine algebra U (sl ). In subsection 3.2, we studythe N-state τ(2)-modelfor an arbitrary
q 2
N. We show that a XXZ chain of U (sl )-cyclic representation with q2N = 1 is equivalent to two
q 2
b
copies of thesame N-state τ(2)-model. Finally we close in section 4 with abrief concluding remark.
Notation: We use standard notations. For a positive integer N greater than one, CN denotes
the vector space of N-cyclic vectors with the canonical base |ni,n ∈ Z (:= Z/NZ). We fix the
N
Nth root of unity ω = e2Nπi, and X,Z, the Weyl CN-operators :
X|ni = |n+1i, Z|ni= ωn|ni (n ∈ Z ),
N
which satisfy XN =ZN =1 and the Weyl relation: XZ = ω−1ZX.
2 N-state τ(2)-model and quantum group U (sl )
q 2
The N-state τ(2)-model [11, 12, 30] (also called the Baxter-Bazhanov-Stroganov model [19]) is the
five-parameter family of L-operators of C2-auxiliary, CN-quantum space with entries expressed
by Weyl operators X,Z:
1−t c X (1 −ω acX)Z
L(t) = b′b b b′b , (2.1)
−t(b1′ − ba′′bcX)Z−1 −tb1′b +ωab′′abcX !
where t is the spectral variable, and a,b,a′,b′,c are non-zero complex parameters. It is known that
the L-operator (2.1) satisfies the YB equation
R(t/t′)(L(t) 1)(1 L(t′)) = (1 L(t′))(L(t) 1)R(t/t′)
aux aux aux aux
O O O O
for the asymmetry six-vertex R-matrix
tω−1 0 0 0
 0 t−1 ω−1 0 
R(t) = . (2.2)
 0 t(ω−1) (t−1)ω 0 
 
 0 0 0 tω−1 
 
 
 
The monodromy matrix, L L (t) where L (t) = L(t) at site ℓ, again satisfies the above YB
ℓ=1 ℓ ℓ
relation, whose ω-twisted trace defines the τ(2)-matrix:
N
L
τ(2)(t) = trC2 Lℓ(ωt), (2.3)
ℓ=1
O
commuting with the spin-shift operator X(:= X ).
ℓ ℓ
ThequantumgroupUq(sl2)istheassociatedQC-algebrageneratedbyK±21,e± withtherelations
1 −1 −1 1
K2K 2 = K 2 K2 = 1 and
K12e±K−21 = q±1e±, [e+,e−] = K −K−1. (2.4)
q−q−1
3
Then the two-by-two matrix with U (sl )-entries defines a one-parameter family of L-operators:
q 2
ρ−1sK−21 −s−1K12 (q−q−1)e−
L(s) = , ρ6= 0 ∈ C, (2.5)
(q−q−1)e+ sK12 −ρs−1K−21 !
which satisfy the YB equation
R (s/s′)(L(s) 1)(1 L(s′))= (1 L(s′))(L(s) 1)R (s/s′). (2.6)
6v 6v
aux aux aux aux
O O O O
for the six-vertex (symmetric) R-matrix [18, 21]:
s−1q−sq−1 0 0 0
 0 s−1−s q−q−1 0 
R (s)= .
6v  0 q−q−1 s−1−s 0 
 
 0 0 0 s−1q−sq−1 
 
 
Indeed,theYBrelation(2.6) forLin(2.5)isthenecessaryandsufficientcondition oftheconstraint
(2.4) in the definition of quantum group U (sl ). Using the local L-operator (2.5), one constructs
q 2
L
the monodromy matrix L L (s) with entries in ( U (sl ))(s),
ℓ=1 ℓ q 2
N L N
A (s) B (s)
L L
L (s)=
ℓ
ℓ=1 CL(s) DL(s) !
O
again satisfying (2.6). The structure of the quantum affine algebra U (sl ) is described by the
q 2
leading and lowest terms of the above entries in the monodromy matrix,
b
A = lim (ρ−1s)−LA (s), A = lim (−s)LA (s),
+ s→∞ L − s→0 L
B± = lims±1→∞(±s)∓(L−1)qB−Lq(−s)1, C± = lims±1→∞(±s)∓(L−1)qC−Lq(−s)1,
D = lim s−LD (s), D = lim (−ρ−1s)LD (s).
+ s→∞ L − s→0 L
One has A = A−1,A = D . Denote T− = B ,S− = B and S+ = C ,T+ = C . The operators
+ − ∓ ± + − + −
k−1 = k = A2 = D2, e = S+, f = S−, e = T−, f = T+
0 1 − + 1 1 0 0
generate the quantum affine algebra U (sl ) with the Hopf-algebra structure
q 2
△(k ) = k ⊗k , b i= 0,1,
i i i
△(e ) = k ⊗e +ρ−1e ⊗k , △(f ) = k ⊗f +ρf ⊗k ,
1 1 1 1 0 1 1 1 1 0
△(e ) = ρ−1k ⊗e +e ⊗k , △(f ) = ρk ⊗f +f ⊗k ,
0 0 0 0 1 0 0 0 0 1
Indeed, the explicit expression of generators of U (sl ) is
q 2
1 1 −1 −1
A− = K2 ⊗···⊗K2, A+ = Kb 2 ⊗···⊗K 2 ,
S± = Li=1K21 ⊗···⊗K12 ⊗e±⊗K−21 ⊗···⊗K−21 ρ∓(L−i),
(2.7)
P i−1 L−i
T± = Li=1|ρ±(i−1){Kz −21 ⊗}···⊗K|−21 ⊗e±⊗{zK12 ⊗·}··⊗K21 .
P i−1 L−i
| {z } | {z }
4
Note that (2.6) is still valid when changing the variable s by αs for a non-zero complex α. Given
a finite-dimensional representation σ : U (sl ) −→ End(Cd), the L-operator of C2-auxiliary and
q 2
Cd-quantum space, L(s) = σ(L(αs)), satisfies the YB relation (2.6). In particular, by setting the
parameter ρ = 1 in (2.5), with α = qd−22 and σ the spin-d−1 (highest-weight) representation of
2
U (sl ) on Cd = ⊕d−1Cek:
q 2 k=0
K21(ek) = qd−12−2kek, e+(ek) = [k]ek−1, e−(ek)= [d−1−k]ek+1,
where [n] = qn−q−n and e+(e0) = e−(ed−1) = 0, one obtains the well-known L-operator of XXZ
q−q−1
chain of spin-d−1 (see, e.g. [20, 26, 27] and references therein).
2
3 The equivalence of τ(2)-models and XXZ chains with cyclic rep-
resentation of U (sl )
q 2
Inthissection, weconsiderthecasewhenq is arootofunity. For aNthrootofunityq,thereexists
a three-parameter family of cyclic representation σφ,φ′,ε of Uq(sl2), labeled by non-zero complex
numbers φ,φ′ and ε, which acts on cyclic CN-vectors by
K21|ni= q−n+φ′2−φ|ni,
e+|ni = qεqφ+n−q−φ−n|n−1i, e−|ni= q−εqφ′−n−q−φ′+n|n+1i, (3.1)
q−q−1 q−q−1
(see, e.g. [13, 14]). The L-operator,
L(s) = σφ,φ′,εL(s) (3.2)
gives rise to the transfer matrix of the XXZ chain with the Uq(sl2)-cyclic representation σφ,φ′,ε,
L
T(s)= (⊗σφ,φ′,ε)(trC2 Lℓ(s)), (3.3)
ℓ=1
O
1 1 1
which commutes with K2(:= ⊗ℓKℓ2, the product of local K2-operators ).
3.1 The identification of the N-state τ(2)-model and the XXZ chain of U (sl )-
q 2
cyclic representation with qN = 1 for odd N
In the subsection, we consider the case N odd, and write N = 2M +1. Let q be the primitive Nth
root-of-unity with q−2 = ω. Then one can express the cyclic representation (3.1) in terms of the
Weyl operators X,Z:
K21 = qφ′2−φZ12, e+ =qε(qφ+1Z−12q−−qq−−φ1−1Z12)X−1, e− = q−ε(qφ′+1Z21−q−q−q−φ′1−1Z−21)X,
hence follow the expression:
K−1 = qφ−φ′Z−1,
K−21e+ = −q−φ2−φ′+ε−1(1−q2φq+−2qZ−−11)X−1, K−21e− = qφ+2φ′−ε+1(1−q−q2−φ′q−−21Z−1)X. (3.4)
5
Using the Fourier basis |ni = 1 Z ω−nj|ji, one may convert the Weyl operator (Z−1,X) to
N j∈ N
(X,Z):
P
c
Z−1|0i,···,Z−1|N −1i = |0i,···,|N −1i X, (3.5)
X X Z
(cid:18) (cid:19) (cid:18) (cid:19)
hence represent K−1,K−21e± icn (3.4) by d c d
K−1 = qφ−φ′X, K−21e+ = −q−φ2−φ′+ε−1(1−q2qφ−+q2−X1)Z−1, K−21e− = qφ+2φ′−ε+1(1−q−q−2φq′−−12X)Z.
By the above expression and introducing the spectral variable t = λ−1s2 for a non-zero complex λ,
−1
the modified L-operator of (3.2), −sK 2 L(s), is gauge equivalent to
1−tλρ−1qφ−φ′X q−ε(1−q−2φ′−2X)Z
,
−tλqε(1−q2φ+2X)Z−1 −tλ+ρqφ−φ′X !
whichisthesameastheL-operator(2.1)ofτ(2)-modelbythefollowingidentification ofparameters:
a = λ−1ρq−φ−φ′−ε, a′ = ρqφ+φ′+ε+2, b= qε, b′ = λ−1q−ε, c= ρ−1qφ−φ′,
(3.6)
equivalently qε = b, q2φ = ωa′c, q2φ′ = b′, ρ2 = ωaa′, λ= 1 .
b ac bb′ bb′
This implies the transfer matrix (3.3) of XXZ chain is equivalent to the τ(2)-transfer matrix (2.3).
Note that the product of local operator K−1’s of XXZ chain is now corresponding to a scalar
multiple of the spin-shift operator X, which commutes with the τ(2)-matrix. The representations
with ε = 0,φ = φ′ in (3.1) form the one-parameter cyclic representation of U (sl ) discussed in
q 2
[28] section 4. In particular, the case φ= φ′ = M is the spin-N−1 highest-weight representation of
2
U (sl ).
q 2
It is known that the degeneracy of τ(2)-model in CPM occurs in the alternating superintegrable
case (see, [30] section 4.3), where the τ(2)-model is characterized by the L-operator (2.1) with the
parameter
a = ωmb′, a′ = ωm′b, c =ωn, (m,m′,n ∈ Z),
or equivalently, with the representation parameters φ,φ′,ε and ρ,λ in (3.6) given by
φ= −(m′+n+1), φ′ = m+n, qε = b, ρ= q−(1+m+m′), λ= 1 . (3.7)
bb′
In this case, there exist the normalized Nth power of S±,T± in (2.7), S±(N) = S±N,T±(N) = S±N.
[N]! [N]!
Asφin(3.7) isan integer, by thealgebraic-Bethe-ansatz technique, onecan show thedegeneracy of
thealternating superintegrableτ(2)-model(forcertain sectors)possessesthesymmetryalgebragen-
erated by S±(N),T±(N), which can be identified with the sl -loop algebra under certain constraints
2
between the integers m,m′ and n, as in the discussion of the homogeneous superintegrable case
in [15, 23, 27]. However, the precise structure about the symmetry algebra for arbitrary m,m′,n
remains unknown to be identified.
6
3.2 XXZ chain of U (sl )-cyclic representation with q2N = 1 as two copies of the
q 2
N-state τ(2)-model for an arbitrary N
Byextendingtheargumentof theprevioussubsection, wenow identify theN-state τ(2)-modelwith
a XXZ chain for an arbitrary N. Let q be a primitive (2N)-th root of unity with q−2 = ω, hence
qN = −1. Consider the three parameter family of Uq(sl2)-cyclic representations σφ,φ′,ε in (3.1) on
the vector space V of 2N-cyclic vectors:
V := {C|ni′|n ∈ Z }.
2N
M
Denote the vectors in V:
|ni:= |ni′+|n+Ni′, (n ∈Z ),
2N
|ni := |ni′−|n+Ni′, (0 ≤ n ≤ N −1).
−
Then |ni = |n+Ni. We define |ni for n ∈ Z by the N-periodic condition: |n+Ni = |ni .
− − −
Hence we can identify {|ni} or {|ni } with the canonical basis of the vector space of N-cyclic
−
vectors, and one has the decomposition of V:
V = V ⊕V , V := C|ni, V := C|ni .
+ − + − −
Z Z
n∈XN n∈XN
The spin-shift operator |ni′ 7→ |n+1i′ of V is decomposed as the sum of X on V and X on V ,
+ −
where
e
|n+1i n 6≡ N −1 (mod N),
X|ni = |n+1i (n ∈ Z ), X|ni = −
N −
( −|n+1i− n ≡ N −1 (mod N).
e
Together with the operator Z: Z|ni= ωn|ni, Z|ni = ωn|ni , one finds (X,Z) , (X,Z) are Weyl
− −
operators of V or V respectively with XN = ZN = 1,XN = −1. Under the cyclic representation
+ −
e
σφ,φ′,ε (3.1) of Uq(sl2), the subspaces V+,V− are interchanged under the generators K12,e±, with
e
the expression:
K21|ni = q−n+φ′2−φ|ni−, K21|ni− = q−n+φ′2−φ|ni (0 ≤ n ≤ N −1),
e+|ni= qεqφ+n−q−φ−n|n−1i (1 ≤ n ≤ N),
q−q−1 −
e−|ni= q−εqφ′−n−q−φ′+n|n+1i (−1 ≤ n ≤ N −2), (3.8)
q−q−1 −
e+|ni = qεqφ+n−q−φ−n|n−1i, e−|ni = q−εqφ′−n−q−φ′+n|n+1i (0 ≤ n ≤ N −1).
− q−q−1 − q−q−1
Using the above formulas, one finds that K−1 and K−21e± are operators of V+ and V−, which can
beexpressed by X,Z, and X,Z respectively such that the relation (3.4) (using q) holds for V , and
+
also valid for V when replacing X by X. By using the V -basis, |nii := qn|ni for 0≤ n ≤ N−1,
− − −
the Weyl pair (X,Z) is coneverted to (q−1X,Z). One can express the operators K−21e± by
e
K−21e+|niie= −q−φ2−φ′+ε(1−q2φq+−2qZ−−11)X−1|nii, K−21e−|nii = qφ+2φ′−ε(1−q−q2φ−′q−−21Z−1)X|nii.
7
UsingtheFourier transform(3.5) ofthebasis{|ni}and{|nii}, bythesameargumentinsubsection
−1
3.1, one can show that −sK 2 L(s) for the L-operator (3.2) is gauge equivalent to two copies of the
sameL-operator (2.1) ofτ(2)-modelwiththeparameter given by(3.6) (replacingqbyq). Therefore
thetransfer matrix (3.3) of XXZ chain is equivalent to the sumof two copies of τ(2)-transfer matrix
(2.3). Note that neither one of the two copies of τ(2)-model inherits the U (sl ) structure of XXZ
q 2
chain, in which one copy is sent to another by (3.8). The symmetry structure of τ(2)-model related
to U (sl ) is different from that in subsection 3.1 for the odd N case. Nevertheless, the τ(2)-model
q 2
withcertain constraintsontheparameteragain possessesthesl -loop-algebra symmetrythatarises
2
from the U (sl )-structure of the corresponding XXZ-model. In particular, in the homogeneous
q 2
superintegrable case, the Onsager-algebra symmetry of the degenerate τ(2)-eigenspace is extended
b
to the symmetry of sl -loop algebra (in certain sectors). The relationship of these two symmetries
2
will hopefully, though not immediately apparent, lead to a solution of the eigenvector problem in
the superintegrable CPM (for an arbitrary N) along the line in [2, 3].
4 Concluding Remarks
By studying the general solution of YB equation of the six-vertex model, we find a one-parameter
L-operators associated to the quantum group U (sl ), which carries a three-parameter family of
q 2
cyclicrepresentation whenq isarootofunity. WehaveshowedthattheXXZchainofU (sl )-cyclic
q 2
representation with q2N = 1 is equivalent to the sum of two copies of a N-state τ(2)-model. In
particular, the Onsager-algebra symmetry of the homogeneous superintegrable CPM is enlarged to
the sl -loop-algebra symmetry induced from the corresponding XXZ-model. When N is odd and q
2
is a primitive Nth root of unity, the N-state τ(2)-model can also be identified with the XXZ chain
of a cyclic representation of U (sl ), hence the quantum space carries the structure of quantum
q 2
affine algebra U (sl ). As a special case, the homogeneous superintegrable τ(2)-model in CPM is
q 2
equivalent to the spin-(N−1)/2 XXZ chain with the canonical U (sl )-structure [28], by which the
q 2
b
sl -loop-algebra symmetry of the superintegrable τ(2)-model was derived in [23, 27]. In the recent
2
b
study of eigenvalues of the chiral Potts model with alternating rapidities [30], the τ(2)-degeneracy
is also found in the alternating superintegrable case. By identifying these τ(2)-models with the
XXZ chains via (3.7), we find the sl -loop-algebra symmetry for the degenerate τ(2)-eigenspace
2
in certain cases. Similar to the homogeneous CPM case [24], some twisted version of Onsager-
algebra symmetry for the τ(2)-model could also possibly exist in the theory of a general alternating
superintegrable chiral Potts model. However, the explicit nature remains hard to know precisely.
The exact relationship and a further symmetry study of general cases are now under consideration.
Acknowledgements
The author wishes to thank Professor S. Kobayashi for the invitation of visiting U. C. Berkeley in
the spring of 2008, where part of this work was carried out. This work is supported in part by
National Science Council of Taiwan under Grant No NSC 96-2115-M-001-004.
8
References
[1] G. Albertini, B. M. McCoy, and J. H. H. Perk, Eigenvalue spectrum of the superintegrable
chiral Potts model, In Integrable system in quantum field theory and statistical mechanics,
Adv. Stud. Pure Math., 19, Kinokuniya Academic, Academic Press, Boston, MA (1989) 1–55.
[2] H.Au-YangandJ.H.H.Perk,Eigenvectors thesuperintegrablemodelI:sl generators, J.Phys.
2
A: Math. Theor. 41 (2008) 275201; arXiv: 0710.5257.
[3] H.Au-YangandJ.H.H.Perk,EigenvectorsinthesuperintegrablemodelII:groundstatesector,
arXiv: 0803.3029,
[4] R. J. Baxter, Calculation of the eigenvalues of the transfer matrix of the chiral Potts model,
Proc. Fourth Asia-Pacific Physics Conference (Seoul, Korea, 1990) Vol 1, World-Scientific,
Singapore (1991) 42–58.
[5] R. J. Baxter, Chiral Potts model: eigenvalues of the transfer matrix, Phys. Lett. A 146 (1990)
110–114.
[6] R. J. Baxter, Chiral Potts model with skewed boundary conditions, J. Stat. Phys. 73 (1993)
461–495.
[7] R. J. Baxter, Interfacial tension of the chiral Potts model, J. Phys. A: Math. Gen. 27 (1994)
1837–1849.
[8] R. J. Baxter, The six and eight-vertex models revisited, J. Stat. Phys. 114 (2004) 43–66;
cond-mat/0403138.
[9] R. J. Baxter, Transfer matrix functional relation for the generalized τ (t ) model, J. Stat.
2 q
Phys. 117 (2004) 1–25; cond-mat/0409493.
[10] R. J. Baxter, Derivation of the order parameter of the chiral Potts model, Phys.Rev.Lett. 94
(2005) 130602; cond-mat/0501227.
[11] R. J. Baxter, V.V. Bazhanov and J.H.H. Perk, Functional relations for transfer matrices of
the chiral Potts model, Int. J. Mod. Phys. B 4 (1990) 803–870.
[12] V.V. Bazhanov and Yu.G. Stroganov, Chiral Potts model as a descendant of the six-vertex
model, J. Stat. Phys. 59 (1990) 799–817.
[13] E.Date, M.Jimbo,K.MikiandT.Miwa, Cyclicrepresentations ofU (sl(n+1, C))atqN = 1,
q
Publ. RIMS, Kyoto Univ. 27 (1991) 347–366.
[14] C.DeConciniandV.G.Kac, Representations ofquantumgroupsatroots ofunity, inOperator
Algebra, Unitary Representations, Enveloping Algebras, and Invariant Theory, Paris (1989)
Progress in Mathematics 92, Birkh¨auser, Boston, Massachusstts (1990) 471– 506.
9
[15] T. Deguchi, K. Fabricius and B. M. McCoy, The sl loop algebra symmetry for the six-vertex
2
model at roots of unity, J. Stat. Phys. 102 (2001) 701–736; cond-mat/9912141.
[16] T. Deguchi: Regular XXZ Bethe states at roots of unity- as highest weight vectors of the sl
2
loop algebra at roots of unity, cond-mat/0503564 v3.
[17] K. Fabricius and B. M. McCoy, Evaluation parameters and Bethe roots for the six vertex
model at roots of unity, Progress in Mathematical Physics Vol 23, eds. M. Kashiwara and T.
Miwa, Birkh¨auser Boston (2002), 119–144; cond-mat/0108057.
[18] L. D. Faddeev, How algebraic Bethe Ansatz works for integrable models, eds. A. Connes, K.
Gawedzki and J. Zinn-Justin, Quantum symmetries/ Symmetries quantiques, Proceedings of
the Les Houches summer school, Session LXIV, Les Houches, France, August 1-September 8,
1995, North-Holland, Amsterdam (1998), 149–219.
[19] G. von Gehlen, N. Iorgov, S. Pakuliak and V. Shadura: Baxter-Bazhanov- Stroganov model:
Separation of variables and Baxter equation, J. Phys. A: Math. Gen. 39 (2006) 7257–7282;
nlin.SI/0603028.
[20] A, N. Kirillov and N. Yu. Reshetikhin, Exact solution of theintegrable XXZ Heisenbergmodel
with arbitrary spin: I. The ground state and the excitation spectrum, J. Phys. A: Math. Gen.
20 (1987) 1565 – 1595.
[21] P. P.Kulish, N.Yu.Reshetikhin and E.K.Sklyanin, YangBaxter equation andrepresentation
theory, Lett. Math. Phys. 5 (1981) 393–403.
[22] B. M. McCoy and S. S. Roan, Excitation spectrum and phase structure of the chiral Potts
model. Phys. Lett. A 150 (1990) 347–354.
[23] A. Nishino and T. Deguchi, The L(sl ) symmetry of the Bazhanov-Stroganov model associ-
2
ated with the superintegrable chiral Potts model, Phys. Lett. A 356 (2006) 366–370 ; cond-
mat/0605551.
[24] S. S. Roan, The Onsager algebra symmetry of τ(j)-matrices in the superintegrable chiral Potts
model, J. Stat. Mech. (2005) P09007; cond-mat/0505698.
[25] S. S. Roan, Bethe ansatz and symmetry in superintegrable chiral Potts model and root-of-
unity six-vertex model, in Nankai Tracts in Mathematics Vol. 10, Differential Geometry and
Physics, eds. Mo-Lin Go and Weiping Zhang, World Scientific, Singapore (2006), 399-409;
cond-mat/0511543.
[26] S. S. Roan, The Q-operator for root-of-unity symmetry in six vertex model, J. Phys. A: Math.
Gen. 39 (2006) 12303-12325; cond-mat/0602375.
[27] S. S. Roan, Fusion operators in the generalized τ(2)-model and root-of-unity symmetry of
the XXZ spin chain of higher spin, J. Phys. A: Math. Theor. 40 (2007) 1481-1511; cond-
mat/0607258.
10