- Email: [email protected]
Three-state chiral Potts models in two dimensions: integrability and symmetry
Physica A 177 (1991) 114-122 North-Holland
Three-state chiral Ports models in two dimensions: integrability and symmetry J.C. Angl6s d'Auriac a, J.M. Maillard b and F.Y. Wu c a CRTBT, 25 avenue des Martyrs, BP 166X, 38084 Grenoble Cedex, France b Laboratoire de Physique Thborique et Hautes Energies, 7bur 16. I ~rbtage, 4 place Jussieu. 75252 Paris Cedex, France c Department of Physics, Northeastern University, Boston, MA 02115, USA
Symmetry and integrability relations for the three-state chiral Ports model on two-dimensional lattices are reviewed. Our detailed and systematic analysis leads to new integrability varieties for specific chiral Potts models including new integrablepoints for the standard antiferromagneticPons model. We also describe a three-state vertex-model representation of the chiral Potts model, and write down the associated fundamental invariants for three-coordinated lattices.
I. Introduction In 1981 Yeomans and Fisher [ 1 ], Ostlund [2], and Huse [3] introduced an axial three-state chiral Potts model as a modelling of c o m m e n s u r a t e - i n c o m m e n s u r a t e transitions ~. The chiral Potts model generalizes the standard Potts model [5] to include interactions with a chiral symmetry, namely, those with strengths depending on the relative positionings of the interacting spins. Unlike the standard Ports model, which is exactly solvable at the critical point, exact results on the chiral Ports model are scarce and hard to obtain. However, a major breakthrough occurred in 1987 when an exactly integrable case of the chiral Potts model was discovered [ 6 ]. This development, which yields the first solvable case with a greater-than-one genus parametrization [6,7], has since led to renewed intense interest on the chiral Potts model (for reviews see, e.g., refs. [ 8 , 9 ] ) . Besides the consideration of integrability, the chiral Ports model also underlines the role played by certain symmetry structure [ 10,11 ] of lattice systems. The exactly soluble chiral Potts models are restricted to special varieties of the parameter space. While these varieties are given in a parametric form in terms of the rapidities of the s u r r o u n d i n g lattice [ 12,13 ], their explicit expressions have also been written down [ 14]. It has been suggested that in the case of the axial three-state chiral Potts model the integrability variety gives rise to the wetting line [ 8 ]. In view of such ~ For a review of commensurate-incommensurate transitions, see ref. [4]. 0378-4371/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
J.C. Anglks d'Auriac et al. / Three-state chiral Potts model in 2D
115
physical significance, it appears useful to re-examine these integrability and symmetry relations. In this paper we present such a study. As we shall see, our detailed and systematic analysis leads to new integrability varieties which had not previously been noted. We also give a vertex-model formulation of the three-state chiral Potts model, and write down the associated fundamental invariants for three-coordinated lattices.
2. Three-state chiral Potts model
Consider an orientedlattice whose lattice sites are occupied by N-state spins and whose lattice edges are directed. Denote the spin state of the ith site by ai= 0, 1, 2, ..., N - 1. For each lattice edge directed from i and j we associate a Boltzmann factor w,j(a~-aj). The chiral Potts model is characterized by wo(a)# wo(-a), wo(a)= w,j(a+N). The partition function is
z = E I-I w,j(~,-aj),
(1)
where the product is over all lattice edges and the summation over all spin states. Clearly, differently oriented lattices generally lead to different partition functions. We confine our considerations to three-state models and regular lattices, and begin by recalling some known results for the checkerboard lattice. Let the edges of a unit cell, say, the black square, of the checkerboard lattice be directed as shown in fig. 1. We write for brevity the four Boltzmann weights w~ for the four edges of the black square as
ai=wi(O), bi=wi(1)=wi(-2), G=wi(2)=wi(-l),
i=1,2,3,4,
(2) which we assume to be positive. Clearly, reversing the arrows of any, say, the ith, edge corresponds to the interchange of b~ and c,. We also see that other regular lattices are recovered by taking
W4 ~
W2
Fig.1.AunitcellofthecheckerboardchiralPottslattice.
116
J.C. Anglks d'Auriac el al. / Three-statechiral Potts modelin 2D ai = a ,
bi = b ,
ci = c ,
a 4 = 1,
b4 = Ca = 0 ,
isotropic square l a t t i c e , triangular lattice,
a4 = b4 = C4 = 1,
(3)
h o n e y c o m b lattice.
The latter two cases c o r r e s p o n d to the contraction and deletion o f the i = 4 edges, a situation shown in fig. 2. We r e m a r k that in physical applications [ 1 ] the Boltzmann factor is often written as
(4)
wi(a) =exp{K, cos[ ~ x ( a + 3 , ) ]] .
The axial chiral Potts model o f refs. [ 1-3 ] corresponds to taking 3~ = 33 = 0, 32 = 34 = 3. The integrability condition for the checkerboard chiral model o f fig. 1 has been given in a p a r a m e t r i c form [ 12,13 ]. Explicitly, the c o n d i t i o n [ 14 ] reads 3 (QI P2P3P4 + Q2PI P3P4 + Q3PI P2P4 + Q4PI P2P3 ) -
-
(5)
(191Q2Q3Q4 + P2Q10304 + P3Q, Q2Q4 + P4Q, Q2Q3) = 0 ,
where Pi = f - 3 h , ,
Q, = f , - 2g, +
f,=aibic,(a3, +b3, +c3) ,
3hi,
(6)
g,=a3b3 +b3c3 +c3a3,
9
~
9
hi=a?b,c?.
We observe that the integrability condition (5) possesses a n S 4 symmetry. Furthermore, the reversal of the orientation of the ith edges, which corresponds to the interchange of hi and c, for any i = 1, 2, 3, 4, does not alter (5). This is a manifestation of an enhanced symmetry at integrability. For the isotropic square lattice model, namely, ai=a, bi=b, c,=c, condition 5)
A
>/
t ~r I
W3
>/ W2
Fig. 2. Triangular (left) and honeycomb (right) lattices derived from the checkerboard lattice by contracting and deleting the w4edges.
J.C. Angl~s d'Auriac et al. / Three-state chiral Potts model in 2D
117
becomes 4 P Q ( 3 P Z - Q 2) = 0 , where P, Q are given in terms off, g, h and a, b, c as in (6). For positive Boltzmann weights we have ~z p > 0, P > Q, and hence v/3 P - Q > 0. It follows that the integrability condition splits into two branches: Q = 0 and x/~ P + Q = 0. Explicitly, other than the trivial condition abc= 0, they are
abc( a3 + b3-I-c 3 ) - 2 ( a3b3 + b3c3 + c3a 3) + 3a2b2c2=O ,
(7a)
( ~f3-t- 1 ) ( a3 + b3 + c 3 ) - 3 (%//3 - 1 ) a b c - 2 ( a2bZc - ' + b2c2a -~ + c2a2b -t ) = 0 . (7b) For the standard Potts model ( b = c = 1 ) b o t h (7a) and (7b) can be factorized resulting in, respectively, ( a - 1) 2 ( a Z - 2 a - 2 ) = 0 ,
(8a)
( a - 1 )2[ ( v / ~ + 1 )a2 + 2 (V/3 - 1 ) a - 2 ] = 0 .
(8b)
In the physical domain a # 1, a > 0, (8a) yields the known ferromagnetic critical point [15] a = l + v / 3 , while (8b) leads to a new integrable point a = w / 3 + 3 v / 2 / 2 , v / 6 / 2 - 2 = 0.628626... for the antiferromagnetic Potts model. For the triangular and honeycomb lattices both P4 and Q4 vanish when using (3). We can however divide (5) throughout by Q4 and take the limit of r=P4/Q4 carefully #3. For isotropic lattices, this leads to the following integrability conditions: 3P 3 + 9 P 2 Q - 3PQ z _ Q 3 = 0 ,
triangular,
( 9a )
3p3-9p2Q-3PQZ+Q3=O,
honeycomb.
(9b)
Note that (9a) and (9b) are related by a negation o f Q. For positive a, b, c we must have, as before, s - Q / P < 1 so that (9a) and (9b) are realized by the branches s = 4 s i n ( n / 1 8 ) - 1 = - 0 . 3 0 5 4 0 7 3 .... s = - 4 c o s ( n / 9 ) - 1 = -4.758710... for the triangular lattice, and s=0.4758710.., and s = 1 - 4 c o s ( 2 n / 9 ) = -2.064177... for the honeycomb lattice. For the standard Ports model (b = c = 1 ), (9a) and (9b) factorize to 8 ( a - 1) 6 ( a 3 + 6 a 2 + 3 a -
1) ( a 3 - 3 a - 1 ) = 0 ,
-8(a-l)6(a3+3a2-1)(a3-3a2-6a-l)=O,
triangular,
(10a)
honeycomb.
(10b)
Setting the last factors in (10a) and (10b) equal to zero leads to the known critical point o f the ferromagnetic Potts model [ 15 ], namely, a = a, - 2 c o s ( n / 9 ) = 1.879385 ....
triangular,
a = a 2 = - l + 2 w / 3 c o s ( n / 1 8 ) = 4 . 4 1 1 4 7 4 ....
honeycomb.
~2Using the inequality x3+y3+z3-3xyz=(x+y+z)[(x-y)2+(y-z)2+(z-x)2]/2>O for x+y+z>O. ~3 In the triangular (honeycomb) case we have r= 1 (r = - 1) obtained by taking the limit of a4--, 1 after first setting b4=c4=0 ( = 1).
118
J. C. Anglbs d'A uriac et al. / Three-state chiral Potts model in 2D
Setting the second-to-last factors equal to zero yields the following integrable points for the antiferromagnetic model: a=l/a2=2x/3sin(2n/9)-2=0.2266816 a = 1/at = 2 c o s ( 2 n / 9 ) - 1 =0.5320888 ....
....
triangular, honeycomb.
Indeed, the triangular variety a 3 + 6 a 2 + 3 a 1 = 0 and its solution at a = 0 . 2 2 6 6 8 1 6 have previously been suggested by Martin and Maillard [ 16] on the basis of numerical analysis and automorphic group structure considerations [ 17 ]. For the three-state antiferromagnetic triangular Ports model, the critical point has been estimated at a = 0.204 [ 18 ]. This supports the conjecture that the free energy is, in fact, regular at the integrable point a = 0.2266816. The integrability variety a 3 + 3a 2_ 1 = 0 for the honeycomb lattice is new and has not been previously noted.
3. Symmetry properties The three-state chiral Potts model possesses a n u m b e r of symmetry properties. First of all, the partition function possesses the obvious symmetry S: bi~-~ci,
all edges i,
( 11 )
corresponding to a reversal of the orientation of all lattice edges. In addition, the following duality relation [ 19 ] exists relating {at, b, c,} of a given edge to those of the dual edge: ( d i = ( a i +b, + c , ) / x f 3 , D: ~ i = ( a , + o ~ b i + o 9 2 c i ) / x / 3 , (O, = ( ai + o92b, + ogci) / x / 3 .
(12)
Here, ~o= exp (2hi/3 ), and the orientation of the dual edge is obtained by rotating the original lattice edge 90 ° counterclockwise (or clockwise) [ 19]. Clearly, applying ( 12 ) twice leads to {di,/~i, c]}= { a , G, bi }. Furthermore, for any lattice whose edge orientation is decomposable into directed cycles, we have the (complex) symmetry I2: w ( a ) --.~o°w(a) for any a. This is so since the product of the Boltzmann factors of each cycle remains unchanged. For three-state models and taking a = 1, 2, this leads to the symmetry g2:
{a, b, c}-~{a, ogb, ~o2c} , {a,b,c}__.{a, ogZb,~oc} "
(13)
Examples of lattices possessing this decomposition property include the checkerboard lattice of fig. 1 as well as the triangular lattice in fig. 2, assuming periodic boundary
J. C Anglbs d'Auriac et al. / Three-state chiral Potts model in 2D
119
conditions (so that cycles can loop around the lattice). Notice that the honeycomb lattice in fig. 2 cannot be so decomposed. Other symmetry properties tend to be lattice and orientation dependent. These are discussed in the following for isotropic lattices: S q u a r e lattice
For the square lattice with orientation shown in fig. 1, we have the symmetries S and ft. In addition, the partition function is invariant under the operation T: a ~ b ~ c
~a .
(14)
This fact can be seen by constructing diagonals in the NW-SE direction and relabelling all spin states on the nth diagonal (counting toward the NE direction) by a ~ a - n. This brings about the change w(tr)--, w ( a + 1 ) and hence the symmetry T. The isotropic square lattice is self-dual and described by the duality relation D. Particularly, D possesses a fixed point b / a = c / a = ( x / 3 - 1 )/2. T r i a n g u l a r lattice
We consider a c o h e r e n t orientation of the triangular lattice shown in fig. 3. Here, an orientation is coherent if arrows around each elementary lattice face point all in the same (cw or ccw) direction. Since the coherently oriented lattice is decomposable into directed cycles of elementary faces (triangles), it possesses the complex symmetry 12 regardless of boundary conditions. The coherent model also possesses the symmetries S and T; the latter fact follows from a relabelling of spin states a--, or- n, n = 1, 2, 3, for the nth sublattice. Finally, since the coherent lattice is obtainable from the lattice in fig. 2 by reversing the arrows on all w~ edges (which results in the interchange b~,--,c~ and leaves ( 5 ) invariant), its integrability condition is the same as that of the triangular lattice in fig. 2, namely, condition (9a).
Fig. 3. Coherent chiral Potts triangular (left) and honeycomb (right) lattices. The two lattices are mutually dual. Orientations of honeycomb edges are obtained by rotating those of the dual triangular edges 90 ° counterclockwise.
120
,L C. AnglOs d'Auriac et al. / Three-state chiral Potts m o d e l in 2D
Honeycomb lattice The honeycomb lattice with the orientation shown in fig. 3 possesses the symmetries S, T a n d £2, while the one shown in fig. 2 possesses the symmetries S and 7, but not £2. In both cases the integrability condition is (9b).
4. Vertex-model formulation The three-state chiral Potts model on any two-dimensional lattice can be formulated as a three-state vertex model on the dual lattice. Starting from a given Ports spin configuration, we draw bonds on the dual lattice as follows: For each edge directed from site i to site j, we draw on its dual edge a dotted line if a,=aj, a heavy line if ~r,=(Tj-I- 1 ( m o d 3), and a thin line if ai=~rj- 1 ( m o d 3). These situations are shown in fig. 4. This procedure leads to a three-state vertex configuration on the dual lattice. Conversely, it is not difficult to see that each allowed vertex configuration on the dual leads to three distinct spin configurations (related by spin-state permutations). Thus, the mapping between the three-state chiral Potts model and the vertex model is 3 to 1. Because of the particular rule that the bonds are drawn on the dual, only certain specific vertex configurations can occur. In the case of the coherent triangular chiral Potts model of fig. 3, for example, the mapping results in a nine-vertex model on the honeycomb lattice with vertex weights shown in fig. 5. In this case the vertex weights are isotropic, and therefore can be written as ~o0~, i+j+k=3, where i, j and k are, respectively, the numbers of dotted, heavy, and thin lines. Explicitly, the vertex weights are o ) 3 0 0 = a 3/2 ,
o ) 0 3 0 = b 3/2 ,
09111 = ( a b c )
0 ) 0 0 3 = c 3/2 ,
3/2 .
(15)
The three-state vertex model has been a topic of increasing recent interest. The partition function o f the three-state vertex model is known to be invariant under an O (3) transformation [20 ] ; explicit expressions of the fundamental algebraic invariants of the O ( 3 ) transformation have also been obtained for three-coordinated lattices [21 ]. Applying the latter findings to the model (15), one finds [22] that the quantities f, g, and h in (6) are indeed intersections of the invariants of O ( 3 ) in the subspace (15). These quantities are therefore the fundamental invariants, the polynomial building blocks of all algebraic invariants, of the three-state chiral Potts model. This finding is consistent with the assumption that the exact criticality of the coherent
3
3
2
1
1
2
0 Fig. 4. The three edge states of a three-state vertex model on the dual and their associated spin state configurations. Black circles denote sites of the original lattice and open circles those of the dual lattice.
J.C. Anglbs d'A uriac et al. / Three-state chiral Potts model in 2D
121
k) ~300
~030
~003
~111
Cdlll
Cdlll
Cdlll
Cdll 1
Cdlll
Fig. 5. Vertex configurations of a three-state nine-vertex model on the honeycomb lattice derived from the coherent triangular chiral Ports model of fig. 3. t r i a n g u l a r c h i r a l P o t t s m o d e l is g i v e n i n t e r m s o f f g, a n d h, w h i c h are f u n c t i o n s o f a, b, a n d c as i n d i c a t e d in ( 6 ) .
Acknowledgements W e w o u l d like t o t h a n k P r o f e s s o r R. R a m m a l for c o m m e n t s a n d s t i m u l a t i n g c o n v e r s a t i o n s , a n d D r . L.H. G w a f o r h e l p f u l d i s c u s s i o n s o n a s p e c t s o f t h e v e r t e x - m o d e l f o r m u l a t i o n . O n e o f us ( F Y W )
is s u p p o r t e d in p a r t b y t h e N a t i o n a l S c i e n c e F o u n -
dation Grant DMR-9015489.
References [ 1 ] J.M. Yeomans and M.E. Fisher, J. Phys. C 14 ( 1981 ) L835. [2] S. Ostlund, Phys. Rev. B 24 ( 1981 ) 398. [3] D.A. Huse, Phys. Rev. B 24 ( 1981 ) 5180. [4] M. den Nijs, in: Phase Transitions and Critical Phenomena, vol. 12, C. Domb and J.L. Lebowitz, eds. (Academic Press, London, 1988). [5] R.B. Potts, Proc. Cambridge Philos. Soc. 48 (1952) 106. [ 6 ] H. Au-Yang, B.M. McCoy, J.H.H. Perk, S. Tang and M.-L. Yan, Phys. Lett. A 123 ( 1987 ) 219. [ 7 ] B.M. McCoy, J.H.H. Perk, S. Tang and C.H. Sah, Phys. Lett. A 125 ( 1987 ) 9. [ 8 ] H. Au-Yang and J.H.H. Perk, Adv. Stud. Pure Math. 19 ( 1989 ) 57. [ 9 ] B.M. McCoy, The chiral Potts model: from physics to mathematics and back, in: Lecture Notes in Mathematics (Springer, Berlin, 1991 ). [ 10] J.M. Maillard, J. Math. Phys. 27 (1986) 2776. [ 11 ] J.M. Maillard, in: Progress in Statistical Mechanics, vol. 3, Chin-Kun Hu, ed. (World Scientific, Singapore, 1988). [ 12 ] R.J. Baxter, J.H.H. Perk and H. Au-Yang, Phys. Len. A 128 ( 1988 ) 138. [ 13] R.J. Baxter, J. Stat. Phys. 52 (1988) 639.
122
J.C. Angl~s d'A uriac et al. / Three-state chiral Potts model in 2D
[ 14 ] J. Avan, J.M. Maillard, M. Talon and C. Viallet, Int. J. Mod. Phys. B 4 (1990) 1743. [ 15 ] A. Hintermann, H. Kunz and F.Y. Wu, J. Star. Phys. 19 ( 1982 ) 623. [16] P. Martin and J.M. Maillard, J. Phys. A 19 (1986) L547. [ 17 ] J.M. Maillard and R. Rammal, J. Phys. A 16 ( 1983 ) 353. [ 18 ] I.G. Enting and F.Y. Wu, J. Stat. Phys. 28 ( t 982 ) 351. [ 19 ] F.Y. Wu and Y.K. Wang, J. Math. Phys. 17 ( 1976 ) 439. [20] F.J. Wegner, Physica 68 (1973) 570. [21 ] L.H. Gwa and F.Y. Wu, preprinl. [22] L.H. Gwa and F.Y. Wu, unpublished.
Three-state chiral Ports models in two dimensions: integrability and symmetry J.C. Angl6s d'Auriac a, J.M. Maillard b and F.Y. Wu c a CRTBT, 25 avenue des Martyrs, BP 166X, 38084 Grenoble Cedex, France b Laboratoire de Physique Thborique et Hautes Energies, 7bur 16. I ~rbtage, 4 place Jussieu. 75252 Paris Cedex, France c Department of Physics, Northeastern University, Boston, MA 02115, USA
Symmetry and integrability relations for the three-state chiral Ports model on two-dimensional lattices are reviewed. Our detailed and systematic analysis leads to new integrability varieties for specific chiral Potts models including new integrablepoints for the standard antiferromagneticPons model. We also describe a three-state vertex-model representation of the chiral Potts model, and write down the associated fundamental invariants for three-coordinated lattices.
I. Introduction In 1981 Yeomans and Fisher [ 1 ], Ostlund [2], and Huse [3] introduced an axial three-state chiral Potts model as a modelling of c o m m e n s u r a t e - i n c o m m e n s u r a t e transitions ~. The chiral Potts model generalizes the standard Potts model [5] to include interactions with a chiral symmetry, namely, those with strengths depending on the relative positionings of the interacting spins. Unlike the standard Ports model, which is exactly solvable at the critical point, exact results on the chiral Ports model are scarce and hard to obtain. However, a major breakthrough occurred in 1987 when an exactly integrable case of the chiral Potts model was discovered [ 6 ]. This development, which yields the first solvable case with a greater-than-one genus parametrization [6,7], has since led to renewed intense interest on the chiral Potts model (for reviews see, e.g., refs. [ 8 , 9 ] ) . Besides the consideration of integrability, the chiral Ports model also underlines the role played by certain symmetry structure [ 10,11 ] of lattice systems. The exactly soluble chiral Potts models are restricted to special varieties of the parameter space. While these varieties are given in a parametric form in terms of the rapidities of the s u r r o u n d i n g lattice [ 12,13 ], their explicit expressions have also been written down [ 14]. It has been suggested that in the case of the axial three-state chiral Potts model the integrability variety gives rise to the wetting line [ 8 ]. In view of such ~ For a review of commensurate-incommensurate transitions, see ref. [4]. 0378-4371/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
J.C. Anglks d'Auriac et al. / Three-state chiral Potts model in 2D
115
physical significance, it appears useful to re-examine these integrability and symmetry relations. In this paper we present such a study. As we shall see, our detailed and systematic analysis leads to new integrability varieties which had not previously been noted. We also give a vertex-model formulation of the three-state chiral Potts model, and write down the associated fundamental invariants for three-coordinated lattices.
2. Three-state chiral Potts model
Consider an orientedlattice whose lattice sites are occupied by N-state spins and whose lattice edges are directed. Denote the spin state of the ith site by ai= 0, 1, 2, ..., N - 1. For each lattice edge directed from i and j we associate a Boltzmann factor w,j(a~-aj). The chiral Potts model is characterized by wo(a)# wo(-a), wo(a)= w,j(a+N). The partition function is
z = E I-I w,j(~,-aj),
(1)
where the product is over all lattice edges and the summation over all spin states. Clearly, differently oriented lattices generally lead to different partition functions. We confine our considerations to three-state models and regular lattices, and begin by recalling some known results for the checkerboard lattice. Let the edges of a unit cell, say, the black square, of the checkerboard lattice be directed as shown in fig. 1. We write for brevity the four Boltzmann weights w~ for the four edges of the black square as
ai=wi(O), bi=wi(1)=wi(-2), G=wi(2)=wi(-l),
i=1,2,3,4,
(2) which we assume to be positive. Clearly, reversing the arrows of any, say, the ith, edge corresponds to the interchange of b~ and c,. We also see that other regular lattices are recovered by taking
W4 ~
W2
Fig.1.AunitcellofthecheckerboardchiralPottslattice.
116
J.C. Anglks d'Auriac el al. / Three-statechiral Potts modelin 2D ai = a ,
bi = b ,
ci = c ,
a 4 = 1,
b4 = Ca = 0 ,
isotropic square l a t t i c e , triangular lattice,
a4 = b4 = C4 = 1,
(3)
h o n e y c o m b lattice.
The latter two cases c o r r e s p o n d to the contraction and deletion o f the i = 4 edges, a situation shown in fig. 2. We r e m a r k that in physical applications [ 1 ] the Boltzmann factor is often written as
(4)
wi(a) =exp{K, cos[ ~ x ( a + 3 , ) ]] .
The axial chiral Potts model o f refs. [ 1-3 ] corresponds to taking 3~ = 33 = 0, 32 = 34 = 3. The integrability condition for the checkerboard chiral model o f fig. 1 has been given in a p a r a m e t r i c form [ 12,13 ]. Explicitly, the c o n d i t i o n [ 14 ] reads 3 (QI P2P3P4 + Q2PI P3P4 + Q3PI P2P4 + Q4PI P2P3 ) -
-
(5)
(191Q2Q3Q4 + P2Q10304 + P3Q, Q2Q4 + P4Q, Q2Q3) = 0 ,
where Pi = f - 3 h , ,
Q, = f , - 2g, +
f,=aibic,(a3, +b3, +c3) ,
3hi,
(6)
g,=a3b3 +b3c3 +c3a3,
9
~
9
hi=a?b,c?.
We observe that the integrability condition (5) possesses a n S 4 symmetry. Furthermore, the reversal of the orientation of the ith edges, which corresponds to the interchange of hi and c, for any i = 1, 2, 3, 4, does not alter (5). This is a manifestation of an enhanced symmetry at integrability. For the isotropic square lattice model, namely, ai=a, bi=b, c,=c, condition 5)
A
>/
t ~r I
W3
>/ W2
Fig. 2. Triangular (left) and honeycomb (right) lattices derived from the checkerboard lattice by contracting and deleting the w4edges.
J.C. Angl~s d'Auriac et al. / Three-state chiral Potts model in 2D
117
becomes 4 P Q ( 3 P Z - Q 2) = 0 , where P, Q are given in terms off, g, h and a, b, c as in (6). For positive Boltzmann weights we have ~z p > 0, P > Q, and hence v/3 P - Q > 0. It follows that the integrability condition splits into two branches: Q = 0 and x/~ P + Q = 0. Explicitly, other than the trivial condition abc= 0, they are
abc( a3 + b3-I-c 3 ) - 2 ( a3b3 + b3c3 + c3a 3) + 3a2b2c2=O ,
(7a)
( ~f3-t- 1 ) ( a3 + b3 + c 3 ) - 3 (%//3 - 1 ) a b c - 2 ( a2bZc - ' + b2c2a -~ + c2a2b -t ) = 0 . (7b) For the standard Potts model ( b = c = 1 ) b o t h (7a) and (7b) can be factorized resulting in, respectively, ( a - 1) 2 ( a Z - 2 a - 2 ) = 0 ,
(8a)
( a - 1 )2[ ( v / ~ + 1 )a2 + 2 (V/3 - 1 ) a - 2 ] = 0 .
(8b)
In the physical domain a # 1, a > 0, (8a) yields the known ferromagnetic critical point [15] a = l + v / 3 , while (8b) leads to a new integrable point a = w / 3 + 3 v / 2 / 2 , v / 6 / 2 - 2 = 0.628626... for the antiferromagnetic Potts model. For the triangular and honeycomb lattices both P4 and Q4 vanish when using (3). We can however divide (5) throughout by Q4 and take the limit of r=P4/Q4 carefully #3. For isotropic lattices, this leads to the following integrability conditions: 3P 3 + 9 P 2 Q - 3PQ z _ Q 3 = 0 ,
triangular,
( 9a )
3p3-9p2Q-3PQZ+Q3=O,
honeycomb.
(9b)
Note that (9a) and (9b) are related by a negation o f Q. For positive a, b, c we must have, as before, s - Q / P < 1 so that (9a) and (9b) are realized by the branches s = 4 s i n ( n / 1 8 ) - 1 = - 0 . 3 0 5 4 0 7 3 .... s = - 4 c o s ( n / 9 ) - 1 = -4.758710... for the triangular lattice, and s=0.4758710.., and s = 1 - 4 c o s ( 2 n / 9 ) = -2.064177... for the honeycomb lattice. For the standard Ports model (b = c = 1 ), (9a) and (9b) factorize to 8 ( a - 1) 6 ( a 3 + 6 a 2 + 3 a -
1) ( a 3 - 3 a - 1 ) = 0 ,
-8(a-l)6(a3+3a2-1)(a3-3a2-6a-l)=O,
triangular,
(10a)
honeycomb.
(10b)
Setting the last factors in (10a) and (10b) equal to zero leads to the known critical point o f the ferromagnetic Potts model [ 15 ], namely, a = a, - 2 c o s ( n / 9 ) = 1.879385 ....
triangular,
a = a 2 = - l + 2 w / 3 c o s ( n / 1 8 ) = 4 . 4 1 1 4 7 4 ....
honeycomb.
~2Using the inequality x3+y3+z3-3xyz=(x+y+z)[(x-y)2+(y-z)2+(z-x)2]/2>O for x+y+z>O. ~3 In the triangular (honeycomb) case we have r= 1 (r = - 1) obtained by taking the limit of a4--, 1 after first setting b4=c4=0 ( = 1).
118
J. C. Anglbs d'A uriac et al. / Three-state chiral Potts model in 2D
Setting the second-to-last factors equal to zero yields the following integrable points for the antiferromagnetic model: a=l/a2=2x/3sin(2n/9)-2=0.2266816 a = 1/at = 2 c o s ( 2 n / 9 ) - 1 =0.5320888 ....
....
triangular, honeycomb.
Indeed, the triangular variety a 3 + 6 a 2 + 3 a 1 = 0 and its solution at a = 0 . 2 2 6 6 8 1 6 have previously been suggested by Martin and Maillard [ 16] on the basis of numerical analysis and automorphic group structure considerations [ 17 ]. For the three-state antiferromagnetic triangular Ports model, the critical point has been estimated at a = 0.204 [ 18 ]. This supports the conjecture that the free energy is, in fact, regular at the integrable point a = 0.2266816. The integrability variety a 3 + 3a 2_ 1 = 0 for the honeycomb lattice is new and has not been previously noted.
3. Symmetry properties The three-state chiral Potts model possesses a n u m b e r of symmetry properties. First of all, the partition function possesses the obvious symmetry S: bi~-~ci,
all edges i,
( 11 )
corresponding to a reversal of the orientation of all lattice edges. In addition, the following duality relation [ 19 ] exists relating {at, b, c,} of a given edge to those of the dual edge: ( d i = ( a i +b, + c , ) / x f 3 , D: ~ i = ( a , + o ~ b i + o 9 2 c i ) / x / 3 , (O, = ( ai + o92b, + ogci) / x / 3 .
(12)
Here, ~o= exp (2hi/3 ), and the orientation of the dual edge is obtained by rotating the original lattice edge 90 ° counterclockwise (or clockwise) [ 19]. Clearly, applying ( 12 ) twice leads to {di,/~i, c]}= { a , G, bi }. Furthermore, for any lattice whose edge orientation is decomposable into directed cycles, we have the (complex) symmetry I2: w ( a ) --.~o°w(a) for any a. This is so since the product of the Boltzmann factors of each cycle remains unchanged. For three-state models and taking a = 1, 2, this leads to the symmetry g2:
{a, b, c}-~{a, ogb, ~o2c} , {a,b,c}__.{a, ogZb,~oc} "
(13)
Examples of lattices possessing this decomposition property include the checkerboard lattice of fig. 1 as well as the triangular lattice in fig. 2, assuming periodic boundary
J. C Anglbs d'Auriac et al. / Three-state chiral Potts model in 2D
119
conditions (so that cycles can loop around the lattice). Notice that the honeycomb lattice in fig. 2 cannot be so decomposed. Other symmetry properties tend to be lattice and orientation dependent. These are discussed in the following for isotropic lattices: S q u a r e lattice
For the square lattice with orientation shown in fig. 1, we have the symmetries S and ft. In addition, the partition function is invariant under the operation T: a ~ b ~ c
~a .
(14)
This fact can be seen by constructing diagonals in the NW-SE direction and relabelling all spin states on the nth diagonal (counting toward the NE direction) by a ~ a - n. This brings about the change w(tr)--, w ( a + 1 ) and hence the symmetry T. The isotropic square lattice is self-dual and described by the duality relation D. Particularly, D possesses a fixed point b / a = c / a = ( x / 3 - 1 )/2. T r i a n g u l a r lattice
We consider a c o h e r e n t orientation of the triangular lattice shown in fig. 3. Here, an orientation is coherent if arrows around each elementary lattice face point all in the same (cw or ccw) direction. Since the coherently oriented lattice is decomposable into directed cycles of elementary faces (triangles), it possesses the complex symmetry 12 regardless of boundary conditions. The coherent model also possesses the symmetries S and T; the latter fact follows from a relabelling of spin states a--, or- n, n = 1, 2, 3, for the nth sublattice. Finally, since the coherent lattice is obtainable from the lattice in fig. 2 by reversing the arrows on all w~ edges (which results in the interchange b~,--,c~ and leaves ( 5 ) invariant), its integrability condition is the same as that of the triangular lattice in fig. 2, namely, condition (9a).
Fig. 3. Coherent chiral Potts triangular (left) and honeycomb (right) lattices. The two lattices are mutually dual. Orientations of honeycomb edges are obtained by rotating those of the dual triangular edges 90 ° counterclockwise.
120
,L C. AnglOs d'Auriac et al. / Three-state chiral Potts m o d e l in 2D
Honeycomb lattice The honeycomb lattice with the orientation shown in fig. 3 possesses the symmetries S, T a n d £2, while the one shown in fig. 2 possesses the symmetries S and 7, but not £2. In both cases the integrability condition is (9b).
4. Vertex-model formulation The three-state chiral Potts model on any two-dimensional lattice can be formulated as a three-state vertex model on the dual lattice. Starting from a given Ports spin configuration, we draw bonds on the dual lattice as follows: For each edge directed from site i to site j, we draw on its dual edge a dotted line if a,=aj, a heavy line if ~r,=(Tj-I- 1 ( m o d 3), and a thin line if ai=~rj- 1 ( m o d 3). These situations are shown in fig. 4. This procedure leads to a three-state vertex configuration on the dual lattice. Conversely, it is not difficult to see that each allowed vertex configuration on the dual leads to three distinct spin configurations (related by spin-state permutations). Thus, the mapping between the three-state chiral Potts model and the vertex model is 3 to 1. Because of the particular rule that the bonds are drawn on the dual, only certain specific vertex configurations can occur. In the case of the coherent triangular chiral Potts model of fig. 3, for example, the mapping results in a nine-vertex model on the honeycomb lattice with vertex weights shown in fig. 5. In this case the vertex weights are isotropic, and therefore can be written as ~o0~, i+j+k=3, where i, j and k are, respectively, the numbers of dotted, heavy, and thin lines. Explicitly, the vertex weights are o ) 3 0 0 = a 3/2 ,
o ) 0 3 0 = b 3/2 ,
09111 = ( a b c )
0 ) 0 0 3 = c 3/2 ,
3/2 .
(15)
The three-state vertex model has been a topic of increasing recent interest. The partition function o f the three-state vertex model is known to be invariant under an O (3) transformation [20 ] ; explicit expressions of the fundamental algebraic invariants of the O ( 3 ) transformation have also been obtained for three-coordinated lattices [21 ]. Applying the latter findings to the model (15), one finds [22] that the quantities f, g, and h in (6) are indeed intersections of the invariants of O ( 3 ) in the subspace (15). These quantities are therefore the fundamental invariants, the polynomial building blocks of all algebraic invariants, of the three-state chiral Potts model. This finding is consistent with the assumption that the exact criticality of the coherent
3
3
2
1
1
2
0 Fig. 4. The three edge states of a three-state vertex model on the dual and their associated spin state configurations. Black circles denote sites of the original lattice and open circles those of the dual lattice.
J.C. Anglbs d'A uriac et al. / Three-state chiral Potts model in 2D
121
k) ~300
~030
~003
~111
Cdlll
Cdlll
Cdlll
Cdll 1
Cdlll
Fig. 5. Vertex configurations of a three-state nine-vertex model on the honeycomb lattice derived from the coherent triangular chiral Ports model of fig. 3. t r i a n g u l a r c h i r a l P o t t s m o d e l is g i v e n i n t e r m s o f f g, a n d h, w h i c h are f u n c t i o n s o f a, b, a n d c as i n d i c a t e d in ( 6 ) .
Acknowledgements W e w o u l d like t o t h a n k P r o f e s s o r R. R a m m a l for c o m m e n t s a n d s t i m u l a t i n g c o n v e r s a t i o n s , a n d D r . L.H. G w a f o r h e l p f u l d i s c u s s i o n s o n a s p e c t s o f t h e v e r t e x - m o d e l f o r m u l a t i o n . O n e o f us ( F Y W )
is s u p p o r t e d in p a r t b y t h e N a t i o n a l S c i e n c e F o u n -
dation Grant DMR-9015489.
References [ 1 ] J.M. Yeomans and M.E. Fisher, J. Phys. C 14 ( 1981 ) L835. [2] S. Ostlund, Phys. Rev. B 24 ( 1981 ) 398. [3] D.A. Huse, Phys. Rev. B 24 ( 1981 ) 5180. [4] M. den Nijs, in: Phase Transitions and Critical Phenomena, vol. 12, C. Domb and J.L. Lebowitz, eds. (Academic Press, London, 1988). [5] R.B. Potts, Proc. Cambridge Philos. Soc. 48 (1952) 106. [ 6 ] H. Au-Yang, B.M. McCoy, J.H.H. Perk, S. Tang and M.-L. Yan, Phys. Lett. A 123 ( 1987 ) 219. [ 7 ] B.M. McCoy, J.H.H. Perk, S. Tang and C.H. Sah, Phys. Lett. A 125 ( 1987 ) 9. [ 8 ] H. Au-Yang and J.H.H. Perk, Adv. Stud. Pure Math. 19 ( 1989 ) 57. [ 9 ] B.M. McCoy, The chiral Potts model: from physics to mathematics and back, in: Lecture Notes in Mathematics (Springer, Berlin, 1991 ). [ 10] J.M. Maillard, J. Math. Phys. 27 (1986) 2776. [ 11 ] J.M. Maillard, in: Progress in Statistical Mechanics, vol. 3, Chin-Kun Hu, ed. (World Scientific, Singapore, 1988). [ 12 ] R.J. Baxter, J.H.H. Perk and H. Au-Yang, Phys. Len. A 128 ( 1988 ) 138. [ 13] R.J. Baxter, J. Stat. Phys. 52 (1988) 639.
122
J.C. Angl~s d'A uriac et al. / Three-state chiral Potts model in 2D
[ 14 ] J. Avan, J.M. Maillard, M. Talon and C. Viallet, Int. J. Mod. Phys. B 4 (1990) 1743. [ 15 ] A. Hintermann, H. Kunz and F.Y. Wu, J. Star. Phys. 19 ( 1982 ) 623. [16] P. Martin and J.M. Maillard, J. Phys. A 19 (1986) L547. [ 17 ] J.M. Maillard and R. Rammal, J. Phys. A 16 ( 1983 ) 353. [ 18 ] I.G. Enting and F.Y. Wu, J. Stat. Phys. 28 ( t 982 ) 351. [ 19 ] F.Y. Wu and Y.K. Wang, J. Math. Phys. 17 ( 1976 ) 439. [20] F.J. Wegner, Physica 68 (1973) 570. [21 ] L.H. Gwa and F.Y. Wu, preprinl. [22] L.H. Gwa and F.Y. Wu, unpublished.