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Surface operator content of the AL face models
Physica A 251 (1998) 132–142
Surface operator content of the AL face models Murray T. Batchelor 1 Department of Mathematics, School of Mathematical Sciences, The Australian National University, Canberra ACT 0200, Australia
Abstract A set of xed boundary weights for both the critical dense and dilute AL face models is constructed from the known boundary weights of the related loop model. The surface operator content and the conformal partition functions then follow from the results obtained via the Bethe c 1998 Elsevier Science B.V. All rights reserved. equations. Keywords: Face models; Loop models; Bethe equations; Surface criticality
1. Introduction The remarkable interplay between critical systems and the symmetries of conformal and modular invariance has been well illustrated by direct calculations on exactly solved models [ 1–3]. One well-known family of such are the critical A–D–E models which are built from the Dynkin diagrams of the simply laced A–D–E Lie algebras [4,5]. These models can all be mapped onto an underlying loop model [ 4 – 6]. Another family of critical models, the dilute A–D–E lattice models [ 7 – 9], can be mapped onto the dilute O(n) loop model [10,11]. The bulk operator content and the modular invariant partition functions of these models are now well understood (see, e.g. Ref. [12]). However, the situation is not so clear for the surface operator content, at least from the perspective of exactly solved models. The particular geometry of interest here is that shown in Fig. 1. We con ne our attention to the AL models. To distinguish between the two families, we refer to them as the dense AL and the dilute AL models, the nomenclature arising naturally from the corresponding loop models. 1
Supported by the Australian Research Council.
c 1998 Elsevier Science B.V. All rights reserved 0378-4371/98/$19.00 Copyright PII S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 6 0 0 - 6
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133
Fig. 1. An open square lattice in the vertical strip geometry. The face models under consideration are de ned on the solid lines. The underlying loop and vertex models are de ned on the dashed lines.
1.1. Face models The bulk face weights of the dense AL models at criticality are [4,5,13] r c Sa Sc W d b = 8 (u) a; c Aa; b Aa; d + 9 (u) b; d Aa;b Ab; c ; Sb Sd a where 8 (u) = 1;
9 =
sin u sin( − u)
(1.1)
(1.2)
and Sa = sin a. The variable u is the spectral parameter and = =h is the crossing parameter with Coxeter number h = L + 1. We are interested here in the boundary between regimes III and IV for which 0¡u¡. The elements of the adjacency matrix A are given by 1; b = a ± 1; Aa; b = (1.3) 0 otherwise: The bulk face weights of the dilute AL models at criticality are [ 7 – 9] c W d b = 1 (u)a;b;c;d + 2 (u)a;b;c Aa;d + 3 (u)a;c;d Aa;b a r r Sa Sc + 4 (u)b; c; d Aa; b + 5 (u)a; b; d Aa; c + 6 (u)a; b c; d Aa; c Sb Sa r Sa Sc +7 (u)a; d c; b Aa; b + 8 (u)a; c Aa; b Aa; d + 9 (u)b; d Aa; b Ab; c ; Sb Sd (1.4)
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M.T. Batchelor / Physica A 251 (1998) 132–142
in which 1 (u) = 1 +
sin u sin(3 − u) ; sin 2 sin 3
2 (u) = 3 (u) =
sin(3 − u) ; sin 3
4 (u) = 5 (u) =
sin u ; sin 3
6 (u) = 7 (u) =
sin u sin(3 − u) ; sin 2 sin 3
8 (u) =
(1.5)
sin (2 − u) sin(3 − u) ; sin 2 sin 3
9 (u) = −
sin u sin( − u) ; sin 2 sin 3
and the generalized Kronecker delta is unity if all its arguments take the same value and zero otherwise. For these models 1; b = a or a ± 1; (1.6) Aa;b = 0 otherwise: The regimes of interest here are regime 1
0¡u¡3
=
L L = 2; 3; : : : ; 4L+1
regime 2
0¡u¡3
=
L+2 L = 3; 4; : : : ; 4L+1
where
h=
L + 2; regime 1; L + 1; regime 2:
(1.7)
1.2. Loop models To obtain a set of integrable boundary weights for the face models we consider the corresponding loop models, which have been solved in the open geometry of Fig. 1 [14,15]. The partition function of the dense O(n) loop model on the dashed lattice of Fig. 1 is de ned by X m m m P 12 8 8 9 9 1010 m (1.8) Z= 12 n ; G
where the bulk weights 8 and 9 are given in Eq. (1.2). The sum is over all con gurations G of non-intersecting closed loops covering all lattice bonds. The two possible
M.T. Batchelor / Physica A 251 (1998) 132–142
135
Fig. 2. The allowed vertices of the loop model along with the possible height con gurations.
loop con gurations at a vertex are numbers 8 and 9 in Fig. 2. In con guration G, mi is the number of occurrences of the vertex of type i, while P is the total number of closed loops of fugacity n = 2 cos . The boundary weights are 10 = 12 = 1 :
(1.9)
The partition function of the dilute O(n) model is de ned by Z=
X G
m13 P 1 m 1 · · · 13 n ;
(1.10)
where now the loops need not cover all lattice bonds and the loop fugacity is given by n = − 2 cos 4. The possible loop con gurations at a vertex are shown in Fig. 2, with a vertex of type i carrying a Boltzmann weight i . The bulk weights are as given in Eq. (1.5). The integrable boundary weights are 10 = 12 = sin [ 12 (3 − u) + ] ; 11 = 13 = sin [ 12 (3 + u) + ] :
(1.11)
There is one set for = 0, the other for = =2. Note that just as there is a loop formulation of the Yang–Baxter equation, ensuring integrability of the loop model in the bulk [16], there is also a loop formulation of the re ection equations, ensuring integrability at a boundary [17]. Rather than seeking explicit solutions of these equations our approach here is via the loop-to-face correspondence.
2. Boundary face weights and exact solutions The face model transfer matrix tD (u) acts between three diagonal rows of the lattice, labelled by the heights {a0 ; a1 ; : : : ; aN }, {b0 ; b1 ; : : : ; bN } and {c0 ; c1 ; : : : ; cN }, with a2k =
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M.T. Batchelor / Physica A 251 (1998) 132–142
b2k and b2k−1 = c2k−1 . Here N is the number of bulk edges in a row, which we take to be even. Thus, tD (u) = K+
c0 b0
b1
×W aN −2
N=2−1
Y
W a2k−2
k =1
bN −1 aN −1
b2k−1 a2k−1
aN K− bN −1
a2k W b2k−1 cN
c2k b2k
b2k+1
:
(2.1)
bN
The bulk face weights of the AL models are as given in Eqs. (1.1) and (1.4). Now, consider the boundary face weights.
2.1. Dense AL model In accord with Fig. 2, we de ne the boundary face weights of the dense AL model to be K+
a a
a ± 1 = 10 ;
K− a ± 1
a a
= 12 ;
(2.2)
where 10 and 12 may be arbitrary height-independent functions. General boundary face weights for this model have been discussed in Refs. [18,19]. The diagonal-to-diagonal transfer matrix tD (u) for the underlying loop model has eigenvalues [14,15] (u) = 10 12
m Y sinh[uj + 12 i(u + )] sinh[uj − 12 i(u + )] ; sinh [uj + 12 i(u − )] sinh[uj − 12 i(u − )] j=1
(2.3)
where uj ( j = 1; 2; : : : ; m) are roots of the Bethe equations "
sinh[uj + 12 i(u − )] sinh [uj − 12 i(u + )] sinh [uj + 12 i(u + )] sinh [uj − 12 i(u − )] =
#N
m Y sinh(uj + uk − i) sinh(uj − uk − i) : sinh(uj + uk + i) sinh(uj − uk + i)
k =1 6= j
(2.4)
M.T. Batchelor / Physica A 251 (1998) 132–142
137
2.2. Dilute AL model Again in accord with Fig. 2 we de ne the boundary face weights of the dilute AL model to be a a K+ a = 11 ; K+ a ± 1 = 10 ; a a K− a ± 1
a a
= 12 ;
K− a
a a
= 13 ;
(2.5)
where the functions 10 –13 are as given in Eq. (1.11). The origin of the factor is seen to be the double periodicity of the more general elliptic boundary weights [20]. With the given normalisation of the weights, the eigenvalues are [15] D (u) = N8 −1 10 12
m Y sinh(uj + i + 12 iu) sinh(uj − i − 12 iu) ; sinh(uj − i + 12 iu) sinh(uj + i − 12 iu) j=1
(2.6)
where the m roots uj satisfy #2 " #N " sinh(uj − 12 i + i) sinh(uj − i − 12 iu) sinh(uj − i + 12 iu) sinh(uj + 12 i + i) sinh(uj + i − 12 iu) sinh(uj + i + 12 iu) =
m Y sinh(uj + uk − 2i) sinh(uj − uk − 2i) sinh(uj + uk + 2i) sinh(uj − uk + 2i)
k =1 6= j
×
sinh(uj + uk + i) sinh(uj − uk + i) : sinh(uj + uk − i) sinh(uj − uk − i)
(2.7)
with again = 0 or =2, depending on the choice of boundary weights. 3. Surface operator content Let 0 be the largest eigenvalue of the transfer matrix. From the explicit calculations on the loop models, we know that the reduced free energy, fN = −N −1 log 0 , scales as fN = f∞ +
f− c f+ + − + o(N −2 ) : N N 24N 2
(3.1)
Here f∞ is the bulk free energy, f± are surface free energies and is a geometric factor. The rest of the eigenspectrum scales as log
0 (x‘ + j) + o(N −1 ) ; = ‘ N
(3.2)
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M.T. Batchelor / Physica A 251 (1998) 132–142
where j = 0; 1; : : : de nes the conformal tower of eigenstates associated with each eigenvalue. Both Eqs. (3.1) and (3.2) are in agreement with the expectation from conformal invariance [ 1–3]. The quantities f∞ and f± are non-universal. Our interest here is in the universal scaling part of the spectrum de ned by the central charge c and the surface scaling dimensions x‘ . Their values can be inferred from the underlying loop models. 3.1. Dense AL model Given the above boundary weights, a comparison of the nite-size transfer matrix eigenspectra of the dense AL model on the one hand, and the dense O(n) loop model on the other, reveals a selection rule on the parameter m appearing in the Bethe equations, Eq. (2.4). This parameter labels the sectors of the transfer matrix of the vertex model representation of the loop model. The restriction on m arises from the mapping of height con gurations to arrow con gurations on the underlying vertex model. Speci cally, m = 12 N − ‘, for ‘ = 0; 1; : : : ; ‘max , where ‘max = b 12 (L − 1)c, the integer part of 12 (L − 1). We con rm numerically that the eigenspectrum of the face model is exactly equivalent to the eigenspectrum of the loop model with the above restriction. However, the equivalence is not one-to-one, as the degeneracies di er. For example, the largest eigenvalue is L-fold degenerate in the face model. There are eigenstates in the ‘6‘max sectors of the loop model which are not eigenstates of the face model. However, these are precisely the same eigenvalues which appear in the sectors with ‘¿‘max . As a result the surface operator content of the face model can be inferred from the known results for the loop model. The central charge and surface scaling dimensions of the loop model are [14,21,22] c=1 −
62 ; ( − )
‘ x‘ = [‘( − ) − ] ;
(3.3)
where ‘ is unrestricted. The Bethe equation calculations also reveal a set of constant scaling dimensions, with x = 2. Now, consider the face model. In terms of the Kac formula [ 1–3] r;s =
[hr − (h − 1)s]2 − 1 ; 4h(h − 1)
(3.4)
we have, recalling that = =h with h = L + 1, c=1 −
6 ; h(h − 1)
x‘ = 1;1+2‘ ;
‘ = 1; : : : ; b 12 (h − 2)c :
(3.5) (3.6)
Along with the constant dimension x = 2, this de nes the surface operator content of the dense AL face model. The rst example is the Ising model with h = 4, c = 12 ; x = 12 and x = 2. The 7 next few cases are h = 5, with c = 10 ; x = 35 ; x = 2, and h = 6, c = 45 ; x = 23 ; x = 2;
M.T. Batchelor / Physica A 251 (1998) 132–142
139
x = 3. Recall that the boundary heights are xed along each boundary, but free to take all allowed values. The operator content induced by other boundary con gurations and di erent geometries have also been considered for this particular family of models [21, 23 – 25]. The modular invariant partition functions can be constructed from the central charge and conformal dimensions through the Virasoro characters [ 1 – 3] r; s =
∞ q−c=24+r; s X n2 h(h−1)+nhr −n(h−1)s q [q − qn(h−1)s+rs ] ; Q(q) n = −∞
= q−c=24+r; s
∞ X
dn (r; s )qn ;
(3.7)
n=0
where Q(q) =
∞ Y
(1 − q n ) :
(3.8)
n=1
The rst few integer coecients dn (r; s ) are tabulated in, e.g., Ref. [3]. From the face model eigenspectra we see that the rst few modular invariant partition functions for the xed boundaries under consideration are h = 4;
Z = 3 1;1 + 3 1; 3 ;
h = 5;
Z = 4 1;1 + 6 1; 3 ;
h = 6;
Z = 5 1;1 + 9 1; 3 + 5 1; 5 :
(3.9)
We can add h = 3; Z = 2 1;1 [21] to begin this list. These relations de ne the asymptotic degeneracies of the eigenspectrum. 3.2. Dilute AL model The transfer matrix eigenspectrum of the dilute AL face model is equivalent to that of the dilute loop model with the restriction m = N − ‘, where ‘ = 0; 1; : : : ; ‘max , with ‘max = L − 1. Again it is precisely the eigenvalues in the ‘¿‘max sectors which do not appear in the eigenspectrum of the face model. The central charge and surface scaling dimensions of the dilute O(n) model with boundary weights, Eq. (1.11) have been calculated from the exact solution of Section 2.2 at the particular value u = [ 26–29]. This is the so-called honeycomb limit of relevance to the surface critical behaviour of self-avoiding random walks [30,27,28,31]. It is straightforward to extend these calculations to the wider range 0¡u¡3. Indeed, the central charge and scaling dimensions are independent of the spectral parameter u. The central charge is c = 1 − 6(g − 1)2 =g ;
(3.10)
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M.T. Batchelor / Physica A 251 (1998) 132–142
where g = 2 − 4. Three sets of “geometric” scaling dimensions have been derived, each corresponding to a choice of the boundary weights. For = 0, the result is [30,26,29] X‘O−O = 14 g‘2 + 12 (g − 1)‘ :
(3.11)
In the language of surface critical phenomena, O–O labels the ordinary surface transition. The choice = =2 (S–S) corresponds to the special surface transition. The other possibility is mixed O–S boundary conditions with = 0 on one side of the strip and = =2 on the other, for which there is also a Bethe solution [28,29]. The extraordinary surface transition has also been recently discussed [32]. We con ne our attention here to the O–O boundaries. The thermal dimension has also been calculated, with in this case x = 2. For regime 1, g = h=(h − 1) with h = L + 2. The central charge, Eq. (3.10) gives the unitary minimal result, Eq. (3.5). The scaling dimensions are given in terms of the Kac formula by X‘O−O = 1+‘;1 ;
‘ = 1; : : : ; h − 3 :
(3.12)
As for the dense model, the Ising case has h = 4 with c = 12 , x = 12 and x = 2. The rst few conformal partition functions are given by h = 4;
Z = 2 1;1 + 2 2;1 ;
h = 5;
Z = 3 1;1 + 4 2;1 + 3 3;1 ;
h = 6;
Z = 4 1;1 + 6 2;1 + 6 3;1 + 4 4;1 :
(3.13)
Since 2;1 = 1; 3 for h = 4 both the dense and dilute results are essentially the same for the Ising case. For regime 2, g = (h − 1)=h with h = L + 1. The central charge is again given by Eq. (3.5) with scaling dimensions X‘O−O = 1;1+‘ ;
‘ = 1; : : : ; h − 2 :
(3.14)
1 ; x = 12 and x = 2. Here the In this regime the Ising results for h = 4 are c = 12 ; x = 16 rst few conformal partition functions are given by
h = 4;
Z = 3 1; 1 + 4 1; 2 + 3 1; 3 ;
h = 5;
Z = 4 1; 1 + 6 1; 2 + 6 1;3 + 4 1; 4 ;
h = 6;
Z = 5 1; 1 + 8 1; 2 + 9 1; 3 + 8 1; 4 + 5 1; 5 ;
again from which the general pattern is readily apparent.
(3.15)
M.T. Batchelor / Physica A 251 (1998) 132–142
141
4. Conclusions A set of particular xed boundary weights has been constructed for both the dense and dilute AL face models from the boundary weights of the related loop model. The transfer matrix eigenspectra of the face models is seen to agree with that of the loop models under restriction of the parameter ‘. The surface operator content and the conformal partition functions then follow from the results obtained via the Bethe equations for the loop models. It will be particularly worthwhile to generalise these Bethe solutions in terms of the elliptic #-functions of the o -critical AL models, as was done originally in the bulk [13,33]. Acknowledgements It is a great pleasure to wish Professor J.M.J. van Leeuwen all the best on this occasion. I look back fondly on my rst postdoc, in Leiden, under his inspirational guidance. Among other things, he taught me to take my rst steps on the ice. This manuscript has bene ted from helpful discussions with Katherine Seaton, Ole Warnaar and Yu-kui Zhou. References [1] C. Itzykson, H. Saleur, J.B. Zuber (Eds.), Conformal Invariance and Applications to Statistical Mechanics, World Scienti c, Singapore, 1988. [2] J.L. Cardy (Ed.), Finite-Size Scaling, North-Holland, Amsterdam, 1988. [3] P. Christe, M. Henkel, Introduction to Conformal Invariance and Its Applications to Critical Phenomena, Springer, Heidelberg, 1993. [4] V. Pasquier, Nucl. Phys. B 285 (1987) 162. [5] V. Pasquier, J. Phys. A 20 (1987) L1229. [6] A.L. Owczarek, R.J. Baxter, J. Stat. Phys. 49 (1989) 1093. [7] S.O. Warnaar, B. Nienhuis, K.A. Seaton, Phys. Rev. Lett. 69 (1992) 710. [8] Ph. Roche, Phys. Lett. B 285 (1992) 49. [9] S.O. Warnaar, B. Nienhuis, K.A. Seaton, Int. J. Mod. Phys. B 7 (1993) 3727. [10] B. Nienhuis, Int. J. Mod. Phys. B 4 (1990) 929. [11] S.O. Warnaar, B. Nienhuis, J. Phys. A 26 (1993) 2301. [12] D.L. O’Brien, P.A. Pearce, J. Phys. A 28 (1995) 4891. [13] G.E. Andrews, R.J. Baxter, P.J. Forrester, J. Stat. Phys. 35 (1984) 193. [14] A.L. Owczarek, R.J. Baxter, J. Phys. A 22 (1989) 1141. [15] C.M. Yung, M.T. Batchelor, Nucl. Phys. B 435 (1995) 430. [16] S.O. Warnaar, B. Nienhuis, H.W.J. Blote, J. Phys. A 26 (1993) 477. [17] M.T. Batchelor, unpublished. [18] R.E. Behrend, P.A. Pearce, D.L. O’Brien, J. Stat. Phys. 84 (1996) 1. [19] C. Ahn, W.M. Koo, Nucl. Phys. B 468 (1996) 461. [20] M.T. Batchelor, V. Fridkin, Y.K. Zhou, J. Phys. A 29 (1996) L61. [21] H. Saleur, M. Bauer, Nucl. Phys. B 320 (1989) 591. [22] M.T. Batchelor, A.L. Owczarek, K.A. Seaton, C.M. Yung, J. Phys. A 28 (1995) 839. [23] J.L. Cardy, Nucl. Phys. B 324 (1989) 581. [24] L. Chim, Int. J. Mod. Phys. A 11 (1996) 4491. [25] P.A. Pearce, unpublished.
142 [26] [27] [28] [29] [30] [31] [32]
M.T. Batchelor / Physica A 251 (1998) 132–142
M.T. Batchelor, J. Suzuki, J. Phys. A 26 (1993) L729. M.T. Batchelor, C.M. Yung, Phys. Rev. Lett. 74 (1995) 2026. M.T. Batchelor, C.M. Yung, J. Phys. A 28 (1995) L421. C.M. Yung, M.T. Batchelor, Nucl. Phys. B 453 (1995) 552. B. Duplantier, H. Saleur, Phys. Rev. Lett. 59 (1987) 539. D. Bennett-Wood, A.L. Owczarek, J. Phys. A 29 (1996) 4755. M.T. Batchelor, J. Cardy, Extraordinary transition in the two-dimensional O(n) model, condmat=9705238. [33] V.V. Bazhanov, B. Nienhuis, S.O. Warnaar, Phys. Lett. B 322 (1994) 198.
Surface operator content of the AL face models Murray T. Batchelor 1 Department of Mathematics, School of Mathematical Sciences, The Australian National University, Canberra ACT 0200, Australia
Abstract A set of xed boundary weights for both the critical dense and dilute AL face models is constructed from the known boundary weights of the related loop model. The surface operator content and the conformal partition functions then follow from the results obtained via the Bethe c 1998 Elsevier Science B.V. All rights reserved. equations. Keywords: Face models; Loop models; Bethe equations; Surface criticality
1. Introduction The remarkable interplay between critical systems and the symmetries of conformal and modular invariance has been well illustrated by direct calculations on exactly solved models [ 1–3]. One well-known family of such are the critical A–D–E models which are built from the Dynkin diagrams of the simply laced A–D–E Lie algebras [4,5]. These models can all be mapped onto an underlying loop model [ 4 – 6]. Another family of critical models, the dilute A–D–E lattice models [ 7 – 9], can be mapped onto the dilute O(n) loop model [10,11]. The bulk operator content and the modular invariant partition functions of these models are now well understood (see, e.g. Ref. [12]). However, the situation is not so clear for the surface operator content, at least from the perspective of exactly solved models. The particular geometry of interest here is that shown in Fig. 1. We con ne our attention to the AL models. To distinguish between the two families, we refer to them as the dense AL and the dilute AL models, the nomenclature arising naturally from the corresponding loop models. 1
Supported by the Australian Research Council.
c 1998 Elsevier Science B.V. All rights reserved 0378-4371/98/$19.00 Copyright PII S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 6 0 0 - 6
M.T. Batchelor / Physica A 251 (1998) 132–142
133
Fig. 1. An open square lattice in the vertical strip geometry. The face models under consideration are de ned on the solid lines. The underlying loop and vertex models are de ned on the dashed lines.
1.1. Face models The bulk face weights of the dense AL models at criticality are [4,5,13] r c Sa Sc W d b = 8 (u) a; c Aa; b Aa; d + 9 (u) b; d Aa;b Ab; c ; Sb Sd a where 8 (u) = 1;
9 =
sin u sin( − u)
(1.1)
(1.2)
and Sa = sin a. The variable u is the spectral parameter and = =h is the crossing parameter with Coxeter number h = L + 1. We are interested here in the boundary between regimes III and IV for which 0¡u¡. The elements of the adjacency matrix A are given by 1; b = a ± 1; Aa; b = (1.3) 0 otherwise: The bulk face weights of the dilute AL models at criticality are [ 7 – 9] c W d b = 1 (u)a;b;c;d + 2 (u)a;b;c Aa;d + 3 (u)a;c;d Aa;b a r r Sa Sc + 4 (u)b; c; d Aa; b + 5 (u)a; b; d Aa; c + 6 (u)a; b c; d Aa; c Sb Sa r Sa Sc +7 (u)a; d c; b Aa; b + 8 (u)a; c Aa; b Aa; d + 9 (u)b; d Aa; b Ab; c ; Sb Sd (1.4)
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M.T. Batchelor / Physica A 251 (1998) 132–142
in which 1 (u) = 1 +
sin u sin(3 − u) ; sin 2 sin 3
2 (u) = 3 (u) =
sin(3 − u) ; sin 3
4 (u) = 5 (u) =
sin u ; sin 3
6 (u) = 7 (u) =
sin u sin(3 − u) ; sin 2 sin 3
8 (u) =
(1.5)
sin (2 − u) sin(3 − u) ; sin 2 sin 3
9 (u) = −
sin u sin( − u) ; sin 2 sin 3
and the generalized Kronecker delta is unity if all its arguments take the same value and zero otherwise. For these models 1; b = a or a ± 1; (1.6) Aa;b = 0 otherwise: The regimes of interest here are regime 1
0¡u¡3
=
L L = 2; 3; : : : ; 4L+1
regime 2
0¡u¡3
=
L+2 L = 3; 4; : : : ; 4L+1
where
h=
L + 2; regime 1; L + 1; regime 2:
(1.7)
1.2. Loop models To obtain a set of integrable boundary weights for the face models we consider the corresponding loop models, which have been solved in the open geometry of Fig. 1 [14,15]. The partition function of the dense O(n) loop model on the dashed lattice of Fig. 1 is de ned by X m m m P 12 8 8 9 9 1010 m (1.8) Z= 12 n ; G
where the bulk weights 8 and 9 are given in Eq. (1.2). The sum is over all con gurations G of non-intersecting closed loops covering all lattice bonds. The two possible
M.T. Batchelor / Physica A 251 (1998) 132–142
135
Fig. 2. The allowed vertices of the loop model along with the possible height con gurations.
loop con gurations at a vertex are numbers 8 and 9 in Fig. 2. In con guration G, mi is the number of occurrences of the vertex of type i, while P is the total number of closed loops of fugacity n = 2 cos . The boundary weights are 10 = 12 = 1 :
(1.9)
The partition function of the dilute O(n) model is de ned by Z=
X G
m13 P 1 m 1 · · · 13 n ;
(1.10)
where now the loops need not cover all lattice bonds and the loop fugacity is given by n = − 2 cos 4. The possible loop con gurations at a vertex are shown in Fig. 2, with a vertex of type i carrying a Boltzmann weight i . The bulk weights are as given in Eq. (1.5). The integrable boundary weights are 10 = 12 = sin [ 12 (3 − u) + ] ; 11 = 13 = sin [ 12 (3 + u) + ] :
(1.11)
There is one set for = 0, the other for = =2. Note that just as there is a loop formulation of the Yang–Baxter equation, ensuring integrability of the loop model in the bulk [16], there is also a loop formulation of the re ection equations, ensuring integrability at a boundary [17]. Rather than seeking explicit solutions of these equations our approach here is via the loop-to-face correspondence.
2. Boundary face weights and exact solutions The face model transfer matrix tD (u) acts between three diagonal rows of the lattice, labelled by the heights {a0 ; a1 ; : : : ; aN }, {b0 ; b1 ; : : : ; bN } and {c0 ; c1 ; : : : ; cN }, with a2k =
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M.T. Batchelor / Physica A 251 (1998) 132–142
b2k and b2k−1 = c2k−1 . Here N is the number of bulk edges in a row, which we take to be even. Thus, tD (u) = K+
c0 b0
b1
×W aN −2
N=2−1
Y
W a2k−2
k =1
bN −1 aN −1
b2k−1 a2k−1
aN K− bN −1
a2k W b2k−1 cN
c2k b2k
b2k+1
:
(2.1)
bN
The bulk face weights of the AL models are as given in Eqs. (1.1) and (1.4). Now, consider the boundary face weights.
2.1. Dense AL model In accord with Fig. 2, we de ne the boundary face weights of the dense AL model to be K+
a a
a ± 1 = 10 ;
K− a ± 1
a a
= 12 ;
(2.2)
where 10 and 12 may be arbitrary height-independent functions. General boundary face weights for this model have been discussed in Refs. [18,19]. The diagonal-to-diagonal transfer matrix tD (u) for the underlying loop model has eigenvalues [14,15] (u) = 10 12
m Y sinh[uj + 12 i(u + )] sinh[uj − 12 i(u + )] ; sinh [uj + 12 i(u − )] sinh[uj − 12 i(u − )] j=1
(2.3)
where uj ( j = 1; 2; : : : ; m) are roots of the Bethe equations "
sinh[uj + 12 i(u − )] sinh [uj − 12 i(u + )] sinh [uj + 12 i(u + )] sinh [uj − 12 i(u − )] =
#N
m Y sinh(uj + uk − i) sinh(uj − uk − i) : sinh(uj + uk + i) sinh(uj − uk + i)
k =1 6= j
(2.4)
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2.2. Dilute AL model Again in accord with Fig. 2 we de ne the boundary face weights of the dilute AL model to be a a K+ a = 11 ; K+ a ± 1 = 10 ; a a K− a ± 1
a a
= 12 ;
K− a
a a
= 13 ;
(2.5)
where the functions 10 –13 are as given in Eq. (1.11). The origin of the factor is seen to be the double periodicity of the more general elliptic boundary weights [20]. With the given normalisation of the weights, the eigenvalues are [15] D (u) = N8 −1 10 12
m Y sinh(uj + i + 12 iu) sinh(uj − i − 12 iu) ; sinh(uj − i + 12 iu) sinh(uj + i − 12 iu) j=1
(2.6)
where the m roots uj satisfy #2 " #N " sinh(uj − 12 i + i) sinh(uj − i − 12 iu) sinh(uj − i + 12 iu) sinh(uj + 12 i + i) sinh(uj + i − 12 iu) sinh(uj + i + 12 iu) =
m Y sinh(uj + uk − 2i) sinh(uj − uk − 2i) sinh(uj + uk + 2i) sinh(uj − uk + 2i)
k =1 6= j
×
sinh(uj + uk + i) sinh(uj − uk + i) : sinh(uj + uk − i) sinh(uj − uk − i)
(2.7)
with again = 0 or =2, depending on the choice of boundary weights. 3. Surface operator content Let 0 be the largest eigenvalue of the transfer matrix. From the explicit calculations on the loop models, we know that the reduced free energy, fN = −N −1 log 0 , scales as fN = f∞ +
f− c f+ + − + o(N −2 ) : N N 24N 2
(3.1)
Here f∞ is the bulk free energy, f± are surface free energies and is a geometric factor. The rest of the eigenspectrum scales as log
0 (x‘ + j) + o(N −1 ) ; = ‘ N
(3.2)
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where j = 0; 1; : : : de nes the conformal tower of eigenstates associated with each eigenvalue. Both Eqs. (3.1) and (3.2) are in agreement with the expectation from conformal invariance [ 1–3]. The quantities f∞ and f± are non-universal. Our interest here is in the universal scaling part of the spectrum de ned by the central charge c and the surface scaling dimensions x‘ . Their values can be inferred from the underlying loop models. 3.1. Dense AL model Given the above boundary weights, a comparison of the nite-size transfer matrix eigenspectra of the dense AL model on the one hand, and the dense O(n) loop model on the other, reveals a selection rule on the parameter m appearing in the Bethe equations, Eq. (2.4). This parameter labels the sectors of the transfer matrix of the vertex model representation of the loop model. The restriction on m arises from the mapping of height con gurations to arrow con gurations on the underlying vertex model. Speci cally, m = 12 N − ‘, for ‘ = 0; 1; : : : ; ‘max , where ‘max = b 12 (L − 1)c, the integer part of 12 (L − 1). We con rm numerically that the eigenspectrum of the face model is exactly equivalent to the eigenspectrum of the loop model with the above restriction. However, the equivalence is not one-to-one, as the degeneracies di er. For example, the largest eigenvalue is L-fold degenerate in the face model. There are eigenstates in the ‘6‘max sectors of the loop model which are not eigenstates of the face model. However, these are precisely the same eigenvalues which appear in the sectors with ‘¿‘max . As a result the surface operator content of the face model can be inferred from the known results for the loop model. The central charge and surface scaling dimensions of the loop model are [14,21,22] c=1 −
62 ; ( − )
‘ x‘ = [‘( − ) − ] ;
(3.3)
where ‘ is unrestricted. The Bethe equation calculations also reveal a set of constant scaling dimensions, with x = 2. Now, consider the face model. In terms of the Kac formula [ 1–3] r;s =
[hr − (h − 1)s]2 − 1 ; 4h(h − 1)
(3.4)
we have, recalling that = =h with h = L + 1, c=1 −
6 ; h(h − 1)
x‘ = 1;1+2‘ ;
‘ = 1; : : : ; b 12 (h − 2)c :
(3.5) (3.6)
Along with the constant dimension x = 2, this de nes the surface operator content of the dense AL face model. The rst example is the Ising model with h = 4, c = 12 ; x = 12 and x = 2. The 7 next few cases are h = 5, with c = 10 ; x = 35 ; x = 2, and h = 6, c = 45 ; x = 23 ; x = 2;
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x = 3. Recall that the boundary heights are xed along each boundary, but free to take all allowed values. The operator content induced by other boundary con gurations and di erent geometries have also been considered for this particular family of models [21, 23 – 25]. The modular invariant partition functions can be constructed from the central charge and conformal dimensions through the Virasoro characters [ 1 – 3] r; s =
∞ q−c=24+r; s X n2 h(h−1)+nhr −n(h−1)s q [q − qn(h−1)s+rs ] ; Q(q) n = −∞
= q−c=24+r; s
∞ X
dn (r; s )qn ;
(3.7)
n=0
where Q(q) =
∞ Y
(1 − q n ) :
(3.8)
n=1
The rst few integer coecients dn (r; s ) are tabulated in, e.g., Ref. [3]. From the face model eigenspectra we see that the rst few modular invariant partition functions for the xed boundaries under consideration are h = 4;
Z = 3 1;1 + 3 1; 3 ;
h = 5;
Z = 4 1;1 + 6 1; 3 ;
h = 6;
Z = 5 1;1 + 9 1; 3 + 5 1; 5 :
(3.9)
We can add h = 3; Z = 2 1;1 [21] to begin this list. These relations de ne the asymptotic degeneracies of the eigenspectrum. 3.2. Dilute AL model The transfer matrix eigenspectrum of the dilute AL face model is equivalent to that of the dilute loop model with the restriction m = N − ‘, where ‘ = 0; 1; : : : ; ‘max , with ‘max = L − 1. Again it is precisely the eigenvalues in the ‘¿‘max sectors which do not appear in the eigenspectrum of the face model. The central charge and surface scaling dimensions of the dilute O(n) model with boundary weights, Eq. (1.11) have been calculated from the exact solution of Section 2.2 at the particular value u = [ 26–29]. This is the so-called honeycomb limit of relevance to the surface critical behaviour of self-avoiding random walks [30,27,28,31]. It is straightforward to extend these calculations to the wider range 0¡u¡3. Indeed, the central charge and scaling dimensions are independent of the spectral parameter u. The central charge is c = 1 − 6(g − 1)2 =g ;
(3.10)
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where g = 2 − 4. Three sets of “geometric” scaling dimensions have been derived, each corresponding to a choice of the boundary weights. For = 0, the result is [30,26,29] X‘O−O = 14 g‘2 + 12 (g − 1)‘ :
(3.11)
In the language of surface critical phenomena, O–O labels the ordinary surface transition. The choice = =2 (S–S) corresponds to the special surface transition. The other possibility is mixed O–S boundary conditions with = 0 on one side of the strip and = =2 on the other, for which there is also a Bethe solution [28,29]. The extraordinary surface transition has also been recently discussed [32]. We con ne our attention here to the O–O boundaries. The thermal dimension has also been calculated, with in this case x = 2. For regime 1, g = h=(h − 1) with h = L + 2. The central charge, Eq. (3.10) gives the unitary minimal result, Eq. (3.5). The scaling dimensions are given in terms of the Kac formula by X‘O−O = 1+‘;1 ;
‘ = 1; : : : ; h − 3 :
(3.12)
As for the dense model, the Ising case has h = 4 with c = 12 , x = 12 and x = 2. The rst few conformal partition functions are given by h = 4;
Z = 2 1;1 + 2 2;1 ;
h = 5;
Z = 3 1;1 + 4 2;1 + 3 3;1 ;
h = 6;
Z = 4 1;1 + 6 2;1 + 6 3;1 + 4 4;1 :
(3.13)
Since 2;1 = 1; 3 for h = 4 both the dense and dilute results are essentially the same for the Ising case. For regime 2, g = (h − 1)=h with h = L + 1. The central charge is again given by Eq. (3.5) with scaling dimensions X‘O−O = 1;1+‘ ;
‘ = 1; : : : ; h − 2 :
(3.14)
1 ; x = 12 and x = 2. Here the In this regime the Ising results for h = 4 are c = 12 ; x = 16 rst few conformal partition functions are given by
h = 4;
Z = 3 1; 1 + 4 1; 2 + 3 1; 3 ;
h = 5;
Z = 4 1; 1 + 6 1; 2 + 6 1;3 + 4 1; 4 ;
h = 6;
Z = 5 1; 1 + 8 1; 2 + 9 1; 3 + 8 1; 4 + 5 1; 5 ;
again from which the general pattern is readily apparent.
(3.15)
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4. Conclusions A set of particular xed boundary weights has been constructed for both the dense and dilute AL face models from the boundary weights of the related loop model. The transfer matrix eigenspectra of the face models is seen to agree with that of the loop models under restriction of the parameter ‘. The surface operator content and the conformal partition functions then follow from the results obtained via the Bethe equations for the loop models. It will be particularly worthwhile to generalise these Bethe solutions in terms of the elliptic #-functions of the o -critical AL models, as was done originally in the bulk [13,33]. Acknowledgements It is a great pleasure to wish Professor J.M.J. van Leeuwen all the best on this occasion. I look back fondly on my rst postdoc, in Leiden, under his inspirational guidance. Among other things, he taught me to take my rst steps on the ice. This manuscript has bene ted from helpful discussions with Katherine Seaton, Ole Warnaar and Yu-kui Zhou. References [1] C. Itzykson, H. Saleur, J.B. Zuber (Eds.), Conformal Invariance and Applications to Statistical Mechanics, World Scienti c, Singapore, 1988. [2] J.L. Cardy (Ed.), Finite-Size Scaling, North-Holland, Amsterdam, 1988. [3] P. Christe, M. Henkel, Introduction to Conformal Invariance and Its Applications to Critical Phenomena, Springer, Heidelberg, 1993. [4] V. Pasquier, Nucl. Phys. B 285 (1987) 162. [5] V. Pasquier, J. Phys. A 20 (1987) L1229. [6] A.L. Owczarek, R.J. Baxter, J. Stat. Phys. 49 (1989) 1093. [7] S.O. Warnaar, B. Nienhuis, K.A. Seaton, Phys. Rev. Lett. 69 (1992) 710. [8] Ph. Roche, Phys. Lett. B 285 (1992) 49. [9] S.O. Warnaar, B. Nienhuis, K.A. Seaton, Int. J. Mod. Phys. B 7 (1993) 3727. [10] B. Nienhuis, Int. J. Mod. Phys. B 4 (1990) 929. [11] S.O. Warnaar, B. Nienhuis, J. Phys. A 26 (1993) 2301. [12] D.L. O’Brien, P.A. Pearce, J. Phys. A 28 (1995) 4891. [13] G.E. Andrews, R.J. Baxter, P.J. Forrester, J. Stat. Phys. 35 (1984) 193. [14] A.L. Owczarek, R.J. Baxter, J. Phys. A 22 (1989) 1141. [15] C.M. Yung, M.T. Batchelor, Nucl. Phys. B 435 (1995) 430. [16] S.O. Warnaar, B. Nienhuis, H.W.J. Blote, J. Phys. A 26 (1993) 477. [17] M.T. Batchelor, unpublished. [18] R.E. Behrend, P.A. Pearce, D.L. O’Brien, J. Stat. Phys. 84 (1996) 1. [19] C. Ahn, W.M. Koo, Nucl. Phys. B 468 (1996) 461. [20] M.T. Batchelor, V. Fridkin, Y.K. Zhou, J. Phys. A 29 (1996) L61. [21] H. Saleur, M. Bauer, Nucl. Phys. B 320 (1989) 591. [22] M.T. Batchelor, A.L. Owczarek, K.A. Seaton, C.M. Yung, J. Phys. A 28 (1995) 839. [23] J.L. Cardy, Nucl. Phys. B 324 (1989) 581. [24] L. Chim, Int. J. Mod. Phys. A 11 (1996) 4491. [25] P.A. Pearce, unpublished.
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