THE DERIVATION OF THE FREE ENERGY OF THE ISING MODEL FROM THAT OF THE EIGHT-VERTEX MODEL D. A. LAVIS Department of Mathematics, King's College, Strand

Size: px

Start display at page:

Download "THE DERIVATION OF THE FREE ENERGY OF THE ISING MODEL FROM THAT OF THE EIGHT-VERTEX MODEL D. A. LAVIS Department of Mathematics, King's College, Strand"

Rachel Preston
5 years ago
Views:

1 THE DERIVATION OF THE FREE ENERGY OF THE ISING MODEL FROM THAT OF THE EIGHT-VERTEX MODEL D. A. LAVIS Department of Mathematics, King's College, Strand, London, WCR LS, England March 5, 99 Abstract The zero-eld eight-vertex model is equivalent to a square lattice Ising model, with a four-spin coupling and second neighbour coupling but no nearest neighbour coupling. When the four-spin coupling is zero the model reduces to two decoupled nearest neighbour Ising models. The standard formula for the free energy of the Ising model is, therefore, implicit in Baxter's expression for the free energy of the zero-eld eight-vertex model. The detailed derivation of this result has not appeared in print. As a footnote to Baxter's work this paper provides the necessary analysis. Introduction The name vertex model is used to denote a lattice model in which the microstates are represented by putting an arrow on each line connecting a pair of nearestneighbour sites. Such models can be constructed on any lattice, but those for the plain square lattice have received the greatest attention. The most general model of this type is the sixteen-vertex model, where the dierent vertex types correspond to all possible directions of the arrows on the four edges meeting at a vertex. This model, which can be shown [] tobeequivalent to an Ising model with two, three and four-site interactions and with an external eld, is unsolved. The eight-vertex model corresponds to the case where the vertex types are restricted to those with an even number of arrows pointing in and out. The vertices in this case, with their corresponding energies, are: e e e 3 e 4 e 5 e e 7 e The six-vertex model, where the vertices 7 and are eliminated by setting e 7 e, is the situation when the same numberofarrows point inandout. This restriction is called the ice rule. The ground-state entropy of this model was obtained To appear: Rep. Math. Phys. 37 (99)

2 by Lieb [], who also derived the free energy [3, 4, 5]. The conditions e e e 3 e 4 e 5 e e 7 e () correspond to to the situation when the vertex energies are unaltered when all the arrows are reversed. By analogy with a ferroelectric model this is referred to as the zero-eld case. Now the model has four independent Boltzmann factors exp(;e k b T ) exp(;e 3 k b T ) exp(;e 5 k b T ) exp(;e 7 k b T ) exp(;e k b T )a exp(;e 4 k b T )b exp(;e k b T )c exp(;e k b T )d: In section the spin formulation of the zero-eld eight-vertex model is described and in section 3 Baxter's derivation of the free energy is summarized. The reduction of that formula to that of the free energy of the Ising model is presented in section 4. Since this work is dependent on the properties of elliptic functions the relevant formulae are given in an appendix together with the derivation of some crucial nome series. The equivalent spin model Given a particular arrow conguration of the eight-vertex model, conguration graphs are drawn consisting of lines on all bonds with arrows pointing to the left or downwards. The restriction of vertex types to those of the eight-vertex model means that an even number of lines are incident ateachvertex the conguration graphs are polygons. If an Ising (s ) spin is placed at the centre of each face then the spin sites form another plane square lattice, which is the dual of the original lattice. If the spins contained within each polygon graph are aligned in the same direction, which is dierent from that of the neighbouring regions, then to every vertex conguration there are two spin congurations. One of the two equivalent relationships between vertices, bonds and spins is ; + 3 ; ; 4 5 ; + 7 () ; + With a four-spin coupling ;K 4, a bottom-left top-right coupling ;K;K, a bottomright top-left coupling ;K+K and a trivial coupling ;K wehavetheidentication a exp (K +K + K 4 ) b exp (K ; K + K 4 ) c exp (K +K ; K 4 ) d exp (K ; K ; K 4 ) between vertex and spin Boltzmann factors []. Because of the two spin congurations corresponding to each vertex conguration the spin partition function is twice the partition function of the zero-eld eight-vertex model. When K > K and K 4 the spin model is ferromagnetic and a b c d. Using the weak-graph transformation [7], it can be shown that, when these conditions apply, the zero-eld eight-vertex model has a transition surface a b + c + d: (4) In terms of the spin couplings this takes the form (K K K 4 ) (5) (3)

3 where (K K K 4 ) tanh(k 4 )cosh (K) + sinh (K) + sinh (K )[tanh(k 4 ) ; ]: () When K K 4 (5)and()givethewell-known formula sinh(k c ) for the critical coupling of the simple Ising model. When K the system is isotropic the critical curve inthek{k 4 plane cuts the K axis at K c. When K!, K 4!, showing that there is no phase transition in a spin- model with a purely four-spin interaction. The critical exponents and, given by ; ; arccosftanh(k 4 )g [ ; arccosftanh(k 4 )g] (7) [] vary as functions of the four-spin coupling. This result is consistent with scaling theory only if K 4 is a marginal coupling. That this is indeed the case follows from the work of Kadano and Wegner [] who established that K 4 scales as r r d. (The scaling dimension of K 4 is the physical dimension, so the scaling exponent is zero.) 3 The free energy of the eight-vertex model An exact expression for the free energy of the zero-eld eight-vertex model was derived by Baxter [, 9, ] from the largest eigenvalue of the appropriate transfer matrix. The rst step of Baxter's analysis was to obtain conditions under which transfer matrices with dierent values of a, b, c and d commute. This procedure is most easily represented in terms of a set of new variables w, w, w 3 and w 4 which satisfy the conditions w w w 3 jw 4 j () The case a > b c d has the spin representation described above andthe critical temperature T c, with K Jk b T, K J k b T, K 4 J 4 k b T, is given as the solution of (4) or (5). Now thevariables w, w, w 3 and w 4 are dened by w (a + b) w w 3 < : (c + d) T <T c, (a ; b) T >T c, < : (a ; b) T <T c, (c + d) T >T c, (9) w 4 (c ; d): and the critical temperature is given by w w 3. Baxter [, 9,] expressed the transfer matrix in terms of Pauli matrices and showed that two transfer matrices, corresponding to two dierent sets of w i values, commute when the ratios (w j ; w k )(w l ; w m), where (j k l m) isany permutation of ( 3 4), are the same for both sets of w i. In fact it is not dicult to see that only two ratios of this form can be chosen independently. Let w ; w 4 w 3 ; w From () and () + ; w ; w w 3 ; w 4 +; ; ; : () ; ; : () Transfer matrices with the same values of and ; commute. For xed values of and ; there remains one degree of freedom in the ratios of the values of w ::: w 4. 3

4 Using the Jacobian elliptic functions cn(j), sn(j) and dn(j) (see Gradshteyn and Ryzhik [] and the appendix below), let w : w : w 3 : w 4 cn(uj`) cn(j`) : dn(uj`) dn(j`) ::sn(uj`) sn(j`) () with, U and ` real numbers and `. It can now be shown using the formulae (A5) that all the ratios (w j ; w k )(w l ; w m) are functions of and `, but not of U and that dn(j`) with the inverse relations sn (j`) + ; ; cn(j`) dn(j`) (3) ` ; ; ; : (4) So when and ` are xed, transfer matrices with dierent values of U commute. In terms of these variables the dimensionless free energy per lattice site, is given [, 9, ] by (w w w 3 w 4 ); ln(w + w ) ; where X n (x n ; q n ) (x n + x ;n ; z n ; z ;n ) nx n ( ; q n )( + x n ) x expf;k(`)g q expf;k(`)k(`)g z expf;uk(`)g: The variable q is the nome of ` dened by (A9). 4 The free energy of the Ising model (5) () From (3), (), (9) and () (K K K 4 ) T <T c, (K K K 4 ) T >T c, (7) ; tanh(k4 ) T <T c, tanh(k 4 )(K K K 4 ) T >T c. The transition surface corresponds to. K K 4, when, from (3), (9), (), (3) and (), () We now consider the case where w 4 U ; K(`): (9) From (), (4) and (7) sinh ` (K) T <T c, sinh ; (K) T >T c, and from () and (9) () z x q 4 expf;k(`)k(`)g: () Using the transformation k p` +` k ; ` +` () 4

5 it follows, from (A5) and (A), that From () K(k )(+`)K(`) K(k ) ( + `)K(`): (3) x expf;k(k )K(k )g (4) is the nome of k. From (3), (9), (5), () and (4) (K) where Since >< >: ;K ; X n ; ln[cosh(k)+]; h n (x) T <T c,(k > K c ), X n h n (x) T >T c,(k < K c ), (5) h n (x) xn ( ; x n ) ( ; x n ) n( ; x n )( + x n : () ) (7) it follows from (), (5) and (A) that where u J >< >: B(K) From () x dh n(x) dx ;+ B(K) K (k ) X n ; sinh(k) + cosh(k) + B(K) K (k ) cosh 3 (K) sinh(k)[sinh (K) ; ] x dh n(x) dx T <T c, X n x n xn + +xn ( + x n ) ; x 3n ( + x n ) 3 x dh n(x) dx T >T c, () (9) ; xn ( + x n ) +x 4n ; 4x4n ( + x 4n ) + 4x 3n ( + x n ) (3) ( + x n ) 3 and hence, from equations (A){(A4), (A7){(A9), X n x dh n(x) dx k K (k ) Since, from () and (), q ( + k ) ; ; k K 3 (k ) : (3) 3 k sinh(k)sech (K) (3) k ; sinh (K) cosh T 7 T c (33) (K) 5

6 by substituting from (3) and (33) into (3) and then into () we obtain, after substitution from (A4) for the integral form of K(k ), u ;J coth(k) 4 + ftanh (K) ; g Z d q ; k sin 3 5 (34) which istheformula for the internal energy of the Ising model on the plane square lattice at all temperatures [5]. Using (3) it is now straightforward to show that (K) ln[cosh(k)] + Z ln + q ; k sin d (35) satises (7) and (34). The expression (35) is one of the many equivalent forms [5] for the free energy of the plane square lattice Ising model. Appendix An advanced treatment of elliptic functions is given by Baxter [] and a comprehensive list of formulae is given by Gradshteyn and Ryzhik []. Complete elliptic integrals of the rst and second kind are denoted by K(m) and E(m) respectively. They satisfy the formulae K() E() E() (A) E(m)K(m )+E(m )H(m) ; K(m)K(m ) dk(m) dm E(m) ; m K(m) mm and we shall also need the integral form dk(m ) dm m K(m ) ; E(m ) mm (A) (A3) K(m) Z d p ; m sin : (A4) The Jacobian elliptic functions sn(ujm), cn(ujm) and dn(jm) are related by cn(ujm) p ; sn (ujm) dn(ujm) p ; m sn (ujm): (A5) The elliptic functions are doubly-periodic ([], p.99). For real u, sn(ujm) and cn(ujm) have period 4K(m) and dn(ujm) has period K(m). For present purposes it is the elliptic function dn(ujm) which is of particular importance. We shall need the following results Z K(m) du dn (ujm) E(m) Z K(m) Z K(m) du Z K(m) dv dn(ujm)dn(u ; vjm)dn(vjm) m K (m)+ du dn(ujm)dn(u + {K(m )jm) p +mf +(; m)k(m)g: (A) (A7) (A) Formula (A) is straightforward to prove using (A5), the substitution x sn(ujm) and the denition of an elliptic integral of the second kind. The results (A7) and

7 (A) are more dicult to prove and need the use of the formulae for dn(u vjm), ([], p.9). The nome x of m is dened by x expf;k(m )K(m)g: From (A) and (A3), dx dm x K (m)mm : In terms of x, dn(ujm) K(m) + K(m) X n x n +x n cos nu K(m) ([], p.9). Substituting u into (A) and using dn(jm) gives X n (A9) (A) (A) x n K(m) +xn ; 4 : (A) By substituting from (A) into (A) and noting that the cosines in (A) are orthogonal over the range of this integral, it follows that X n x n ( + x n ) E(m)K(m) ; : (A3) In a similar way, by substituting from (A) into (A7), X n x 3n ( + x 3n ) m K 3 (m) + K(m) ; : (A4) We now use a modied version m ; m +m m q ; m p m +m (A5) of the Landen transformation, for which K(m ) ( + m )K(m) E(m ) E(m)+m K(m) +m : (A) ([], p.9). It follows from (A9) that x is the nome of m. Replacing m by m in (A), setting u {K(m ) and using dn( {K(m )jm ) p +m,gives X x n ( + x n ) +x 4n K(m p ) +m ; n r +m K(m) ; (A7) and replacing m by m in (A3) gives X x 4n E(m )K(m ) ; ( + x 4n ) n 4 K(m)fE(m)+m K(m)g; : (A) Final, by replacing m by m in (A), substituting from (A) and performing the integration, it can be shown that X n x 3n ( + x n ) ( + x 4n ) 4 K(m ) p +m f + ( ; m )K(m )g; 4 r +m 4 K(m) f +m K(m)g; 4 : (A9) 7

8 Acknowledgements I am grateful to G.S. Joyce for many valuable discussions. References [] Suzuki M. and Fisher M.E., J.Math.Phys., (97), 35. [] Lieb E.H., Phys.Rev., (97),. [3] Lieb E.H. in Lectures in Theoretical Physics, Vol. XI (Eds. K.T. Mantappa and W.E. Brittin), Gordon and Breach, N.Y. (99). [4] Lieb E.H. and Wu F.W. in Phase Transitions and Critical Phenomena, Vol. (Eds. C. Domb and M.S. Green), Academic Press, London, (97). [5] Bell G.M. and Lavis D.A., Statistical Mechanics of Lattice Models: Volume, Ellis Horwood, Chichester, (99). [] Kadano L.P. and Wegner F.J., Phys.Rev. B4, (97), 399. [7] Wu F.Y., Phys.Rev. 3, (99), 4. [] Baxter R.J.,Exactly Solved Models in Statistical Mechanics, Academic Press, London, (9). [9] Baxter R.J., Phys.Rev.Lett., (97), 3. [] Baxter R.J., Ann.Phys. 7, (97), 93. [] Gradshteyn I.S. and Ryzhik I.M.,Table of Integrals, Series, and Products, 4th Ed. Academic Press, London, (9).

The 6-vertex model of hydrogen-bonded crystals with bond defects

arxiv:cond-mat/9908379v [cond-mat.stat-mech] Oct 999 The 6-vertex model of hydrogen-bonded crystals with bond defects N. Sh. Izmailian, Chin-Kun Hu and F. Y. Wu Institute of Physics, Academia Sinica, Nankang,