J . Phys. Chem. 1987. 91, 4970-4980
4970
and it is not clear whether the similarity is accidental. Finally, it is worth emphasizing that the conductivity .(w) of Figure 8 is an exact (with 5 . It is as well worth pointing out that it is due to the existence of relations of the form (2.7) that most of the quantities relevant to the solution of the model could actually be computed. What plays a crucial role in making the calculations both interpretable in combinatorial sense and accessible is indeed a set of identities, which are referred to as a R~gers-Ramanujan’s~~ (in that they include and generalize some well-known identities of combinatorics, due to these two authors). The importance of the latter is that they allow to write infinite sums-typically defining generating functions-as products (whose logarithm can then be easily handled). It is a recent result by Lepowsky, Milne, and Wilson25that all of these identities can be interpreted in terms of the representation theory of the Euclidean Kac-Moody Lie algebra Ai. More precisely their product side is essentially the principally specialized character of a standard module of this algebra. Resorting to the “vertex” differential operatorial realization26 of the action of the infinite-dimensional Heisenberg subalgebra 7f of Ai, to write the specialized character of the space of highest weight vectors of 7 f , the sum side of the identities turns out to be essentially the generating function for the dimensions of the homogeneous components of such a space. A second aspect of interest of the class of models considered here is that all of them (as, indeed, most of the two-dimensional classical and one-dimensional quantum exactly solvable lattice models) provide nontrivial solutions to the Yang-Baxter factorization equation^.^,^^ This interplay between the solution of a model and of its factorization equation has far-reaching geometric as well as algebraic roots. For a general n-vertex model, a configuration is defined by assigning to each bond of the (square) lattice a “color” out of a finite set of choices, and to each vertex, say a, an energy $ ( a ) , depending on the color of the bonds (iJ,k,l in clockwise order) issuing from it, out of a set of n choices. In general for a two-color (f)‘case, n = 24 = 16; however, in the cases whose solution is known, further restrictions, consisting of setting to zero the energy of some configurations, reduce this number to 6 or 8, and the number D of parameters, namely of actually different energies, to 6 or lower. The partition function for a lattice of N vertical and M horizontal rows is given by Z(p) = Tr (TM) where the 2” X 2’ matrix T is the transfer matrix
(2.9)
whose elements are labeled by the configurations ( p ) = (p(l),. . . , p (N ) )and (p’) = (p’(l),...,p’(N)j of the vertical bonds (of t y p e j and I , respectively). In (2.10) the sum is over all configurations of the horizontal bonds (of type i , k ) are the Boltzmann weights for all sites a l r..., aN in a horizontal row; periodical boundary conditions were assumed. Since changing all the Boltzmann weights by the same multiplicative factor does not change the free energyf= -(l/PMN) In Z in a significant way, one can think of the vector x whose components are these weights as a point in the projective space Pkl, and of T as a function over such space, T = T(x). Now, eq 2.9 implies that in order to solve the problem one should diagonalize T (indeed, in view of the thermodynamic limit, N,M m , find its maximum eigenvalue). It was Baxter’s fundamental insight to ask for deformations from x to x’ in the space of parameters, such that
-
(24) An extensive review of the partition identities can be found in Andrews, G. E. The Theory of Partitions, Encyclopaedia of Mathematics and its Applications; Rota, G. C., Ed.; Addison Wesley: Reading, MA, 1976. Generalized identities are in: Andrews, G.E. Pacific J . Math. 1984, 114, 267. (25) Lepowsky, J.; Wilson, R. L. Adc. Math. 1982, 45, 21; Inuent. Math. 1984, 77, 199. (26) For a review of these equations and their role in exactly solvable models see, e.g.: Thacker, H. B. Rec. Mod. Phys. 1981, 53, 253.
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987 4913
Artificial Intelligence in Computational Physics PYX), T(x’)l = 0
(2.12)
This leads to the Yang-Baxter equations
c
s![;]:[;p,ql]”l:](8+8’, q;]@p’) =
k(l ) , W )
c
k(l),k(Z)
sY[;]Y{;p’)q]i[;](fi+W si{;yi[;](w(2.13)
where the sums are over the bond colors, and 8,8’ are parameters. The idea is that the latter parametrize a curve (more precisely a variety), embedded in CP&’,such that when x, d a r e restricted to it the resulting transfer matrices commute as required by (2.13). When this curve exists,fcan be computed as a function of 8 on it by standard methods. This has led to the belief that the complete integrability of a model can be identified with the possibility of solving its factorization equationsz7(even though in principle there through which no such curves passes may exist points in CPD-’ at which the model is solvable). Once more this has far-reaching consequences. If one defines a lattice A, = Z TZ, using T as complex unit, T E H; it is a standard procedure in geometry to identify an elliptic curve with the complex torus E , = C/A7. If a,b E (l/n)Z, the 8-functions with rational characteristics (2.4) form a basis for the vector space of entire functions invariant under the operators S,, T,, where for real r,s and holomorphic f(z)
+
(SSncZ)
= fb + $1
( T r n ( z ) = exp(imrz + i2~rz)J(z + r ~ )
(2.14)
such that
fi [!I ( 2 , ~ )= (SbTafi r 8 1 k ) (2.15) It is a known result in geometry that if (ai, bi) is a set of coset representatives of [(14n)Z/Z]z in [(l/n)Zl2, 0 I i I nz - 1 , then the map @,:E, P“ defined by Z (..., 8[f[i]](nz, T), ...) (2.16)
--
is an holomorphic embedding of the elliptic curve into PflL1.The set @.,(E,) is an algebraic subvariety defined by a set of homogeneous polynomials, which generalize Riemann’s 8 formulae. It has been proved in a number of cases (in particular the Zz symmetric model, corresponding to n = 2) and conjectured for the general Z, symmetric model, that these formulae are precisely the Baxter-Yang equation^.^^*^^ On the other hand, the Riemann’s identities also have a group theoretical significance19in that there is a 8 identity corresponding to every rational orthogonal 2n X 2n matrix, and those equations which are independent (i.e,, inequivalent under the generic operation of the group of inner automorphisms of the group of rational orthogonal matrices) define by intersection just the curve of integrability for the model. One of the methods utilized in deriving the results discussed so far, in particular those concerning the vertex models, is the so-called corner transfer matrix t e c h n i q ~ ewhich ,~ generalizes the concepts used in getting to (2.9) and (2.10) by a different constructive definition of T. The lattice is divided into four adjacent quadrants, and a corner transfer matrix A(k),k = 1, ..., 4 is set up on each of them (A(k)){o,,{o\ =
c
IT
Sff(ct)
(2.17)
{configurations plaquettes internal spins u(cl)l ( i j , k , O
where the matrix elements are labeled by the configurations of the m spins on the quadrant boundary, (CY)and (u’),and they are set equal to zero if the central spin, shared by the two components of the boundary is not defined unambiguously, Le., u1 # ul’. Then (2.9) is replaced by Z = T r (A(’)A(z)A(3)A(4)] (2.18) What makes this approach particularly suggestive is that the explicit construction of the leads to expressing them in the form of a product of matrices Ui, i = 1, ..., m each acting only (27) Tracy, C. A. Physica D 1985, 140,253. (28) Cherednik, I. V. SOL..J . Nucl. Phys. 1982, 36, 320. (29) Tracy, C. A. Physica D 1985, IdD, 203.
on the ith plaquette, and equivalent to the identity everywhere else (a similar structure can be recognized also in the customary transfer matrix approach, but we refer here to the case having the most general and promising properties). The operators Vi depend on the parameter 8, identifying points in the space of parameters, and satisfy the following commutation relations:
Ui(8) U,(S’)= Uj(8’) Ui(8) V 8,8’ and li -f > 2 Ui(8) ui+l(d+8’)Vi($‘)= Uj+I(8’)Ui(8+S’) Ui+1(8) (i = 1, ..., m -1) (2.19) The second equation in (2.19) is essentially equivalent to (2.13). These commutations are intriguing for several reasons. The first one is that all solutions worked out so far to the factorization equations are such that Ui(0) = I. (2.19) shows that a different “boundary condition” for these equations could be that, for 19 = 8’ = 0, the U;s coincide with the generators of the braid group, which also satisfy the resulting commutation relation^.^" A byproduct of this observation is that a solution to eq 2.19 could derive from finding a representation of the Ui’s in an algebra related with the braid group a1geb1-a.~’ An example of this is the following solution of (2.19) Ui(8) = [I
+ J(fi,S)eil
(2.20)
where ei, i = 1, ..., m are the projection operators of the (finite dimensional) Hecke-von N e ~ m a n nalgebra, ~~ whose defining relations are eiej = ejei
if li - jl
>2
e; = ei eiei+’ei- 6ei = ei+leiei+l- 6ei+l
(2.21)
and the functionsA6,6) are the solutions of the functional equation
which are easily found in the form
(6=
-1
1/41
+ (1 - 46)1/2 - (1 - (1 - 46)’/2) exp(c(1 - 46)1/20) (6 < 1/41 tan ((c/2)(46
- 1)’/28)
f(8”) = 2(46 - 1)llz - tan ((c/2)(46 - 1)1&.9)
(6 > 74) (2.23)
where c is an arbitrary constant fixingf(0). Equations 2.20-2.23 are immediately recognized as the known T e m ~ e r l e y - L i e b ~ ~ representation of the square lattice Potts and staggered Ice-type (6-vertex) models. Analogous results hold for the general q-state Potts models.34 Indeed by this approach one can probably hope to find a general answer to the intriguing property, supported so far purely by numerical studies, that the real zeroes of the partition function for the latter models cluster around limit points. These limits, at which the Potts models are associated with the leading discrete (30) Artin, E. Abh. Math. Sem. Uniu. Hamburg 1925, 4 , 41. (31) Rasetti, M. ”Braid Group and Euclidean Lie Algebra in Statistical Mechanics of Spin Systems” in Group Theoretical Methods in Physics; Denardo, G., Ghirardi, G., Weber, T. Eds.; Springer Verlag: Berlin, 1984. Rasetti, M.; D’Ariano, G. ‘The Ising Model on Finitely Generated Groups and the Braid Group”; In Differential Geometric Methods in Mathematical Physics; Doebner, H. D., Henning, J. D., Eds.; Springer Verlag: Berlin, 1985. (32) Jones, V. F. R. Inuent. Math. 1983, 72, 1; Bull. A m . Math. S o t . 1985, 12, 103. (33) Temperley, H. N. V.; Lieb, E. H . Proc. R. SOC.London A 1971, 322, 251.
(34) Martin, P. P.; Rasetti, M., private communication, paper in preparation.
4974
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987
infinity of conformal field theories in two dimensions, occur when the q value corresponds to a generalized Beraha number;3s Le.
Now the representation (2.21) of the Hecke-von Neumann algebra turns out to be reducible just at the generalized Beraha numbers (including irrational^),^^ in which case the generating function for the dimension of the irreducible representation (in the therm) is a modular function! These issues modynamic limit, m all require further analysis and clarification (especially in view of the appearance in the set (2.24) of noninteger q’s) which can be only gained making reference to finite examples, to be worked out explicitly. Since the dimension of the irreducible representation of (2.21) for finite m is given by
-
4m - 2 c, = C,_,, m + l
CI = 1
(2.25)
one can see that already the almost trivial case m = 4 requires manipulation of matrices 14 X 14. A detailed analysis of any larger lattice cannot but resort to symbolic manipulation. It should be also noted in passing that the partition function 2 for a q-state Potts model is closely related to the dicromatic polynomial of the planar graph determined by the lattice.’ This opens two more tantalizing questions: on the one hand, writing the model in its antiferromagnetic version (negative coupling constant) and letting a), Z turns into the cromatic the temperature go to zero ( p polynomial for the graph: if, assuming q = 4, one could show that the positive contributions to Z in this case are numerically larger than the negative ones, one should have an alternative proof of the four-color problem t h e ~ r e m . ~ It ’ has been further shown by K a ~ f f m a nand ~ ~J o n e ~ ~that ~ , ~the ’ cromatic and dicromatic polynomials can be thought of as topological invariants for knots and links having the lattice graph as “universe”, i.e., as their planar projection not distinguishing between over- and underpasses. This leads us into the topological implications. Topology enters the picture in a somewhat subtler way than number theory, geometry, and algebra, and in a way encompasses all of them. First it controls the combinatorial structure of the models. The classical example, which will be thoroughly discussed apropos of the toy model to be solved later on, is the Ising model, for which the evaluation of the partition function is reconducted to finding the generating function for the number of closed loops one can draw on the lattice (in any dimension). More deeply, however, it provides the basic structure for higher dimensional models. Configurations for the latter are in one to one correspondence with spin structures for high (infinite in the thermodynamic limit) genus Riemann surfaces.40 Under the action of global diffeomorphisms of such surfaces, different classes of spin structures (there are 22g of them) cannot mix, whereas all the spin structures in each class mix. It is necessary for the Gibbs measure for the model to be invariant under global diffeomorphisms, Le., modular transformations (because the lattice lives in a Riemann surface). This on the one hand identifies the partition function as a combination of 6-functions, and on the other hand the requirement
-
(35) Friedan, D.; Qiu, Z.; Shenker, S . Phys. Reu. L e f t . 1984, 52, 1575. Cappelli, A.; Itzykson, C.; Zuber, J. B. Nucl. Phys. B 1987,28O[FS 181,445 Gepner, G.; Witten, E. Nucl. Phys. B 1986, 278, 493. (36) Martin, P. P. J . Phys. A Lett.: Math. Gen. 1987, 20, 539. (37) Appel, K.; Haken, W. Bull. A m . Math. Soc. 1976, 82, 71 1; Illinois J . Math. 1977, 21, 429. Appel, K.; Haken, W.; Koch, J. Illinois J . Math. 1977, 21, 491. (38) Kauffmann, L. Formal Knot Theory; Princeton University Press: Princeton, NJ. 1983. (39) Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W. B. R.; Millet, K. C.; Ocneanu, A. Bull. A m . Math. SOC.1985, 12, 239. (40) Rasetti, M. “Exact Partition Function for a Class of Statistical Mechanical Problems on Lattices Homogeneous under Finitely Presented Groups”, in NonPerrurbative Aspects of Quantum Field Theory: Julve, J., Ramon-Medrano, M., Eds.; World Scientific: Singapore, 1982. Rasetti, M. “Ising Model and Dimer Covering of Lattices Embedded in Manifolds with Genus > I ; in Selected Topics in Statistical Mechanics: Bogolubov, Jr., N N.. Plechko, V. N., Eds.; J.I.N.R.: Dubna, USSR, 1981.
Jacucci and Rasetti of invariance points out to the group of diffeomorphisms of the surface modulo the normal subgroup of diffeomorphisms isotopic to the identity4I (which is referred to as the mapping class group of the surface4*)as the natural tool to compute such functions (which can be written as determinants of an elliptic operator related to the transfer matrix). Not much is known about the mapping class group 9; however, much progress has been made recently toward obtaining a finite (minimal) presentation for it. In particular, resorting to its representations provided by the action on the homology of the surface,43 it has been shown that a presentation exists, in terms of the canonical basis of the first homology group of 9 assumed as generators, characterized by a set of defining relations which are all based on subsurfaces of genus at most t ~ o . ~ The ’ , ~general theory for the Ising model4I in this case reconducts the evaluation of the loop-generating function 3 (or of the partition function, which is simply proportional to the latter) to that of a determinant (more precisely the Pfaffian) of an opertor M which can be written as a linear combination of the Cegular representations of the generators of a central extension 9 of 9. Then In 3 =
Y2 In det M
=
Y2 Tr In M
9 = exp{y2Tr In M ] = exp(y2Tr In Ex&]
(2.26) (2.27)
k
where ( x k Jis a set of parameters depending on the coupling constants of the model as well as on the lattice coordination structure, and_gk denotes the regular representation of the kth generator of 9. Recalling that in the regular representation only elements of 5 equivalent to fI have a nonvanishing trace, at the right-hand side of (2.27) the logarithm can be formally expanded, and it turns out that only those terms in the expansion corresponding to words in the generators gkequivalent to the identity give a nonvanishing contribution to 3. Indeed, for finite lattice Tr In Ex& can be resummed to a (numerical) logarithmic function (a well-known result in enumerative graph theory).45 This maps the physical problem into a well-known question of combinatorial topology: given a group in terms of its finite presentation (namely a finite set of generators and relations), set up a closed finite algorithm allowing to identify and enumerate all words in the group equivalent, upon reduction by the defining relations, to the identity. This is known as the Dehn’s word problem.46 We want to point out here a few interesting features of this point of view, which make it especially appealing for handling by automatic manipulation. First of all the word problem structure is not typical only of the last category of models discussed; a finer, and more sophisticated, analysis of the results presented above, in particular the number-theoretical and geometric features of the thermodynamic functions and order parameters, would show that in all cases the supporting structure is that of a group-theoretical word equivalence. We like to remark that a fascinating and challenging characteristic of the approach thus emerging is its structure of an harmonic analysis performed without resorting to any explicit matrix representation of the group. The solution of the word problem starting from the group presentation by symbolic manipulation is in a way a new form of representation, based on ab initio definitions, of the group: a whole new realm of utilization (41) Rasetti, M. ”Ising Model on Finitely Presented Groups”, in Group Theoretical Methods in Physics: Serdaroglu, M., Inonu, E. Eds.; Springer Verlag: Berlin, 1983. (42) Birman, J.; Hilden, H. “OnMapping Class Groups of Closed Surfaces as Covering Spaces”, in Advances in the Theory of Riemann Surfaces; Princeton University Press, Princeton, NJ; Ann. Math. Sludies 1971, 66. Zieschang, H. Finite Groups of Mapping Classes of Surfaces; Springer Lecture Notes in Mathematics 875; Springer Verlag: Berlin, 1981. (43) Birman, J. Braids, Links and Mapping Class Groups; Princeton University Press: Princeton, NJ; Ann. Math. Studies 1975, 82. Miller, E. Y. J . DfJ Geometry 1986, 24, 1. (44) Hatcher, A.; Thurston, W. Topology 1980, 19, 221. Wajnryb, B. Isr. J . Math. 1983, 45, 157. (45) Harary, F., Ed. Graph Theory and Theoretical Physics, Academic: London, 1967. (46) Dehn, M. Math. Ann. 1911, 71, 116.
Artificial Intelligence in Computational Physics of computers in the field of group theory, going much beyond the applications in physics! Finally there seems to emerge as well a tremendous pattern of unification among the various models considered. A nontrivial analysis of the minimal presentation of the mapping class for instance, shows that there exists an exact sequence of groups which implies that 9 is indeed the extension of a braid group (of order 2g - 1) by a permutation group Szg. Thus the braiding structure characteristic of the vertex model factorization equations, and of the corner transfer matrices of the Potts models, is inherent in the three-dimensional Ising model as well. To conclude, we would like to mention, even without substantiating the relevant notions, a different context in which a physical problem has been translated into a word problem for some finitely presented group, to be thought of as a group of transformations of compact Riemann surfaces. It has been shown that not only the orbits of dynamical systems which correspond to geodesic flows on surfaces of negative curvature can be coded as words in a group acting transitively on the surface (a result due to Morse4'), but that such systems exhibit chaotic behavior if the word problem for a group which is the extension of by the fundamental group of the surface (and is a subgroup of the mapping class group) is undecidable in the sense of formal logic (Godel-Post-Church theorem4*). This observation opens new possible applications of the computer-aided word analysis to the field of deterministically chaotic dynamical systems.
3. An Explicit Example: the Ising Model on a Cayley Lattice The construction of models of physical systems exhibiting phase transitions has been, as mentioned before, one of the recurrent themes of statistical mechanics over a span of time which is now several decades. The main mathematical feature which characterizes a phase transition is the appearance of singularities in the thermodynamic functions vs. the state parameters. The corresponding mathematical models need therefore to be based on a deep underlying structure capable of relating the combinatorial complexityresponsible for the buildup of the singularity-to the physical properties which, in turn, from the mathematical point of view, are related to the topological and algebraic functions describing the system and its dynamical observables. The problem is so complex that physicists have often adopted the philosophy of preferring a model not too realistic but exactly solvable (from which to learn just the ways to complexity and singular behavior) to one describing very accurately physical reality but impossible to solve except by a number of approximations. It is from this sort of approach to phase transition theory that models such as the king model49(describing two-component alloys or spin ferromagnetic systems), the vertex modelsSo(crystals with hydrogen bonding, ferroelectricity), Potts and Ashkin-Teller models5' (higher spin ferromagnets, many component alloys), were generated.9 Statistical physics has indeed seen recently a very productive interplay between the notions coming from the exact results concerning such models and more heuristic types of approach, such as the renormalization group methodlo or the results coming from numerical computer calculations on systems large on a microscopic scale but still not of macroscopic size.S2 It is, e.g., (47) Morse, M. Symbolic Dynamics; mimeographed notes by R. Oldenburger of lectures given in 1937-1938; The Institute for Advanced Study: Princeton. 1966. ~ ~., NJ. ~ ~ . . ~, ..~ (48) Rasetti, M. Chaos as Godelian Undecidability; The Institute for Advanced Study: Princeton, NJ 1986, Agnes, c,;Rasetti, M. "World Problem Undecidability and Chaos", in Chaotic Behauiour in Nonlinear Systems; Vulpiani, A,, et al., Eds.; World Scientific: Singapore, 1987. (49) Ising, E. Z . Phys. 1925, 31, 253. (50) Lieb, E. H. Phys. Rev. 1967, 162, 162; Phys. Reu. Lett. 1967, 18, 1046; Phys. Reu. Lett. 1967, 19, 108. Sutherland, B. Phys. Rev. Lett. 1967, 19, 103; J . Math. Phys. 1970, 1 1 , 3183. Fan, C; Wu, F. Y . Phys. Rev. E 1970, 2,723. Baxter, R. J. Phys. Rev. Lett. 1971,26, 832; Ann. Phys. (N.Y.)1972, 70, 193; 1973, 76, 1, 25, 48. (51) Potts, R. B. Proc. Cambridge Philos. Soc. 1952,48, 106. Ashkin, J.; Teller, E. Phys. Rev. 1943, 64, 178. ~
~
~
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987 4975 due to this interplay that such relevant concepts (or conjectures) as scaling and universality have entered into play and are today the backbone of our understanding of critical phenomena, as discussed in previous sections. It is in the above perspective that we intend to show here how a computer apporach to some of these models, resorting to symbolic manipulation rather than numerical computation, can provide a new insight into the structure of phase transition theory. The model on which we shall focus our attention is the Ising m0de1.I~ The latter can be thought of as a model of a magnet made up of N (assumed even) molecules, each a microscopic magnetic dipole, constrained to lie on the sites x'(i), i = 1, ..., N, of a regular lattice A of coordination number q, and to point on either one of two opposite directions. Thus each molecule has two possible configurations labeled by a spin variable s:(~), i = 1, ..., N , ?(i) E A with values f 1. Let s = {s:(~)}:~denote a configuration of the whole magnet-there are 2N of them-and 4
c c Jas:(,)Sa(,)+ &
H = H ( s ) = -(l/q)
.F(i)fA
u=l
(3.1)
the corresponding energy, where interaction is assumed to occur only between nearest-neighbor spins (the vectors Z,'s characterize all the nearest-neighbor pairs of sites in the lattice). The model describes a ferromagnet, if J, > 0 for all CY'S. Notice that the J,'s are not necessarily different for different a's. The basic problem of statistical mechanics is to compute the partition function = Z(T) = exp[-H(s)/KgT] (3.2)
z
c S
where KBis Boltzmann's constant and T the temperature, from which most of the equilibrium properties can be derived. We will denote by K , the dimensionless quantities J,IKBT. We shall focus our attention on a method whereby the calculation of Z can be reconducted to a different algorithmic expression than its bare definition, which is particularly effective in the case of lattices which can be embedded in surfaces of topological genus zero. The first step based on the observation that exp(Kgf(,)s,(,!+,a) = cosh (K,) + sinh ( K , ) S ~ ( ~ ~reconducts ~ ( ~ ) + ~Z, to the generating function C((z,)) for closed loops on the lattice:
where n, denotes the total number of bonds on the lattice characterized by &, N((1,))is the number of closed, simple (no edge repeated) possibly disconnected loops of perimeter p = C,l, that one can draw on A having 1, sides parallel to ,Z, and t , = tanh K,. The second step, based on the construction of the so-called decorated latticeS3Ad, a lattice obtained from A by replacing all of its sites by an island of 3(q - 2) points arranged in a planar pattern of connected triangles respecting the bond configuration of A, consists in giving first to the bonds of Ad the orientation prescribed by Kasteleyn's theorem54(that will be briefly reviewed in the sequel), defining then th_eweighted incidence matrix M for the oriented decorated lattice &, the absolute value of the weights being 1 for the bonds of &\A, t , for those in Ad n A, with the signs induced by the orientation, and writing finally 4
Z = IPf MI
n (cosh K,)".
n= . I -
(3.5)
where Pf denotes the Pfaffian of M , which is well-defined since M is manifestly antisymmetric. Kasteleyn's theorem states that any finite graph 6 not including loops of multiple edges can be oriented in such a way that for a (52) For a review of some of the numerical results see e.g.: Domb, C.; Green, M. S., Eds.; Phase Transition and Critical Phenomena; Academic: New York, 1986; in particular Vol. 3, 5B, 6. (53) Fisher, M. E. J . Math. Phys. 1966, 7, 1776. (54) Kasteleyn, P. W. J . Math. Phys. 1963, 4 , 287.
4976
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987
Jacucci and Rasetti
V H
(6')
Figure 2. The graph of A. A possible labeling of the sites by elements y is shown in parentheses. Notice that the presentation of r is r = (A, BJA*,B3,(AB)').
Figure 1. The lattice A
-
& / A 3In , axonometric projection
given vertex u E V (Vis the set of vertices of the graph), every vertex of V - (0)has an odd number of edges of the oriented graph G wit! a positive end on it. There follows that for every circuit 6 C G the number of edges oriented in a given sense may be made of opposite parity to the number of vertices of G enclosed by Moreover this latter orientation guarantees that all te_rms of the expansion of the Pfaffian of the incidence matrix of G have the same sign. In the evaluation of Pf M one has to face two main difficulties: the first one, affecting all systems, whatever their dimension, is connected with the thermodynamic limit, namely the necessity of dealing eventually with infinite matrices. The second one is that for three-dimensional systems (notice that in deriving (3.3) and (3.4) no use was made of the dimensionality of A, and they hold in general), the analysis leading to (3.5) can indeed be repeated, but with the following changes. The lattice A can still be thought of as being embedded in a two-dimensional surface; however, the latter has topological genus g higher than zero. Kasteleyn's theorem holds even in such a case with a major modification: there are 22g inequivalent orientations of A, each of which contributes to the generating function G with its own Pfaffian. The main difficulty here is that in the thermodynamic limit also g typically becomes infinitely large, so that one is led to having to evaluate infinitely many determinants (IPf MI = (det of infinite matrices. It is in view of these tremendous difficulties that recently emphasis was given to finding solutions to the Ising model defined over lattices that are the Cayley graphs of some finitely presented group A,55 This has the advantage that on the one hand the group structure, by the tools of either harmonic analysis or the combinatorial algorithm connected with the word problem (which will be discussed with more detail further on), allows the reduction of rank of the matrices whose Pfaffian is to be computed to the dimensions of the irreducible representations of r (which are finite if r is finite). On the other hand in the three-dimensional cases there is a natural extension r' of r, which reconducts the calculation of the partition function Z to that of a single Pfaffian. I" is the discrete version of the mapping class group of the closed orientable surface S, of genus g in which A is embedded: the elements of r are orientation-preserving self-homeomorphism of S , modulo i ~ o t o p y . ~ '
e.
Figure 3. The decorated lattice & . associated to A, oriented according
to Kasteleyn's rule. Recent results on the presentation of I" and its intimate relation to the braid g r o ~ pallow ~ ~ serious , ~ ~ hopes about the possibility of obtaining in the not too distant future exact results concerning three-dimensional models. In the latter context it is the word problem approach which plays a crucial role. We shall not enter here in any detail about the 3-D case and will exemplify the methods discussed on a relatively simple planar finite case. The lattice A selected is isomorphic with the factor group of the modular group A (the group of unimodular fractional linear transformations on the upper complex plane (the Lobachevskij space H = (z 6 C Iz = x + iy, y > 0) endowed with metric ds2 = (dx2 dy2)/y2))by its normal subgroup of the principal congruence of index 3: A3= (y = (: 5) t AI a = d = A 1 (mod r = & / A 3has coordination q = 3); c = b = O(mod 3)]. A 3 and consists of 12 vertices, 18 bonds, and 8 faces (4 of which are triangles and 4 hexagons) (see Figure 1). It is interesting to notice that A is isomorphic with the decorated lattice of a tetrahedral lattice. A is embedded in a sphere S (genus g = 0); its sites Py can be labeled by the elements y of r (see Figure 2). Ad is obtained by replacing each Py by a triple of points a= a, b, c forming small geodesic triangles in S. 4,oriented according to Kasteleyn's rule (see Figure 3) is $1 transitive with respect to a group, namely the central extension J? of I', whose presentation reads
+
e,
f=
( 5 s ) (a) Lund, F.; Rasetti, M.; Regge, T. Commun. Math. Phys. 1976,51, 15; Teor. Mat. Fir. 1977, 11, 246. (b) Rasetti, M.; Regge. T. Riu. Nuooo Cimento 1981, 4, 1.
-
Recalling that
(A,BI - A 2 , - B3, - ( A B ) 3 )
(3.6)
is a discrete subgroup of the spatial rotation group
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987 4977
Artificial Intelligence in Computational Physics
which is readily checked to be identical with (3.9) and (3.10). The second method is based on the observation that the determinant in (3.8) can be written
TABLE I 3 6 7 8
3 3 4 4 5
0 3 3 4 3
4 4 12 3 12
9 10
11 12
5 6 6 7 8
12 4 18 12 3
4 3 4 4 4
+ tb2)d"R det ( A - W ( B + B 1 ) A + Z(B - B1)) (3.12) where w = ( t b ) / ( 1 + tb2), z = taw and for the sake of simplicity 2 = (1
we dropped the symbol 2: it is of course implied that when operators A , E, or B' are written one should think of their regular representation. Then, by the identity utilized in eq 26 of section 1,
TABLE I1 no. nonconnected
P = EaPe
components
I,
lb
N(la,lb)
2 3 2
6 9 6 7 8 9 12
0 0 3 3 3 3 0
6 4 4 12 12 4 1
6=3+3 9=3+3+3 9=6+3 10=7+3 11=8+3 12=9+3 12= 3 3 3 + 3
2 2 2 4
+ +
(3.7)
The first approach to (3.8) is based on the harmorjc analysis56 using the fermionic irreducible representations of I?-there are three of them inequivalent, all of dimension 2-and the determinant at the right-hand side of (3.8) is reconducted to a product of powers of three 2 X 2 determinants, and one finds
+ 4ta3RS3
2t:S4]
(3.9)
R = 1 + tb3,? ! A = fb(1
+ tb)
(3.10)
+
+ +
+
G(l,,tb) = 1 + 4?b3+ 6fb6 + 4ta3t2 + 12t;t: 3t,4tb4 12?;tb5 4t,9 8ta3tb6 i2t2tb5 12ta3tb7 18t;tb6 + i2ta3t2 12t2tb7 + t b I Z 4ta3tb9+ 3f,4tb8 (3.11)
+
+ +
[A-
+ B 1 ) A+ z ( B - E l ) ] ' )(3.13)
Tr [ A - w(E
+ B ' ) A + z ( E - E ' ) ] =/
Tr (An(l)C(2).,.An(Zi-l)e(ZI)...An(k-1)Cnfk)) (3.14)
MJ)I
where (n(i)l is the set of ordered partitions of the integer I, k (even) the (variable) number of elements of one such partitions, whose elements n(i) and n(k), and only those, are allowed to have value zero, and
C = z ( E - E ' ) - w(E Now, due to (3.7) A"O) = (-)"'O)A
= (-)"'"I
+ B')A
(3.15)
if n(j) = 2N(j)+ 1 is odd if n(j) = 2NG) is even
whereas
c"= (-w)rh'Zm(2i),rh'2m(%-l)(E Im(i)l
- ~ l ) m ( l ) [+ ( ~~ l ) ~ ] m ( 2 ) . . .
1-1
'=I
+
( E - B')m(2i-1)[(E B')AIm"') ... ( E - B1)m(h-')[(E B1)A]m(h)(3.16)
+
where (m(i))denotes here the set of ordered partitions of the integer n, and h (even) the number of elements of each partition (once more m ( 1 ) and m(h) can be zero). Moreover, m
We shall use the result (3.9) and (3.10) as a reference for checking the validity of those obtained by the two alternative computeraided methods we shall briefly discuss now. The first of these is implemented in two steps: (i) the direct enumeration of the connected closed loops on Ad, following the hybrid strategy, based on the classical backtracking search enhanced by a selective updated bound function, trimming in an optional way the search tree, discussed in ref 57, and (ii) the exhaustive enumeration by the same method of all possible loop configurations, including nonsimply connected loops, by logical union (superposition) of connected closed loops. The results are summarized in Tables I and 11. Table I shows the nonzero numbers of connected loops of perimeter p = 1, lb, with 1, sides of type A, and /b sides of type B. Table I1 gives the number of closed nonsimply connected loops (the sum over a! ranges over the nonconnected components; the effective structure of each component can be easily derived by comparing the first column of Table I1 with the possibilities listed in the first three columns of Table I). Inserting in (3.4), one finds
+
/=o
p=l
where
+
+
+ B-')A z(B - B')l)
On the other hand
tatb[%(B) - %?(B1)])11/2(3.8)
+
1/pC(-)(''')(DTr
w(E
+ tb2) %(A) - tb[%(B) + % ( B ' ) ] % ( A )+
G(ta,fb) = [R4
[A- W(B
P
+ tb2)(1/2)dlmRexp(-XC
Writing then the exelicit form of M in terms of the regular representation 2 of I', one has G(ta,fb) = ldet ((1
+ tb2)(1/2)d1mR eXp('/, Tr In -
= (1
0(3), and that U(3) = SU(2)/Z2, where Z2 = ( f l ) , is the cyclic group of order 2, one can immediately realize that the central extension we are interested in leads but to discrete subgroups of the spinorial bidimensional rotation group SU(2). We shall solve the Ising model on A in three different ways, two of which lend themselves quite naturally to a symbolic manipulation computer-aided approach. Denoting, for simplicity, by K,, Kb the coupling constants for bonds generated by generators A and E, respectively, we notice first that (3.3) and (3.4) are written in the presence case
z = (cash Ka cosh2 Kb)* G(ta,tb)
G(ta,tb) = (1
( 5 6 ) Guiot, C . A f t i Accad. Sci. Torino,C1. Sci. Mat. Fis. 1983, 117, 195.
( E - Bl)m =
(-)"-9'S(;)E'
(3.17)
= 2q - m - 3s; 0 5 t I2
(3.18)
q=O
where
s = [[(2q - m)/3]],
t
[[x]] denoting the largest integer Ix, and
[ ( E + B')Alm' = C ( E A ) " ( ~ ) ( B ~ .A.) ~. ( ~E) ~ ) ~ ( ~ ~ - ~ ) ( ~ ~ ~ ) ~ ( ~ ~ ) . . . l"(1)l
(3.19) where (u(i)J is the set of ordered partitions of the integer m', and j (even) the number of elements of each partition (u( 1) and u ( j ) are allowed to be zero). Finally, (EA)" = (-)[["/3118' ( B I A ) O= E',
(3.20)
r = u - 3[[u/3]]; 0 lr l 2
The whole reduction procedure can of course be iteratively repeated over and o_ver again. In the framework of the regular representation of r, only the identity has nonvanishing trace (Tr I = dim B ) ,while any other element has trace equal to zero. Then only the factors equivalent to the identity survive in the sum of the right-handed side (rhs) of (3.14). This maps the evaluation of (3.13) into the word problem for r. Notice that the procedure need not to be repeated for all integers I in (3.13, 3.14), because the sum over I at rhs of (3.13) has to be equal to thepth power
4978
The Journal of Physical Chemistry, Vol. 91, No. 19, I987
Jacucci and Rasetti enumeration of the connected loops on A, Le., point (i) mentioned in section 3, yielding the connected loops contribution to G(t,,t,). The exhaustive enumeration of all loop configurations for this lattice, including non-simply-connected loops, remains to be done. The mapping of the calculation of G(t,,tb) into the word problem for f , that will be addressed first in the following, is viable only for substantially smaller lattices,s8at the present level of development of the computational strategies involved. 4.1. Typographic Arithmetics and the Word Problem. The rhs of (3.13) of section 3 gives G(t,,t,) as an exponential function of a logarithmic power series in t,,tb while (3.11) shows that G(ta,tb)is a polynomial of finite order in the same variables. In fact, the power series can be resummed to logarithmic functions, as mentioned at the end of section 2, by collecting terms belonging to power series expansions of expressions corresponding to the traces of all possible nonperiodic words in A,B belonging to the identity class of the group. Furthermore, contributions from long words cancel exactly, leaving the polynomial expression 3.1 1. An upper limit to the value of I in (3.14) is then set by the number of lattice vertices. -In mapping the evaluation of (3.13) into the word problem for r, one therefore needs to consider all words in the equivalence class of the identity of well-defined maximum length. No restriction to irreducible words applies, yet words corresponding to periodic circuits on the graph must be avoided, as already included in (3.13) via the resummation of the logarithm. The symbolic manipulation approach is based on group generators and relators only, Le., does not rely on representations of the group. The exhaustive enumeration of the words belonging to the identity equivalence class I is achieved in two elementary steps: generation of all words, and testing of each word to decide whether it belongs to class I. The necessary completeness of the first step entails a combinatorial explosio? with word length. Existing works8 on the 12-vertex group r treated in section 3-limited however to irreducible words, yielding ca. 32 000 different ones-suggests that lattices with more than 30 vertices are yet out of reach. The decision algorithm consists in a finite sequence of elementary operations producing an answer to the question: “Is the word W in the class I?”. These operations are of a “typographic” kind, e.g., substitutions of a subset of symbols in the word with a different one@ . ’ The word-forming symbols are alphabetic letters representing the group generators. The decidability of the identity equivalence class, or Dehn’s problem, is solved by the equivalence principle of Magnus et for the finitely presented groups of interest here. It is based on operations that do not alter the equivalence class of words: insertion of a relator or of a trival word (e.g., AA-I) anywhere in the word, and/or removal of any identity word contained therein. A modification of Magnus principle, of decisive importance for an efficient implementation on a computer of these formal logics theorems, shows that the testing algorithm can be constructed on operations that reduce, or at least do not increase, the word length.58 In essence, the relators themselves are not used by their introduction anywhere in the word, followed by attempts to remove any identity word thus formed inside it, but certain substitution operators derived from them are utilized that achieve the same goal, yet never increase the word length. An efficient implementation of the principle requires that substitution operators be derived not only from the fundamental relators of the group, but also from all words of the class I shorter than the one under consideration. The most efficient substitution operators are of course those that substitute long strings of characters with short ones. A large body of knowledge is generated during the computer elaboration and must be managed, stored, retrieved, and updated. An “expert system” architecture is useful to this end, and the PROLOG language is appropriate to code the relative program.s8 An elucidating example is worked out in detail in ref 58 for the group T6 = ( A , B ;A3,B2,( A B ) Z )clearly , showing the combina-
0’47
Figure 4. The specific heat C, vs. temperature T, corresponding to the choice of coupling constants J, = KB,Jb= 0.02 KB. The broken line (- - -) shows the contribution to C, coming from the case J, = 0 (the lattice is the disjoint union of four triangles). The dot-broken line is the difference between C, and the latter. (-.-e-)
of a polynomial in z and w . We expect that the algorithm described should be conveniently implemented in terms of computer-aided symbolic manipulation. Preliminary results indicate that once more (3.9, 3.10) are recovered. It may be worth concluding with a few remarks pointing out where is the physical interest of the problem discussed above. Manifolds of negative curvature such as the Riemann surfaces on which we thought our lattices embedded, cannot be isometrically mapped on a Euclidean plane but at the expense of introducing various sorts of defects (such as dislocations or disclination lines etc.). Vice versa, one can, in several cases, think of disordered systems, such as amorphous solids or disordered structures, as regular ordered systems in a non-Euclidean space. The method described provides then a tool to deal with disordered systems by a technique formally very similar to that adopted for regular periodic structures. On the other hand, the specific example solved above has a characteristic feature which can be of sensible physical interest. The specific heat shows indeed an unexpected double-maximum feature when plotted vs. temperature (Figure 4). We expect that such a feature should disappear if one considered finer and finer triangulations consistent with the tetrahedral symmetry, in that the two maxima should merge in the thermodynamic limit. Yet this behavior of the specific heat can be an interesting clue to understanding phenomenological data for finite clusters.
4. Computer-Aided Approach to the Ising Model: Typographic Arithmetics and the Heuristic Search Approach to the Loop-Counting Problem Let us now turn our attention to the computational aspects of using each one of the two alternative computer-aided methods formulated in section 3. We shall discuss in some detail the logical structure of the problem as exhibited by each formulation, as well as related program implementation features appropriate to a computer problem solving approach. However, research in this field is at a preliminary stage and formidable computational problems, posed by lattices with large numbers of vertices, are far from solved. At the present state of the art, the largest lattice tractable has 60 v e r t i c e ~ . It~ has ~ ~ been ~ ~ partly solved by direct (57) Nicolussi, T.; Jacucci, G.; Rasetti, M. Counting Loops on Lattice Graphs: an A I Application to Computational Physics, submitted to the International Joint Conference of Artificial Intelligence, Milano, Italy, 1987. (58) Nicolussi, T . Thesis, Department of Physics, University of Trento, Italy, 1987. (59) Magnus, W.; Karrass, A,; Solitar D. Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, Dover: New York, 1976.
(60) Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden Braid Basic Books: 1979.
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987 4979
Artificial Intelligence in Computational Physics
38 10
37 36 15
18
28
Figure 5. Two views of A, the Cayley graph of the icosahedral group. Figure 6. Search tree, labels in accord with Figure 5.58
torial explosion with word length of the number of words to be tested. It should be pointed out that, when searching for irreducible words only, a major drawback of this approach, not shared by the topological approach based on graph representation to follow, is the apparent lack, for the time being, of an algorithm capable of identifying sterile words, Le., words that will not generate any irreducible word in I by addition of letters at their end. Absence of such an algorithm forces the computer into useless search. 4.2. Loop-Counting Solution for the Cayley Graph of Icosahedral Group. The calculation of the generating function G({t,)) is the central problem in this approach. It essentially reduces to the calculation of the numbers “((I,]) for a given lattice. A relatively simple planar finite case, yet well out of reach of the pencil and paper approach used in section 3, has been solved with computer along this route. In the following we shall heavily draw from that work. We shall specialize eq 1 3 to the case of equal J,. Then the generating function G is
G(t,l) = C N ( 1 ) t‘ I20
(4.1)
where N(1) is the number of different closed loops with 1 bonds that can be drawn on the lattice A with no repetitions, with the convention that “(0) = 1. Furthermore, we shall choose A to be the Cayley graph of the icosahedral group (Figure 5). It can be shown that there are 231different loop coverings of this graph,55bor -2 X lo9, Le., a very large number; only the slightly less laborious problem has been s ~ l v e d of ~ ~finding ~ ~ * the contribution G,(t,l) to G(t,l),defined by restricting the sum in eq 4.1 to connected loops only
G,(t,I) = CN,(I) t‘
(4.2)
I20
where N,(I) is the number of different closed connected loops with I bonds that can be drawn on the lattice A with no repetitions, with the convention that NC(O)= 0. This quantity can be used to obtain a good numerical estimate of G(t,l) by the following approximate relation+’
G(t,l)
-
exp(G,(t,O)
(4.3)
It will turn out in the present case that the number of connected loops is also very large, a few percent of the total, so that the exact solution should require an increase in effort of one order of magnitude only. To find all closed connected loops that can be drawn on the graph of Figure 5, and hence the 60 numbers Nc(1),I = 1, ..., 60, proceed as follows. The graph has 60 vertices connected by 90 near-neighbor bonds. Choose any vertex and assign to it the number 1 as a label. Find all paths starting from, and ending in, 1, touching any other vertex point at most once. Finally assign a multiplicity p to each such loop, with = 60/1, and I the number of points in the loop (to allow for the correct counting of loops insisting on all other vertices). A search technique is needed for the exhaustive enumeration of all closed paths to and from 1. An iterative back-tracking search was used57$58(favored with respect to recursion for need of a -~ ~~
~
(61) Uhlenbeck, G. E.; Ford, G. W . Lecture in Statistical Mechanics; American Mathematical Society: Providence, RI, 1963; Vol 1, p 40.
Figure 7. A path that cannot be closed.58
transparent control over search-interrupt and status-saving). The search visits a simple binary tree (see Figure 6), because each vertex of the graph is the venue of three bonds: one in and two out as the path proceeds. Label all other vertices on the graph with natural numbers 2, ..., 60. A path on the graph is identified by the vector whose natural number components describe the succession of labels of the vertex points visited, in the order, by the path. Sixty is the maximum dimension of the vector. Valid vectors do not possess repetitions of any vertex label in their components. However, closed paths are characterized by the last component being equal to the first, equal to 1. The search technique employs a pointer to the index of the component presently under consideration for addition to the vector. The allowed values of the component to be added coincide with the labels of the two vertices reachable by one of the two outgoing bonds from the vertex identified by the last added component of the vector. The computer code restricts the choice of the value of the component to be added using matrix data structures. The technique employed is a sort of indexed addressing typical to microprocessor assembler. The resulting pointer arithmetics has been most naturally implemented in C language. The search is conducted using “hybrid strategies”,62Le., a classical backtracking technique augmented by a suitable “bound functionn to trim the search tree. The bound function is applied following a “selective updating” scheme, in which a trade-off balance is reached between the time required to decide on branch cutting, and time spent in visiting sterile branches. One of the following two situations should limit the growth process of the vector describing a path: a. The label 1 is added as Ith component to the vector; this closes the path, and a new solution is found. No valid vector can have the first I components coinciding with the vector under consideration; as a consequence, the search should stop its way toward the leaves of the tree, and it should backtrack one or more steps toward the root to look for other solutions. b. The vertex identified by the presently added label cannot be connected to 1 via any path comprising only yet untouched vertices. We are therefore on a sterile branch from which we ~~
( 6 2 ) Pearl, J. Heuristic, Intelligent Search Strategies f o r Computer Problem Soluing; Addison-Wesley: Reading, MA, 1984.
4980
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987
1
2 3 4 5 6 7 8 9 IO 11
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
0 0 0 0 12 20 0 0 60 30 60 90 180 300 412 870 1320 2 700 4 920 9 138 16800 26 850 45 180 72 430 114300 180 960 274 860 417 030 630840 944 080 1371 120
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1 980600 2 855 300 3 990 570 5 475 312 7 387 650 9 726 060 12 420 990 15474 100 18806916 22 149 180 25 337 490 28 027 860 29 938 920 30 771 900 30391 500 28 570 560 25 492 450 21 561 240 I7 170806 12 717 500 8 745 660 5 5 15 800 3110880 1 547 880 656 820 226 990 60 390 10 320 1090 total 374 237 206
should backtrack (see Figure 7). While situation (a) is immediately identified, situation (b) requires the definition of a special identification procedure: a bound function. The bound function is of course useful only if it is capable of deciding whether there indeed is at least one solution attached to the present branch, without having to explicitly visit it all to the leaves through its various ramifications. A good bound function for this problem has been found57~58 by using the connectivity property of the subgraph obtained by considering the collection C of labels of yet unvisited vertices. In practice one removes from the collection C in succession the labels of those neighbors of vertex 1 that appear in C, then the labels of the neighbors of these appearing in C, and so forth as far as possible. Call x the label of the last component of the vector presently under consideration. If x has not been removed from C in the process, there is no path from 1 to x through unvisited vertices only, and the branch is sterile. Of course the procedure can be stopped with affirmative answer as soon as x is removed from C. When backtracking from a sterile branch, the bound function is called at each backtracking step toward the root. This bound function is useful because knowledge of the connectivity of a graph is a low-level information, of O(I),or even lower, albeit dependent on the distance of x from 1, if the removing procedure is stopped before its completion. The exhaustive search is of 0 (exp(l)) instead. However, since the time cost of evaluating the bound function is nonnegligible, it is necessary to employ a technique of selective updating,62 which avoids updating the function at each step. First of all, calls to the bound function are
Jacucci and Rasetti avoided whenever the exhaustive visit to the leaves of the branches attached to the node costs less than the evaluation of the bound function itself. Furthermore, the function, in general, is evaluated only after m steps of the tree search from the last success (event of type (a) above). Of course the event success is monitored at each step. The optimal value of m has been chosen operationally by monitoring the time needed to find the first 5000 different closed connected paths, using various values of m . The best value of m found is 300. The time gain with respect to evaluating the bound function at each step is estimated to be a factor of 10. Large savings of computer time are guaranteed by this technique especially in the case where the solutions are not evenly distributed in the search space but rather are condensed in groups. The result of the search consists in the enumeration of all different loops, each one with its particular form and weights. Table 111 summarizes the number of different connected loops of given length, I on the graph A.57258 These numbers, together with their partitions into the numbers of type A and type B bonds, permit an accurate numerical estimate of Z , through eq 4.1 and 4.3, and hence of the thermodynamic properties of the model for any desired temperature. The total number of loops found, -374 X lo6, has not come without some amazement on the part of the scientific community of mathematical physicits involved. The first few lines in Table 111 are easily verified by hand. The computer codes were written in LISP for clarity and then translated in C for fast execution. The search required 15 days on a station SUN 2/120. 5. Conclusions The present work is the first part of a kind of “manifesto” favoring “noble” uses, Le., other than numeric, of automatic computation in physics. A wealth of computer science languages and techniques are available to this end, often also enhancing clarity and simplicity. We have shown how in certain fields the mathematical complexity of physical problems naturally lends itself to a computer-aided approach featuring symbolic manipulation for the word problem or combinatorial enumeration in topology. We have also worked out in detail applications of this approach to the solution of the Ising model on a finite 3-Dlattice homogeneous under a finitely presented group. In so doing, the combinatorial complexity of the calculation fully reveals itself, requiring the implementation of sophisticated knowledge-based techniques to solve the problem for a reasonably large number of lattice sites. From this point of view, the topological approach based on the enumeration of closed loops on the graph has the advantage, over the typographic arithmetical treatment of the word problem, of more readily accessible information on the consequences of the deep-lying group structure. This information turns out to be invaluable in efficiently trimming the search tree. However, we feel that the present state of this research is but a good starting point for the quest of more efficient problem-solving strategies needed for more complex problems.
Acknowledgment. The examples of implementation on the computer schematically described in the text are drawn from the unpublished thesis work, ref 58, of Tullio Nicolussi to whom we express our gratitude.
