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Highly symmetric cellular automata and their symmetry-breaking patterns
Physica D 148 (2001) 255–271
Highly symmetric cellular automata and their symmetry-breaking patterns Hendrik Moraal∗ Institute for Theoretical Physics, University of Cologne, D-50937 Cologne, Germany Received 21 July 1999; received in revised form 2 May 2000; accepted 9 October 2000 Communicated by F.H. Busse
Abstract A highly symmetric cellular automaton (in one dimension) is defined by two properties: (i) its local rule is a completely symmetric function of the n cells of its surrounding and (ii) it is invariant with respect to a regular, Abelian group acting on its M-letter alphabet, which can then be identified with this group. The motivation for studying such systems is provided by the unsolved problem of partial symmetry breaking in spin models with such an Abelian symmetry. It is shown that the symmetries (i) and (ii) do not contradict each other if and only if n and M have no common factor. As examples, the four-letter alphabets, which may be identified either with the cyclic group C (4) or with the Klein group K(4) ∼ = S (2) ⊗ S (2) are studied for the standard n = 3 surrounding. It is shown that these automata show complicated patterns of broken symmetries. These give information on the corresponding spin models, the chiral 4-state clock model and the (symmetric and general) Ashkin–Teller models, respectively. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Cellular automata; Symmetry; Phase transitions; Spin models
1. Introduction Cellular automata (CAs) have been studied for a long time as models for complex dynamical systems (in discrete time) in physics, chemistry and biology [1–3]. These automata are defined by the simple rule that the state of cell at time t + 1 depends (arbitrarily) on the states of a limited number of neighbouring ones at time t. Nonetheless, such simple rules can be used to model very many aspects of nature [4], including certain approaches to artificial life [5]. Also, there are many CAs which are capable of universal computation [6]. Therefore, it may be expected that CAs can also be used to obtain information on the model systems of statistical physics. In the present article, this will be attempted for the class of classical, discrete spin models such as the Ising [7], Potts [8] or Ashkin–Teller [9] models. This type of model has a finite number M of states and is defined in such a way, that its ground state is M-fold degenerate. Therefore, it is expected that at low temperatures symmetry-breaking phase transitions occur. The symmetry group G of the pair spin interaction function E(i, j ) is given by the set of ∗
Tel.: +49-221-4704300; fax: +49-221-4705159. E-mail address: [email protected] (H. Moraal). 0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 0 ) 0 0 1 9 5 - 0
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all permutations g of the M states (i.e., g is an element of the set S(M), known as the symmetric group, of all M! permutations of M objects), which leave the energy function invariant: g ∈ G ↔ g ∈ S(M),
E(g(i), g(j )) = E(i, j )
(1.1)
for all states i and j . If the group G is transitive on the states, i.e., if for every pair of states i and j there is an element g ∈ G with g(i) = j , the M-fold degeneracy of the ground state follows [10]. If the symmetry group contains nontrivial proper subgroups, various phases with partly broken symmetry may intervene between the high-temperature phase (which has at least the full symmetry of the model) and the low-temperature one (in which no symmetry remains). Some information on these intermediate phases has been obtained from high-temperature expansions of the field-free free energy [11], but by the nature of this approach, the symmetries of the intermediate phases cannot be elucidated. In the following, some examples of this type of spin model will be given. More information on the construction and properties of such models are collected in Appendix A. The simplest one and the prototype of all is the Ising model [7], the simplest model of a ferromagnet or antiferromagnet described by localized spins. Here there are only two states (denoted by 0 and 1) and the symmetry group consists of the unit element and of the interchange of the states. This is the group S(2) of all permutations of two objects. The energy function can be chosen so that one has E(0, 0) = E(1, 1) = 0 and E(0, 1) = E(1, 0) = E as follows from Eq. (1.1). It is usually more convenient to work with the matrix of Boltzmann factors: 1 for i = j, Ω(i, j ) = exp[−βE(i, j )] = (1.2) ω for i 6= j, where ω = exp(−βE) and β = 1/kT with the temperature T and Boltzmann’s constant k. For ω < 1, this describes the ferromagnetic case of the model, in which the spins tend to have the same value; in the antiferromagnetic case ω > 1, pairs of interacting spins tend to be in different states. This same form for this matrix, but now for an arbitrary number M of states, is obtained from a first generalization, the Potts model [8]. This has exactly two different energies, depending only on (in)equality of the states; the symmetry group is then the group of all permutations of the states, S(M). The ferromagnetic and antiferromagnetic cases are again described as for the Ising model. In the special case of M = 4, for instance, the matrix of Boltzmann factors is (see also Appendix A for the constructions necessary to obtain Eqs. (1.3)–(1.6)): 1 ω ω ω ω 1 ω ω (1.3) ΩPotts = ω ω 1 ω . ω ω ω 1 The (4-state) Ashkin–Teller models [9] comprise the symmetric and the general one. In the first case, the symmetry group is chosen as the group of permutations of the corners of a square, which leave this invariant. This group D(4) consists, therefore, of the rotations over multiples of 90◦ [as permutations of the four states, these are the unit e, (0123), (02)(13) and (0321)] and of the reflections with respect to the diagonals [(0)(2)(13) and (1)(3)(02)]. The notation D(M) is used generally to denote the symmetry group of all rotations and reflections of a symmetric polygon with M vertices, called a dihedral group. Similarly, the rotations alone define the cyclic group, denoted by C(M). The symmetric Ashkin–Teller model (SAT) has two independent Boltzmann factors: 1 ω1 ω2 ω1 ω1 1 ω1 ω2 (1.4) ΩSAT = ω2 ω1 1 ω1 . ω1
ω2
ω1
1
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The general Ashkin–Teller model (GAT) has a subgroup of D(4) as its symmetry group: {e, (01)(23), (02)(13), (03)(12)}. This group (called the Klein group) K(4) is the same as the symmetry group of a pair of uncoupled Ising spins: K(4) ∼ = S(2) ⊗ S(2), where the direct product sign ⊗ expresses the independence of the spins. This model has three independent Boltzmann factors: 1 ω1 ω2 ω3 ω1 1 ω3 ω2 (1.5) ΩGAT = ω2 ω3 1 ω1 . ω3
ω2
ω1
1
It is possible to interpret this model in terms of two (pair-interacting) Ising models with Boltzmann factors ω1 and ω2 , respectively, which are coupled to each other by a four-spin interaction which is related to ω3 . A model with a cyclic group can have a nonsymmetric energy or Boltzmann factor matrix; such models are called chiral M-state clock models. For the case of the cyclic group C(4), there are again three independent Boltzmann factors, but now arranged in an asymmetric fashion: 1 ω1 ω2 ω3 ω3 1 ω1 ω2 (1.6) ΩChiral = ω2 ω3 1 ω1 . ω1
ω2
ω3
1
In all of the three last cases, there are nontrivial subgroups (all isomorphic to the Ising model group S(2)), which may lead to intermediate phases, see also Appendix A. In the cases of the SAT and of the chiral 4-state clock model, the same subgroup {e, (02)(13)} occurs, whereas for the GAT model there are in addition the subgroups {e, (01)(23)} and {e, (03)(12)}. Once the Boltzmann factor matrix of a spin model is known, the field-free partition function of this model on a finite part LN of a lattice with N spins is given by XY Ω(il , im ), (1.7) Z0 (LN ) = {ik } [l,m]
where the summation is over all possible values of the N spins and the product is over all pairs of interacting spins, e.g., all pairs of nearest neighbours. In case there is also an external, homogeneous field given by a function F (i), which is essentially arbitrary, this corresponds to a Boltzmann factor A(i) = exp[−βF (i)] and the partition function becomes ZF (LN ) =
XY
Ω(il , im )
{ik } [l,m]
N Y
A(in ).
(1.8)
n=1
The free energy f per spin is, in the thermodynamic limit, given by −βf = lim N −1 ln ZF (LN ).
(1.9)
N →∞
As is well known, all thermodynamic functions can in principle be obtained from this by differentiation with respect to the temperature or to the field parameters. The mean value of the density of state i, for instance, is given by N Y X X Y δ(i, is ) Ω(il , im ) A(in ) (1.10) ρ(i) = lim lim N −1 ZF (LN )−1 F →0 N→∞
{ik }
s∈LN
[l,m]
n=1
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Fig. 1. The phase diagram of the SAT model on the square lattice, schematic after Ref. [11]. The points marked I are Ising-like phase transitions, P denotes the 4-state Potts model phase transition. The straight line IP is the self-dual one.
in the limit of zero external field. The deviations of these densities from their common high-temperature value ρ0 (i) = M −1 are the order parameters of the spin model; of these, M − 1 are linearly independent. Higher order distribution functions may be defined similarly. The symmetry group of a phase consists of all permutations from S(M), which leave all distribution functions invariant. All of the models that have been studied in some detail are such that their symmetry group is an extension of an Abelian group A which has exactly M elements (i.e., which is regular), since this implies the existence of a duality transformation [10]. This is an exact transformation mapping the low-temperature part of the phase diagram onto the high-temperature part and vice versa. It follows that, if there is only one phase transition boundary, this must be the self-dual line (or, more generally, a hyperplane), which is left invariant by duality. In the examples above, the cyclic group C(4) of the chiral 4-state clock model and the Klein group K(4) of the GAT model are already regular and Abelian, whereas the groups D(4) and S(4) of the SAT and of the 4-state Potts models, respectively, contain both of these regular groups. An example of the fact that part of the self-dual line is a phase boundary, see Fig. 1, which shows the phase diagram of the SAT model obtained from a high-temperature expansion. In view of the above, the aim of the present article is to study CAs with rules that are invariant with respect to such an Abelian group and obtain possible broken symmetries from their long-time evolution. This, however, implies that these CAs must be defined for an M-letter alphabet (to be identified with A), so that there will be a gigantic number of rules to inspect. Although the proposed invariance will reduce the number of rules, this will seldom suffice to make a full enumeration of the possible rules feasible. Therefore, the rules must be restricted further by another type of symmetry, which can only be due to the invariance of the rule with respect to a rearrangement of its arguments. Maximal symmetry in this latter sense is achieved by taking the rule as completely independent of the sequence of its arguments. The totalistic [12] rules are in this class. This paper is organized as follows. In Section 2, the question as to the existence of CAs with rules which are invariant under all permutations of their n arguments and also with respect to an Abelian, regular (and then necessarily transitive) group A, which can be identified with the alphabet, is answered in the affirmative if n and M have no common factors. This phenomenon is illustrated by two examples. The number of free parameters for such a CA rule is given generally and shown to be rather small if n and M are not too large. A number of more technical results needed for these statements is collected in Appendix B. It is also shown that the existence of a nontrivial group of automorphisms of A can reduce the number of rules that have to be studied even further. In
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Section 3, the highly symmetric CAs for n = 3 and M = 4 are considered as examples. There are two possible choices of the group A here. The corresponding spin models are the 4-state chiral clock model on the one hand and the symmetric and GAT ones on the other. Also, the 4-state Potts model is obtained as a special case. These models have already been used above as examples of the general case. It is shown that the subgroup structure of the symmetric group S(4), together with the knowledge of the automorphism groups of these subgroups, leads to only 172 nonrelated rules. The question as to the invertibility of these CAs is solved by an algorithmic technique. In Section 4, the symmetries (if any) remaining after a large number of iterations of these rules are given for all these possibilities. These results have been obtained by simulation. They are, insofar this is possible, compared to the symmetry-breaking patterns for the corresponding spin models. This allows for the certain identification of the intermediate phases of the Ashkin–Teller models. The chiral 4-state clock model promises to be very interesting, since the associated CAs show a rich symmetry-breaking pattern.
2. Highly symmetric cellular automata The time evolution of a general CA is given by a local rule of the form ai (t + 1) = ϕ(ai−r (t), ai−r+1 (t), . . . , ai+s−1 (t), ai+s (t)).
(2.1)
A special case of this are the popular r = s = 1 rules of the form ai (t + 1) = ϕ(ai−1 (t), ai (t), ai+1 (t)),
(2.2)
where the state of a cell at time t + 1 depends on its own state and those of its immediate neighbours at time t, i.e., ai (t) is the state of cell i at time t, which is an element of a finite alphabet A with M letters. Since it is the purpose here to find rules which are invariant with respect to a regular, Abelian group, it is convenient to identify this group with A. As an example, let the alphabet be simply {0, 1} and let the symmetry group be the Ising model one, S(2), the only nontrivial element of which exchanges the two states everywhere. This means that if ϕ(0, 0, 1) = 1, say, is a rule, then also the form with 0 and 1 exchanged: ϕ(1, 1, 0) = 0. This can be expressed concisely as ϕ(a, b, c) = 1 + ϕ(1 + a, 1 + b, 1 + c),
(2.3)
where all additions are done modulo 2. If one writes σ for the interchange of 0 and 1, this can be written alternatively as ϕ(a, b, c) = σ [ϕ(σ (a), σ (b), σ (c))].
(2.4)
Generally, the invariance of a rule with respect to a permutation σ ∈ S(M) of the alphabet is described by (here n = r + s + 1 is the extent of the surrounding of a cell which influences its state at the next time step) σ −1 [ϕ(σ (a1 ), σ (a2 ), . . . , σ (an ))] = ϕ(a1 , a2 , . . . , an ).
(2.5)
For the special case of the Abelian group A this becomes ϕ(a1 + α, a2 + α, . . . , an + α) = α + ϕ(a1 , a2 , . . . , an ),
(2.6)
where the (commutative) group operation is written as addition. By the transitivity of A, this implies that the free parameters of the local rule can be chosen as the values ϕ(0, b1 , . . . , bn−1 ). In particular, it will always be assumed
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that there is a special state, to be identified with the unit element 0 ∈ A, such that ϕ(0, 0, . . . , 0) = 0 holds; this is a slight generalization of part of the notion of legality as defined by Wolfram [13]. It follows that one has ϕ(a, a, . . . , a) = a
∀a ∈ A.
(2.7)
The second ingredient of a highly symmetric CA is the complete symmetry of the rule with respect to an arbitrary permutation of the arguments: ϕ(aπ(1) , aπ(2) , . . . , aπ(n) ) = ϕ(a1 , a2 , . . . , an ) ∀π ∈ S(n).
(2.8)
For the case of r = s = 1, this means that the following chain of equalities must hold: ϕ(a, b, c) = ϕ(a, c, b) = ϕ(b, a, c) = ϕ(b, c, a) = ϕ(c, a, b) = ϕ(c, b, a).
(2.9)
As shown in Appendix B, a highly symmetric CA can exist only if M and n have no common factors, expressed by the fact that the greatest common divisor (M, n) = 1. This phenomenon is already evident for two simple examples. In both of these, the alphabet is simply {0, 1}, identified with the group of the Ising model S(2): 0 + 0 = 0, 0 + 1 = 1 + 0 = 1, 1 + 1 = 0. If n = 2, one has: 1. Eq. (2.7) implies ϕ(0, 0) = 0, ϕ(1, 1) = 1; 2. the symmetry with respect to the exchange of the arguments (Eq. (2.8)) implies ϕ(0, 1) = ϕ(1, 0); 3. the S(2)-symmetry of Eq. (2.6) gives ϕ(0, 1) = 1 + ϕ(1, 0), which is an obvious contradiction. For the case n = 3, however, one finds: 1. ϕ(0, 0, 0) = 0 and ϕ(1, 1, 1) = 1 by Eq. (2.7), 2. a = ϕ(0, 0, 1) = ϕ(0, 1, 0) = ϕ(1, 0, 0) and b = ϕ(1, 1, 0) = ϕ(1, 0, 1) = ϕ(0, 1, 1) by Eq. (2.8) and 3. Eq. (2.6) is now compatible with this provided that b = a + 1 holds. Therefore, there are only two highly symmetric CAs for this case, corresponding to a = 0, 1. It is now assumed that (M, n) = 1 holds; the number of independent parameters is then severely restricted by Eqs. (2.6) and (2.8). From the explicit form of Eq. (B.3) as derived in Appendix B, the results for the first few values of n are collected in Table 1. It follows that the smallest Abelian symmetry groups with nontrivial subgroups which can be studied for even values of n have M = 9, 15, 21, . . . , elements, since for M a prime, no partially broken symmetries are possible, see Appendix A. Since little is known about the corresponding spin models (a few results for M = 9 only are available [10,11]), the cases for even n will not be considered in the following. For n = 3 and M = 2, the CAs are the standard ones studied already by Wolfram [13,16]. There are only two highly symmetric ones as derived in the examples above, which are rules 150 and 232 in Wolfram’s code. Of these, rule 150 describes an invertible CA, which preserves the S(2) symmetry, whereas rule 232 corresponds to a symmetry-breaking, noninvertible one [13]. Since the Ising model in more than one dimension also shows a symmetry-breaking phase transition, rule 232 could be called “the” CA rule (with n = 3) of the Ising model. The case n = 3, M = 4, the next simplest case with the possibility of partial symmetry breaking, is treated extensively in the next section. Simple complete symmetry breaking also occurs in the cases n = 5, M = 2 (four special totalistic rules [12]) and n = 4, Table 1 []The numbers of highly symmetric CA rules for different extents of the surrounding n
N(n, M)
2
1 2 (M − 1) 1 6 (M − 1)(M + 4) 1 2 24 (M − 1)(M + 7M + 18) 1 2 120 (M − 1)(M + 6)(M + 5M
3 4 5
Restriction M odd (M, 3) = 1 M odd + 16)
(M, 5) = 1
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M = 3 (81 rules). These are not further studied here. Other partially broken symmetries can only be obtained for cases where there is, again, a plethora of rules. If a highly symmetric CA is given, then associated ones are defined by an arbitrary permutation σ ∈ S(M) as (see also Eq. (2.5)): ϕσ (a1 , . . . , an ) = σ (ϕ(σ −1 (a1 ), . . . , σ −1 (an ))).
(2.10)
Such an associated CA is also obviously symmetric with respect to all permutations of its arguments, but not necessarily with respect to A. It is not difficult to see that this latter symmetry is left invariant if and only if σ is an automorphism of A, i.e., σ (a + b) = σ (a) + σ (b)
∀a, b ∈ A
(2.11)
holds. Therefore, each highly symmetric CA has, in general, a number of associated ones, the number of which is given by the length of the orbit of this CA under the action of the automorphism group Aut(A). For a spin model with a transitive symmetry group containing A, these automorphisms give rise to symmetries of the partition function [10]. For the GAT model, for instance, this means that the partition function is a completely symmetric function of the three Boltzmann factors, whereas for the 4-state chiral clock model, the partition function is symmetric with respect to the interchange of ω1 and ω3 , see also Appendix A. Now all associated CAs of Eq. (2.10) are rather trivially related to the original CA. In particular, the set of configurations obtained after applying the global rule implied by ϕσ () simply consists of the σ -transformed configurations obtained from ϕ() and this holds also for an arbitrary number of iterations of the global rules. Therefore, it suffices to study symmetry breaking, invertibility and other global properties for only one of the highly symmetric CAs from such an orbit. 3. The highly symmetric CAs for the case M = 4 and n = 3 For the case M = 4, there are two regular, Abelian groups to be considered, namely the cyclic group C(4) and the Klein group K(4) ∼ = S(2) ⊗ S(2). As discussed in Section 1, the spin model corresponding to C(4) is the chiral 4-state clock model; if the interaction is symmetric (the nonchiral subcase), this reduces to the SAT model [9] with group D(4). The spin model with the Klein group as symmetry is the GAT model [9]. Both of these groups contain nontrivial subgroups isomorphic to S(2), which are necessarily nontransitive, so that partially broken symmetries are possible. From the second equation of Table 1 it follows that the number of highly symmetric CAs for the smallest possible surrounding, n = 3, is equal to 256 for both choices, so that a full study of these CAs is possible. (Actually, only 172 nonrelated CA rules have to be considered, see below.) The Ashkin–Teller models show three types of phases as found from high-temperature series for the square lattice [11]. From special cases and duality [17] as well as from the explicit solutions available for these models on Cayley trees [10], one expects these to correspond to phases with: (i) the full symmetry of the model or an even higher one in the high-temperature region of the phase diagram, (ii) completely broken symmetry in its low-temperature region and (iii) some kind of partially broken symmetry in certain in-between regions, related to the S(2) subgroup(s) of the model. The phase diagram of the SAT model is shown schematically in Fig. 1 in the plane of the two independent Boltzmann factors ω1 and ω2 for this model as defined in Eq. (1.4). For the GAT model with three independent Boltzmann factors (see Eq. (1.5)), the phase diagram is much more involved, but the three phases with their different symmetries, K(4) or larger at high temperatures, three intermediate pockets of partially broken symmetry and {e} for low temperature are again found [11]. In order to obtain the number 172 for the total number of unrelated CA rules, it is necessary to consider the transitive subgroups of S(4), whose subgroup structure is shown as in Fig. 2. The three realizations of C(4) are
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H. Moraal / Physica D 148 (2001) 255–271
Fig. 2. The subgroup structure of the transitive groups on four letters.
generated by the permutations: g1 = (0123), g2 = (0231) and g3 = (0312); these generate the groups Ci (4), i = 1, 2, 3, respectively. Of these, only one has to be studied, say C1 (4), since the others are conjugate groups, which lead to associated CAs (it suffices to take the σ ’s of Eq. (2.10) equal to the δ’s of Eq. (3.6)). The only nontrivial automorphism σ of this group maps g1 onto its inverse, so that one has σ = (0)(2)(13). The four parameters necessary to describe a highly symmetric CA based on C1 (4) are taken as ai = ϕ(0, 0, i),
i = 1, 2, 3,
b = ϕ(0, 1, 2).
(3.1)
To provide an effective coding for these CA rules, a rule given by the parameters (a1 , a2 , a3 , b) will be given the number a1 + 4a2 + 16a3 + 64b.
(3.2)
All ϕ(i, j, k) entries not directly given by those of Eq. (3.1) can be obtained from Eq. (2.6), where summation is now modulo 4, and (2.8). It is not difficult to see that the automorphism σ maps the parameters of Eq. (3.1) onto corresponding (primed) ones given as a10 = σ (a3 ),
a20 = σ (a2 ),
a30 = σ (a1 ),
b0 = σ (b) + 2 mod 4.
(3.3)
It follows that there are 16 invariant models given by (a1 , a3 ) = (0, 0), (1, 3), (2, 2) or (3, 1),
a2 = 0 or 2,
b = 1 or 3.
(3.4)
These σ -invariant models then have the D1 (4)-symmetry of the SAT model, see Fig. 2. Since these will also be obtained from K(4) symmetric rules (again, see Fig. 2), there are exactly 120 rules with C(4)-symmetry that have to be studied, one from each orbit of length 2 with respect to σ . These orbits, which all correspond to the chiral 4-state clock model, are listed in Table 2, ordered with respect to their broken symmetries as explained in the next section. The Klein group is normal in S(4); in terms of permutations it is given as K(4) = {e, (01)(23), (02)(13), (03)(12)}.
(3.5)
By its normality, its automorphism group is isomorphic to S(3), i.e., any permutation of its nonunit elements is an automorphism. When written additively, its elements are either the pairs (00), (01), (10) and (11) with addition modulo 2 in both entries or, equivalently, the numbers 0, 1, 2 and 3, with as addition the (bitwise) exclusive-or
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Table 2 The orbits of the C(4)-symmetric highly symmetric CA rules and the corresponding symmetries after many iterations Subgroup
Orbits
{e}
(0,128) (8,136) (17,179) (35,146) (53,189) (68,76) (193,240)
(1,176) (12,132) (18,163) (36,142) (57,185) (71,92) (194,224)
(2,160) (13,180) (32,130) (48,129) (65,112) (77,116) (195,208)
(3,144) (15,148) (33,178) (49,177) (66,96) (79,84) (196,204)
(4,140) (16,131) (34,162) (50,161) (67,80) (117,125) (210,227)
S(2)
(55,157)
(59,153)
(63,149)
(82,99)
S(2) ⊗ S(2) (non-transitive)
(9,184) (26,171) (43,154) (75,88)
(10,168) (27,155) (56,137) (90,107)
(11,152) (40,138) (58,169) (202,232)
(24,139) (41,186) (73,120) (218,235)
(25,187) (42,170) (74,104)
C(4)
(5,188) (69,124) (205,244)
(21,191) (86,111) (207,212)
(51,145) (197,252) (213,255)
(52,141) (198,236) (214,239)
(60,133) (203,216) (229,254)
D(4) (no long-range order)
(6,172) (22,175) (31,151) (45,182) (62,165) (87,95) (101,126) (201,248) (221,247) (237,246)
(7,156) (23,159) (37,190) (46,166) (70,108) (89,123) (102,110) (206,228) (222,231) (245,253)
(14,164) (28,135) (38,174) (47,150) (78,100) (93,119) (105,122) (209,243) (225,242)
(19,147) (29,183) (39,158) (54,173) (81,115) (94,103) (109,118) (215,223) (230,238)
(20,143) (30,167) (44,134) (61,181) (85,127) (97,114) (199,220) (217,251) (233,250)
operation, denoted by ⊕. Both these representations mirror the fact that this group is the symmetry group of a pair of noninteracting Ising spins. In the latter notation, the five nontrivial automorphisms are: σ1 = (3)(12),
σ2 = (1)(23),
σ3 = (2)(13),
δ2 = δ1−1 = (132).
δ1 = (123),
(3.6)
The five models associated with a rule ϕ() are numbered as ϕk (), k = 1, . . . , 5. The four free parameters can again be chosen as in Eq. (3.1); all other rule entries follow from Eq. (2.6), with addition understood as ⊕, and (2.8). The rules will again be coded as in Eq. (3.2). There can, in general, be no confusion with this coding arrangement, since the basic group will always be made explicit. If it is necessary to emphasize that a rule occurs as the same number in the two codes, this will be indicated explicitly or, in tables, by an asterisk. (k) The five related models have parameters ai , b(k) given by (1)
a2 = σ1 (a1 ),
(2)
b(2) = σ2 (b) ⊕ 1,
a1 = σ1 (a2 ), a3 = σ2 (a2 ),
(1)
(5)
a1 = δ1 (a3 ), (5)
a2 = δ2 (a3 ),
b(1) = σ1 (b),
(5)
(2)
a1 = σ2 (a1 ),
(3)
a2 = σ3 (a2 ),
(4)
a3 = δ1 (a2 ),
a1 = σ3 (a3 ),
(4)
b(3) = σ3 (b) ⊕ 2, a1 = δ2 (a2 ),
(1)
a3 = σ1 (a3 ),
a2 = δ1 (a1 ),
a3 = δ2 (a1 ),
(3)
(4)
b(5) = δ2 (b) ⊕ 1.
(2)
a2 = σ2 (a3 ),
(3)
a3 = σ3 (a1 ), b(4) = δ1 (b) ⊕ 2, (3.7)
From these, the following types of orbits are found to exist: 1. There are two orbits of length 1; these two models have numbers 192 and 249, respectively, and are also produced as D1 (4)-models based on C1 (4) (with the same rule numbers). These CAs have the full 4-state Potts model symmetry S(4).
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2. There is one length 2 orbit, consisting of the pair (222, 231). These are invariant with respect to the alternating group A(4) ∼ = K(4)C(3), the semidirect product of the Klein group with the subgroup of automorphisms generated by δ1 , see Eq. (3.6) and Fig. 2 or, alternatively, the group of all even permutations of four states. No spin model can have this symmetry group, since it is doubly transitive and immediately implies the Potts model symmetry [10], see also Appendix A. 3. There are 14 orbits of length 3 with D(4)-symmetry, all corresponding to the SAT model. The results are listed in Table 3. 4. Finally, there are 35 orbits of length 6 with the K(4)-symmetry of the GAT model. These are listed in Table 4. Altogether, the number of orbits, from which a representative has to be studied is 120(C(4)) + 2(S(4)) + 1(A(4)) + 14(D(4)) + 35(K(4)) = 172.
(3.8)
The invertibility of these CAs, i.e., the property that the global rule maps the complete set A? of words built from the alphabet onto itself, has been checked with the algorithm described in Ref. [18]. The only invertible ones are rules 249 (Potts model symmetry S(4)) and (54, 91, 173) (SAT D(4)-symmetry). Table 3 The D(4)-type orbits of the highly symmetric CAs based on K(4). The starred numbers are the ones in D1 (4), which are the same as the corresponding ones based on C1 (4). For these, the S(2) ⊗ S(2) broken symmetry contains the permutation (02)(13). The symmetries after many iterations of the global rules are indicated Subgroup
Orbits
{e} S(2) ⊗ S(2) (nontransitive) D(4) (no LRO) S(4) (no LRO, SRO)
(0,64∗ ,128) (48,72∗ ,129) (63,106∗ ,149) (6,83∗ ,172) (219∗ ,237,246) (54,91∗ ,173)
(9,113∗ ,184) (57,121∗ ,185) (193,200∗ ,240) (198,211∗ ,236)
(15,98∗ ,148) (207,212,226∗ ) (213,234∗ ,255) (201,241∗ ,248)
Table 4 The orbits of the K(4)-symmetric highly symmetric CA rules and the symmetries remaining after applying the global rules many times Subgroup
Orbits
{e}
(1,8,65,112,136,176) (3,12,66,96,132,144) (16,32,68,76,130,131) (18,36,71,92,142,163) (21,42,85,127,170,191) (25,41,117,125,186,187) (29,43,101,126,154,183) (194,195,196,204,208,224) (203,205,216,225,242,244)
(2,4,67,80,140,160) (7,14,82,99,156,164) (17,40,69,124,138,179) (20,34,79,84,143,162) (24,33,77,116,139,178) (28,35,78,100,135,146) (49,56,73,120,137,177) (199,206,210,220,227,228)
S(2) ⊗ S(2) (nontransitive)
(5,10,81,115,168,188) (26,37,93,119,171,190) (50,52,75,88,141,161) (53,58,89,123,169,189) (197,202,209,232,243,252)
(11,13,97,114,152,180) (27,45,109,118,155,182) (51,60,74,104,133,145) (59,61,105,122,153,181) (217,233,245,250,251,253)
K(4) (no LRO)
(22,38,87,95,174,175) (31,47,102,110,150,151) (214,215,223,230,238,239)
(23,46,86,111,159,166) (55,62,90,107,157,165)
D(4) (no LRO)
(19,44,70,108,134,147) (218,221,229,235,247,254)
(30,39,94,103,158,167)
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4. Broken, unbroken and extra symmetries For all orbits, one of the equivalent CAs has been simulated to ascertain the symmetries (if any) remaining after many iterations of the rule. It is assumed that the limits for the short- and long-range pair distributions exist: S(α, β) = lim hδai ,α δai+1 ,β i,
(4.1)
t→∞
where the average h·i is the spatial average over i, δa,b is the Kronecker delta, and L(α, β) = lim lim hδai ,α δai+k ,β i.
(4.2)
t→∞ k→∞
This then implies for the average state densities: X X S(α, β) = L(α, β). ρ(α) = lim hδai ,α i = t→∞
β∈A
(4.3)
β∈A
In practice, the simulations use a uniformly random start configuration and an adjustable number of iterations, after which it is assumed that the infinite time limit is practically reached; this relaxation phase must contain up to 20 000 iterations for some of the rules. After this phase, a time average over 250 iterations and a space average over 500 cells is performed to obtain both the short- and long-range distributions. For these latter, k in Eq. (4.2) is taken in the range 100–300, whereas the total number of cells is of the order of 1000. In this way, the symmetries can, in general, be read off unambiguously. If this is not the case, the number of initial iterations is increased and a numerical limit (5%) for the possible symmetries (all of which are assumed to have been found from the unambiguous cases) is used until a consistent result is obtained, which is correct within this limit. The results obtained in this way for the 120 orbits corresponding to the C1 (4)-symmetric rules (the group is generated by g1 = (0123)) are pertinent for the chiral 4-state clock model. They are listed in Table 2 and can be described as follows: C1: If the symmetry is completely broken, denoted by {e} in the “Subgroup” column of Table 2, all entries in the vector ρ(α) of Eq. (4.3) can be different. This immediately implies that the matrices S() and L() have (more or less randomly distributed) differing entries, so that there is short- as well as long-range order in this case. These CAs simulate the low-temperature behaviour of the spin model. C2: If only the symmetry g12 = (02)(13) is conserved, the matrices S() and L() both have the structure
a d S, L() = a c
c b d b
a c a d
d b , c b
a 6= b, c 6= d.
(4.4)
This is the S(2)-entry in Table 2. The vector of densities has the form (p, q, p, q),
p 6= q.
(4.5)
This is an intermediate phase for the 4-state clock model with symmetry a subgroup of the original one. C3: If, in addition to g12 = (02)(13), the permutations (02) and (13), which do not belong to C1 (4), are also symmetries of S() and L(), then these form a nontransitive symmetry group isomorphic to S(2) ⊗ S(2), denoted as such in Table 2. The pair distribution matrices are now symmetric, i.e., obtained from Eq. (4.4) by setting c = d. The vector of densities has the same form as Eq. (4.5). If this behaviour also occurs for the 4-state clock model, it corresponds to a different intermediate phases.
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C4: The symmetry of the pair distribution matrices is full C(4)-symmetry as in Eq. (1.6): a b c d d a b c S, L() = c d a b , b 6= d. b c d a
(4.6)
In this case, the vector of densities is simply ( 41 , 41 , 41 , 41 ), but there is chiral long-range order. This is a very interesting behaviour, which would correspond to a third intermediate phase of the 4-state clock model. C5: The short-range order shows D(4)-symmetry, i.e., it has the structure: a b c b b a b c S() = (4.7) c b a b , b 6= c. b
c
b
a
This structure is the same as that of the SAT model, see Eq. (1.4). If this is the case, no long-range order occurs, 1 for all α, β. In this case, the symmetry group of the short-range pair distribution again is larger L(α, β) = 16 than the basic group. These CAs simulate the high-temperature phase of the spin model. Similarly, the (SAT) D(4)-type orbits corresponding to K(4) show four types of symmetries as listed in Table 3: D1: Completely broken symmetry, {e}, the same as case C1, simulating the low-temperature behaviour of the spin model. D2: Nontransitive S(2) ⊗ S(2) symmetry as in case C3. Depending on the rule, these groups can also be realized as the conjugate subgroups {e, (01), (23), (01)(23)} or as {e, (03), (12), (03)(12)}. These CAs simulate an intermediate phase. D3: Short-range D(4) symmetry without long-range order as in case C5, the “high-temperature” CAs. D4: Complete absence of short- and long-range order for the invertible rules in the orbit (54, 91, 173). This is the only case for which extra symmetries occur in short-range order (full S(4)-symmetry). These CAs probably do not have an equivalent (“ultra-high” temperature) spin model phase. The CAs corresponding to the GAT model have K(4)-type orbits and fall in one of the four classes, see Table 4: K1: Completely broken symmetry (the low temperature case), the same as cases C1 and D1. K2: Nontransitive S(2)⊗S(2) symmetry with three possibilities as in case D2. Again, this group contains elements not in the original group and corresponds to an intermediate phase. K3: The short-range order matrix shows K(4)-symmetry (just as the GAT model, see Eq. (1.5)): a b c d b a d c S() = (4.8) c d a b , b, c, d all different, d
c
b
a
but there is no long-range order (high-temperature phase). K4: There is short-range D(4)-order without long-range order as in cases C5 and D3. There are again extra symmetries in the short-range order distribution function. This is probably not realizable in the Ashkin–Teller model. Finally, there remain a few orbits not listed in the tables: S1: Rule 192 has full S(4)-symmetry of the Potts model, but its time evolution breaks all as in cases C1, D1 and K1.
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S2: Rule 249 also has the Potts model symmetry; there is no short- or long-range order in this invertible case, as in case D4. This seems to be a general feature of invertible, highly symmetric CAs, since it also holds for Wolfram’s rule 150, see Section 2. A: Rules (222, 231) have A(4)-symmetry and do not correspond to a spin model. There is no long-range order for these, but the short-range order has the (larger) symmetry of the 4-state Potts model, obtained from Eq. (4.7) by setting a 6= b = c. This is the same structure as shown for the Potts model in Eq. (1.3). In this special case, however, it is not yet possible to distinguish between A(4)- and S(4)-symmetric short-range order, since a matrix with the former symmetry automatically has the larger one. Therefore, it is necessary to consider the three-cell distribution function T (α, β, γ ) = lim hδai ,α δai+1 ,β δai+2 ,γ i.
(4.9)
t→∞
The group S(4) is transitive on all triples (α, β, γ ) of pairwise different letters, whereas A(4) has two orbits, which are related by an odd permutation. A numerical check of the function of Eq. (4.9) shows conclusively that the values on these orbits are different, so that the short-range order symmetry is really only A(4). The possible combinations of short-range (SRO) and long-range (LRO) symmetry groups as found above are listed in Table 5. It should be emphasized, that this classification is an experimental one; there is as yet no insight into the behaviour of a rule from first principles. Such an insight may well be impossible to achieve in a computational fashion, see Ref. [19]. The entries describing the typical time evolution are observed for the majority of the corresponding rules, but there are always some cases not conforming to this scheme. Therefore, Table 5 is not meant to represent an extension of Wolfram’s classes [13]. The phase separation observed for most of the CAs with partially broken symmetry is to be understood as follows: there are two well-separated regions, one populated by a mixture of equal parts of two of the four states, the other by a similar mixture of the other two states. The boundaries between these two regions fluctuate in a chaotic fashion, but do not interpenetrate. An example is shown in Fig. 3. The absence of long-range order explains the chaotic behaviour of the corresponding rules. Only in the last case, however, the time evolution is purely random, as would be expected for invertible, highly symmetric CAs. Cases D1–D4, S1 and S2 can be compared directly with the phase diagram of the SAT model, Fig. 1. Cases D1 and S1 correspond to the low-temperature phase, case D3 to the high-temperature one (no long-range order). The intermediate phase marked “S(2)” in Fig. 1 then must correspond to the nontransitive S(2) ⊗ S(2) case D2. The phase separation observed for the CAs in this class is also expected to occur in this spin model. It is due to the fact that for ω2 > ω1 , states which differ by ±2 mod 4, e.g., (02) and (13) (as in Fig. 3) for D1 (4), tend to mix, because of the lower energy for such pairs as compared to the ones which differ by ±1 mod 4 (these are the pairs (01), (12), (23) and (30) in the example). This also explains why the broken symmetry appears where it does in Fig. 1. Cases D4 and S2 can only be realized in the extreme high-temperature limit, ω1 = ω2 = 1. Table 5 The combinations of SRO and LRO symmetries SRO group
LRO group
Cases
Typical time evolution
{e} S(2) S(2) ⊗ S(2) C(4) K(4) D(4) A(4) S(4)
{e} S(2) S(2) ⊗ S(2) C(4) S(4) S(4) S(4) S(4)
C1, D1, K1, S1 C2 C3, D2, K2 C4 K3 C5, D3, K4 A D4, S2
Periodic, short cycles Periodic, long cycles Phase separation Periodic, long cycles Chaotic Chaotic Chaotic Randomly chaotic
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Fig. 3. An example of the phase separation associated with a broken symmetry. The rule number is 9 based on C(4). The four panels show the distribution of states in the order 0, 1, 2, 3. The time direction is from top to bottom.
Similar considerations apply to the GAT model. Case K1 corresponds to the completely broken low-temperature phase, case K3 to the high-temperature one without long-range order. Since case K2 gives three realizations of the intransitive partially broken S(2) ⊗ S(2)-symmetry, the three pockets of intermediate phases found by high-temperature series expansions [11] must have these symmetries. Finally, case A could correspond to the high-temperature phase of the 4-state Potts model (all three ω’s equal), but the fact that its short-range order shows a smaller symmetry makes this identification somewhat dubious. This seems to be a case of a CA without spin model parallel. The above shows that results obtained from the symmetry-breaking patterns of highly symmetric CAs can contribute to an understanding of these patterns in the corresponding spin models. Unfortunately, no results from high-temperature series are available for a chiral C(4) spin model. This is regrettable, since cases C2 and C4, which show chiral asymmetry, are of particular interest in this context, see, e.g., Ref. [20]. This holds especially for the case C4, which has a chiral asymmetry which does not show itself in the state density vector.
Appendix A. Results on spin models To make this paper as self-contained as possible, a number of results concerning spin models that are used in the body of the text are collected in this appendix. More details can be found in the book [10]. Given a transitive group G operating on M states, a spin model can be constructed as follows: 1. Since E(i, i), the energy function for a pair of spins in identical states, does not depend on i [E(i, i) = E(g(i), g(i)) = E(j, j ) for some g ∈ G by transitivity], one can set E(i, i) = 0 for all i by a trivial shift of the zero of energy. This implies Ω(i, i) = 1 for all i. 2. Now one takes a directed or undirected “edge” (0, k) with k 6= 0 and assigns it an energy Ek . Then all edges obtained from this one by the permutations of G must have the same energy. Repeating this step, until no more edges (0, l) remain, gives the necessary graphs for reading off the number of independent energy parameters or Boltzmann factors. These graphs form an edge-disjunct decomposition of the graph with all possible edges (a nondirected edge can be identified with a pair of oppositely directed parallel edges), the complete graph K(M). The above procedure is illustrated in Fig. 4 for the cases treated in Section 1: 1. Starting with the edge (0, 1) (directed or undirected), the group S(4) of the 4-state Potts model produces all possible edges, so that there is only one energy parameter or Boltzmann factor Ω(i, j ) = ω for all i 6= j , compare Eq. (1.3).
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Fig. 4. Examples of the derivation of the number of independent energy function values or Boltzmann factors for a number of spin models with four states. See the text for details.
2. Starting with the undirected edge (0, 1) again, the group D(4) of the SAT model produces the square (0123); there is a second energy parameter related to the diagonals of the square, which are obtained from (0, 2). This model has then two independent energy parameters, see also Eq. (1.4). 3. For the Klein group K(4) of the GAT model, all three edges (0, 1), (0, 2) and (0, 3) produce different graphs, so that this model has three independent Boltzmann factors as in Eq. (1.5). Since all three graphs are isomorphic, the automorphism group of K(4) consists of all permutations of these three graphs, i.e., it is isomorphic to S(3). 4. Starting with the directed edge (0, 1), the cyclic group C(4) of the chiral 4-state clock model produces a directed square (0123). The (undirected) diagonals and the square with all directions reversed are produced from (0, 2) and (0, 3), respectively, resulting in Eq. (1.6). The two directed squares are again isomorphic, signalling that their exchange is an automorphism. It is to be noted that the symmetry group of the spin model constructed as above can be larger than the starting group. In particular, if the group G is doubly transitive, i.e., such that for every pair of pairs (i, j ) and (k, l) there is a g ∈ G with g(i) = k and g(j ) = l, then all edges are produced starting from (0, 1), so that the corresponding spin model is the Potts model. Now the alternating groups A(M) of the even permutations of M objects have this property, so that these cannot occur as spin model groups. Groups that can occur as spin model groups are called permissible if only undirected edges are used in their construction, else chirally permissible. For example, C(4) is chirally permissible, but its corresponding permissible group is D(4). It can be shown that partially broken symmetries are possible only for spin models with a nontransitive, normal subgroup H, i.e., a subgroup for which g −1 Hg = H holds for all g ∈ G. Since the lengths of the orbits of H must divide M for such a subgroup, no partially broken symmetries are possible if M is a prime. If A is regular and Abelian, then every proper subgroup is nontransitive and normal. This implies the results for C(4) and K(4) noted in Section 1. The SAT group D(4) has only the 180◦ rotations as a normal, nontransitive subgroup, i.e., the permutations {e, (0)(2)(13)}.
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Appendix B. Results for highly symmetric CAs In this appendix, two rather technical discussions concerning the existence and the number of free parameters of highly symmetric CAs are collected. Combination of Eqs. (2.6) and (2.8) give relations between the independent ϕ(0, b1 , . . . , bn−1 ) parameters: ϕ(0, aπ(2) − aπ(1) , . . . , aπ(n) − aπ(1) ) = ϕ(0, a2 − a1 , . . . , an − a1 ) + a1 − aπ(1) .
(B.1)
In order for a highly symmetric CA to be possible, this equation must not lead to a contradiction for all π ∈ S(n). Such a contradiction is only possible if the following set of equations has a solution for all k: aπ(k) = ak + β
for all k = 2, 3, . . . , n with β = aπ(1) − a1 6= 0.
(B.2)
A specific k will be in a cycle of π; let the length of this cycle be u. Applying Eq. (B.2) u times, i.e., for k, π(k), π 2 (k), etc., gives the condition that uβ = 0 holds. This must hold for all cycles of π , so that A must contain an element of order dividing the greatest common divisor of these cycle lengths. But such common divisors must be divisors of n, whereas the order of an element of A is a divisor of M. Therefore, a contradiction can only ensue if M and n have common factors. In fact, it will not be certainly possible to define a highly symmetric CA if this is the case: it is a well-known fact that every finite group G with order a multiple of a prime p, |G| = gpm , contains subgroups of orders p a , a = 1, . . . , m (for a = m, these are the Sylow subgroups, see, e.g., [14,15]). Now if M and n have a prime p as common factor, A contains an element of order p and there is certainly a permutation in S(n) with n/p cycles of length p. This shows that CAs satisfying Eqs. (2.6) and (2.8) are constructible if and only if the greatest common divisor of M and n is 1, (M, n) = 1. In order to find the number of free parameters in case (M, n) = 1 holds, let there be l different letters occurring in the n arguments of the rule, such that there are l1 groups containing a single letter, l2 groups of pairs of equal letters, . . . , ln−1 groups of (n − 1) equal letters, l = l1 + l2 + · · · + ln−1 . The case for index n, ln , does not have to be considered here, since there is no free parameter associated with the case that all entries are equal, see Eq. (2.7). The group S(n) does not mix different choices of the li, so that there are l!/(l1 ! · · · ln−1 !) possible realizations. M The number of ways to choose l different letters is . Since the invariance with respect to the Abelian group l provides a factor 1/M, the total number of free parameters is given by N (n, M) =
X 0 (M − 1)(M − 2) · · · (M − l + 1) , l1 ! · · · ln−1 !
(B.3)
l1 ,... ,ln−1
where the prime on the summation indicates the restriction to n arguments: n−1 X ili = n.
(B.4)
i=1
The results for n ≤ 5 are listed in Table 1. References [1] J. von Neumann, in: A. Burks (Ed.), Theory of Self-reproducing Automata, University of Illinois Press, Champaign, IL, 1966. [2] S. Wolfram, Theory and Applications of Cellular Automata, Advanced Series in Complex Systems, Vol. 1, World Scientific, Singapore, 1986. [3] H. Gutowitz, Cellular Automata, MIT Press, Cambridge, MA, 1991.
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Highly symmetric cellular automata and their symmetry-breaking patterns Hendrik Moraal∗ Institute for Theoretical Physics, University of Cologne, D-50937 Cologne, Germany Received 21 July 1999; received in revised form 2 May 2000; accepted 9 October 2000 Communicated by F.H. Busse
Abstract A highly symmetric cellular automaton (in one dimension) is defined by two properties: (i) its local rule is a completely symmetric function of the n cells of its surrounding and (ii) it is invariant with respect to a regular, Abelian group acting on its M-letter alphabet, which can then be identified with this group. The motivation for studying such systems is provided by the unsolved problem of partial symmetry breaking in spin models with such an Abelian symmetry. It is shown that the symmetries (i) and (ii) do not contradict each other if and only if n and M have no common factor. As examples, the four-letter alphabets, which may be identified either with the cyclic group C (4) or with the Klein group K(4) ∼ = S (2) ⊗ S (2) are studied for the standard n = 3 surrounding. It is shown that these automata show complicated patterns of broken symmetries. These give information on the corresponding spin models, the chiral 4-state clock model and the (symmetric and general) Ashkin–Teller models, respectively. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Cellular automata; Symmetry; Phase transitions; Spin models
1. Introduction Cellular automata (CAs) have been studied for a long time as models for complex dynamical systems (in discrete time) in physics, chemistry and biology [1–3]. These automata are defined by the simple rule that the state of cell at time t + 1 depends (arbitrarily) on the states of a limited number of neighbouring ones at time t. Nonetheless, such simple rules can be used to model very many aspects of nature [4], including certain approaches to artificial life [5]. Also, there are many CAs which are capable of universal computation [6]. Therefore, it may be expected that CAs can also be used to obtain information on the model systems of statistical physics. In the present article, this will be attempted for the class of classical, discrete spin models such as the Ising [7], Potts [8] or Ashkin–Teller [9] models. This type of model has a finite number M of states and is defined in such a way, that its ground state is M-fold degenerate. Therefore, it is expected that at low temperatures symmetry-breaking phase transitions occur. The symmetry group G of the pair spin interaction function E(i, j ) is given by the set of ∗
Tel.: +49-221-4704300; fax: +49-221-4705159. E-mail address: [email protected] (H. Moraal). 0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 0 ) 0 0 1 9 5 - 0
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all permutations g of the M states (i.e., g is an element of the set S(M), known as the symmetric group, of all M! permutations of M objects), which leave the energy function invariant: g ∈ G ↔ g ∈ S(M),
E(g(i), g(j )) = E(i, j )
(1.1)
for all states i and j . If the group G is transitive on the states, i.e., if for every pair of states i and j there is an element g ∈ G with g(i) = j , the M-fold degeneracy of the ground state follows [10]. If the symmetry group contains nontrivial proper subgroups, various phases with partly broken symmetry may intervene between the high-temperature phase (which has at least the full symmetry of the model) and the low-temperature one (in which no symmetry remains). Some information on these intermediate phases has been obtained from high-temperature expansions of the field-free free energy [11], but by the nature of this approach, the symmetries of the intermediate phases cannot be elucidated. In the following, some examples of this type of spin model will be given. More information on the construction and properties of such models are collected in Appendix A. The simplest one and the prototype of all is the Ising model [7], the simplest model of a ferromagnet or antiferromagnet described by localized spins. Here there are only two states (denoted by 0 and 1) and the symmetry group consists of the unit element and of the interchange of the states. This is the group S(2) of all permutations of two objects. The energy function can be chosen so that one has E(0, 0) = E(1, 1) = 0 and E(0, 1) = E(1, 0) = E as follows from Eq. (1.1). It is usually more convenient to work with the matrix of Boltzmann factors: 1 for i = j, Ω(i, j ) = exp[−βE(i, j )] = (1.2) ω for i 6= j, where ω = exp(−βE) and β = 1/kT with the temperature T and Boltzmann’s constant k. For ω < 1, this describes the ferromagnetic case of the model, in which the spins tend to have the same value; in the antiferromagnetic case ω > 1, pairs of interacting spins tend to be in different states. This same form for this matrix, but now for an arbitrary number M of states, is obtained from a first generalization, the Potts model [8]. This has exactly two different energies, depending only on (in)equality of the states; the symmetry group is then the group of all permutations of the states, S(M). The ferromagnetic and antiferromagnetic cases are again described as for the Ising model. In the special case of M = 4, for instance, the matrix of Boltzmann factors is (see also Appendix A for the constructions necessary to obtain Eqs. (1.3)–(1.6)): 1 ω ω ω ω 1 ω ω (1.3) ΩPotts = ω ω 1 ω . ω ω ω 1 The (4-state) Ashkin–Teller models [9] comprise the symmetric and the general one. In the first case, the symmetry group is chosen as the group of permutations of the corners of a square, which leave this invariant. This group D(4) consists, therefore, of the rotations over multiples of 90◦ [as permutations of the four states, these are the unit e, (0123), (02)(13) and (0321)] and of the reflections with respect to the diagonals [(0)(2)(13) and (1)(3)(02)]. The notation D(M) is used generally to denote the symmetry group of all rotations and reflections of a symmetric polygon with M vertices, called a dihedral group. Similarly, the rotations alone define the cyclic group, denoted by C(M). The symmetric Ashkin–Teller model (SAT) has two independent Boltzmann factors: 1 ω1 ω2 ω1 ω1 1 ω1 ω2 (1.4) ΩSAT = ω2 ω1 1 ω1 . ω1
ω2
ω1
1
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The general Ashkin–Teller model (GAT) has a subgroup of D(4) as its symmetry group: {e, (01)(23), (02)(13), (03)(12)}. This group (called the Klein group) K(4) is the same as the symmetry group of a pair of uncoupled Ising spins: K(4) ∼ = S(2) ⊗ S(2), where the direct product sign ⊗ expresses the independence of the spins. This model has three independent Boltzmann factors: 1 ω1 ω2 ω3 ω1 1 ω3 ω2 (1.5) ΩGAT = ω2 ω3 1 ω1 . ω3
ω2
ω1
1
It is possible to interpret this model in terms of two (pair-interacting) Ising models with Boltzmann factors ω1 and ω2 , respectively, which are coupled to each other by a four-spin interaction which is related to ω3 . A model with a cyclic group can have a nonsymmetric energy or Boltzmann factor matrix; such models are called chiral M-state clock models. For the case of the cyclic group C(4), there are again three independent Boltzmann factors, but now arranged in an asymmetric fashion: 1 ω1 ω2 ω3 ω3 1 ω1 ω2 (1.6) ΩChiral = ω2 ω3 1 ω1 . ω1
ω2
ω3
1
In all of the three last cases, there are nontrivial subgroups (all isomorphic to the Ising model group S(2)), which may lead to intermediate phases, see also Appendix A. In the cases of the SAT and of the chiral 4-state clock model, the same subgroup {e, (02)(13)} occurs, whereas for the GAT model there are in addition the subgroups {e, (01)(23)} and {e, (03)(12)}. Once the Boltzmann factor matrix of a spin model is known, the field-free partition function of this model on a finite part LN of a lattice with N spins is given by XY Ω(il , im ), (1.7) Z0 (LN ) = {ik } [l,m]
where the summation is over all possible values of the N spins and the product is over all pairs of interacting spins, e.g., all pairs of nearest neighbours. In case there is also an external, homogeneous field given by a function F (i), which is essentially arbitrary, this corresponds to a Boltzmann factor A(i) = exp[−βF (i)] and the partition function becomes ZF (LN ) =
XY
Ω(il , im )
{ik } [l,m]
N Y
A(in ).
(1.8)
n=1
The free energy f per spin is, in the thermodynamic limit, given by −βf = lim N −1 ln ZF (LN ).
(1.9)
N →∞
As is well known, all thermodynamic functions can in principle be obtained from this by differentiation with respect to the temperature or to the field parameters. The mean value of the density of state i, for instance, is given by N Y X X Y δ(i, is ) Ω(il , im ) A(in ) (1.10) ρ(i) = lim lim N −1 ZF (LN )−1 F →0 N→∞
{ik }
s∈LN
[l,m]
n=1
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Fig. 1. The phase diagram of the SAT model on the square lattice, schematic after Ref. [11]. The points marked I are Ising-like phase transitions, P denotes the 4-state Potts model phase transition. The straight line IP is the self-dual one.
in the limit of zero external field. The deviations of these densities from their common high-temperature value ρ0 (i) = M −1 are the order parameters of the spin model; of these, M − 1 are linearly independent. Higher order distribution functions may be defined similarly. The symmetry group of a phase consists of all permutations from S(M), which leave all distribution functions invariant. All of the models that have been studied in some detail are such that their symmetry group is an extension of an Abelian group A which has exactly M elements (i.e., which is regular), since this implies the existence of a duality transformation [10]. This is an exact transformation mapping the low-temperature part of the phase diagram onto the high-temperature part and vice versa. It follows that, if there is only one phase transition boundary, this must be the self-dual line (or, more generally, a hyperplane), which is left invariant by duality. In the examples above, the cyclic group C(4) of the chiral 4-state clock model and the Klein group K(4) of the GAT model are already regular and Abelian, whereas the groups D(4) and S(4) of the SAT and of the 4-state Potts models, respectively, contain both of these regular groups. An example of the fact that part of the self-dual line is a phase boundary, see Fig. 1, which shows the phase diagram of the SAT model obtained from a high-temperature expansion. In view of the above, the aim of the present article is to study CAs with rules that are invariant with respect to such an Abelian group and obtain possible broken symmetries from their long-time evolution. This, however, implies that these CAs must be defined for an M-letter alphabet (to be identified with A), so that there will be a gigantic number of rules to inspect. Although the proposed invariance will reduce the number of rules, this will seldom suffice to make a full enumeration of the possible rules feasible. Therefore, the rules must be restricted further by another type of symmetry, which can only be due to the invariance of the rule with respect to a rearrangement of its arguments. Maximal symmetry in this latter sense is achieved by taking the rule as completely independent of the sequence of its arguments. The totalistic [12] rules are in this class. This paper is organized as follows. In Section 2, the question as to the existence of CAs with rules which are invariant under all permutations of their n arguments and also with respect to an Abelian, regular (and then necessarily transitive) group A, which can be identified with the alphabet, is answered in the affirmative if n and M have no common factors. This phenomenon is illustrated by two examples. The number of free parameters for such a CA rule is given generally and shown to be rather small if n and M are not too large. A number of more technical results needed for these statements is collected in Appendix B. It is also shown that the existence of a nontrivial group of automorphisms of A can reduce the number of rules that have to be studied even further. In
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Section 3, the highly symmetric CAs for n = 3 and M = 4 are considered as examples. There are two possible choices of the group A here. The corresponding spin models are the 4-state chiral clock model on the one hand and the symmetric and GAT ones on the other. Also, the 4-state Potts model is obtained as a special case. These models have already been used above as examples of the general case. It is shown that the subgroup structure of the symmetric group S(4), together with the knowledge of the automorphism groups of these subgroups, leads to only 172 nonrelated rules. The question as to the invertibility of these CAs is solved by an algorithmic technique. In Section 4, the symmetries (if any) remaining after a large number of iterations of these rules are given for all these possibilities. These results have been obtained by simulation. They are, insofar this is possible, compared to the symmetry-breaking patterns for the corresponding spin models. This allows for the certain identification of the intermediate phases of the Ashkin–Teller models. The chiral 4-state clock model promises to be very interesting, since the associated CAs show a rich symmetry-breaking pattern.
2. Highly symmetric cellular automata The time evolution of a general CA is given by a local rule of the form ai (t + 1) = ϕ(ai−r (t), ai−r+1 (t), . . . , ai+s−1 (t), ai+s (t)).
(2.1)
A special case of this are the popular r = s = 1 rules of the form ai (t + 1) = ϕ(ai−1 (t), ai (t), ai+1 (t)),
(2.2)
where the state of a cell at time t + 1 depends on its own state and those of its immediate neighbours at time t, i.e., ai (t) is the state of cell i at time t, which is an element of a finite alphabet A with M letters. Since it is the purpose here to find rules which are invariant with respect to a regular, Abelian group, it is convenient to identify this group with A. As an example, let the alphabet be simply {0, 1} and let the symmetry group be the Ising model one, S(2), the only nontrivial element of which exchanges the two states everywhere. This means that if ϕ(0, 0, 1) = 1, say, is a rule, then also the form with 0 and 1 exchanged: ϕ(1, 1, 0) = 0. This can be expressed concisely as ϕ(a, b, c) = 1 + ϕ(1 + a, 1 + b, 1 + c),
(2.3)
where all additions are done modulo 2. If one writes σ for the interchange of 0 and 1, this can be written alternatively as ϕ(a, b, c) = σ [ϕ(σ (a), σ (b), σ (c))].
(2.4)
Generally, the invariance of a rule with respect to a permutation σ ∈ S(M) of the alphabet is described by (here n = r + s + 1 is the extent of the surrounding of a cell which influences its state at the next time step) σ −1 [ϕ(σ (a1 ), σ (a2 ), . . . , σ (an ))] = ϕ(a1 , a2 , . . . , an ).
(2.5)
For the special case of the Abelian group A this becomes ϕ(a1 + α, a2 + α, . . . , an + α) = α + ϕ(a1 , a2 , . . . , an ),
(2.6)
where the (commutative) group operation is written as addition. By the transitivity of A, this implies that the free parameters of the local rule can be chosen as the values ϕ(0, b1 , . . . , bn−1 ). In particular, it will always be assumed
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that there is a special state, to be identified with the unit element 0 ∈ A, such that ϕ(0, 0, . . . , 0) = 0 holds; this is a slight generalization of part of the notion of legality as defined by Wolfram [13]. It follows that one has ϕ(a, a, . . . , a) = a
∀a ∈ A.
(2.7)
The second ingredient of a highly symmetric CA is the complete symmetry of the rule with respect to an arbitrary permutation of the arguments: ϕ(aπ(1) , aπ(2) , . . . , aπ(n) ) = ϕ(a1 , a2 , . . . , an ) ∀π ∈ S(n).
(2.8)
For the case of r = s = 1, this means that the following chain of equalities must hold: ϕ(a, b, c) = ϕ(a, c, b) = ϕ(b, a, c) = ϕ(b, c, a) = ϕ(c, a, b) = ϕ(c, b, a).
(2.9)
As shown in Appendix B, a highly symmetric CA can exist only if M and n have no common factors, expressed by the fact that the greatest common divisor (M, n) = 1. This phenomenon is already evident for two simple examples. In both of these, the alphabet is simply {0, 1}, identified with the group of the Ising model S(2): 0 + 0 = 0, 0 + 1 = 1 + 0 = 1, 1 + 1 = 0. If n = 2, one has: 1. Eq. (2.7) implies ϕ(0, 0) = 0, ϕ(1, 1) = 1; 2. the symmetry with respect to the exchange of the arguments (Eq. (2.8)) implies ϕ(0, 1) = ϕ(1, 0); 3. the S(2)-symmetry of Eq. (2.6) gives ϕ(0, 1) = 1 + ϕ(1, 0), which is an obvious contradiction. For the case n = 3, however, one finds: 1. ϕ(0, 0, 0) = 0 and ϕ(1, 1, 1) = 1 by Eq. (2.7), 2. a = ϕ(0, 0, 1) = ϕ(0, 1, 0) = ϕ(1, 0, 0) and b = ϕ(1, 1, 0) = ϕ(1, 0, 1) = ϕ(0, 1, 1) by Eq. (2.8) and 3. Eq. (2.6) is now compatible with this provided that b = a + 1 holds. Therefore, there are only two highly symmetric CAs for this case, corresponding to a = 0, 1. It is now assumed that (M, n) = 1 holds; the number of independent parameters is then severely restricted by Eqs. (2.6) and (2.8). From the explicit form of Eq. (B.3) as derived in Appendix B, the results for the first few values of n are collected in Table 1. It follows that the smallest Abelian symmetry groups with nontrivial subgroups which can be studied for even values of n have M = 9, 15, 21, . . . , elements, since for M a prime, no partially broken symmetries are possible, see Appendix A. Since little is known about the corresponding spin models (a few results for M = 9 only are available [10,11]), the cases for even n will not be considered in the following. For n = 3 and M = 2, the CAs are the standard ones studied already by Wolfram [13,16]. There are only two highly symmetric ones as derived in the examples above, which are rules 150 and 232 in Wolfram’s code. Of these, rule 150 describes an invertible CA, which preserves the S(2) symmetry, whereas rule 232 corresponds to a symmetry-breaking, noninvertible one [13]. Since the Ising model in more than one dimension also shows a symmetry-breaking phase transition, rule 232 could be called “the” CA rule (with n = 3) of the Ising model. The case n = 3, M = 4, the next simplest case with the possibility of partial symmetry breaking, is treated extensively in the next section. Simple complete symmetry breaking also occurs in the cases n = 5, M = 2 (four special totalistic rules [12]) and n = 4, Table 1 []The numbers of highly symmetric CA rules for different extents of the surrounding n
N(n, M)
2
1 2 (M − 1) 1 6 (M − 1)(M + 4) 1 2 24 (M − 1)(M + 7M + 18) 1 2 120 (M − 1)(M + 6)(M + 5M
3 4 5
Restriction M odd (M, 3) = 1 M odd + 16)
(M, 5) = 1
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M = 3 (81 rules). These are not further studied here. Other partially broken symmetries can only be obtained for cases where there is, again, a plethora of rules. If a highly symmetric CA is given, then associated ones are defined by an arbitrary permutation σ ∈ S(M) as (see also Eq. (2.5)): ϕσ (a1 , . . . , an ) = σ (ϕ(σ −1 (a1 ), . . . , σ −1 (an ))).
(2.10)
Such an associated CA is also obviously symmetric with respect to all permutations of its arguments, but not necessarily with respect to A. It is not difficult to see that this latter symmetry is left invariant if and only if σ is an automorphism of A, i.e., σ (a + b) = σ (a) + σ (b)
∀a, b ∈ A
(2.11)
holds. Therefore, each highly symmetric CA has, in general, a number of associated ones, the number of which is given by the length of the orbit of this CA under the action of the automorphism group Aut(A). For a spin model with a transitive symmetry group containing A, these automorphisms give rise to symmetries of the partition function [10]. For the GAT model, for instance, this means that the partition function is a completely symmetric function of the three Boltzmann factors, whereas for the 4-state chiral clock model, the partition function is symmetric with respect to the interchange of ω1 and ω3 , see also Appendix A. Now all associated CAs of Eq. (2.10) are rather trivially related to the original CA. In particular, the set of configurations obtained after applying the global rule implied by ϕσ () simply consists of the σ -transformed configurations obtained from ϕ() and this holds also for an arbitrary number of iterations of the global rules. Therefore, it suffices to study symmetry breaking, invertibility and other global properties for only one of the highly symmetric CAs from such an orbit. 3. The highly symmetric CAs for the case M = 4 and n = 3 For the case M = 4, there are two regular, Abelian groups to be considered, namely the cyclic group C(4) and the Klein group K(4) ∼ = S(2) ⊗ S(2). As discussed in Section 1, the spin model corresponding to C(4) is the chiral 4-state clock model; if the interaction is symmetric (the nonchiral subcase), this reduces to the SAT model [9] with group D(4). The spin model with the Klein group as symmetry is the GAT model [9]. Both of these groups contain nontrivial subgroups isomorphic to S(2), which are necessarily nontransitive, so that partially broken symmetries are possible. From the second equation of Table 1 it follows that the number of highly symmetric CAs for the smallest possible surrounding, n = 3, is equal to 256 for both choices, so that a full study of these CAs is possible. (Actually, only 172 nonrelated CA rules have to be considered, see below.) The Ashkin–Teller models show three types of phases as found from high-temperature series for the square lattice [11]. From special cases and duality [17] as well as from the explicit solutions available for these models on Cayley trees [10], one expects these to correspond to phases with: (i) the full symmetry of the model or an even higher one in the high-temperature region of the phase diagram, (ii) completely broken symmetry in its low-temperature region and (iii) some kind of partially broken symmetry in certain in-between regions, related to the S(2) subgroup(s) of the model. The phase diagram of the SAT model is shown schematically in Fig. 1 in the plane of the two independent Boltzmann factors ω1 and ω2 for this model as defined in Eq. (1.4). For the GAT model with three independent Boltzmann factors (see Eq. (1.5)), the phase diagram is much more involved, but the three phases with their different symmetries, K(4) or larger at high temperatures, three intermediate pockets of partially broken symmetry and {e} for low temperature are again found [11]. In order to obtain the number 172 for the total number of unrelated CA rules, it is necessary to consider the transitive subgroups of S(4), whose subgroup structure is shown as in Fig. 2. The three realizations of C(4) are
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Fig. 2. The subgroup structure of the transitive groups on four letters.
generated by the permutations: g1 = (0123), g2 = (0231) and g3 = (0312); these generate the groups Ci (4), i = 1, 2, 3, respectively. Of these, only one has to be studied, say C1 (4), since the others are conjugate groups, which lead to associated CAs (it suffices to take the σ ’s of Eq. (2.10) equal to the δ’s of Eq. (3.6)). The only nontrivial automorphism σ of this group maps g1 onto its inverse, so that one has σ = (0)(2)(13). The four parameters necessary to describe a highly symmetric CA based on C1 (4) are taken as ai = ϕ(0, 0, i),
i = 1, 2, 3,
b = ϕ(0, 1, 2).
(3.1)
To provide an effective coding for these CA rules, a rule given by the parameters (a1 , a2 , a3 , b) will be given the number a1 + 4a2 + 16a3 + 64b.
(3.2)
All ϕ(i, j, k) entries not directly given by those of Eq. (3.1) can be obtained from Eq. (2.6), where summation is now modulo 4, and (2.8). It is not difficult to see that the automorphism σ maps the parameters of Eq. (3.1) onto corresponding (primed) ones given as a10 = σ (a3 ),
a20 = σ (a2 ),
a30 = σ (a1 ),
b0 = σ (b) + 2 mod 4.
(3.3)
It follows that there are 16 invariant models given by (a1 , a3 ) = (0, 0), (1, 3), (2, 2) or (3, 1),
a2 = 0 or 2,
b = 1 or 3.
(3.4)
These σ -invariant models then have the D1 (4)-symmetry of the SAT model, see Fig. 2. Since these will also be obtained from K(4) symmetric rules (again, see Fig. 2), there are exactly 120 rules with C(4)-symmetry that have to be studied, one from each orbit of length 2 with respect to σ . These orbits, which all correspond to the chiral 4-state clock model, are listed in Table 2, ordered with respect to their broken symmetries as explained in the next section. The Klein group is normal in S(4); in terms of permutations it is given as K(4) = {e, (01)(23), (02)(13), (03)(12)}.
(3.5)
By its normality, its automorphism group is isomorphic to S(3), i.e., any permutation of its nonunit elements is an automorphism. When written additively, its elements are either the pairs (00), (01), (10) and (11) with addition modulo 2 in both entries or, equivalently, the numbers 0, 1, 2 and 3, with as addition the (bitwise) exclusive-or
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Table 2 The orbits of the C(4)-symmetric highly symmetric CA rules and the corresponding symmetries after many iterations Subgroup
Orbits
{e}
(0,128) (8,136) (17,179) (35,146) (53,189) (68,76) (193,240)
(1,176) (12,132) (18,163) (36,142) (57,185) (71,92) (194,224)
(2,160) (13,180) (32,130) (48,129) (65,112) (77,116) (195,208)
(3,144) (15,148) (33,178) (49,177) (66,96) (79,84) (196,204)
(4,140) (16,131) (34,162) (50,161) (67,80) (117,125) (210,227)
S(2)
(55,157)
(59,153)
(63,149)
(82,99)
S(2) ⊗ S(2) (non-transitive)
(9,184) (26,171) (43,154) (75,88)
(10,168) (27,155) (56,137) (90,107)
(11,152) (40,138) (58,169) (202,232)
(24,139) (41,186) (73,120) (218,235)
(25,187) (42,170) (74,104)
C(4)
(5,188) (69,124) (205,244)
(21,191) (86,111) (207,212)
(51,145) (197,252) (213,255)
(52,141) (198,236) (214,239)
(60,133) (203,216) (229,254)
D(4) (no long-range order)
(6,172) (22,175) (31,151) (45,182) (62,165) (87,95) (101,126) (201,248) (221,247) (237,246)
(7,156) (23,159) (37,190) (46,166) (70,108) (89,123) (102,110) (206,228) (222,231) (245,253)
(14,164) (28,135) (38,174) (47,150) (78,100) (93,119) (105,122) (209,243) (225,242)
(19,147) (29,183) (39,158) (54,173) (81,115) (94,103) (109,118) (215,223) (230,238)
(20,143) (30,167) (44,134) (61,181) (85,127) (97,114) (199,220) (217,251) (233,250)
operation, denoted by ⊕. Both these representations mirror the fact that this group is the symmetry group of a pair of noninteracting Ising spins. In the latter notation, the five nontrivial automorphisms are: σ1 = (3)(12),
σ2 = (1)(23),
σ3 = (2)(13),
δ2 = δ1−1 = (132).
δ1 = (123),
(3.6)
The five models associated with a rule ϕ() are numbered as ϕk (), k = 1, . . . , 5. The four free parameters can again be chosen as in Eq. (3.1); all other rule entries follow from Eq. (2.6), with addition understood as ⊕, and (2.8). The rules will again be coded as in Eq. (3.2). There can, in general, be no confusion with this coding arrangement, since the basic group will always be made explicit. If it is necessary to emphasize that a rule occurs as the same number in the two codes, this will be indicated explicitly or, in tables, by an asterisk. (k) The five related models have parameters ai , b(k) given by (1)
a2 = σ1 (a1 ),
(2)
b(2) = σ2 (b) ⊕ 1,
a1 = σ1 (a2 ), a3 = σ2 (a2 ),
(1)
(5)
a1 = δ1 (a3 ), (5)
a2 = δ2 (a3 ),
b(1) = σ1 (b),
(5)
(2)
a1 = σ2 (a1 ),
(3)
a2 = σ3 (a2 ),
(4)
a3 = δ1 (a2 ),
a1 = σ3 (a3 ),
(4)
b(3) = σ3 (b) ⊕ 2, a1 = δ2 (a2 ),
(1)
a3 = σ1 (a3 ),
a2 = δ1 (a1 ),
a3 = δ2 (a1 ),
(3)
(4)
b(5) = δ2 (b) ⊕ 1.
(2)
a2 = σ2 (a3 ),
(3)
a3 = σ3 (a1 ), b(4) = δ1 (b) ⊕ 2, (3.7)
From these, the following types of orbits are found to exist: 1. There are two orbits of length 1; these two models have numbers 192 and 249, respectively, and are also produced as D1 (4)-models based on C1 (4) (with the same rule numbers). These CAs have the full 4-state Potts model symmetry S(4).
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2. There is one length 2 orbit, consisting of the pair (222, 231). These are invariant with respect to the alternating group A(4) ∼ = K(4)C(3), the semidirect product of the Klein group with the subgroup of automorphisms generated by δ1 , see Eq. (3.6) and Fig. 2 or, alternatively, the group of all even permutations of four states. No spin model can have this symmetry group, since it is doubly transitive and immediately implies the Potts model symmetry [10], see also Appendix A. 3. There are 14 orbits of length 3 with D(4)-symmetry, all corresponding to the SAT model. The results are listed in Table 3. 4. Finally, there are 35 orbits of length 6 with the K(4)-symmetry of the GAT model. These are listed in Table 4. Altogether, the number of orbits, from which a representative has to be studied is 120(C(4)) + 2(S(4)) + 1(A(4)) + 14(D(4)) + 35(K(4)) = 172.
(3.8)
The invertibility of these CAs, i.e., the property that the global rule maps the complete set A? of words built from the alphabet onto itself, has been checked with the algorithm described in Ref. [18]. The only invertible ones are rules 249 (Potts model symmetry S(4)) and (54, 91, 173) (SAT D(4)-symmetry). Table 3 The D(4)-type orbits of the highly symmetric CAs based on K(4). The starred numbers are the ones in D1 (4), which are the same as the corresponding ones based on C1 (4). For these, the S(2) ⊗ S(2) broken symmetry contains the permutation (02)(13). The symmetries after many iterations of the global rules are indicated Subgroup
Orbits
{e} S(2) ⊗ S(2) (nontransitive) D(4) (no LRO) S(4) (no LRO, SRO)
(0,64∗ ,128) (48,72∗ ,129) (63,106∗ ,149) (6,83∗ ,172) (219∗ ,237,246) (54,91∗ ,173)
(9,113∗ ,184) (57,121∗ ,185) (193,200∗ ,240) (198,211∗ ,236)
(15,98∗ ,148) (207,212,226∗ ) (213,234∗ ,255) (201,241∗ ,248)
Table 4 The orbits of the K(4)-symmetric highly symmetric CA rules and the symmetries remaining after applying the global rules many times Subgroup
Orbits
{e}
(1,8,65,112,136,176) (3,12,66,96,132,144) (16,32,68,76,130,131) (18,36,71,92,142,163) (21,42,85,127,170,191) (25,41,117,125,186,187) (29,43,101,126,154,183) (194,195,196,204,208,224) (203,205,216,225,242,244)
(2,4,67,80,140,160) (7,14,82,99,156,164) (17,40,69,124,138,179) (20,34,79,84,143,162) (24,33,77,116,139,178) (28,35,78,100,135,146) (49,56,73,120,137,177) (199,206,210,220,227,228)
S(2) ⊗ S(2) (nontransitive)
(5,10,81,115,168,188) (26,37,93,119,171,190) (50,52,75,88,141,161) (53,58,89,123,169,189) (197,202,209,232,243,252)
(11,13,97,114,152,180) (27,45,109,118,155,182) (51,60,74,104,133,145) (59,61,105,122,153,181) (217,233,245,250,251,253)
K(4) (no LRO)
(22,38,87,95,174,175) (31,47,102,110,150,151) (214,215,223,230,238,239)
(23,46,86,111,159,166) (55,62,90,107,157,165)
D(4) (no LRO)
(19,44,70,108,134,147) (218,221,229,235,247,254)
(30,39,94,103,158,167)
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4. Broken, unbroken and extra symmetries For all orbits, one of the equivalent CAs has been simulated to ascertain the symmetries (if any) remaining after many iterations of the rule. It is assumed that the limits for the short- and long-range pair distributions exist: S(α, β) = lim hδai ,α δai+1 ,β i,
(4.1)
t→∞
where the average h·i is the spatial average over i, δa,b is the Kronecker delta, and L(α, β) = lim lim hδai ,α δai+k ,β i.
(4.2)
t→∞ k→∞
This then implies for the average state densities: X X S(α, β) = L(α, β). ρ(α) = lim hδai ,α i = t→∞
β∈A
(4.3)
β∈A
In practice, the simulations use a uniformly random start configuration and an adjustable number of iterations, after which it is assumed that the infinite time limit is practically reached; this relaxation phase must contain up to 20 000 iterations for some of the rules. After this phase, a time average over 250 iterations and a space average over 500 cells is performed to obtain both the short- and long-range distributions. For these latter, k in Eq. (4.2) is taken in the range 100–300, whereas the total number of cells is of the order of 1000. In this way, the symmetries can, in general, be read off unambiguously. If this is not the case, the number of initial iterations is increased and a numerical limit (5%) for the possible symmetries (all of which are assumed to have been found from the unambiguous cases) is used until a consistent result is obtained, which is correct within this limit. The results obtained in this way for the 120 orbits corresponding to the C1 (4)-symmetric rules (the group is generated by g1 = (0123)) are pertinent for the chiral 4-state clock model. They are listed in Table 2 and can be described as follows: C1: If the symmetry is completely broken, denoted by {e} in the “Subgroup” column of Table 2, all entries in the vector ρ(α) of Eq. (4.3) can be different. This immediately implies that the matrices S() and L() have (more or less randomly distributed) differing entries, so that there is short- as well as long-range order in this case. These CAs simulate the low-temperature behaviour of the spin model. C2: If only the symmetry g12 = (02)(13) is conserved, the matrices S() and L() both have the structure
a d S, L() = a c
c b d b
a c a d
d b , c b
a 6= b, c 6= d.
(4.4)
This is the S(2)-entry in Table 2. The vector of densities has the form (p, q, p, q),
p 6= q.
(4.5)
This is an intermediate phase for the 4-state clock model with symmetry a subgroup of the original one. C3: If, in addition to g12 = (02)(13), the permutations (02) and (13), which do not belong to C1 (4), are also symmetries of S() and L(), then these form a nontransitive symmetry group isomorphic to S(2) ⊗ S(2), denoted as such in Table 2. The pair distribution matrices are now symmetric, i.e., obtained from Eq. (4.4) by setting c = d. The vector of densities has the same form as Eq. (4.5). If this behaviour also occurs for the 4-state clock model, it corresponds to a different intermediate phases.
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C4: The symmetry of the pair distribution matrices is full C(4)-symmetry as in Eq. (1.6): a b c d d a b c S, L() = c d a b , b 6= d. b c d a
(4.6)
In this case, the vector of densities is simply ( 41 , 41 , 41 , 41 ), but there is chiral long-range order. This is a very interesting behaviour, which would correspond to a third intermediate phase of the 4-state clock model. C5: The short-range order shows D(4)-symmetry, i.e., it has the structure: a b c b b a b c S() = (4.7) c b a b , b 6= c. b
c
b
a
This structure is the same as that of the SAT model, see Eq. (1.4). If this is the case, no long-range order occurs, 1 for all α, β. In this case, the symmetry group of the short-range pair distribution again is larger L(α, β) = 16 than the basic group. These CAs simulate the high-temperature phase of the spin model. Similarly, the (SAT) D(4)-type orbits corresponding to K(4) show four types of symmetries as listed in Table 3: D1: Completely broken symmetry, {e}, the same as case C1, simulating the low-temperature behaviour of the spin model. D2: Nontransitive S(2) ⊗ S(2) symmetry as in case C3. Depending on the rule, these groups can also be realized as the conjugate subgroups {e, (01), (23), (01)(23)} or as {e, (03), (12), (03)(12)}. These CAs simulate an intermediate phase. D3: Short-range D(4) symmetry without long-range order as in case C5, the “high-temperature” CAs. D4: Complete absence of short- and long-range order for the invertible rules in the orbit (54, 91, 173). This is the only case for which extra symmetries occur in short-range order (full S(4)-symmetry). These CAs probably do not have an equivalent (“ultra-high” temperature) spin model phase. The CAs corresponding to the GAT model have K(4)-type orbits and fall in one of the four classes, see Table 4: K1: Completely broken symmetry (the low temperature case), the same as cases C1 and D1. K2: Nontransitive S(2)⊗S(2) symmetry with three possibilities as in case D2. Again, this group contains elements not in the original group and corresponds to an intermediate phase. K3: The short-range order matrix shows K(4)-symmetry (just as the GAT model, see Eq. (1.5)): a b c d b a d c S() = (4.8) c d a b , b, c, d all different, d
c
b
a
but there is no long-range order (high-temperature phase). K4: There is short-range D(4)-order without long-range order as in cases C5 and D3. There are again extra symmetries in the short-range order distribution function. This is probably not realizable in the Ashkin–Teller model. Finally, there remain a few orbits not listed in the tables: S1: Rule 192 has full S(4)-symmetry of the Potts model, but its time evolution breaks all as in cases C1, D1 and K1.
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S2: Rule 249 also has the Potts model symmetry; there is no short- or long-range order in this invertible case, as in case D4. This seems to be a general feature of invertible, highly symmetric CAs, since it also holds for Wolfram’s rule 150, see Section 2. A: Rules (222, 231) have A(4)-symmetry and do not correspond to a spin model. There is no long-range order for these, but the short-range order has the (larger) symmetry of the 4-state Potts model, obtained from Eq. (4.7) by setting a 6= b = c. This is the same structure as shown for the Potts model in Eq. (1.3). In this special case, however, it is not yet possible to distinguish between A(4)- and S(4)-symmetric short-range order, since a matrix with the former symmetry automatically has the larger one. Therefore, it is necessary to consider the three-cell distribution function T (α, β, γ ) = lim hδai ,α δai+1 ,β δai+2 ,γ i.
(4.9)
t→∞
The group S(4) is transitive on all triples (α, β, γ ) of pairwise different letters, whereas A(4) has two orbits, which are related by an odd permutation. A numerical check of the function of Eq. (4.9) shows conclusively that the values on these orbits are different, so that the short-range order symmetry is really only A(4). The possible combinations of short-range (SRO) and long-range (LRO) symmetry groups as found above are listed in Table 5. It should be emphasized, that this classification is an experimental one; there is as yet no insight into the behaviour of a rule from first principles. Such an insight may well be impossible to achieve in a computational fashion, see Ref. [19]. The entries describing the typical time evolution are observed for the majority of the corresponding rules, but there are always some cases not conforming to this scheme. Therefore, Table 5 is not meant to represent an extension of Wolfram’s classes [13]. The phase separation observed for most of the CAs with partially broken symmetry is to be understood as follows: there are two well-separated regions, one populated by a mixture of equal parts of two of the four states, the other by a similar mixture of the other two states. The boundaries between these two regions fluctuate in a chaotic fashion, but do not interpenetrate. An example is shown in Fig. 3. The absence of long-range order explains the chaotic behaviour of the corresponding rules. Only in the last case, however, the time evolution is purely random, as would be expected for invertible, highly symmetric CAs. Cases D1–D4, S1 and S2 can be compared directly with the phase diagram of the SAT model, Fig. 1. Cases D1 and S1 correspond to the low-temperature phase, case D3 to the high-temperature one (no long-range order). The intermediate phase marked “S(2)” in Fig. 1 then must correspond to the nontransitive S(2) ⊗ S(2) case D2. The phase separation observed for the CAs in this class is also expected to occur in this spin model. It is due to the fact that for ω2 > ω1 , states which differ by ±2 mod 4, e.g., (02) and (13) (as in Fig. 3) for D1 (4), tend to mix, because of the lower energy for such pairs as compared to the ones which differ by ±1 mod 4 (these are the pairs (01), (12), (23) and (30) in the example). This also explains why the broken symmetry appears where it does in Fig. 1. Cases D4 and S2 can only be realized in the extreme high-temperature limit, ω1 = ω2 = 1. Table 5 The combinations of SRO and LRO symmetries SRO group
LRO group
Cases
Typical time evolution
{e} S(2) S(2) ⊗ S(2) C(4) K(4) D(4) A(4) S(4)
{e} S(2) S(2) ⊗ S(2) C(4) S(4) S(4) S(4) S(4)
C1, D1, K1, S1 C2 C3, D2, K2 C4 K3 C5, D3, K4 A D4, S2
Periodic, short cycles Periodic, long cycles Phase separation Periodic, long cycles Chaotic Chaotic Chaotic Randomly chaotic
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Fig. 3. An example of the phase separation associated with a broken symmetry. The rule number is 9 based on C(4). The four panels show the distribution of states in the order 0, 1, 2, 3. The time direction is from top to bottom.
Similar considerations apply to the GAT model. Case K1 corresponds to the completely broken low-temperature phase, case K3 to the high-temperature one without long-range order. Since case K2 gives three realizations of the intransitive partially broken S(2) ⊗ S(2)-symmetry, the three pockets of intermediate phases found by high-temperature series expansions [11] must have these symmetries. Finally, case A could correspond to the high-temperature phase of the 4-state Potts model (all three ω’s equal), but the fact that its short-range order shows a smaller symmetry makes this identification somewhat dubious. This seems to be a case of a CA without spin model parallel. The above shows that results obtained from the symmetry-breaking patterns of highly symmetric CAs can contribute to an understanding of these patterns in the corresponding spin models. Unfortunately, no results from high-temperature series are available for a chiral C(4) spin model. This is regrettable, since cases C2 and C4, which show chiral asymmetry, are of particular interest in this context, see, e.g., Ref. [20]. This holds especially for the case C4, which has a chiral asymmetry which does not show itself in the state density vector.
Appendix A. Results on spin models To make this paper as self-contained as possible, a number of results concerning spin models that are used in the body of the text are collected in this appendix. More details can be found in the book [10]. Given a transitive group G operating on M states, a spin model can be constructed as follows: 1. Since E(i, i), the energy function for a pair of spins in identical states, does not depend on i [E(i, i) = E(g(i), g(i)) = E(j, j ) for some g ∈ G by transitivity], one can set E(i, i) = 0 for all i by a trivial shift of the zero of energy. This implies Ω(i, i) = 1 for all i. 2. Now one takes a directed or undirected “edge” (0, k) with k 6= 0 and assigns it an energy Ek . Then all edges obtained from this one by the permutations of G must have the same energy. Repeating this step, until no more edges (0, l) remain, gives the necessary graphs for reading off the number of independent energy parameters or Boltzmann factors. These graphs form an edge-disjunct decomposition of the graph with all possible edges (a nondirected edge can be identified with a pair of oppositely directed parallel edges), the complete graph K(M). The above procedure is illustrated in Fig. 4 for the cases treated in Section 1: 1. Starting with the edge (0, 1) (directed or undirected), the group S(4) of the 4-state Potts model produces all possible edges, so that there is only one energy parameter or Boltzmann factor Ω(i, j ) = ω for all i 6= j , compare Eq. (1.3).
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Fig. 4. Examples of the derivation of the number of independent energy function values or Boltzmann factors for a number of spin models with four states. See the text for details.
2. Starting with the undirected edge (0, 1) again, the group D(4) of the SAT model produces the square (0123); there is a second energy parameter related to the diagonals of the square, which are obtained from (0, 2). This model has then two independent energy parameters, see also Eq. (1.4). 3. For the Klein group K(4) of the GAT model, all three edges (0, 1), (0, 2) and (0, 3) produce different graphs, so that this model has three independent Boltzmann factors as in Eq. (1.5). Since all three graphs are isomorphic, the automorphism group of K(4) consists of all permutations of these three graphs, i.e., it is isomorphic to S(3). 4. Starting with the directed edge (0, 1), the cyclic group C(4) of the chiral 4-state clock model produces a directed square (0123). The (undirected) diagonals and the square with all directions reversed are produced from (0, 2) and (0, 3), respectively, resulting in Eq. (1.6). The two directed squares are again isomorphic, signalling that their exchange is an automorphism. It is to be noted that the symmetry group of the spin model constructed as above can be larger than the starting group. In particular, if the group G is doubly transitive, i.e., such that for every pair of pairs (i, j ) and (k, l) there is a g ∈ G with g(i) = k and g(j ) = l, then all edges are produced starting from (0, 1), so that the corresponding spin model is the Potts model. Now the alternating groups A(M) of the even permutations of M objects have this property, so that these cannot occur as spin model groups. Groups that can occur as spin model groups are called permissible if only undirected edges are used in their construction, else chirally permissible. For example, C(4) is chirally permissible, but its corresponding permissible group is D(4). It can be shown that partially broken symmetries are possible only for spin models with a nontransitive, normal subgroup H, i.e., a subgroup for which g −1 Hg = H holds for all g ∈ G. Since the lengths of the orbits of H must divide M for such a subgroup, no partially broken symmetries are possible if M is a prime. If A is regular and Abelian, then every proper subgroup is nontransitive and normal. This implies the results for C(4) and K(4) noted in Section 1. The SAT group D(4) has only the 180◦ rotations as a normal, nontransitive subgroup, i.e., the permutations {e, (0)(2)(13)}.
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Appendix B. Results for highly symmetric CAs In this appendix, two rather technical discussions concerning the existence and the number of free parameters of highly symmetric CAs are collected. Combination of Eqs. (2.6) and (2.8) give relations between the independent ϕ(0, b1 , . . . , bn−1 ) parameters: ϕ(0, aπ(2) − aπ(1) , . . . , aπ(n) − aπ(1) ) = ϕ(0, a2 − a1 , . . . , an − a1 ) + a1 − aπ(1) .
(B.1)
In order for a highly symmetric CA to be possible, this equation must not lead to a contradiction for all π ∈ S(n). Such a contradiction is only possible if the following set of equations has a solution for all k: aπ(k) = ak + β
for all k = 2, 3, . . . , n with β = aπ(1) − a1 6= 0.
(B.2)
A specific k will be in a cycle of π; let the length of this cycle be u. Applying Eq. (B.2) u times, i.e., for k, π(k), π 2 (k), etc., gives the condition that uβ = 0 holds. This must hold for all cycles of π , so that A must contain an element of order dividing the greatest common divisor of these cycle lengths. But such common divisors must be divisors of n, whereas the order of an element of A is a divisor of M. Therefore, a contradiction can only ensue if M and n have common factors. In fact, it will not be certainly possible to define a highly symmetric CA if this is the case: it is a well-known fact that every finite group G with order a multiple of a prime p, |G| = gpm , contains subgroups of orders p a , a = 1, . . . , m (for a = m, these are the Sylow subgroups, see, e.g., [14,15]). Now if M and n have a prime p as common factor, A contains an element of order p and there is certainly a permutation in S(n) with n/p cycles of length p. This shows that CAs satisfying Eqs. (2.6) and (2.8) are constructible if and only if the greatest common divisor of M and n is 1, (M, n) = 1. In order to find the number of free parameters in case (M, n) = 1 holds, let there be l different letters occurring in the n arguments of the rule, such that there are l1 groups containing a single letter, l2 groups of pairs of equal letters, . . . , ln−1 groups of (n − 1) equal letters, l = l1 + l2 + · · · + ln−1 . The case for index n, ln , does not have to be considered here, since there is no free parameter associated with the case that all entries are equal, see Eq. (2.7). The group S(n) does not mix different choices of the li, so that there are l!/(l1 ! · · · ln−1 !) possible realizations. M The number of ways to choose l different letters is . Since the invariance with respect to the Abelian group l provides a factor 1/M, the total number of free parameters is given by N (n, M) =
X 0 (M − 1)(M − 2) · · · (M − l + 1) , l1 ! · · · ln−1 !
(B.3)
l1 ,... ,ln−1
where the prime on the summation indicates the restriction to n arguments: n−1 X ili = n.
(B.4)
i=1
The results for n ≤ 5 are listed in Table 1. References [1] J. von Neumann, in: A. Burks (Ed.), Theory of Self-reproducing Automata, University of Illinois Press, Champaign, IL, 1966. [2] S. Wolfram, Theory and Applications of Cellular Automata, Advanced Series in Complex Systems, Vol. 1, World Scientific, Singapore, 1986. [3] H. Gutowitz, Cellular Automata, MIT Press, Cambridge, MA, 1991.
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