Damage spreading in non-frustrated phase of a triangular antiferromagnet

PHYSICA ELSEVIER

Physica A 232 (1996) 162-170

Damage spreading in non-frustrated phase of a triangular antiferromagnet M. A n t o n i u k * ,

K. Kuiakowski

Faculty of Physics and Nuclear Techniques, University of Minino and Metalluroy, al. Mickiewicza 30, 30-059 KrakSw, Poland

Received 4 March 1996

Abstract We investigate a first-order phase transition from a non-magnetic phase to a non-frustrated phase of a two-dimensional triangular lattice with antiferromagnetic bonds. A cellular automaton is defined, which chooses a magnetic state of subsequent lattice sites, where an interphase boundary arrives, as to minimise the local energy. The ground state magnetic state is found to be unstable with respect to the point defects of magnetic structure.

1. Introduction It is known that some computations can be performed only step by step; no shorter way is possible to preview final results [1]. Problems which lead to such computations are termed "computationally irreducible". It is also known that there is an interesting analogy between physical processes and computations [2]. As for this analogy, one can ask if there are also "computationally irreducible" processes. Simulation o f these process would be the only way to investigate them theoretically. There is no general criterion for deciding whether a given process is computationally irreducible or not. To look for an experimental example o f a candidate o f such a process is an attractive task. Cellular automata (CA) are known to be an efficient tool to simulate discrete processes. Any classification o f (CA) is, then, valid at least for a class o f physical processes. There is a class o f CA which is known to be computationally irreducible (Class IV) [3]. An important feature of this class is the presence o f propagating structures. These structures can be observed when a perturbation is applied to initial configuration. * Corresponding author. 0378-4371/96/$15.00 Copyright (~ 1996 Published by Elsevier Science B.V. All rights reserved PII $0378-4371(96)00135-5

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The difference between the time evolution of the perturbed and unperturbed systems is called a damage. Spreading of perturbations or damages [4] in many-body systems composed of interacting units is an interdisciplinary, quite general problem [5]. The physics of magnetism is a field where lattice problems are well established and formulated in maybe the most precise terms. It also offers several models which are very convenient to formulate questions important in other fields of research [6, 7]. However, only a few of these questions have analytic answers, and computer simulation methods are of common use [8]. This is true, in particular, if one deals with dynamical phase transitions [9]. Recently, new magnetic phases of a frustrated Ising system were described for the first time [10, 11]. The experimental reference system was RMn2, where R is rare earth. There, magnetic moments of Mn ions were found to be unstable with respect to small changes of neighbouring sites configuration. In some phases all these moments disappeared, in some other phases only a part of the Mn ions carried magnetic moment. The latter phases were called "mixed phases". The phase diagram was determined [ 11 ] for equilibrium states of an antiferromagnetic triangular lattice, which included nine phases. This richness was a result of a frustration of magnetic bonds. We should add that if all ions are magnetic, the ground state is complicated. If some atoms lose magnetic moments, the simple periodic structure was shown to have minimal energy. In this paper we are interested in a first-order phase transition between two phases of a triangular lattice, from a non-magnetic one to a mixed phase, where some atoms are magnetic. This particular phase does not contain frustrated bonds. It is termed "II" in [10], and "VI" in [11]. In the non-magnetic phase, Mn atoms do not carry any magnetic moment. In the mixed phase, magnetic atoms of Mn form a hexagonal pattern (see Fig. 1). The transition is expected to occur by the growing of clusters of the new phase. We propose a deterministic algorithm to describe the kinetics of this process. The rule is to choose a magnetic moment at each site as to minimise local energy. The important trick is to update sites sequentially, as they are reached by an interphase boundary. We believe that such a description can reflect some features of an experimental situation, if the process is quick enough and the temperature is sufficiently low. This is the case of the so-called "drift driven" process [12]. For a more thorough discussion of kinetic processes see [12-14] and the references therein. The algorithm is equivalent to a cellular automaton with local rules. We investigate damages of the mixed phases, introduced by faults of the algorithm. The resulting damaged state can be seen as a pseudostable state [13], whereas the pure mixed state is a ground state. The goal of this paper is that the mixed phase is extremely sensitive to some kinds of defects. The mixed phase itself is doubly degenerated, i.e. two kinds of domains are possible. We call these domains A and B. Let us suppose that the transition occurs from the non-magnetic phase to the state A. If an interphase boundary meets a defect, a domain of state B is created and spreads on the lattice. The state A can be restored if a defect is found in state B, and so on. The whole lattice is, then bistable in the sense that defects switch on and off alternative states of the mixed phase. Moreover, in the thermodynamic limit the density of domain walls is quite high.

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M. Antoniuk, K. Kutakowski/Physica A 232 (1996) 162-170

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o

~

.

1

-

~

/\!\/\/\/" .~ ~

i \ / \

0

z\

~

~

0

÷

/\

Fig. 1. State A of the mixed phase of a triangular lattice with antiferromagnetic bonds. State B can be obtained by a change of sign of all spins or by a rotation of the lattice around any site by ~/3.

A question arises as to whether a structure exists (perhaps disordered) with physical properties which are stable if perturbed by one-site, mutually independent damages? If the range of a damage is finite, such a structure can be found within a standard procedure of partition function formalism. However, we find that the range of damages is infinite. Point damages can be statistically independent, but the energy of their effects is by no means additive; in this sense they interact. Our conclusion is that the physical properties of a damaged lattice cannot be determined within, say, Gibbs statistics. Instead, a step-by-step calculation is necessary. In other words, the problem is computationally irreducible. In the next section the rules of the automaton are given. Subsequently, we show some kinds of domain walls, which have been found during the simulation. A final discussion closes the paper.

2. The rules The rules are defined on three previously updated nearest neighbours of a site in a triangular lattice. Formerly, sites are non-magnetic. As the interphase boundary arrives, magnetic moment "up", "down" or "zero" is to be determined at each site. We assume that this boundary is a straight line, which forms a small positive angle with the horizontal axis, and the boundary moves downwards. Then the succession of sites, where magnetic moments are formed, is such as this one of reading letters in any English text. Antiferromagnetic bonds force a moment to be antiparallel to those neighbouring to the site. There are three such neighbours in a triangular lattice: two sites in the row above and one site at the left. In Table 1, a formal prescription is given of a spin at a given site, for all possible configurations of the environment. Note that some configurations never occur. The spreading of a new phase from a point (nucleus) can be described by a similar algorithm as well. In this case, the interphase boundary is convex, and the rules should be generalized to include the case of only two previously updated neighbours. If the antiferromagnetic bond is strong enough, we get the XOR (exclusive OR) rule [15].

M. Antoniuk, K. KutakowskilPhysica A 232 (1996) 162-170

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Table 1 The value of spin at given sites, for all possible configurations of the nearest neighbourhood (note that antiferromagnetic bond is never frustrated in this phase; in the last column, the conditions are given in terms of the Hamiltonian H + H', which are necessary to get the output as in the second column) Neighbourhood

Spin

Conditions

000, + 0 - , + - 0 , 0 + - , - 0 + , - + 0 , 0--+, + - + , - - + - 00-, 0 - 0 , -00, - - 0 00+, 0 + 0 , +00, +0+

0

A > 0, A+2K > 0, A+3K > J

+ -

A + K < J, A + 2 K < 2J A + K < J, A + 2 K < 2J

The succession o f updating could be determined b y a distance from a site to a centre o f nucleation, which should not be centred at a point o f symmetry. However, for large values o f the radius o f an interphase boundary this deterministic prescription becomes unphysical. W e limit our considerations to the case when an interphase boundary is a straight line. The algorithm given in Table 1 can be summarised as follows: "no parallel nearest neighbours, as many antiparallel bonds as possible". Within this rule, one degree o f freedom is left, i.e. the output o f a non-magnetic neighbourhood "000". This spin could be " + " or " - " , without frustrating antiferromagnetic bonds. N o w we set the spin to zero in this case; the other possibilities are treated separately below. Still, the conclusions remain unchanged. The mixed phase presented in Fig. 1 was obtained in [10, 1 1] as a ground state for the Hamiltonian [16, 17]

H=JZsisj+
AZs~, i

(I)

where J is an exchange integral, d is a one-ion anisotropy constant, and si = • 1 or 0 is a spin at an ith site. However, the conditions at the interphase boundary are different than within a domain o f a given magnetic phase. To obtain the mixed phase (Fig. 1), we have to add one more term to the Hamiltonian. This added term is the biquadratic exchange interaction [18] H ' = K Z SiSj, 22
(2)

where K is a constant. This term can be interpreted as a two-ion magnetovolume effect []9,20], which can be particularly important for unstable Mn ions. If the total Hamiltonian consists both of H and H', the prescription given in Table I minimises local energy for the case 0 < A < K + A < J < 3K + A. This condition summarises the conditions in the last column in Table I. The automaton defined on the neighbourhood described above, if compared to usual l-dimensional automata [3], has a special feature which we would like to mention briefly here. Changing the state of one site, we can expect that some other sites can be

M. Antoniuk, K. KutakowskilPhysica A 232 (1996) 162-170

166

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+O-+O-+O-+-+02',Ox+-O+-O+-O+-O+ 0-+0-+0-+0 )O-+O~(~+-O+-O+-O++O-+O-+O--F -+0-+02'~,+-0+0+-0 0-+0-+0-+0 0-+0-+0~,,~+-0+-0+ +0-+0-+0-+ +O-+O-+O~ON+-O+0-+O-+O-+O: O-+O-+O-+OX, Q+-O +0-+0-+0-'+ +O-+O-+O-+O:NDN+ 0-+0-+0-+0 i O-+O-+O-+O-+OX, x," +0-+0-+0-+ +0-+0-+0-+0-+ 0-+0-+0-+0: 0-+0-+0-+0-+0 +0-+0-+0-+ +0-+0-+0-+00-+0-+0-+0~ 0-+0-+0-+0-+ +0-+0-+0-+ +0-+0-+0-+0 0-+0-+0-+0 0-+0-+0-+0+0-+0-+0+0-+0-+0-+

A

B

A

B

Fig. 2(a). Superposition of two defects in the first row. The defects are: "0" instead of "+", and one "0" is skipped. The defects are marked by arrows. Domain walls are marked by solid lines.

influenced. The set of these potentially influenced sites forms a triangle ("light cone") for usual automata, with an angle of n/3. For our rule, this angle is 2n/3, with right arm horizontal. If we treat the process of forming of one row of spins as one step, infinitely many spins can be influenced by a change of only one spin in every step. Below we demonstrate that this is not the case for our rules. Still, we will see that sometimes damages exceed the angle n/3.

3. The results

We have checked the consequences of several kinds of point defects, introduced to the lattice during formation of the mixed phase. Some typical cases are shown in Fig. 2. Fig. 2(a) shows the result of the superposition of two point damages. In the first row, initial perfect structure A with elemental cell "0 + - " is broken two times. Once we

M. Antoniuk, K. KutakowskilPhysica A 232 (1996) 162-170

+-0+-0 O+-O++-O+-O 0+-O++-O+-O 0+-O++-O+-O 0+-o++-0+-0 o+-0++-o+-0 o+-o++-0+-0 0+-0++-o+-0 o+-o++-0+-0 o+-o++-0+-0 0+-0++-0+-0 0+-O+ A

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0-+O~+-O+ ~+O~0+-O+-O+-O+O+-O++ 0 - + O~q~+- 0 ~ - + O ~ + - O +- 0 +- O + - O +- O 0 - + 0 - + O ~q~+ ~+ O - + O~4~+- 0 + - O +- O + - 0 + + O- + O- + O ~ q ~ - + O- + O - ~ + - 0 +- O +- O + 0-+0-+ 0-+OO+~-+0-+O~+-O+-O+-0 +O-+O-+O-+ -Oq~-+ O-+O~+-O+-O+ 0-+O-+O-+0 3+-O~-+O-+OX~+-O +A i+ O - + O- + O- + - O + - O q ~ - + O - + O ~q~+- O 0-+ 0-+0-+0 3+- 0+- Oq~- + 0-+ 0:~+ +O-+O-+O--F-O+-O+-O+NON-+O-+O-'N D-+O-+O-+O D+-O+- O+-Oq~-+O-+ ~+0-+0-+0-+ -O+-O+-O+-O+N0-+O b - + o - + o - + o ~) + - o + - o + - o + - o " ~ Q B +o-+o-+o--~ -o+-o+-o+-o+-o+~ ~-+o-+o-+o o+-o+-o+-o+-o+\ +o-+o-+o--~ -o+-o+-o+-o+-o )-+o-+o-+o )+-o+-o+-o+-o+ +O-+O-+O--F -O+-O+-O+-O+3-+O-+O-+O 0+-O+-0+-O+-O +0-+0-+0-4 -O+-0+-O+-0+ B

A

Fig. 2(b). Superposition of two defects in the first row. The defects are: "0" instead of "+", another "0" instead of " - " . The defects are marked by arrows. Domain walls are marked by solid lines.

find "0" instead of "+", and in another place one "0" is skipped. First damage initialises two domain walls, forming an angle n/6. We find the structure B between these domain walls. Second damage initialises the vertical domain wall. At some point, two domain walls cross. Roughly speaking, they do not interact. Part of the vertical wall is within structure A, another part is within B. The phases A and B differ by spatial orientation of magnetic moments, but not by structure. In Fig. 2(b) two defects superpose as well, but here twice a magnetic atom is substituted by a non-magnetic one is the first row. As we see, damages can cancel themselves, i.e. damage of structure B leads back to structure A. Figs. 2(a) and (b) show that damages are additive, except some small shift of a domain wall. In Fig. 3, results are presented on the same initial situation as in Fig. 2(a), but here the rules are slightly different. What is changed is the output of one environment, "000", which is now " - " instead of "0". Such a change could be introduced if the phase transition takes place in the presence of a weak external magnetic field. The

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M. Antoniuk, K. KutakowskilPhysica A 232 (1996) 162-170

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+-o+-op-x+-o+-o+

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B

+-0+-0+-0+-0+-0+-0+-0 30+-0+-0+-0+-0+-0+-0+ +-0+-0+-0+-0+-0+-0+30+-0+-0+-0+-0+-0+-0 +-0+-0+-0+-0+-0+-0+ 30+-0+-0+-0+-0+-0++-0+-0+-0+-0+-0+-0 00+-0+-0+-0+-0+-0+ +-0+-0+-0+-0+-0+00+-0+-0+-0+-0+-0 +-0+-0+-0+-0+-0+ 00+-0+-0+-0+-0++-0+-0+-0+-0+-0 00+-0+-0+-0+-0+ +-O+-O+-O+-O+O0+-O+-O+-O+°O +-0+-0+-0+-0+ OO+-O+-O+-O++-0+-0+-0+-0 OO+-O+-O+-O+ +-O+-O+-O+OO+-O+-O+-O A

Fig. 3. Initial defects are the same as those in Fig. 2(a), but the algorithm is changed to simulate weak magnetic field directed downwards (see text). result is that one domain wall vanishes; new structure B is limited to the area between two vertical lines.

4. Discussion There is an essential difference between investigations of damage spreading for deterministic and probabilistic automata. In the former case, statistical tools do not work well; the correlation function is 1 or - 1 and does not decay for long distances, the correlation time is short or infinite. In the latter case, calculations are repeated for different initial conditions to get good statistics [21], and the results are formulated with the help of probability theory, which offers the link to statistical mechanics. Sometimes, this is the only way to perform calculations, as in the case of the famous Kauffman model [6]. This model can be formulated as the superposition of all possible cellular automata [21]. There are two conditions for "good statistics": the system should be

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sufficiently large and the time of calculations should be as long as to omit transient effects. These conditions are in contradiction, because the calculation time increases with system volume. Fortunately, this latter increase seems to be linear with the number of sites for the case of small connectivity K [21,22]. We have demonstrated that spreading of damages in the mixed phase of an antiferromagnetic lattice can be of infinite range. In terms of [3] our algorithm is "complex"; we observe propagating damages throughout a whole lattice. We believe that the kinetics of the phase transition described above should be classified as belonging to class IV in the Wolfram's classification [3], or to sensitive ones in terms of Binder's classification [23]. It seems to be the consequence of a complex magnetic structure of the mixed phase. As the interphase boundary moves, the magnetic moments of atoms are determined as sensitive to variations of the magnetic state of each of the neighbouring atoms. Such sensitivity is absent in ferromagnetic structures, where any damages are averaged out. One can ask as to what a typical environment of a site with spin " + " or " - " is. Such a question is relevant for a distribution of hyperfine fields, measured by means of Mossbauer technique. This question could be answered with an application of Gibbs statistical mechanics, where weights of point defects should be assigned to domain walls. In the thermodynamic limit, the energy of one point defect vanishes, but domain wall energy remains finite. In this sense the ground state is unstable with respect to point defects. An infinitely small concentration of point defects leads to a finite concentration of domain walls. On the other hand, some defects cancel domain walls. The question could be if there is some "damage invariant" structure, i.e. a structure which characterises an infinite sample with a given density of point defects. It seems to us that this problem is computationally irreducible. Concluding, the magnetic structure of an antiferromagnetic triangular lattice with unstable magnetic moments (e.g. Mn ions) is expected to be perturbed by infinitely long-range damages, which can change its macroscopic properties. These damages are strongly dependent on sample history. The effect of the movement of an interphase boundary in discrete magnetic structures is found to be relevant for computation theory.

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[9] B. Derrida, in: Fundamental Problems in Statistical Mechanics VII, ed. H. van Beijeren (Elsevier, Amsterdam, 1990) p. 273. [10] R. Ballou, C. Lacroix and M.D. Nunez-Regueiro, Phys. Rev. Lett. 66 (1991) 1910. [11] C. Lacroix, M.D. Ntmez-Regueiro and R. Ballou, Int. J. Mod. Phys. B 7 (1993) 1004. [12] K. Binder, Rep. Prog. Phys. 50 (1987) 783. [13] B. Fultz, in: Ordering and Disordering in Alloys, ed. A.R. Yavari (Elsevier, London, 1992) p. 31. [14] J.D. Gunton and M. Droz, Introduction to the Theory of Metastable and Unstable States (Springer, Berlin, 1983). [15] H. Hartman and G.Y. Vichniac, in: Disordered Systems and Biological Organization, eds. E. Bienenstock, F. Fogelman Soulie and G. Weisbuch (Springer, Berlin, 1986) p. 53. [16] M. Blume, Phys. Rev. 141 (1966) 517. [17] H.W. Capel, Physica B 32 (1966) 966. [18] J. Adler, J. Oitmaa and A.M. Stewart, Physica B 86-88 (1977) 1109. [19] S.N. Gadekaar and T.V. Ramakrishnan, J. Phys. C: Solid State Phys. 13 (1980) L957. [20] B.K. Chakrabarti, N. Bhattacharyya and S.K. Sinha, J. Phys. C: Solid State Phys. 15 (1982) L777. [21] D. Stauffer, J. Stat. Phys. 74 (1994) 1293. [22] K. Kulakowski and M. Antoniuk, Acta Phys. Pol. B (1996), in print. [23] P.-M. Binder, Complex Systems 7 (1993) 241.