, (1)First-order phase transition is investigated from a non-magnetic phase of triangular lattice to a mixed phase, which is a ground state of the Blume-Capel Hamiltonian. The phase transition is assumed to be athermal, i.e. magnetic moments at an interphase boundary are determined as to minimize local energy. This transition can be described by deterministic cellular automata, which depend on the parameters of the Hamiltonian. Metastable magnetic structures are found which depend on initial states of a substrate. Damages of some of these structures spread through the whole lattice. The results can be relevant for magnetic compounds Rare Earth - Mn2.
Phase transitions is an attractive field for computational physics, both for applications and basic theory. Continuous phase transitions is a classical subject of various simulations because of their universality [1]. Much less is known on discontinuous (first-order) phase transitions. However, they can be investigated numerically in a more natural way, because the range of many-body correlations remains finite. This is true in particular when we deal with kinetic problems. In this case, computer simulation is the tool to investigate, and may help to justify our model assumptions. On the other hand, solid media can be treated within two schemes: continuous or discrete. Respectively, differential equations [2-4] or lattice calculations [5] have been applied. Whereas in the former case truncation errors of numerical method are annoying, in the latter they are absent. A powerful technique to simulate kinetic phenomena on lattices is the cellular automata formalism [6]. It is applied widely to physics of magnetism [5,7], theoretical biology [8], neural network problems [9] and many other fields. The famous Ising model [10] offers a link to these fields from statistical mechanics; that is why we often call "spin" any discrete variable. To introduce temperature to the simulation scheme is often a nontrivial task. However, at low temperature a first-order phase transition can occur in athermal way. These are so-called glissile transitions [11]. This kind of process is known in metallurgy [12] rather than in magnetism; however, we note that the rule of minimum of energy was applied [13] to simulation of dynamics of magnetic domain walls. In such a case, the current in state space is limited to the "drift term" in Binder's terminology [4]. It is an assumption of this paper that local magnetic configuration at an interphase boundary is chosen as to minimize local energy. In this way, cellular automaton is determined by the system energy. Dynamical phase diagram is obtained, if a cellular automaton is given for any set of model parameters. Such a diagram would be rather poor, if applied to ferromagnetic phase. However, complex magnetic structures of a triangular lattice with antiferromagnetic bonds were found recently [14] to be relevant for magnetic compounds RareEarth-Mn2. The goal was to find that magnetic moments of Mn atoms were unstable and some of them disappeared. In the ground state, magnetic and nonmagnetic atoms formed a hexagonal pattern, with nonmagnetic atom at centres of hexagons. None bond was frustrated in this state, termed as "VI" in [14]. Here we adopt this notation.
The system was described in [14] by the Blume-Capel Hamiltonian [15,16]. In our recent paper [17] we got some preliminary results on damage spreading in metastable phases of the system. We supplemented the Blume-Capel Hamiltonian by a biquadratic term, the last term in Eq.1. The aim was to stabilize the phase VI as a metastable one; however, such a term could be justified as describing a change of volume when an atom of Mn became magnetic. The Zeeman term is omitted for brevity.
, (1)
where J is an exchange integral, delta is a crystal field constant, K is two-site magnetoelastic constant, Si= (+-)1 or 0 is a spin at an i-th site. The sums are performed over nearest neighbours only. In the Table 1, the dynamic phase diagram is given for the Hamiltonian H, i.e. cellular automaton is defined for each point of the plane (x=J/delta, y=K/delta). Actual phase which is formed as an interphase boundary moves, depends on the automaton and the magnetic state of atoms just behind the boundary. The latter depends on the state of atoms behind them and so forth. In [17] we argued that the problem of determining the phase in the presence of defects was computationally irreducible. The automaton which stabilized the phase VI was found to propagate the defects over the whole lattice; therefore, it could be classified as belonging to the class IV of the famous Wolfram's classification [6]. The interphase boundary is assumed to move, say, downwards, forming a small positive angle with horizontal axis. Below the boundary, the phase is non-magnetic (phase "IX" in [14]). The sequence of sites updated is, then, the same as the sequence of reading any English text. Every such a site has three neighbours updated in the past and three neighbours which will be updated in the future: still they are not. Energy of nonmagnetic sites is zero.
In the Table 1, the automata A,B,C,D and E are defined. To define an automaton, we have to determine output for every configuration of three neighbours. As energy is a sum of pair interactions, only votation rules are taken into account. Then we have to give the outputs of 10 configurations: 000, +++, ---, +00, -00, ++0, --0, +-0, ++- and --+. Here we have limited ourselves to x>0, y>0. In equilibrium, the phase VI is stable below the curve y=x-2/3. This line does not coincide with the curves y=x-1/2, y=x-1 in dynamic phase diagram, because for the latter, minimum of energy is searched for an atom at the interphase boundary. There, the conditions are different than in a "bulk" phase, which is a plane in our two-dimensional case.
Below we give some examples of damage spreading within the structures, which we find to be self-propagating, when the automata C,D and E are active. Note that the phase VI can be thermodynamically stable in all three cases.
Fig.1. For the automaton C, the solution is non-magnetic for any boundary conditions.
Fig.2. For the automaton D, the structure VI is self-propagating. However, damages propagate also within a cone. Spins of damaged domain are reversed: instead of sequence"+-0" we observe "-+0". Initial damage at first line is marked as bold "o".
Fig.3. For the automaton E, the self-propagating structure is different from the structure VI. Damages propagate along lines: we can observe a pair of two parallel spins at each line.
Concluding, the idea of dynamic phase diagram is described here for the first time. Magnetic structures formed during a movement of an interphase boundary can be very different from those which are thermodynamically stable. In particular, phase VI of triangular antiferromagnetic lattice is found to be very sensitive for point defects. Actual structure of such a lattice with a given amount of damages can be compared to a spin glass. It is determined by complex "calculation" performed by Nature, which seems to be computationally irreducible.
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