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Multicritical Behavior of Two Coupled Ising Models in the Presence of a Random Field

A system defined by two coupled Ising models, with a bimodal random field acting in one of them, is investigated. The interactions among variables of each Ising system are infinite-ranged, a limit where mean field becomes exact. This model is studied at zero temperature, as well as for finite temperatures, representing physical situations which are appropriate for describing real systems, such as plastic crystals. A very rich critical behavior is found, depending directly on the particular choices of the temperature, couplings, and random-field strengths. Phase diagrams exhibiting ordered, partially-ordered, and disordered phases are analyzed, showing the sequence of transitions through all these phases, similarly to what occurs in plastic crystals. Due to the wide variety of critical phenomena presented by the model, its usefulness for describing critical behavior in other substances is also expected.

where the lower-temperature phase presents such a partial orientational order, like the plastic-crystal of Fig. 1. The property of translational invariance makes the plastic crystals much simpler to be studied from both analytical and numerical methods, becoming very useful towards a proper understanding of the glass transition [12][13][14]. In some plasticcrystal models one introduces a coupling between two Ising models, associating each of these systems respectively, to the translational and rotational degrees of freedom [17][18][19][20], as a proposal for explaining satisfactorily thermodynamic properties of the plastic phase. Accordingly, spin variables {t i } and {r i } are introduced in such a way to mimic translational and rotational degrees of freedom of each molecule i, respectively. The following Hamiltonian is considered [17][18][19][20], where ij represents a sum over distinct pairs of nearest-neighbor spins. In the first summation, the Ising variables t i = ±1 may characterize two lattices A and B (or occupied and vacant sites). One notices that the rotational variables r i could be, in principle, continuous variables, although the fact that the minimization of the coupling contribution αt i r i is achieved for t i r i = 1 (α > 0), or for t i r i = −1 (α < 0), suggests the simpler choice of binary variables (r i = ±1) to be appropriate, based on the energy minimization requirement.
In the present model the variables t i and r i represent very different characteristics of a molecule. Particularly, the rotational variables r i are expected to change more freely than the translational ones; for this reason, one introduces a random field acting only on the rotational degrees of freedom. In fact, the whole contribution i (αt i + h i )r i is known to play a fundamental role for the plastic phase of ionic plastic crystals, like the alkalicyanides KCN, NaCN and RbCN. In spite of its simplicity, the above Hamiltonian is able to capture the most relevant features of the plastic-crystal phase, as well as the associated phase transitions, namely, liquid-plastic and plastic-solid ones [15][16][17][18][19][20][21].
A system described by a Hamiltonian slightly different from the one of Eq. (1), in which the whole contribution i (αt i + h i )r i was replaced by i α i t i r i , i.e., with no random field acting on variable r i separately, was considered in Ref. [20]. In such a work one finds a detailed analysis of the phase diagrams and order-parameter behavior of the corresponding model. However, to our knowledge, previous investigations on the model defined by Eq. (1) have not considered thoroughly the effects of the random field h i , with a particular attention to the phase diagrams for the case of a randomly distributed bimodal one, h i = ±h 0 ; this represents the main motivation of the present work. In the next section we define the model, determine its free-energy density, and describe the numerical procedure to be used. In Section III we exhibit typical phase diagrams and analyze the behavior of the corresponding order parameters, for both zero and finite temperatures; the ability of the model to exhibit a rich variety of phase diagrams, characterized by multicritical behavior, is shown. Finally, in Section IV we present our main conclusions.

II. THE MODEL AND FREE-ENERGY DENSITY
Based on the discussion of the previous section, herein we consider a system composed by two interacting Ising models, described by the Hamiltonian where (ij) represent sums over all distinct pairs of spins, a limit for which the meanfield approximation becomes exact. Moreover, τ i = ±1 and σ i = ±1 (i = 1, 2, · · · , N) depict Ising variables, D stands for a real parameter, whereas both J σ and J τ are positive coupling constants, which will be restricted herein to the symmetric case, J σ = J τ = J > 0. Although this later condition may seem as a rather artificial simplification of the Hamiltonian in Eq. (1), the application of a random field h i acting separately on one set of variables, will produce the expected distinct physical behavior associated with {τ i } and {σ i }. The random fields {h i } will be considered as following a symmetric bimodal probability distribution function, The infinite-range character of the interactions allows one to write the above Hamiltonian in the form from which one may calculate the partition function associated with a particular configuration of the fields {h i }, where β = 1/(kT ) and Tr ≡ Tr {τ i ,σ i =±1} indicates a sum over all spin configurations. One can now make use of the Hubbbard-Stratonovich transformation [22,23] to linearize the quadratic terms, where H i (τ, σ) depends on the random fields {h i }, as well as on the spin variables, being given by Performing the trace over the spins and defining new variables, related to the respective order parameters, one obtains where Now, one takes the thermodynamic limit (N → ∞), and uses the saddle-point method to where the free-energy density functional f (m τ , m σ ) results from a quenched average of g i (m τ , m σ ) in Eq. (10), over the bimodal probability distribution of Eq. (3), with The extremization of the free-energy density above with respect to the parameters m τ and m σ yields the following equations of state, where In the following section we present numerical results for the order parameters and phase diagrams of the model, at both zero and finite temperatures. All phase diagrams are represented by rescaling conveniently the energy parameters of the system, namely, kT /J, h 0 /J and D/J. Therefore, for given values of these dimensionless parameters, the equations of state [Eqs. (14) and (15) • Continuous (second order) critical frontier: continuous line; • First-order critical frontier: dotted line; • Tricritical point: located by a black circle; • Ordered critical point: located by a black asterisk; • Triple point: located by an empty triangle.   The critical frontier shown in Fig. 3(b), separating the two low-temperature phases, is characteristic of an Ising ferromagnet in the presence of a bimodal random field [3]. The black circle denotes a tricritical point, where the higher-temperature continuous frontier meets the lower-temperature first-order critical line. The type of phase diagram exhibited in Fig. 3(b) will be referred herein as topology I.  Fig. 4, where one sees that the topology I [ Fig. 3(b)] goes through substantial changes, as shown in Fig. 4(a) (to be called herein as topology II). As expected from the behavior presented in Fig. 3(a), one notices that the border of the P phase (a continuous frontier) is shifted to higher temperatures. However, the most significant difference between topologies I and II consists in the low-temperature frontier separating the ordered and partially-ordered phases. Particularly, the continuous frontier, as well as the tricritical point shown in Fig. 3(b), give place to an ordered critical point [24], at which the low-temperature first-order critical frontier terminates. Such a topology has been found also in some random magnetic systems, like the Ising and Blume-Capel models, subject to random fields and/or dilution [5,9,[25][26][27][28]. In the present model, we i.e., a discontinuity, corresponding to a change to the partially-ordered phase; on the other hand, the magnetization m σ remains unaffected when going through this critical frontier. For higher temperatures, |m τ | becomes very small, but still finite, turning up zero only at the P boundary; in fact, the whole region around the ordered critical point is characterized by a finite small value of |m τ |. Another unusual effect is presented in Fig. 4(d), for which (h 0 /J) = 0.65, i.e., slightly to the right of the first-order critical frontier: the order parameter m τ is zero for low temperatures, but becomes nonzero by increasing the temperature, as one becomes closer to the critical ordered point. This rather curious phenomenon is directly related to the correlation between the variables {σ i } and {τ i }: since for (kT /J) ≈ 0.5 the magnetization m σ is still very close to its maximum value, a small value for |m τ | is induced, so that both order parameters go to zero together only at the P frontier.
Behind the results presented in Figs. 4(a)-(d) one finds a very interesting feature, namely, the possibility of going continuously from the ordered phase to the partiallyordered phase by circumventing the ordered critical point. This is analogous to what happens in many substances, e.g., water, where one goes continuously (with no latent heat) from the liquid to the gas phase by circumventing a critical end point [1,2]. In Fig. 5 the free-energy density of Eq. (12) is analyzed at two different points along the first-order critical frontier of Fig. 4(a), namely, a low-  Fig. 3(b)], namely, the ordered (|m τ | = 1) and partially-ordered (m τ = 0) phases. However, the higher-temperature point typifies the phenomenon discussed in  Fig. 6(b)]. Changes similar to those occurring herein between topologies II and III, as well as topologies III and IV, were found also in some magnetic systems, like the Ising and Blume-Capel models, subject to random fields and/or dilution [5,9,[25][26][27][28]. Particularly, the splitting of the low-temperature first-order critical frontier into two higher-temperature first-order lines that terminate in the ordered and tricritical points, respectively [as exhibited in Fig. 6(a)], is consistent with results found in the Blume-Capel model under a bimodal random magnetic, by varying the intensity of the crystal field [28]. In Fig. 7 we analyze the behavior of the m τ and m σ for topology III, in the region of multicritical points of the phase diagram for (D/J) = 0.5, along three typical thermodynamic paths, as shown in Fig. 7(a). In Fig. 7 From all these cases, only topology IV typifies a well-known phase diagram, characterized by a tricritical point, where the higher-temperature continuous frontier meets the lowertemperature first-order critical line. This phase diagram is qualitatively similar to the one found for the Ising ferromagnet in the presence of a bimodal random field [3], and it does not present the herein physically relevant partially-ordered phase. For (D/J) ≥ 1/2, even though the random field is applied only in the τ -system, the correlations lead the σ-system to follow a qualitatively similar behavior.
The phase diagrams referred as topologies I and II exhibit all three phases. In the later case we have found a first-order critical line terminating at an ordered critical point, leading to the potential physical realization of going continuously from the ordered phase to the partially-ordered phase by circumventing this critical point. In these two topologies, the sequence of transitions from the disordered to the partially-ordered, and then to the ordered phase, represents the physical situation that occurs in plastic crystals. For conveniently chosen thermodynamic paths, i.e., varying temperature and random field appropriately, one may go from the liquid phase (m σ = m τ = 0), to a plastic-crystal phase (m σ = 0; m τ = 0), where the rotational degrees of freedom are found in a disordered state, and then, to an ordered crystal phase (m σ = 0; m τ = 0).
From the point of view of multicritical behavior, topology III [(D/J) = 1/2] corresponds to the richest type of phase diagram, being characterized by several critical lines and multicritical points; one finds its most complex criticality around (h 0 /J) = 1, signaling a great competition among the different types of orderings. Although the partiallyordered phase does not appear in this particular case, one has also the possibility of circumventing the ordered critical point, such as to reach a region of the phase diagram along which |m τ | becomes very small, resembling a partially-ordered phase.
Since the infinite-range interactions among variables of each Ising system correspond to a limit where mean-field approach becomes exact, an immediate question concerns whether some of the results obtained above represent an artifact of such limit. Certainly, such a relevant point is directly related with the existence of some of these features in the associated short-range three-dimensional magnetic models. For example, the tricritical point found in topologies III and IV is essentially the same that appears within the meanfield approach of the Ising model in the presence of a bimodal random field. This later model has been extensively investigated on a cubic lattice through different numerical approaches, where the existence of this tricritical point is still very controversial. On the other hand, a first-order critical frontier terminating at an ordered critical point, and the fact that one can go from one phase to another by circumventing this point, represents a typical physical situation that occurs in real substances. The potential for exhibiting such a relevant feature represents an important advantage of the present model.
Finally, we emphasize that the rich critical behavior presented in the phase diagrams corresponding to topologies II and III suggest the range 0 < (D/J) ≤ 1/2 as appropriate for describing plastic crystals. The potential of exhibiting successive transitions from the ordered to the partially-ordered and then to the disordered phase should be useful for a better understanding of these systems. Furthermore, the characteristic of going continuously from the ordered phase to the partially-ordered phase by circumventing an ordered critical point represents a typical physical situation that occurs in many substances, and opens the possibility for the present model to describe a wider range of materials.