Bicritical Central Point of 𝐽_FN−𝐽_SN Ising Model Phase Diagram

Abstract

We deal with a 2D half occupied square lattice with repulsive interactions between first and second neighboring particles. Despite the intensive studies of the present model the central point of the phase diagram for which the ratio of the two interaction strengths 𝑅=𝐽_SN/𝐽_FN=0.5 is still open. In the present paper we show, using standard Monte Carlo calculations, that the situation corresponds to a phase of mixed ordered structures quantified by an “algebraic” order parameter defined as the sum of densities of the existing ordered clusters. The introduced grandeur also characterizes the transitions towards the known pure ordered phases for the other values of 𝑅 as mentioned by the agreement of our results with those of the literature. The computation of the Cowley short range order parameter against 𝑅 suggests that the central point is bicritical and is a state to cross when passing between the two pure phases.

1. Introduction

The study of frustrated magnetic systems always arouses the interest [1–7]. It is known that the nature of the ordering at low temperatures regime of these materials depends on the concentration of the magnetic species and on the relative magnitude of competing energy scales [8, 9]. The 𝐽_FN-𝐽_SN Ising model offers a valuable framework for dealing with the phenomena involved as has been confirmed by recent discovery of high-𝑇_𝑐 superconductivity in pnictides [10, 11] (the model helps in dealing with the magnetism of the square Fe sublattice). However, when the interactions between first and second neighboring particles are repulsive in a square lattice the approximations adopted to solve the Ising model give rise to a controversial phase diagram [12–18]. In fact nowadays everyone agrees that the transition line from the disordered to the ordered 𝑝(2×2) phase is continuous when the ratio between the second “𝐽_SN" and the first “𝐽_FN" neighboring interactions 𝑅=𝐽_SN/𝐽_FN<0.5. The situation 𝑅>0.5 where the degenerate 𝑝(2×1)/𝑝(1×2) ordered phase holds is still under debate, particularly the extent of interaction rate for which the transition is first order [1, 2, 14–17, 19]. In both cases, the characterization of phase transition is done by means of order parameters that quantify the breakdown of lattice occupation symmetry. This cannot be applied to the system with 𝑅=0.5 as to this interaction strength corresponds three ground states: the two ordered phases (𝑝(2×2) and 𝑝(2×1)) and a third “2d-honeycomb 𝑝(4×2)/𝑝(2×4)" structure joining a central particle to one first and to two second neighboring particles.

Numerical calculations using relaxation processes based on the jump of unique particles are far from being sufficient to alter the stability of clusters formed. To overcome the status quo and reach a stable low temperature pure phase, it was suggested to improve the jump algorithm by considering parallel tempering method [20]. Here the finite size scaling analysis on the obtained effective critical temperatures concludes an absence of phase transition in the thermodynamic limit [14, 15]. In the present, we argue that no pure phase holds rather there are clusters with different ordered structures and sizes that interpenetrate each other well. In other words, the low temperature phase lets appear a kind of long range order with mixed structures. By expecting the local environment of a given site, one easily recognizes the ordered structure's the site belongs to and calculates the density of each ordered structure. The sum of densities of the existing structures is a nonconservative stochastic variable we introduce to play the role of an order parameter. Standard Monte Carlo calculations using the Metropolis algorithm are performed and conclude the existence of a nonvanishing critical temperature when freezing the initial disordered lattice. We then generalize the application of the present grandeur to all values of 𝑅 ratio and discuss the agreement with results of the literature. Next we consider the Cowley Short Range Order Parameter “SROP" that we compute versus the strength of the competing interaction energies at fixed temperatures. The calculations lead to the same results except for the central point where the discontinuity in the behavior of SROP reflects that the phase transition is first order. We notice here that the bicriticality does not originate from mixture of different particles as in references [21, 22].

2. Model and Method

The study is performed in the framework of lattice gas model. We consider a bidimensional square lattice with lateral size 𝐿 for which interaction energy is insured by the following Hamiltonian of static interactions:1𝐻=−2𝐽_FN𝑖,𝑗𝑛_𝑖𝑛_𝑗−12𝐽_{SN⟨⟨𝑖,𝑘⟩⟩}𝑛_𝑖𝑛_𝑘−𝜇_𝑖𝑛_𝑖. (1) “𝑛_𝑖" is a Boolean occupation number, 𝑅 denotes the ratio between interaction energies of second “𝐽_SN” and first “𝐽_FN” neighboring particles, and 𝜇 is the chemical potential and number of angle brackets indicates number of the neighbors.

Adsorption/desorption processes on periodically bounded lattices are insured meaning standard Metropolis transition rate applied to the state of single site of the lattice𝑊=min1,𝑒^{−𝛽Δ𝐻}. (2)𝛽denotes the inverse reduced temperature and Δ𝐻 corresponds to the energy difference between the final and the initial configurations. We recall that in the Metropolis algorithm a site is allowed to change its state if the global energy of the system is reduced or as a response to a local deformation of the lattice represented by the comparison of the adsorption/desorption length to a number chosen at random. For each given size of the lattice (𝐿=40,60,80) the relaxation is controlled by computing at fixed temperature the energy, the average concentration, and the introduced algebraic order parameter “AOP". The later grandeur characterizes the sum of densities of the ordered clusters on the lattice 𝜌=∑_𝑖𝜌_𝑖, where “𝑖" designates the structure and 𝜌_𝑖=∑𝑛^𝜌_𝑟 / 𝐿². Here the summation is performed once constraints relative to establishment of a type of order are fulfilled (e.g., a site belongs to the 𝑝(2×2) ordered structure whenever its first neighboring sites are all empty)𝑖≡2×2=𝑛^𝜌_𝑟=1;_⟨𝜌𝑟^⟩𝑛^𝜌_𝑟=0,_⟨𝜌𝑟^⟩𝑛^𝜌_𝑟,𝑛=4𝑖≡2×1=^𝜌_𝑟=1;_{⟨𝑟⟩⟂}𝑛^𝜌_𝑟=2;_{⟨𝑟⟩‖}𝑛^𝜌_𝑟=2,_{𝑟⟨⟨⟩⟩}𝑛^𝜌_𝑟,𝑛=0𝑖≡4×2=^𝜌_𝑟=1;_⟨𝑟⟩𝑛^𝜌_𝑟=1;_{𝑟‖⟨⟨⟩⟩}𝑛^𝜌_𝑟=2,_{𝑟⟨⟨⟩⟩}𝑛^𝜌_𝑟=2. (3) Brackets designate the neighbours and the number of the brackets the number of the neighbours and the symbols ⟂ and ‖ denote perpendicular and parallel direction of the given site.

3. Results and Discussions

Figure 1 reports adsorption isotherm as well as the behavior of AOP for different temperature regimes. For high temperatures adsorption isotherm presents the known Langmuir gas behavior. Absence of ordering is characterized by weak values of AOP. This can be understood by existence of ordered clusters of limited sizes as a consequence of the equal probability of occupying lattice sites.

Figure 1 

Variation of adsorption isotherms and density of ordered structures versus chemical potential for different interaction regimes at 𝑅=0.5. Size of the lattice is fixed to 𝐿=50.

To the transition to ordering marked by plateau in adsorption isotherm corresponds a clear increase of AOP values. All sites of the lattice contribute to the appearance of the order as can be seen by the collapse of both adsorption isotherm and AOP curves in the interval corresponding to the coverage fixed at 𝜃=0.5. Signature of this kind of long range order constituted by mixture of different types of ordered clusters is also revealed by the behavior of the susceptibility of the system. The later is defined as a measure of the autocorrelation function of the AOP: 𝜒_𝑇≡1𝐾_𝐵𝑇𝜌²−𝜌². (4) In Figure 2 we report the susceptibility for different sizes of the lattice as well as the finite size scaling analysis joining the effective critical temperatures (the inset). Maxima of the curve are related according to the formula𝑇_𝐿=𝑇_𝐶+𝐴⋅𝐿^−1/𝜈. (5) This leads to a nonvanishing critical temperature at the thermodynamic limit (𝑇_𝐶=0.039). The obtained critical exponents suggest that the phase transition is nonuniversal (𝜈=0.94 and 𝛾=1.56±0.13).

Figure 2 

Validity of algebraic order parameter 𝜌 hypothesis: behavior of the susceptibility defined as fluctuations of ordered structures density versus reduced temperature for different sizes of the lattice and finite size scaling analysis (the inset: we notice that the error bars have the same size of the symbols used in the graph).

Another proof of the validity of our hypothesis lies in characterizing the phase transition to all 𝑅 values. The phase diagram of Figure 3 shows that obtained critical temperatures agree with the results of studies within Monte Carlo technique [12–15, 23]. Once again finite size scaling analysis confirms an Ising-like behavior with universal critical exponents of the system with 𝑅<0.5 and that order/disorder phase transition for 𝑅>0.5 is nonuniversal (𝜈=0.70 and 𝛾=1.726±0.08 for 𝑅=0.75 for example, the value has to be compared with the one in [12, 13]).

Figure 3 

Phase Diagram of the half full lattice obtained by means of the algebraic order parameter fluctuations.

Now that transition to ordered phase at 𝑅=0.5 has nonnull temperature in the thermodynamic limit, it would be interesting to characterize transition from 𝑝(2×2) ordered phase to 𝑝(2×1)/𝑝(1×2) one as a response of varying the relative magnitude of competing interaction energies. For this reason we compute the Cowley [24] short range order parameter 𝛼=(⟨𝑛_𝑖𝑛_𝑗⟩−𝜃²)/𝜃(1−𝜃) versus intensity of 𝑅 ratio for different temperatures of the system. The situation corresponds to crossing over the phase diagram. One distinguishes plateaus in the low temperature regime (Figure 4) that quantify the appareance of ordered structures in the lattice with a 𝑝(2×2) structure when 𝑅<0.5 and a 𝑝(2×1)/𝑝(1×2) phase for 𝑅>0.5.

Figure 4 

Behavior of short range order parameter while crossing the phase diagram. The curve shows bicritical character of the remarkable phase diagram central point.

Transition to the disordered phase is signalled by the intersection between the critical curve and the line of constant temperature. Extent of the plateaus increases by cooling while phase with mixed structures constitutes remarkable point. Jump in short range order parameter reflects the transition is first order. Then the central point presents bicritical behaviour and constitutes the only one way between the two ordered pure phases.

4. Conclusion

We can conclude from the results obtained that within the definition of the AOP it is possible to envisage—for the frustrated square lattice with repulsive interactions—a transition to an ordered phase with a mixture of structures constituted by ground state configurations of a same energy. This phase transition presents a bicritical behavior depending on whether the establishment phase is carried out by cooling at 𝑅=0.5 or by varying the ratio of exchange constants at fixed temperature. AOP grandeur can also be applied to characterize ordinary ordered/disordered phase transition. The critical universal behavior for 𝑅<0.5 is found. For 𝑅>0.5 the calculations lead to nonuniversal critical behavior and support then the absence of any first-order phase transition.

Acknowledgment

The authors are grateful to Pr. J. Prost for fruitful discussions and suggestions.