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Journal of Physics A Mathematical and General

Multiple re-entrant phenomenon on decorated square lattice Ising models

Roberto J. V. dos Santos, F. C. Sá Barreto, S. Coutinho

Published Jun 21, 1990• 7 citations
Publisher

Abstract

The re-entrant phenomenon is studied in the antiferromagnetic Ising model on a square lattice with the bonds decorated with two and three-coupled spin 1/2 variables (dimer and trimer) in two different scenarios. In the first one ferromagnetic competing decorated bonds are anneal diluted in the system. The exact phase diagrams of the critical temperature as a function of the concentration and of the competition parameter are obtained. In contrast to the single spin bond decoration, a double re-entrant behaviour is observed. The bond percolation threshold P c =(1-1/2)/2 is achieved and it is independent of the decorating variable. In the second scenario the pure axial decorated system is studied. The analytical temperature against competition parameter phase diagram and the specific heat are calculated and discussed. In this latter case multiple re-entrances are observed for a certain range of the competition parameter. The nature of this re-entrant phenomenon is discussed.

DOI
10.1088/0305-4470/23/12/032

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The re-entrant phenomenon is studied in the antiferromagnetic Ising model on a square lattice with the bonds decorated with two and three-coupled spin 1/2 variables (dimer and trimer) in two different scenarios. In the first one ferromagnetic competing decorated bonds are anneal diluted in the system. The exact phase diagrams of the critical temperature as a function of the concentration and of the competition parameter are obtained. In contrast to the single spin bond decoration, a double re-entrant behaviour is observed. The bond percolation threshold P c =(1-1/2)/2 is achieved and it is independent of the decorating variable. In the second scenario the pure axial decorated system is studied. The analytical temperature against competition parameter phase diagram and the specific heat are calculated and discussed. In this latter case multiple re-entrances are observed for a certain range of the competition parameter. The nature of this re-entrant phenomenon is discussed.
The phase diagram and the thermodynamics of the axial decorated Ising model on a square lattice with m-dimensional bond spins are studied for the m=1 (Ising), m=2 (XY), m=3 (Heisenberg) and m= infinity cases. Competing interactions between the site and decorating bond spins have been considered. The expressions for the partition function, the thermodynamics and the pair correlation functions of the decorated Ising chain have been analytically obtained. The effect of the dimensionality of the decorating bond spin on the thermodynamic behaviour of both models is analyzed.
The two-dimensional Ising model defined on square lattices with diamond-type bond-decorations is employed to study the nature of the ferromagnetic phase transitions of inhomogeneous systems. The model is studied analytically under the bond-renormalization scheme. For an n-level decorated lattice, the long-range ordering occurs at the critical temperature given by the fitting function (kBTc/J)n=1.6410+(0.6281)exp[−(0.5857)n], and the local ordering inside n-level decorated bonds occurs at the temperature given by the fitting function (kBTm/J)n=1.6410−(0.8063)exp[−(0.7144)n]. The critical amplitude Asing(n) of the logarithmic singularity in specific heat characterizes the width of the critical region, and it varies with the decoration-level n as Asing(n)=(0.2473)exp[−(0.3018)n], obtained by fitting the numerical results. The cross over from a finite-decorated system to an infinite-decorated system is not a smooth continuation. For the case of infinite decorations, the critical specific heat becomes a cusp with the height c(n)=0.639852. The results are compared with those obtained in the cell-decorated Ising model.
A system of crossed Ising chains that can be interpreted as a multiply decorated square Ising lattice is considered and the local spontaneous magnetization which 'hangs through' between the crossing points is discussed. With growing chain distance the coupling constant along the decorated horizontal bonds is allowed to increase such that the critical temperature of the second-order transition remains constant. A narrowing of the critical region with increasing chain distance is observed; in the limit of infinite distance spontaneous magnetization, specific heat and susceptibility exhibit a first-order transition. Thus the model serves as an example for the continuous variation of the character of a phase transition from second to first kind as a function of the 'external parameter' chain distance.
In this paper, we performed the comprehensive studies of magnetic frustration properties in the Ising model on a decorated square lattice in the framework of an exact analytical approach based on the Kramers-Wannier transfer matrix method. The assigned challenge consisted in finding rigorous formulas for residual entropies of all possible cases of degeneration of the system without exception and also elucidating the influence of frustrations on the behavior of the thermodynamic functions of the system under discussion. The accomplished results turned out to be very rich and intricate, especially in comparison with the original undecorated square lattice, in which frustration properties are completely absent. Such an abundance of properties is due to the multiparametric problem of decorated lattice with four exchange interactions J_{x}, J_{y}, J_{dx}, and J_{dy} taking on arbitrary continuous values and different signs: positive (ferromagnetic) and negative (antiferromagnetic), as well as with two discrete variables d_{x} and d_{y} (multiplicities of decorating spins in both lattice directions), corresponding to infinite natural series of numbers.