International Journal of Modern Physics B

TWO-DIMENSIONAL ISING MODEL IN ANNEALED RANDOM FIELDS

L.L. Gonçalves, Tomoya Horiguchi

Published Aug 1, 1990• 2 citations

Abstract

The critical behaviour of the two-dimensional Ising model in annealed random fields is studied. Two different kinds of random field are considered and exact results obtained for the transition temperature when the field distribution is symmetric. In the first case the random field is unrestricted and the explicit results obtained for discrete and gaussian field distributions show the annealed field favours the ordering of the system. In the second case the random field satisfies a constraint and explicit results are also obtained for discrete and gaussian field distributions. In the discrete distribution the field assumes the values ±h0,0 and it is shown that there is a critical field hc such that for h0>hc the system presents a reentrant phase transition. In this case the ferromagnetic ground state becomes unstable and there is the appearance of a superantiferromagnetic phase. For h0<hc the ferromagnetic state is stable and the transition temperature decreases due to the presence of the random field.

DOI: 10.1142/s021797929000084x

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International Journal of Modern Physics B

TWO-DIMENSIONAL ISING MODEL IN ANNEALED RANDOM FIELDS

L.L. Gonçalves, Tomoya Horiguchi

Published Aug 1, 1990• 2 citations

Publisher

Abstract

DOI: 10.1142/s021797929000084x

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{p}_{0}

,${p}_{\ifmmode\pm\else\textpm\fi{}}$, and a parameter \ensuremath{\mu} is introduced which is determined by specifying

{p}_{0}

. Exact results are obtained by mapping the model onto an effective Ising model on the rectangular lattice without field, and the probabilities

{p}_{0}

,${p}_{\ifmmode\pm\else\textpm\fi{}}$ are expressed in terms of the magnetization and two-spin correlations of the effective model. It is shown that the long-range order is not destroyed by the random field and the transition temperature increases as

{p}_{0}

decreases. For a given

{p}_{0}

the critical temperature increases with the field strength and these combined effects show that the annealed random field favors the ordering of the system, therefore presenting behavior opposite to that expected for the system in a quenched random field.

({J}_{x})

in the

x

direction and antiferromagnetic interaction

({J}_{y})

in the

y

direction. The phase diagram in the $T\text{\ensuremath{-}}H$ plane is obtained for the particular case

{J}_{x}={J}_{y}

. Special focus is given in the low-temperature region of the phase diagram, where a first-order phase transition is observed using the mean-field approximation, which is in disagreement with the linear chain approximation (LCA). Our results indicate a second-order phase transition for all values of

H∕J∊[0,2]

, with the presence of a reentrant behavior only observed in the BP approximation in accordance with the results of the LCA and exact solution. The null field critical temperature is an increasing function of

r={J}_{y}∕{J}_{x}

, and in the $r\ensuremath{\rightarrow}0$ limit we have found, using BP and EFT, the approximate form ${k}_{B}{T}_{N}∕{J}_{x}\ensuremath{\simeq}A∕\mathrm{ln}(1∕r)$ in accordance with the exact result of Onsager.