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Physical review. B, Condensed matter

Two-dimensional Ising model in an annealed random field: Exact results

L.L. Gonçalves, Robin B. Stinchcombe

Published Apr 1, 1986• 3 citations
Publisher

Abstract

The effect of a discrete distribution of annealed random fields on the properties of the two-dimensional Ising model is studied. The random field is allowed to assume the values 0,\ifmmode\pm\else\textpm\fi{}h with the probabilities ${p}_{0}$,${p}_{\ifmmode\pm\else\textpm\fi{}}$, 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.

DOI
10.1103/physrevb.33.4762

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The effect of a discrete distribution of annealed random fields on the properties of the two-dimensional Ising model is studied. The random field is allowed to assume the values 0,\ifmmode\pm\else\textpm\fi{}h with the probabilities p0{p}_{0}p0​,p\ifmmode±\else\textpm\fi{p}_{\ifmmode\pm\else\textpm\fi{}}p\ifmmode±\else\textpm\fi​, and a parameter \ensuremath{\mu} is introduced which is determined by specifying p0{p}_{0}p0​. Exact results are obtained by mapping the model onto an effective Ising model on the rectangular lattice without field, and the probabilities p0{p}_{0}p0​,p\ifmmode±\else\textpm\fi{p}_{\ifmmode\pm\else\textpm\fi{}}p\ifmmode±\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 p0{p}_{0}p0​ decreases. For a given p0{p}_{0}p0​ 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.
The critical behavior of the two-dimensional Ising model (square lattice, exchange constant J) in an uniform field, and in an annealed random field is considered. The random field is generated by decorating the horizontal and vertical bonds of the lattice, and it satisfies an arbitrary distribution which is imposed by introducing a pseudo-chemical potential. By decimating the decorating variables the model can be mapped onto a homogeneous Ising model with effective exchange constant J′ and effective external field h′, dependent on the temperature. These parameters, which satisfy a set of coupled equations, depend on the spin average and nearest-neighbor two-spin correlation, and are obtained numerically. For the symmetric field distribution [Formula: see text] the mapping of the critical frontier on the (K′=βJ′,H′=βh′) plane onto the (K=β J,H=βh) plane is determined and, as in the model introduced by Essam and Place, there is a region on the (K, H) plane which cannot be reached from any real values of (K′, H′). The critical exponents are determined numerically, and it is shown that they do not satisfy renormalization relations obtained for their model.
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.
We study the equilibrium properties of an Ising model on a disordered random network where the disorder can be quenched or annealed. The network consists of fourfold coordinated sites connected via variable length one-dimensional chains. Our emphasis is on nonuniversal properties and we consider the transition temperature and other equilibrium thermodynamic properties, including those associated with one-dimensional fluctuations arising from the chains. We use analytic methods in the annealed case, and a Monte Carlo simulation for the quenched disorder. Our objective is to study the difference between quenched and annealed results with a broad random distribution of interaction parameters. The former represents a situation where the time scale associated with the randomness is very long and the corresponding degrees of freedom can be viewed as frozen, while the annealed case models the situation where this is not so. We find that the transition temperature and the entropy associated with one-dimensional fluctuations are always higher for quenched disorder than in the annealed case. These differences increase with the strength of the disorder up to a saturating value. We discuss our results in connection to physical systems where a broad distribution of interaction strengths is present.
The effect of quenched random fields and local perturbations of critical temperature on the critical behavior at phase transitions is studied within the framework of an exactly solvable model that takes into account interaction of fluctuations with equal and opposite momenta. Using the replica method the dimensional reduction by 2 for systems with finite-range interaction and quenched random fields is explicitly shown. For interaction of the infinite range the model demonstrates the mean-field critical asymptotics independently of dimensionality or the presence of random fields. \textcopyright{} 1996 The American Physical Society.
We have studied the critical phenomena of the Ising model with long-range interaction and the Ising model with spatially correlated random field, both of which are expected to change the effective dimension of the system continuously.The investigation was conducted from the numerical viewpoint by using the Markov chain Monte Carlo method.Second order phase transitions can be classified into the universality classes and the dimensionality of the Ising model changes the universality class: the one-dimensional Ising model has no long-range order at finite temperatures (lower critical dimension), four or higher dimensional model has the mean-field universality (upper critical dimension), two-dimensional Ising model belongs to different universality from either of them, and so on.Although both of the Ising models with long-range interaction and the Ising model with spatially correlated random field are believed to change their critical behavior from its real spatial dimension into different effective dimensions, their detailed behavior has not been known well because of the notorious difficulties in analysis.In this work, we have obtained the detailed behavior of these models by introducing and developing the various methods.To investigate the Ising model with long-range interaction, we introduced the Fukui-Todo method, which drastically reduces the computational cost from O(N 2 ) to O(N ), where N is the number of spins, and developed the generalized Ewald summation, the improved estimator, and the combined Binder ratio technique, which improves the convergence of the estimation.These techniques lead to the successful estimation of the boundary of the universality classes.We also investigated the Ising model with correlated random field by introducing the method for correlated random number generation and the various techniques for the precise analysis.Although the present numerical investigations suggested that the correlated random field causes a shift of effective dimensionality, they also revealed the difficulty for stronger random-field correlations because of the huge finite-size effects.i Prof. Synge
The statistical mechanics of finite $L\ifmmode\times\else\texttimes\fi{}L$ Ising square lattices in a field $\ifmmode\pm\else\textpm\fi{}h$ of random sign is investigated numerically by a modified recursive transfer matrix method for 6< L< 166<~L<~166< L< 16. Our results are consistent with the absence of a spontaneous magnetization for $h\ensuremath{\ne}0$ even in the ground state. The singularities occurring at T=0T=0T=0 in the range 0< h< 4J0<~h<~4J0< h< 4J, JJJ being the exchange constant, are discussed in terms of a cluster expansion. For nonzero TTT, less than the critical temperature of the pure two-dimensional Ising model, and hhh sufficiently small, the system exhibits a nonzero spin-glass order parameter of the Mattis type although there is no magnetization. The ferromagnetic correlation function becomes long ranged for $h\ensuremath{\rightarrow}0$ and is calculated from the domain-wall density.