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Physical review. B, Condensed matter

Fluctuation effects in the Ising model with reduced interaction and quenched disorder

Demetris Nicolaides, Alexander A. Lisyansky

Published Apr 1, 1996• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.53.8168

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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.
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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​,${p}_{\ifmmode\pm\else\textpm\fi{}}$, 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​,${p}_{\ifmmode\pm\else\textpm\fi{}}$ 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 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.
We investigate dimensional reduction, the property that the critical behavior of a system in the presence of quenched disorder in dimension ddd is the same as that of its pure counterpart in $d\ensuremath{-}2$, and its breakdown in the case of the random-field Ising model in which both the interactions and the correlations of the disorder are long ranged, i.e., power-law decaying. To some extent the power-law exponents play the role of spatial dimension in a short-range model, which allows us to probe the theoretically predicted existence of a nontrivial critical value separating a region where dimensional reduction holds from one where it is broken, while still considering the physical dimension d=3d=3d=3. By extending our recently developed approach based on a nonperturbative functional renormalization group combined with a supersymmetric formalism, we find that such a critical value indeed exists, provided one chooses a specific relation between the decay exponents of the interactions and of the disorder correlations. This transition from dimensional reduction to its breakdown should therefore be observable in simulations and numerical analyses, if not experimentally.
We consider the influence of a quenched random field on the critical behavior of a quantum spherical model at the zero-temperature quantum phase transition. We find a complete solution of the model for arbitrary translationally invariant pair interactions. It turns out that the critical behavior for zero as well as finite temperatures is dominated by static random field fluctuations rather than by quantum or thermal fluctuations. Therefore the critical behavior close to the zero-temperature quantum phase transition is identical to that close to a finite-temperature transition. The system does not show a crossover from quantum to classical behavior. \textcopyright{} 1996 The American Physical Society.
The critical properties of a ddd-dimensional Bose model in the presence of a random quenched field are investigated by renormalization-group techniques via the replica trick. To first order in $\ensuremath{\epsilon}=6\ensuremath{-}d$ it is found that both at nonzero- and zero-temperature regimes the grand-canonical critical exponents are the same as those of the corresponding classical model so that thermal and quantum fluctuations appear to be irrelevant. Furthermore, at zero temperature, the scaling laws already established for the pure model are retrieved with the replacement of ddd with $d\ensuremath{-}4$, but the random critical exponents are not derivable from the pure ones with the dimensional shift $d\ensuremath{\rightarrow}d\ensuremath{-}4$.
A variety of phenomena is observed experimentally in random-field (RF) systems realized by the application of an external field to diluted antiferromagnets. At low temperatures, infinitely long hysteretic effects are manifested by the history dependence of the final states: long-range order is observed if the field is applied after cooling, while domain states are reached when field cooled. While no indications for critical fluctuations are detected in 2-D systems, scaling behavior, for both the correlation length and the specific heat, is observed in 3-D systems over an intermediate range of temperatures. The related critical properties seem to be well described by the corresponding ones in the 2-D pure Ising model. The renormalization-group approach, which yields for the equilibrium critical exponents their values of the pure model in d−2 dimensions, is reviewed. A generalization of the dimensional-reduction approach, which accounts self-consistently for renormalized responses of the RF system, is presented. The dynamical effects are implicitly incorporated through the variation in the critical response between the local and the global regimes, associated with short- and long-time scales, respectively. In both regimes the lower critical dimension is found to be d=2 in accordance with stability arguments. The short-time critical behavior indicates a dimensional reduction by one for the 3-D thermal exponents, in agreement with the experimental results.