On the missing scaling relation in random field systems

Schwartz, Moshe; Gofman, M. N.; Natterman, Thomas

doi:10.1016/0378-4371(91)90071-j

Cited by 36 publications

(27 citation statements)

References 14 publications

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“…If this were exact, then we would conclude that GϪ scales like 2 , and hence that ␥ ϭ2␥. Furthermore, this would also imply that 23 Aϵ lim…”

Section: ͑12͒mentioning

confidence: 99%

“…6 Theoretical arguments in favor of ¯ϭ2 are summarized in Ref. 23 In contrast, an alternative approach 26 -29 starts from a droplet picture and maintains that is a new independent exponent, so that one needs three independent exponents to describe the critical behavior. For instance, Bray and Moore 27 derived scaling laws for the RFIM, based on the idea that the thermal phase transition is controlled by the zero-temperature fixed point.…”

Section: ͑12͒mentioning

confidence: 99%

“…In addition, the value of A was found to be quite close to unity in all dimensions, as was suggested on theoretical grounds. 23 The purpose of this paper is to give the details of the construction of these series and their analysis, the results of which were summarized previously. 54 This avenue of research is presently continuing.…”

Section: ͑12͒mentioning

confidence: 99%

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Critical behavior of the random-field Ising model

et al. 1996

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We study the critical properties of the random field Ising model in general dimension d using hightemperature expansions for the susceptibility, χ=∑ j [〈σ i σ j ⟩ T -〈σ i ⟩ T 〈σ j ⟩ T ] h and the structure factor, G=∑ j [〈σ i σ j ⟩ T ] h , where 〈⟩ T indicates a canonical average at temperature T for an arbitrary configuration of random fields and [ ] h indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each h i =±h 0 and a Gaussian distribution in which each hi has variance h 0 2 . We obtained series for χ and G in the form ∑ n=1,15 a n (g,d)(J/T) n , where J is the exchange constant and the coefficients a n (g,d) are polynomials in g≡h 0 2 /J 2 and in d. We assume that as T approaches its critical value, T c , one has χ~(T-T c ) −γ and G~(T-T c ) −γ . For dimensions above d=2 we find a range of values of g for which the critical exponents obtained from our series seem not to depend on g. For large values of g our results show a g dependence which is attributable to either a tricritical point or a first-order transition. All our results for critical exponents suggest that γ¯=2γ, in agreement with the two-exponent scaling picture. In addition we have also constructed series for the amplitude ratio, A=(G/χ 2 )(T 2 )/(gJ 2 ). We find that A approaches a constant value as T→T c (consistent with γ¯=2γ) with A~1. It appears that A is somewhat larger for the bimodal than for the Gaussian model, in agreement with a recent analysis at high d. Disciplines Physics CommentsAt the time of publication, author A. Brooks Harris was also affiliated with Tel Aviv University, Tel Aviv, Israel. Currently, he is a faculty member in the Physics Department at the University of Pennsylvania. We study the critical properties of the random field Ising model in general dimension d using hightemperature expansions for the susceptibility, ϭ ͚ j ͓͗ i j ͘ T Ϫ͗ i ͘ T ͗ j ͘ T ͔ h and the structure factor, Gϭ͚ j ͓͗ i j ͘ T ͔ h , where ͗͘ T indicates a canonical average at temperature T for an arbitrary configuration of random fields and ͓͔ h indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each h i ϭϮh 0 and a Gaussian distribution in which each h i has variance h 0 2 . We obtained series for and G in the form ͚ nϭ1,15 a n (g,d)(J/T) n , where J is the exchange constant and the coefficients a n (g,d) are polynomials in gϵh 0 2 /J 2 and in d. We assume that as T approaches its critical value, T c , one has ϳ(TϪT c ) Ϫ␥ and Gϳ(TϪT c ) Ϫ␥ . For dimensions above dϭ2 we find a range of values of g for which the critical exponents obtained from our series seem not to depend on g. For large values of g our results show a g dependence which is attributable to either a tricritical point or a first-order transition. All our results for critical exponents suggest that ␥ ϭ2␥, in agreement with the two-exponent scaling picture. In addition we have also constructed series for the amplitude ratio, Aϭ(G/ 2 )(T 2 )/(gJ 2 ). We find that ...

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“…If this were exact, then we would conclude that GϪ scales like 2 , and hence that ␥ ϭ2␥. Furthermore, this would also imply that 23 Aϵ lim…”

Section: ͑12͒mentioning

confidence: 99%

Section: ͑12͒mentioning

confidence: 99%

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Critical behavior of the random-field Ising model

et al. 1996

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“…A set of analytical arguments by Schwartz et al [252,253,254,255] These are values for the binary h i = ±h distribution of the random fields, however, those obtained for a continuous (Gaussian) distribution are not significantly different [233]. In addition Dayan et al [77] performed a real space renormalization group calculation that gave identical values for η and η as those listed in (25).…”

Section: Random Field Systemsmentioning

confidence: 99%

Monte Carlo Studies of Ising Spin Glasses and Random Field Systems

Rieger

1995

Annual Reviews of Computational Physics II

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We review recent numerical progress in the study of finite dimensional strongly disordered magnetic systems like spin glasses and random field systems. In particular we report in some details results for the critical properties and the non-equilibrium dynamics of Ising spin glasses. Furthermore we present an overview over recent investigations on the random field Ising model and finally of quantum spin glasses.

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“…[7] proposed the additional relation @=2', which also implies y=2y and 8=2 -rl. In fact, a detailed proof [8,9] In the second stage, we combined our recently developed e%cient three dimensional visualization methods [21] with several analysis algorithms [19], which allow nonanalytic conAuent corrections to scaling, to study series for g and gd in the above g windows. Values of the critical values K, (g) at selected g values are given in Table I, and we discuss the exponent values below.…”

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confidence: 99%