Evidence for two exponent scaling in the random field Ising model

Gofman, M. N.; Adler, Joan; Aharony, Amnon; Harris, A. B.; Schwartz, Moshe

doi:10.1103/physrevlett.71.1569

Cited by 79 publications

(116 citation statements)

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“…Therefore, if there are three independent exponents we may take the third one to be either or ¯. The most important result of the present work, a brief summary of which was given previously, 54 is to establish that the critical point of the random field model is described by twoexponent scaling, through the relation ¯ϭ2 .…”

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confidence: 95%

“…In fact, the latter study led us to find an error in the last 2 terms of the series as reported in Ref. 54. The correct terms are given below, and all the series were reanalyzed yielding somewhat revised estimates for the exponents and amplitude ratio, as listed below.…”

Section: ͑12͒mentioning

confidence: 99%

“…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. Elsewhere 53 we will describe a study in high dimension which complements some of the results given here.…”

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confidence: 99%

“…The corrections do not change the basic qualitative conclusions of Ref. 54. Briefly, this paper is organized as follows.…”

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confidence: 99%

“…In the third stage, we addressed the issue of two versus three independent exponents, 54 by studying the amplitude ratio, A of Eq. ͑13͒.…”

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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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Section: ͑12͒mentioning

confidence: 95%

Section: ͑12͒mentioning

confidence: 99%