Exact Inequality for Random Systems: Application to Random Fields

Schwartz, Moshe; Soffer, Avy

doi:10.1103/physrevlett.55.2499

Cited by 122 publications

(119 citation statements)

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“…Our analysis shows that all physical nonanalytic fixed points satisfying the Schwartz-Soffer inequality 19 have many unstable modes. In addition to the nonanalytic fixed points, we find four analytic fixed points given in Eq.…”

Section: Functional Renormalization Group For Large N Modelsmentioning

confidence: 99%

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Effect of second-rank random anisotropy on critical phenomena of a random-fieldO(N)spin model in the largeNlimit

2005

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We study the critical behavior of a random field O(N ) spin model with a second-rank random anisotropy term in spatial dimensions 4 < d < 6, by means of the replica method and the 1/N expansion. We obtain a replica-symmetric solution of the saddle-point equation, and we find the phase transition obeying dimensional reduction. We study the stability of the replica-symmetric saddle point against the fluctuation induced by the second-rank random anisotropy. We show that the eigenvalue of the Hessian at the replica-symmetric saddle point is strictly positive. Therefore, this saddle point is stable and the dimensional reduction holds in the 1/N expansion. To check the consistency with the functional renormalization group method, we obtain all fixed points of the renormalization group in the large N limit and discuss their stability. We find that the analytic fixed point yielding the dimensional reduction is practically singly unstable in a coupling constant space of the given model with large N . Thus, we conclude that the dimensional reduction holds for sufficiently large N .

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Section: Functional Renormalization Group For Large N Modelsmentioning

confidence: 99%

“…However, this fixed point solution is unphysical because it does not satisfy the Schwartz-Soffer inequality 2η ≥η. 19 This inequality requires a = 1 + O(1/N ). Other physical lower-branch fixed points satisfying the Schwartz-Soffer inequality have many relevant modes of O(N ).…”

Section: R =mentioning

confidence: 99%

Effect of second-rank random anisotropy on critical phenomena of a random-fieldO(N)spin model in the largeNlimit

2005

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“…Some exact inequalities among exponents were obtained by Schwartz and Soffer. 19 For the critical exponent they found у 4Ϫd 2 ͑10͒ and ¯р2 . ͑11͒…”

Section: ͑7͒mentioning

confidence: 99%

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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“…[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%