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 ...
