Critical behavior of the random-field Ising model

Abstract

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 ...

Critical aspects of the random-field Ising model

“…This latter estimate suggests, through the relationη = 4−γ/ν, the valueη = 1.026(2) for the critical exponent that describes the power-law decay of the disconnected correlation function of the RFIM. Overall, both values of β/ν and γ/ν are refinements of previously obtained estimates (see Tab. 1 for details and a direct comparison among previous works), and moreover, the value ofη is in agreement with the valueη = 1, that can be predicted from elementary considerations in the ordered phase [56] and the results of high-temperature series expansions of the RFIM [25,26].…”

“…The criteria for determining the order of the phase transition and its dependence on the field distribution have been discussed throughout the years [15][16][17][18][19][20][21][22][23][24][25][26]. In fact, different results have been proposed for different field distributions, like the existence of a tricritical point at the strong disorder regime of the system, present only in the bimodal case [15][16][17]20].…”

Quasi-Long Range Order in Glass States of Impure Liquid Crystals, Magnets, and Superconductors

“…Even the simplest problem, random-field Ising model, 32 is far from being completely understood. Recent results 84,85,46 make it hopeful that a satisfactory theory of phase transitions in the random-field Ising model will be developed soon. 2) where the const = max q D{h}|dP (h)/dh(q)| 2 /P (h).…”

Barkhausen noise in the random field Ising magnetNd2Fe14B

“…The original experimental studies of the RFIM followed a proposal by Imry and Ma, 12 in which a site-diluted Ising antiferromagnet in a large static magnetic field forms a realization of the RFIM Hamiltonian. While this approach proved fruitful for studying quantities such as the thermodynamic critical exponents 13,14 and correlation lengths, 15 the lack of a net long wavelength moment in antiferromagnets limits the potential probes and hence the set of physical questions that can be addressed. Uniaxial relaxor ferroelectrics were subsequently shown to be realizations of the RFIM 16,17 and similarly have provided insights into the critical and scaling behavior.…”