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 high-temperature expansions for the susceptibility, \ensuremath{\chi}=${\ensuremath{\sum}}_{\mathit{j}}$ [〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$${\mathrm{〉}}_{\mathit{T}}$-〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{〉}}_{\mathit{T}}$〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$${\mathrm{〉}}_{\mathit{T}}$${]}_{\mathit{h}}$ and the structure factor, G=${\ensuremath{\sum}}_{\mathit{j}}$[〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$${\mathrm{〉}}_{\mathit{T}}$${]}_{\mathit{h}}$, where 〈${\mathrm{〉}}_{\mathit{T}}$ indicates a canonical average at temperature T for an arbitrary configuration of random fields and [${]}_{\mathit{h}}$ indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each ${\mathit{h}}_{\mathit{i}}$=\ifmmode\pm\else\textpm\fi{}${\mathit{h}}_{0}$ and a Gaussian distribution in which each ${\mathit{h}}_{\mathit{i}}$ has variance ${\mathit{h}}_{0}^{2}$. We obtained series for \ensuremath{\chi} and G in the form ${\ensuremath{\sum}}_{\mathit{n}=1,15}$${\mathit{a}}_{\mathit{n}}$(g,d)(J/T${)}^{\mathit{n}}$, where J is the exchange constant and the coefficients ${\mathit{a}}_{\mathit{n}}$(g,d) are polynomials in g\ensuremath{\equiv}${\mathit{h}}_{0}^{2}$/${\mathit{J}}^{2}$ and in d. We assume that as T approaches its critical value, ${\mathit{T}}_{\mathit{c}}$, one has \ensuremath{\chi}\ensuremath{\sim}(T-${\mathit{T}}_{\mathit{c}}$${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ and G\ensuremath{\sim}(T-${\mathit{T}}_{\mathit{c}}$${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$. For dimensions above d=2 we find a range of values of g for which the critical exponents obtained from our series seem not to depend on g.For large values of g our results show a g dependence which is attributable to either a tricritical point or a first-order transition. All our results for critical exponents suggest that \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}, in agreement with the two-exponent scaling picture. In addition we have also constructed series for the amplitude ratio, A=(G/${\mathrm{\ensuremath{\chi}}}^{2}$)(${\mathit{T}}^{2}$)/(${\mathit{gJ}}^{2}$). We find that A approaches a constant value as T\ensuremath{\rightarrow}${\mathit{T}}_{\mathit{c}}$ (consistent with \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}) with A\ensuremath{\approxeq}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. \textcopyright{} 1996 The American Physical Society.