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Physical Review Letters

Evidence for two exponent scaling in the random field Ising model

Michael Gofman, Joan Adler, Amnon Aharony A, A. Brooks Harris, Moshe Schwartz

Published Sep 6, 1993• 77 citations
Publisher

Abstract

Novel methods were used to generate and analyze new 15 term high temperature series for both the (connected) susceptibility \ensuremath{\chi} and the structure factor (disconnected susceptibility) ${\mathrm{\ensuremath{\chi}}}_{\mathit{d}}$ for the random field Ising model with dimensionless coupling K=J/kT, in general dimension d. For both the bimodal and the Gaussian field distributions, with mean square field ${\mathit{J}}^{2}$g, we find that (${\mathrm{\ensuremath{\chi}}}_{\mathit{d}}$-\ensuremath{\chi})/${\mathit{K}}^{2}$g${\mathrm{\ensuremath{\chi}}}^{2}$=1 as T\ensuremath{\rightarrow}${\mathit{T}}_{\mathit{c}}$(g), for a range of [${\mathit{h}}^{2}$]=${\mathit{J}}^{2}$g and d=3,4,5. This confirms the exponent relation \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma} (where ${\mathrm{\ensuremath{\chi}}}_{\mathit{d}}$\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$, \ensuremath{\chi}\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$, t=T-${\mathit{T}}_{\mathit{c}}$) providing that random field exponents are determined by two (and not three) independent exponents. We also present new accurate values for \ensuremath{\gamma}.

DOI
10.1103/physrevlett.71.1569

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Novel methods were used to generate and analyze new 15 term high temperature series for both the (connected) susceptibility \ensuremath{\chi} and the structure factor (disconnected susceptibility) \ensuremathχd{\mathrm{\ensuremath{\chi}}}_{\mathit{d}}\ensuremathχd​ for the random field Ising model with dimensionless coupling K=J/kT, in general dimension d. For both the bimodal and the Gaussian field distributions, with mean square field J2{\mathit{J}}^{2}J2g, we find that (\ensuremathχd{\mathrm{\ensuremath{\chi}}}_{\mathit{d}}\ensuremathχd​-\ensuremath{\chi})/K2{\mathit{K}}^{2}K2g\ensuremathχ2{\mathrm{\ensuremath{\chi}}}^{2}\ensuremathχ2=1 as T\ensuremath{\rightarrow}Tc{\mathit{T}}_{\mathit{c}}Tc​(g), for a range of [h2{\mathit{h}}^{2}h2]=J2{\mathit{J}}^{2}J2g and d=3,4,5. This confirms the exponent relation \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma} (where \ensuremathχd{\mathrm{\ensuremath{\chi}}}_{\mathit{d}}\ensuremathχd​\ensuremath{\sim}t\ensuremath−\ensuremathγ\ifmmode\elseˉ\textasciimacron\fi{\mathit{t}}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}t\ensuremath−\ensuremathγ\ifmmode\elseˉ​\textasciimacron\fi, \ensuremath{\chi}\ensuremath{\sim}t\ensuremath−\ensuremathγ{\mathit{t}}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}t\ensuremath−\ensuremathγ, t=T-Tc{\mathit{T}}_{\mathit{c}}Tc​) providing that random field exponents are determined by two (and not three) independent exponents. We also present new accurate values for \ensuremath{\gamma}.
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}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$〉T{\mathrm{〉}}_{\mathit{T}}〉T​-〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$〉T{\mathrm{〉}}_{\mathit{T}}〉T​〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$〉T{\mathrm{〉}}_{\mathit{T}}〉T​]h{]}_{\mathit{h}}]h​ and the structure factor, G=${\ensuremath{\sum}}_{\mathit{j}}$[〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$〉T{\mathrm{〉}}_{\mathit{T}}〉T​]h{]}_{\mathit{h}}]h​, where 〈〉T{\mathrm{〉}}_{\mathit{T}}〉T​ indicates a canonical average at temperature T for an arbitrary configuration of random fields and []h{]}_{\mathit{h}}]h​ indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each hi{\mathit{h}}_{\mathit{i}}hi​=\ifmmode\pm\else\textpm\fi{}h0{\mathit{h}}_{0}h0​ and a Gaussian distribution in which each hi{\mathit{h}}_{\mathit{i}}hi​ has variance h02{\mathit{h}}_{0}^{2}h02​. We obtained series for \ensuremath{\chi} and G in the form ${\ensuremath{\sum}}_{\mathit{n}=1,15}$an{\mathit{a}}_{\mathit{n}}an​(g,d)(J/T)n{)}^{\mathit{n}})n, where J is the exchange constant and the coefficients an{\mathit{a}}_{\mathit{n}}an​(g,d) are polynomials in g\ensuremath{\equiv}h02{\mathit{h}}_{0}^{2}h02​/J2{\mathit{J}}^{2}J2 and in d. We assume that as T approaches its critical value, Tc{\mathit{T}}_{\mathit{c}}Tc​, one has \ensuremath{\chi}\ensuremath{\sim}(T-Tc{\mathit{T}}_{\mathit{c}}Tc​${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ and G\ensuremath{\sim}(T-Tc{\mathit{T}}_{\mathit{c}}Tc​${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$. 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/${\mathrm{\ensuremath{\chi}}}^{2}$)(T2{\mathit{T}}^{2}T2)/(gJ2{\mathit{gJ}}^{2}gJ2). We find that A approaches a constant value as T\ensuremath{\rightarrow}Tc{\mathit{T}}_{\mathit{c}}Tc​ (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.
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The ferromagnet-to-paramagnet transition of the four-dimensional random-field Ising model with Gaussian distribution of the random fields is studied. Exact ground states of systems with sizes up to 324{32}^{4}324 are obtained using graph theoretical algorithms. The magnetization, the disconnected susceptibility, the susceptibility, and a specific-heat-like quantity are calculated. Using finite-size scaling techniques, the corresponding critical exponents are obtained: $\ensuremath{\beta}=0.13(5),$ $\overline{\ensuremath{\gamma}}=2.83(50),$ $\ensuremath{\gamma}=1.42(20).$ For the specific heat both $\ensuremath{\alpha}=0$ (logarithmic divergence) and an algebraic divergence with $\ensuremath{\alpha}=0.26(5)$ are compatible with the data. Furthermore, values for the critical randomness hc=4.18(1){h}_{c}=4.18(1)hc​=4.18(1) and the correlation-length exponent $\ensuremath{\nu}=0.78(10)$ were found. These independently obtained exponents are compatible with all (hyper)scaling relations and support the two-exponent scenario $(\overline{\ensuremath{\gamma}}=2\ensuremath{\gamma}).$
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