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Physical review. B, Condensed matter

High-temperature series and the random-field Ising model

A. Houghton, Anil Khurana, F. J. Seco

Published Aug 1, 1986• 26 citations
Publisher

Abstract

A formalism is given whereby high-temperature series for the random-field Ising model are obtained by partitioning the vertices of the pure Ising series' diagrams; for any given order in the high-temperature variable, the random fields are thereby treated exactly. Series for the susceptibility up to seventh order in the high-temperature variable have been obtained for an arbitrary distribution of the random fields. Series for a bimodal and a Gaussian distribution were analyzed by Dlog-Pad\'e methods. The analysis predicts that, for a bimodal distribution of random fields, the tricritical point predicted by mean-field theory persists in all dimensions above the lower critical dimension, but moves to weaker fields as the dimension is reduced. For the Gaussian distribution, the transition is predicted to be second order over the whole phase boundary for dimension d\ensuremath{\ge}4; however, for d<4 the analysis suggests that fluctuations drive the transition first order for sufficiently strong fields implying that d=4 is a ``critical'' dimension for this model. Above four dimensions the value of the susceptibility exponent determined from the series is consistent with the predictions of the d\ensuremath{\rightarrow}d-2 rule for both distributions.

DOI
10.1103/physrevb.34.1700

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A formalism is given whereby high-temperature series for the random-field Ising model are obtained by partitioning the vertices of the pure Ising series' diagrams; for any given order in the high-temperature variable, the random fields are thereby treated exactly. Series for the susceptibility up to seventh order in the high-temperature variable have been obtained for an arbitrary distribution of the random fields. Series for a bimodal and a Gaussian distribution were analyzed by Dlog-Pad'e methods. The analysis predicts that, for a bimodal distribution of random fields, the tricritical point predicted by mean-field theory persists in all dimensions above the lower critical dimension, but moves to weaker fields as the dimension is reduced. For the Gaussian distribution, the transition is predicted to be second order over the whole phase boundary for dimension d\ensuremath{\ge}4; however, for d<4 the analysis suggests that fluctuations drive the transition first order for sufficiently strong fields implying that d=4 is a ``critical'' dimension for this model. Above four dimensions the value of the susceptibility exponent determined from the series is consistent with the predictions of the d\ensuremath{\rightarrow}d-2 rule for both distributions.
A formalism is given whereby high-temperature series for the random-field Ising model on a ddd-dimensional hypercubic lattice is obtained by a partitioning of the vertices of the pure-Ising-series diagrams. For a bimodal distribution of quenched random fields we determine the series for the susceptibility to seventh order. Order by order the disorder is treated exactly. DlogDlogDlog Pad\'e analyses give a susceptibility exponent $\ensuremath{\gamma}$ in d=3d=3d=3 which crosses over from 1.24 in the pure limit to 1.40 as disorder increases.
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.
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 J2{\mathit{J}}^{2}J2g, we find that (${\mathrm{\ensuremath{\chi}}}_{\mathit{d}}$-\ensuremath{\chi})/K2{\mathit{K}}^{2}K2g${\mathrm{\ensuremath{\chi}}}^{2}$=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 ${\mathrm{\ensuremath{\chi}}}_{\mathit{d}}$\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-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}.
The form of the field distribution, in a random-field Ising model, is known to play an important role in determining the order of the phase transition and the existence of a tricritical point. We have studied a random-field Ising model with a trimodal distribution (sum of three \ensuremath{\delta} functions), which simulates a Gaussian distribution for a certain range of its parameters, within the mean-field renormalization-group scheme. We have considered clusters of one, two, and four spins, and the recursion relations for the determination of the fixed points are numerically analyzed. Using this analysis we have constructed the phase diagram of the system in the space of coupling (K), field (h), and zero-field probability (p). It is shown that the line of tricritical points ends for values of p near (2/3, very different from that obtained in the usual mean-field approximation. Mean-field renormalization-group (p\ensuremath{\simeq}(2/3) results are found to be consistent with the equivalence of the trimodal and Gaussian (absence of tricritical points) distributions until the fourth-order moments for p=(2/3.
In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(hr{\mathit{h}}_{\mathit{r}}hr​/T)2{)}^{2})2 being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.
Mean-field theory and renormalization-group arguments are used to show that the phase transition in a system with a random ordering field becomes first order at sufficiently low transition temperature, provided the (symmetric) random-field distribution function has a minimum at zero field. The first-order region is separated from the second-order region by a tricritical point. Both the critical and the tricritical exponents at d>4d>4d>4 dimensions are shown to be the same as for the pure system at $d\ensuremath{-}2$ dimensions. The relevance to spin glasses and other systems is discussed. The new tricritical point is very different from all previously studied tricritical points, as it deviates from mean-field theory at d=5d=5d=5, and not at d=3d=3d=3. Although quantitative results are calculated only at $d=5\ensuremath{-}\ensuremath{\epsilon}$ dimensions, the qualitative results are expected to apply at d=3d=3d=3.