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Physical review. B, Condensed matter

Experimental verification of activated critical dynamics in the d=3 random-field Ising model

A. E. Nash, A. R. King, Vincent Jaccarino

Published Jan 1, 1991• 18 citations
Publisher

Abstract

ac susceptibility studies of the d=3 random-field Ising-model (RFIM) system ${\mathrm{Fe}}_{0.46}$${\mathrm{Zn}}_{0.54}$${\mathrm{F}}_{2}$ in the 5\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$\ensuremath{\leqslant}\ensuremath{\omega}/2\ensuremath{\pi}\ensuremath{\leqslant}${10}^{5}$ Hz frequency range show the \ensuremath{\omega} scaling of the peak value [${\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}^{\ensuremath{'}}$(\ensuremath{\omega})${]}_{\mathrm{p}}$ and the characteristic rounding temperature ${\mathrm{t}}^{\mathrm{*}}$(\ensuremath{\omega}) are consistent with activated, but not conventional, dynamic scaling. We find [${\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}^{\ensuremath{'}}$(\ensuremath{\chi})${]}_{\mathrm{p}}$\ensuremath{\sim}log|log(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$)| and ${\mathrm{t}}^{\mathrm{*}}$(\ensuremath{\omega})\ensuremath{\sim}|log(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$${\mathrm{|}}^{\mathrm{\ensuremath{-}}1/\mathrm{y}}$ with y=1.05\ifmmode\pm\else\textpm\fi{}0.16 and ${\mathrm{\ensuremath{\omega}}}_{0}$\ensuremath{\simeq}${10}^{8}$ ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$ in agreement with the random-exchange Ising-model (REIM) dynamic frequency at the REIM-to-RFIM crossover. The relation between these results and the Villain-Fisher theories is discussed.

DOI
10.1103/physrevb.43.1272

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ac Susceptibility Study of the d=3 Random-Field Critical Dynamics

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The dynamic critical behavior in a $d=3$ random-field Ising-model system is studied with ac susceptibility $\ensuremath{\chi}(\ensuremath{\omega})$ techniques on ${\mathrm{Fe}}_{0.46}$${\mathrm{Zn}}_{0.54}$${\mathrm{F}}_{2}$. Extraordinary critical slowing down is seen in the $\ensuremath{\omega}$ dependence of ${\ensuremath{\chi}}^{\ensuremath{'}}(\ensuremath{\omega})$ at very low frequencies. In the range 2 Hz to 8.5 kHz, the peak value ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}$ decreases and the width ${t}^{*}(\ensuremath{\omega})$ increases with increasing $\ensuremath{\omega}$. A conventional dynamicscaling interpretation with ${t}^{*}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\frac{1}{z\ensuremath{\nu}}}$ gives ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}\ensuremath{\sim}|\mathrm{ln}\ensuremath{\omega}|$ and yields $z\ensuremath{\nu}\ensuremath{\simeq}14$. An equally good fit is obtained with ${t}^{*}(\ensuremath{\omega})\ensuremath{\sim}{|\mathrm{ln}\ensuremath{\omega}|}^{\ensuremath{-}1}$ and ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}\ensuremath{\sim}\mathrm{ln}|\mathrm{ln}\ensuremath{\omega}|$, consistent with an activated dynamic scaling.

Physical Review Letters•Jun 9, 1986•62 citations

Critical behavior of the random-field Ising model

Misha Gofman, Joan Adler, Amnon Aharony A +2 more

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.

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Activated dynamic scaling in the random-field Ising model: A nonperturbative functional renormalization group approach

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The random-field Ising model shows an extreme critical slowdown that has been described by activated dynamic scaling: The characteristic time of relaxation toward equilibrium diverges exponentially with the correlation length $ln\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{\psi}}/T$, with $\ensuremath{\psi}$ an a priori unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions $d$ less than a critical value ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, also associated with dimensional-reduction breakdown, $\ensuremath{\psi}=\ensuremath{\theta}$ with $\ensuremath{\theta}$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For $d>{d}_{\text{DR}}$, on the other hand, $\ensuremath{\psi}=\ensuremath{\theta}\ensuremath{-}2\ensuremath{\lambda}$, where $\ensuremath{\theta}=2$ and $\ensuremath{\lambda}>0$ an additional exponent. At the upper critical dimension $d=6,\ensuremath{\lambda}=1$, so that $\ensuremath{\psi}=0$, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches, and droplets. We also propose a way to check the two regimes by computer simulations of long-range one-dimensional systems.

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ac susceptibility studies of the d=3 random-field Ising-model (RFIM) system {\mathrm{Fe}}_{0.46}$${\mathrm{Zn}}_{0.54}$${\mathrm{F}}_{2} in the 5\ifmmode\times\else\texttimes\fi{}10\ensuremath−3{10}^{\mathrm{\ensuremath{-}}3}10\ensuremath−3\ensuremath{\leqslant}\ensuremath{\omega}/2\ensuremath{\pi}\ensuremath{\leqslant}105{10}^{5}105 Hz frequency range show the \ensuremath{\omega} scaling of the peak value [\ensuremathχc\ensuremath′{\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}^{\ensuremath{'}}\ensuremathχc\ensuremath′​(\ensuremath{\omega})]p{]}_{\mathrm{p}}]p​ and the characteristic rounding temperature t∗{\mathrm{t}}^{\mathrm{*}}t∗(\ensuremath{\omega}) are consistent with activated, but not conventional, dynamic scaling. We find [\ensuremathχc\ensuremath′{\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}^{\ensuremath{'}}\ensuremathχc\ensuremath′​(\ensuremath{\chi})]p{]}_{\mathrm{p}}]p​\ensuremath{\sim}log|log(\ensuremath{\omega}/\ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​)| and t∗{\mathrm{t}}^{\mathrm{*}}t∗(\ensuremath{\omega})\ensuremath{\sim}|log(\ensuremath{\omega}/{\mathrm{\ensuremath{\omega}}}_{0}$${\mathrm{|}}^{\mathrm{\ensuremath{-}}1/\mathrm{y}} with y=1.05\ifmmode\pm\else\textpm\fi{}0.16 and \ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​\ensuremath{\simeq}108{10}^{8}108 s\ensuremath−1{\mathrm{s}}^{\mathrm{\ensuremath{-}}1}s\ensuremath−1 in agreement with the random-exchange Ising-model (REIM) dynamic frequency at the REIM-to-RFIM crossover. The relation between these results and the Villain-Fisher theories is discussed.
The dynamic critical behavior in a d=3d=3d=3 random-field Ising-model system is studied with ac susceptibility $\ensuremath{\chi}(\ensuremath{\omega})$ techniques on Fe0.46{\mathrm{Fe}}_{0.46}Fe0.46​Zn0.54{\mathrm{Zn}}_{0.54}Zn0.54​F2{\mathrm{F}}_{2}F2​. Extraordinary critical slowing down is seen in the $\ensuremath{\omega}$ dependence of ${\ensuremath{\chi}}^{\ensuremath{'}}(\ensuremath{\omega})$ at very low frequencies. In the range 2 Hz to 8.5 kHz, the peak value ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}$ decreases and the width ${t}^{*}(\ensuremath{\omega})$ increases with increasing $\ensuremath{\omega}$. A conventional dynamicscaling interpretation with ${t}^{*}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\frac{1}{z\ensuremath{\nu}}}$ gives ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}\ensuremath{\sim}|\mathrm{ln}\ensuremath{\omega}|$ and yields $z\ensuremath{\nu}\ensuremath{\simeq}14$. An equally good fit is obtained with ${t}^{*}(\ensuremath{\omega})\ensuremath{\sim}{|\mathrm{ln}\ensuremath{\omega}|}^{\ensuremath{-}1}$ and ${[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}\ensuremath{\sim}\mathrm{ln}|\mathrm{ln}\ensuremath{\omega}|$, consistent with an activated dynamic scaling.
We study the critical properties of the random field Ising model in general dimension d using high-temperature expansions for the susceptibility, \ensuremath{\chi}=${\ensuremath{\sum}}_{\mathit{j}}$ [〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\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.
The random-field Ising model shows an extreme critical slowdown that has been described by activated dynamic scaling: The characteristic time of relaxation toward equilibrium diverges exponentially with the correlation length $ln\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{\psi}}/T$, with $\ensuremath{\psi}$ an a priori unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions ddd less than a critical value ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, also associated with dimensional-reduction breakdown, $\ensuremath{\psi}=\ensuremath{\theta}$ with $\ensuremath{\theta}$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For d>dDRd>{d}_{\text{DR}}d>dDR​, on the other hand, $\ensuremath{\psi}=\ensuremath{\theta}\ensuremath{-}2\ensuremath{\lambda}$, where $\ensuremath{\theta}=2$ and $\ensuremath{\lambda}>0$ an additional exponent. At the upper critical dimension $d=6,\ensuremath{\lambda}=1$, so that $\ensuremath{\psi}=0$, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches, and droplets. We also propose a way to check the two regimes by computer simulations of long-range one-dimensional systems.
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.
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In this thesis classical disordered spin systems, in particular, the random field Ising model (RFIM), are studied using numerical methods. The previous experimental and theoretical research on the RFIM is reviewed. Critical exponents of the RFIM are extracted at zero temperature. The heat capacity exponent α is found to be near zero. The ground states within a certain region with varying external field and disorder are portrayed, and it is speculated that if related quantities are scaled properly, ground state portraits at different system sizes are statistically similar. The performance of the Wang-Landau (WL) algorithm, which is used in our positive temperature study of the RFIM, is measured by the mean first-passage time. It is proved that the mean first, passage time of the mean field Ising model scales as N 2 log N, where N is the number of spins. The logarithmic correction term to the unbiased Markovian random walk is due to the boundaries. When applied to the Ising model with nearest neighbor interactions, the WL algorithm suffers from power law slowing down. Carefully designed cluster updating, however, can accelerate the WL algorithm and eliminate the slowing down. The WL algorithm is used in our finite temperature study of the RFIM. Physical quantities, such as the specific heat and the susceptibility, are obtained by the WL algorithm over a broad range of temperature. It is discovered that for some realizations with system size larger than 8 3, sharp peaks are present in these physical quantities. These sharp peaks result from flipping a large domain, and furthermore, the critical domains are strongly correlated to the domains found at the zero temperature transition. This observation suggests a strong version of the zero temperature fixed point picture. The MKRG approximation is employed to test the correlation between the critical states and the ground states in much larger systems. It is found that the fraction of correlated realizations drops as system size grows, while there are still significant number of correlated realizations even for systems as large as 10243. Whether substantial correlations will remain in the thermodynamic limit is unclear from our present data.