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Physical Review Letters

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

A. R. King, John A. Mydosh, Vincent Jaccarino

Published Jun 9, 1986• 62 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.56.2525

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

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

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Response of Ising systems to oscillating and pulsed fields: Hysteresis, ac, and pulse susceptibility

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Physical review. B, Condensed matter•Sep 1, 1995•232 citations
The dynamic critical behavior in a d=3d=3d=3 random-field Ising-model system is studied with ac susceptibility \ensuremathχ(\ensuremathω)\ensuremath{\chi}(\ensuremath{\omega})\ensuremathχ(\ensuremathω) techniques on {\mathrm{Fe}}_{0.46}$${\mathrm{Zn}}_{0.54}$${\mathrm{F}}_{2}. Extraordinary critical slowing down is seen in the \ensuremathω\ensuremath{\omega}\ensuremathω dependence of \ensuremathχ\ensuremath′(\ensuremathω){\ensuremath{\chi}}^{\ensuremath{'}}(\ensuremath{\omega})\ensuremathχ\ensuremath′(\ensuremathω) at very low frequencies. In the range 2 Hz to 8.5 kHz, the peak value [\ensuremathχc\ensuremath′(\ensuremathω)]p{[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}[\ensuremathχc\ensuremath′​(\ensuremathω)]p​ decreases and the width t∗(\ensuremathω){t}^{*}(\ensuremath{\omega})t∗(\ensuremathω) increases with increasing \ensuremathω\ensuremath{\omega}\ensuremathω. A conventional dynamicscaling interpretation with t∗(\ensuremathω)\ensuremath∼\ensuremathω1z\ensuremathν{t}^{*}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\frac{1}{z\ensuremath{\nu}}}t∗(\ensuremathω)\ensuremath∼\ensuremathωz\ensuremathν1​ gives [\ensuremathχc\ensuremath′(\ensuremathω)]p\ensuremath∼∣ln\ensuremathω∣{[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}\ensuremath{\sim}|\mathrm{ln}\ensuremath{\omega}|[\ensuremathχc\ensuremath′​(\ensuremathω)]p​\ensuremath∼∣ln\ensuremathω∣ and yields z\ensuremathν\ensuremath≃14z\ensuremath{\nu}\ensuremath{\simeq}14z\ensuremathν\ensuremath≃14. An equally good fit is obtained with t∗(\ensuremathω)\ensuremath∼∣ln\ensuremathω∣\ensuremath−1{t}^{*}(\ensuremath{\omega})\ensuremath{\sim}{|\mathrm{ln}\ensuremath{\omega}|}^{\ensuremath{-}1}t∗(\ensuremathω)\ensuremath∼∣ln\ensuremathω∣\ensuremath−1 and [\ensuremathχc\ensuremath′(\ensuremathω)]p\ensuremath∼ln∣ln\ensuremathω∣{[{\ensuremath{\chi}}_{c}^{\ensuremath{'}}(\ensuremath{\omega})]}_{p}\ensuremath{\sim}\mathrm{ln}|\mathrm{ln}\ensuremath{\omega}|[\ensuremathχc\ensuremath′​(\ensuremathω)]p​\ensuremath∼ln∣ln\ensuremathω∣, consistent with an activated dynamic scaling.
ac susceptibility studies of the d=3 random-field Ising-model (RFIM) system Fe0.46{\mathrm{Fe}}_{0.46}Fe0.46​Zn0.54{\mathrm{Zn}}_{0.54}Zn0.54​F2{\mathrm{F}}_{2}F2​ in the 5\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$\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 [${\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}^{\ensuremath{'}}$(\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 [${\mathrm{\ensuremath{\chi}}}_{\mathrm{c}}^{\ensuremath{'}}$(\ensuremath{\chi})]p{]}_{\mathrm{p}}]p​\ensuremath{\sim}log|log(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$)| and t∗{\mathrm{t}}^{\mathrm{*}}t∗(\ensuremath{\omega})\ensuremath{\sim}|log(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$${\mathrm{|}}^{\mathrm{\ensuremath{-}}1/\mathrm{y}}$ with y=1.05\ifmmode\pm\else\textpm\fi{}0.16 and ${\mathrm{\ensuremath{\omega}}}_{0}$\ensuremath{\simeq}108{10}^{8}108 ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$ in agreement with the random-exchange Ising-model (REIM) dynamic frequency at the REIM-to-RFIM crossover. The relation between these results and the Villain-Fisher theories is discussed.
We study the critical properties of the random field Ising model in general dimension d using high-temperature expansions for the susceptibility, \ensuremath{\chi}=${\ensuremath{\sum}}_{\mathit{j}}$ [〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\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.
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We have studied, using Monte Carlo (MC) simulation for ferromagnetic Ising systems in one to four dimensions and solving numerically the mean-field (MF) equation of motion, the nature of the response magnetization m(t) of an Ising system in the presence of a periodically varying external field [h(t)=h0{\mathit{h}}_{0}h0​cos(\ensuremath{\omega}t)]. From these studies, we determine the m-h loop or hysteresis loop area A(=\ensuremath{\oint}mdh) and the dynamic order parameter Q(=\ensuremath{\oint}mdt) and investigate their variations with the frequency (\ensuremath{\omega}) and amplitude (h0{\mathit{h}}_{0}h0​) of the applied external magnetic field and the temperature (T) of the system. The variations in A are fitted to a scaling form, assumed to be valid over a wide range of parameter (\ensuremath{\omega},h0{\mathit{h}}_{0}h0​,T) values, and the best-fit exponents are obtained in all three dimensions (D=2,3,4). The scaling function is Lorentzian in the MF case and is exponentially decaying, with an initial power law, for the MC cases. The dynamic phase boundary (in the h0{\mathit{h}}_{0}h0​-T plane) is found to be frequency dependent and the transition (from Q\ensuremath{\ne}0 for low T and h0{\mathit{h}}_{0}h0​ to Q=0 for high T and h0{\mathit{h}}_{0}h0​) across the boundary crosses over from a discontinuous to a continuous one at a tricritical point. These boundaries are determined in various cases.We find that the response can be generally expressed as m(t)=P(\ensuremath{\omega}(t-${\mathrm{\ensuremath{\tau}}}_{\mathrm{eff}}$)) where P denotes a periodic function with the same frequency \ensuremath{\omega} of the perturbing field and ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$(h0{\mathit{h}}_{0}h0​,\ensuremath{\omega},T) denotes the effective delay. We established that this effective delay ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ of the response is the crucial term and it practically determines all the above observations for A, Q, etc. Investigating the nature of the in-phase (\ensuremath{\chi}\ensuremath{'}) and the out-of-phase (\ensuremath{\chi}\ensuremath{'}\ensuremath{'}) susceptibility, defined as \ensuremath{\chi}\ensuremath{'}=(m0{\mathit{m}}_{0}m0​/h0{\mathit{h}}_{0}h0​)cos(\ensuremath{\varphi}) and \ensuremath{\chi}\ensuremath{'}\ensuremath{'}=(m0{\mathit{m}}_{0}m0​/h0{\mathit{h}}_{0}h0​)sin(\ensuremath{\varphi}); \ensuremath{\varphi}=\ensuremath{\omega}${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ [and m0{\mathit{m}}_{0}m0​ is the amplitude of m(t)], we find that the loop area A is directly given by \ensuremath{\chi}\ensuremath{'}\ensuremath{'} and also the temperature variation of \ensuremath{\chi}\ensuremath{'}\ensuremath{'}, at fixed \ensuremath{\omega} and h0{\mathit{h}}_{0}h0​, gives a prominent peak at the dynamic transition point. We have also studied the behavior of the response magnetization by the application of a short-duration (compared with the relaxation time) pulsed magnetic field.Here, we observed (both in the MC and MF cases) that the width ratio (of the half-width and the width of the response magnetization and of the pulsed field, respectively) and the susceptibility (the ratio of excess magnetization over its equilibrium value and the height of the pulsed field) both show sharp peaks at the order-disorder (ferromagnetic-paramagnetic) transition point. We have also studied similar response behavior of Ising systems in the presence of time-varying longitudinal and transverse fields, solving numerically the mean-field equation of motion. We have again studied here the nature of the dynamic phase transition and the behavior of the ac susceptibility (both longitudinal and transverse) across the dynamic phase boundary. For a short-duration pulse of the transverse field, the width ratio and the pulse susceptibility are again seen to diverge at the order-disorder transition point.