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Physical Review Letters

Critical Behavior of the Three-Dimensional Dilute Ising Antiferromagnet in a Field

Andrew T. Ogielski, David A. Huse

Published Mar 24, 1986• 140 citations
Publisher

Abstract

We report results of large Monte Carlo simulations of an experimentally realizable random-field system: the three-dimensional dilute Ising antiferromagnetic in a field. We find that the correlation time diverges dramatically as $T\ensuremath{\rightarrow}{T}_{c}$; the results are consistent with a proposed new type of activated dynamic scaling. The transition appears to be continuous, with effective critical exponents $\ensuremath{\eta}=0.5\ifmmode\pm\else\textpm\fi{}0.1$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}=\ensuremath{-}1.0\ifmmode\pm\else\textpm\fi{}0.3$ for strong fields, away from the weak-field regime where crossover effects distort exponent estimates. These strong-field exponents satisfy recently derived inequalities.

DOI
10.1103/physrevlett.56.1298

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We report results of large Monte Carlo simulations of an experimentally realizable random-field system: the three-dimensional dilute Ising antiferromagnetic in a field. We find that the correlation time diverges dramatically as T\ensuremath→TcT\ensuremath{\rightarrow}{T}_{c}T\ensuremath→Tc​; the results are consistent with a proposed new type of activated dynamic scaling. The transition appears to be continuous, with effective critical exponents \ensuremathη=0.5\ifmmode±\else\textpm\fi0.1\ensuremath{\eta}=0.5\ifmmode\pm\else\textpm\fi{}0.1\ensuremathη=0.5\ifmmode±\else\textpm\fi0.1 and \ensuremathη\ifmmode~\else˜\fi=\ensuremath−1.0\ifmmode±\else\textpm\fi0.3\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}=\ensuremath{-}1.0\ifmmode\pm\else\textpm\fi{}0.3\ensuremathη\ifmmode~\else˜\fi​=\ensuremath−1.0\ifmmode±\else\textpm\fi0.3 for strong fields, away from the weak-field regime where crossover effects distort exponent estimates. These strong-field exponents satisfy recently derived inequalities.
We perform intensive numerical simulations of the three-dimensional site-diluted Ising antiferromagnet in a magnetic field at high values of the external applied field. Even if data for small lattice sizes are compatible with second-order criticality, the critical behavior of the system shows a crossover from second-order to first-order behavior for large system sizes, where signals of latent heat appear. We propose ``apparent'' critical exponents for the dependence of some observables on the lattice size for a generic (disordered) first-order phase transition.
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We present a field-theoretical treatment of the critical behavior of a three-dimensional weakly diluted quenched Ising model. To this end we analyze in the replica limit $\stackrel{\ensuremath{\rightarrow}}{n}0$ the five-loop renormalization-group functions of the ${\ensuremath{\varphi}}^{4}$ theory with O(n)O(n)O(n)-symmetric and cubic interactions [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)]. The minimal subtraction scheme allows one to develop either the $\sqrt{\ensuremath{\varepsilon}}$-expansion series or to proceed within the three-dimensional approach, performing expansions in terms of renormalized couplings. Doing so, we compare both perturbation approaches and discuss their convergence and possible Borel summability. To study the crossover effect we calculate the effective critical exponents. We report resummed numerical values for the effective and asymptotic critical exponents. The results obtained within the three-dimensional approach agree pretty well with recent Monte Carlo simulations. $\sqrt{\ensuremath{\varepsilon}}$ expansion does not allow reliable estimates for d=3.d=3.d=3.
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A variety of phenomena is observed experimentally in random-field (RF) systems realized by the application of an external field to diluted antiferromagnets. At low temperatures, infinitely long hysteretic effects are manifested by the history dependence of the final states: long-range order is observed if the field is applied after cooling, while domain states are reached when field cooled. While no indications for critical fluctuations are detected in 2-D systems, scaling behavior, for both the correlation length and the specific heat, is observed in 3-D systems over an intermediate range of temperatures. The related critical properties seem to be well described by the corresponding ones in the 2-D pure Ising model. The renormalization-group approach, which yields for the equilibrium critical exponents their values of the pure model in d−2 dimensions, is reviewed. A generalization of the dimensional-reduction approach, which accounts self-consistently for renormalized responses of the RF system, is presented. The dynamical effects are implicitly incorporated through the variation in the critical response between the local and the global regimes, associated with short- and long-time scales, respectively. In both regimes the lower critical dimension is found to be d=2 in accordance with stability arguments. The short-time critical behavior indicates a dimensional reduction by one for the 3-D thermal exponents, in agreement with the experimental results.