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Journal of Physics C Solid State Physics

Self-similar replication approach to random-fields systems

Yonathan Shapir

Published Oct 30, 1984• 14 citations
Publisher

Abstract

Very long relaxation times are present in random-fields systems near their critical points. As the correlation length grows, already above the transition it is necessary to differentiate between the short-time (local) regime and the long-time (global) one. A self-similar replication procedure is proposed in order to separate iteratively between slow and fast modes. Recursion relations among effective exponents on successive time scales are obtained and apparent reduced dimensions, in both regimes, are deduced. For the short time behaviour of Ising systems we the authors find a dimensional reduction by one, both in d=3 and in d=2 (the lower critical dimension). The results are in agreement with the measurements of the specific heat and the correlation length performed on random antiferromagnets by linear birefringence and neutron scattering respectively.

DOI
10.1088/0022-3719/17/30/001

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Very long relaxation times are present in random-fields systems near their critical points. As the correlation length grows, already above the transition it is necessary to differentiate between the short-time (local) regime and the long-time (global) one. A self-similar replication procedure is proposed in order to separate iteratively between slow and fast modes. Recursion relations among effective exponents on successive time scales are obtained and apparent reduced dimensions, in both regimes, are deduced. For the short time behaviour of Ising systems we the authors find a dimensional reduction by one, both in d=3 and in d=2 (the lower critical dimension). The results are in agreement with the measurements of the specific heat and the correlation length performed on random antiferromagnets by linear birefringence and neutron scattering respectively.
The critical behavior of random-field systems is characterized by exponentially long relaxation times. They may differ by orders of magnitude and slow modes have to be considered quenched with respect to the faster ones. This requires a drastic modification of the renormalization process in which the degrees of freedom are integrated out. A new replica-space renormalization procedure, to carry out the coarse-graining of the time intervals, is presented. We relate by recursion-relations the effective reduced dimensions of consecutive time scales. Their stable fixed-points yield the apparent dimensionalities for the longest and shortest time scales, which are insensitive to the exact behavior on intermediate scales. For Ising systems we find that d = 2 is the lower critical dimension in both regimes. The thermal exponents for d = 3, in the short-time observable regime, are related to those of the pure system in d = 2. This is consistent with the observations in field-cooled random antiferromagnets of the correlation length exponent nu approx. = 1 (by neutron scattering) and of the symmetric logarithmic divergence in the specific-heat (by linear birefringence). 17 refs., 3 figs.
The thermal critical properties of random-field systems in the observable, linear, local-response regime are considered. A consistent nonperturbative approach yields an effective reduced dimension $\overline{d}=\frac{[d+\frac{1}{\ensuremath{\nu}(\overline{d})}]}{2}$ for the thermal exponents if $\ensuremath{\alpha}(\overline{d})<~0$. The consequences for Ising systems are particularly reliable at the upper and lower critical dimensions (du.c.d.=6{d}_{\mathrm{u}.\mathrm{c}.\mathrm{d}.}=6du.c.d.​=6 and dl.c.d.=2{d}_{\mathrm{l}.\mathrm{c}.\mathrm{d}.}=2dl.c.d.​=2, respectively) as well as at d=3d=3d=3 where $\ensuremath{\alpha}=0$ (logarithmic divergence). The results are in agreement with measurements on random antiferromagnets of the specific heat (by linear birefringence) and the correlation length (by neutron scattering).
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.
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.
A simple scaling description of the ordering transition in random-field Ising systems is developed and supported by renormalization-group arguments in terms of a zero-temperature critical fixed point. The main prediction is that the characteristic relaxation time \ensuremath{\tau} will diverge extremely rapidly as the critical point is approached: \ensuremath{\tau}\ensuremath{\sim}exp(${\ensuremath{\xi}}^{\mathrm{theta}}$) with \ensuremath{\xi} the correlation length and theta the ``violation of hyperscaling'' exponent (d-theta)\ensuremath{\nu}=2-\ensuremath{\alpha}. Recent experiments which exhibit onset of hysteresis in a very narrow temperature range are discussed.
The authors study the Ising ferromagnet in a random field following the renormalisation of the distribution of random fields. The disorder in the local field induces fluctuations in the exchange couplings which are taken into account. The method allows a straightforward determination of the exponents associated with the zero-temperature fixed point which governs the critical behaviour along the critical line and in particular the critical slowing down. They find the correlation length exponent v=1.0 which is in very good agreement with numerical calculations. They also obtain the crossover exponent for small random fields.