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Physical review. B, Condensed matter

Dynamics of spin systems with correlated random fields

Bosiljka Tadić, R. Pirc

Published Dec 1, 1984• 1 citation
Publisher

Abstract

We study the dynamic critical behavior of spin systems in the presence of quenched random magnetic fields with long-range correlations of the type ${[h(0)h(R)]}_{\mathrm{av}}\ensuremath{\sim}{R}^{\ensuremath{-}(d\ensuremath{-}\ensuremath{\theta})}$. We assume that the dynamics is described by a Langevin equation without conservation of the order parameter. It is shown in both the $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}$ expansion ($\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}=6+\ensuremath{\theta}\ensuremath{-}d$) to order ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}^{2}$ and $\frac{1}{N}$ expansion to leading order for $4+\ensuremath{\theta}<d<6+\ensuremath{\theta}$ that the dynamic critical exponent $z$ can be expressed in terms of static exponent $\ensuremath{\eta}$ as $z=2+g(\ensuremath{\theta})\ensuremath{\eta}$. The dynamic scaling form of the spin-spin correlation function at $T={T}_{c}$ is obtained to order ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}^{2}$.

DOI
10.1103/physrevb.30.6763

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We study the dynamic critical behavior of spin systems in the presence of quenched random magnetic fields with long-range correlations of the type [h(0)h(R)]av\ensuremath∼R\ensuremath−(d\ensuremath−\ensuremathθ){[h(0)h(R)]}_{\mathrm{av}}\ensuremath{\sim}{R}^{\ensuremath{-}(d\ensuremath{-}\ensuremath{\theta})}[h(0)h(R)]av​\ensuremath∼R\ensuremath−(d\ensuremath−\ensuremathθ). We assume that the dynamics is described by a Langevin equation without conservation of the order parameter. It is shown in both the \ensuremathϵ\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}\ensuremathϵ\ifmmode~\else˜\fi​ expansion (\ensuremathϵ\ifmmode~\else˜\fi=6+\ensuremathθ\ensuremath−d\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}=6+\ensuremath{\theta}\ensuremath{-}d\ensuremathϵ\ifmmode~\else˜\fi​=6+\ensuremathθ\ensuremath−d) to order \ensuremathϵ\ifmmode~\else˜\fi2{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}^{2}\ensuremathϵ\ifmmode~\else˜\fi​2 and 1N\frac{1}{N}N1​ expansion to leading order for 4+\ensuremathθ<d<6+\ensuremathθ4+\ensuremath{\theta}<d<6+\ensuremath{\theta}4+\ensuremathθ<d<6+\ensuremathθ that the dynamic critical exponent zzz can be expressed in terms of static exponent \ensuremathη\ensuremath{\eta}\ensuremathη as z=2+g(\ensuremathθ)\ensuremathηz=2+g(\ensuremath{\theta})\ensuremath{\eta}z=2+g(\ensuremathθ)\ensuremathη. The dynamic scaling form of the spin-spin correlation function at T=TcT={T}_{c}T=Tc​ is obtained to order \ensuremathϵ\ifmmode~\else˜\fi2{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}^{2}\ensuremathϵ\ifmmode~\else˜\fi​2.
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We study the crossover from pure to random critical behavior in ddd-dimensional ($d=4\ensuremath{-}\ensuremath{\epsilon}$) classical spin systems which is induced by quenched random fields with spatial correlations such that ${[h(0)h(R)]}_{\mathrm{av}}\ensuremath{\sim}\frac{{h}_{0}^{2}}{{R}^{d+\ensuremath{\theta}}}$, where $\ensuremath{\theta}=2\ensuremath{-}{\ensuremath{\epsilon}}_{d}$. An effective critical exponent for the magnetic susceptibility ${\ensuremath{\gamma}}_{\mathrm{eff}}=\frac{\ensuremath{-}\ensuremath{\partial}\mathrm{ln}\ensuremath{\chi}}{\ensuremath{\partial}\mathrm{ln}t}$ is calculated with the use of a renormalization-group approcah and a double $\ensuremath{\epsilon}$,$\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}$ expansion to linear order in $\ensuremath{\epsilon}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}\ensuremath{\equiv}\ensuremath{\epsilon}+{\ensuremath{\epsilon}}_{d}$. The crossover scaling function for the susceptibility is obtained and the width of the crossover region is estimated.
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