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Journal of Physics C Solid State Physics

Critical dynamics of long-range random-field models in a 1/N expansion

Reiner Kree, Ion Stancu

Published Jan 20, 1988• 0 citations
Publisher

Abstract

The authors calculate the dynamical critical exponent z for an N-vector model with long-range exchange couplings in random magnetic fields with long-range correlations using a 1/N expansion.
DOI
10.1088/0022-3719/21/2/001

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Random-field models are considered with long-range exchange, J(r) approximately 1/r d+ sigma , and/or long-range correlated random fields, (h(r)h(r')) approximately 1/ mod r-r' mod d- rho . A general scaling analysis is presented, in which the existence of three independent critical exponents is emphasized. Four different universality classes are identified, according to whether the fixed point controlling the critical behaviour is of short-range or long-range character for the exchange and/or the random-field correlations. The lines in the rho - sigma plane at which the long-range couplings first become relevant are expressed in terms of the exponents eta and eta of the corresponding short-range problem: long-range exchange is relevant for sigma <2- eta , long-range field correlations for rho >2 eta - eta . The scaling results are confirmed and amplified by explicit calculations of three independent exponents of the n-component spin model, correct to order 1/n. The results reveal errors in published expansions around the upper critical dimension.
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 1N\frac{1}{N}N1​ expansion to leading order for $4+\ensuremath{\theta}<d<6+\ensuremath{\theta}$ that the dynamic critical exponent zzz 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=TcT={T}_{c}T=Tc​ is obtained to order ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}^{2}$.
The effect of the long-range correlated random magnetic fields which behave like 〈h(x)h(y)〉〈h(x)h(y)〉〈h(x)h(y)〉 $\ensuremath{\sim}\frac{1}{{|x\ensuremath{-}y|}^{d\ensuremath{-}\ensuremath{\sigma}}}$ on the critical phenomena is discussed with the use of the nonlinear $\ensuremath{\sigma}$ model. The crossover between the two critical behaviors dominated by the long-range and short-range disorder fixed points is shown to occur at $(m\ensuremath{-}2)\ensuremath{\sigma}\ensuremath{-}d+4=0$ (mmm is the spin dimensionality). The critical exponents calculated at the long-range disorder fixed point around the lower critical dimensionality $4+\ensuremath{\sigma}$ are not the same as those in $d\ensuremath{-}2$ expansions in the pure system.
The authors solve exactly the large-n limit of the n-vector model with a quenched random field, allowing for the possibilities of both long-range correlations in the random field and long-range exchange interactions. It should be emphasised that, although the ideal Bose gas is frequently assumed identical to the large-n limit of the n-vector model, their identity has only been demonstrated for models with full translational symmetry (Gunton and Buckingham 1968, Stanley 1968). Indeed, without such symmetry, there are examples where several kinds of chemical potential must be introduced into the Bose gas in order to preserve the relationship with the n-vector model (Knops 1973, Fishman and Ziman 1982). In fact, the authors find that the two models behave equivalently. In addition, they are able to extend earlier results in the various ways necessary to present a comparison of the large-n limit with recent work (Chang and Abrahams 1984, Kardar et al. 1983) using epsilon -expansions about the lower and upper critical dimension some discrepancies emerge.