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Physical review. B, Condensed matter

Dimensional reduction with correlated random fields. A superspace renormalization-group calculation

Mehran Kardar A, Bruce McClain, Cyrus C. Taylor

Published May 1, 1983• 24 citations
Publisher

Abstract

We consider phase transitions in the presence of random magnetic fields, with long-range correlations such that $〈h(0)h(r)〉\ensuremath{\sim}\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. The upper critical dimension is $6+\ensuremath{\sigma}$. A superspace renormalization-group calculation is carried out to order ${\ensuremath{\epsilon}}^{2}$, where $\ensuremath{\epsilon}=6+\ensuremath{\sigma}\ensuremath{-}d$. To order $\ensuremath{\epsilon}$, the critical properties are that of a pure system in $d\ensuremath{-}2\ensuremath{-}\ensuremath{\sigma}$ dimensions. This dimensional reduction is also justified by physical arguments concerning the lower critical dimensionality, and the hyperscaling relationship. However, it fails at order ${\ensuremath{\epsilon}}^{2}$.

DOI
10.1103/physrevb.27.5875

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We consider phase transitions in the presence of random magnetic fields, with long-range correlations such that 〈h(0)h(r)〉\ensuremath∼1rd\ensuremath−\ensuremathσ〈h(0)h(r)〉\ensuremath{\sim}\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}〈h(0)h(r)〉\ensuremath∼rd\ensuremath−\ensuremathσ1​. The upper critical dimension is 6+\ensuremathσ6+\ensuremath{\sigma}6+\ensuremathσ. A superspace renormalization-group calculation is carried out to order \ensuremathϵ2{\ensuremath{\epsilon}}^{2}\ensuremathϵ2, where \ensuremathϵ=6+\ensuremathσ\ensuremath−d\ensuremath{\epsilon}=6+\ensuremath{\sigma}\ensuremath{-}d\ensuremathϵ=6+\ensuremathσ\ensuremath−d. To order \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ, the critical properties are that of a pure system in d\ensuremath−2\ensuremath−\ensuremathσd\ensuremath{-}2\ensuremath{-}\ensuremath{\sigma}d\ensuremath−2\ensuremath−\ensuremathσ dimensions. This dimensional reduction is also justified by physical arguments concerning the lower critical dimensionality, and the hyperscaling relationship. However, it fails at order \ensuremathϵ2{\ensuremath{\epsilon}}^{2}\ensuremathϵ2.
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The possibility of long-range order under quenched random magnetic fields can be deduced from strong-coupling rescaling behavior. The lower critical dimensions of dl=2{d}_{l}=2dl​=2 and 4 are indicated, respectively, for Ising and nnn-component spin models with n>1n>1n>1.
The random-field Ising model is one of the few disordered systems where the perturbative renormalization group can be carried out to all orders of perturbation theory. This analysis predicts dimensional reduction, i.e., that the critical properties of the random-field Ising model in D dimensions are identical to those of the pure Ising ferromagnet in D-2 dimensions. It is well known that dimensional reduction is not true in three dimensions, thus invalidating the perturbative renormalization group prediction. Here, we report high-precision numerical simulations of the 5D random-field Ising model at zero temperature. We illustrate universality by comparing different probability distributions for the random fields. We compute all the relevant critical exponents (including the critical slowing down exponent for the ground-state finding algorithm), as well as several other renormalization-group invariants. The estimated values of the critical exponents of the 5D random-field Ising model are statistically compatible to those of the pure 3D Ising ferromagnet. These results support the restoration of dimensional reduction at D=5. We thus conclude that the failure of the perturbative renormalization group is a low-dimensional phenomenon. We close our contribution by comparing universal quantities for the random-field problem at dimensions 3≤D<6 to their values in the pure Ising model at D-2 dimensions, and we provide a clear verification of the Rushbrooke equality at all studied dimensions.
We show that the hyperscaling law is modified whenever the relevant fixed point of the renormalization-group transformation is at zero temperature (infinite coupling). Instead of 2-\ensuremath{\alpha}=d\ensuremath{\nu}, the hyperscaling law now reads 2-\ensuremath{\alpha}=d'\ensuremath{\nu} where d' is the anomalous dimension which is equal to d-\ensuremath{\Vert}y\ensuremath{\Vert} and y is the exponent for the leading irrelevant operator that flows to zero temperature. We discuss the random-field Ising model in detail. We also perform a numerical domain-wall renormalization-group analysis for the d=3 model. For a Gaussian random-field distribution, our results are consistent with a second-order transition at low temperatures, and \ensuremath{\Vert}y\ensuremath{\Vert} is found to be 1.3--1.7. For a \ifmmode\pm\else\textpm\fi{}h distribution, the transition appears to be first order at T=0. .AE