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Journal of Physics A Mathematical and General

Migdal-Kadanoff renormalization group with long-range disorder

Xintian Wu, Zhilei Huang

Published Aug 7, 1994• 1 citation
Publisher

Abstract

Phase transitions in a two-dimensional q-state Potts model with long-range disorder in which the correlation decays as a power law approximately r -a for separation r are studied analytically within a Migdal-Kadanoff renormalization group. Correlation terms are introduced into the parameter space of the renormalization group and the recursion relations are derived. We expand q near q 0 , where the specific-heat exponent of the pure system alpha p vanishes, and a near d=2. For small alpha p and delta =2-a<<1, we find three fixed points: 'pure','short-range disorder' and 'long-range disorder'. In our calculation the disorder is relevant if av-2<0 for a<d, where the v are the correlation-length exponents of the fixed points. The correlation-length exponent for the 'long-range disorder' fixed point is v long =2/a. These are consistent with field-theoretic renormalization group results.

DOI
10.1088/0305-4470/27/15/012

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Phase transitions in a two-dimensional q-state Potts model with long-range disorder in which the correlation decays as a power law approximately r -a for separation r are studied analytically within a Migdal-Kadanoff renormalization group. Correlation terms are introduced into the parameter space of the renormalization group and the recursion relations are derived. We expand q near q 0 , where the specific-heat exponent of the pure system alpha p vanishes, and a near d=2. For small alpha p and delta =2-a<<1, we find three fixed points: 'pure','short-range disorder' and 'long-range disorder'. In our calculation the disorder is relevant if av-2<0 for a<d, where the v are the correlation-length exponents of the fixed points. The correlation-length exponent for the 'long-range disorder' fixed point is v long =2/a. These are consistent with field-theoretic renormalization group results.
In this paper we provide new analytic results on two-dimensional q q -Potts models ( q ≥ 2 q ≥ 2 ) in the presence of bond disorder correlations which decay algebraically with distance with exponent a a . In particular, our results are valid for the long-range bond disordered Ising model ( q=2 q = 2 ). We implement a renormalization group perturbative approach based on conformal perturbation theory. We extend to the long-range case the RG scheme used in [V. Dotsenko et al., Nucl. Phys. B 455 701-23] for the short-range disorder. Our approach is based on a 2 2 -loop order double expansion in the positive parameters (2-a) ( 2 − a ) and (q-2) ( q − 2 ) . We will show that the Weinrib-Halperin conjecture for the long-range thermal exponent can be violated for a non-Gaussian disorder. We compute the central charges of the long-range fixed points finding a very good agreement with numerical measurements.
The one-dimensional q-state Potts model with ferromagnetic pair interactions which decay with the distance r as is considered. We calculate, through a real-space renormalization group technique using Kadanoff blocks of length b, the critical temperature and the correlation length critical exponent as a function of for different values of q. Some of the very few known rigorous results for general q are reproduced by our approach. Several asymptotic behaviours are derived analytically for q = 2, 3 in the limit. We also obtain extrapolated critical temperatures for arbitrary values of and for q = 2, 3, 4, which we believe approximate the exact ones well, except in the region near . Furthermore, the use of another extrapolation procedure suitable only in the vicinity of led us to conjecture that the exact critical temperature is the same for any value of q . We also verify that , which is consistent with a recent conjecture of Tsallis.
As a model for a phase transition in an inhomogeneous system, we consider a system where the local transition temperature varies in space, with a correlation function obeying a power law $\ensuremath{\sim}{x}^{\ensuremath{-}a}$ for large separations xxx. We extend the Harris criterion for this case, finding that for a<da<da<d (where ddd is the spatial dimension) the disorder is irrelevant if $a\ensuremath{\nu}\ensuremath{-}2>0$, while if a>da>da>d we recover the usual Harris criterion: The disorder is irrelevant if $d\ensuremath{\nu}\ensuremath{-}2=\ensuremath{-}\ensuremath{\alpha}>0$. An mmm-vector system of this type is studied with the use of a renormalization-group expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and $\ensuremath{\delta}=4\ensuremath{-}a$. We find a new long-range-disorder fixed point in addition to the short-range-disorder and pure fixed points found previously. The crossover between fixed points is found to follow the extended Harris criterion. The new fixed point has complex eigenvalues, leading to oscillating corrections to scaling, and has a correlation-length exponent $\ensuremath{\nu}=\frac{2}{a}$. We argue that this new scaling relation is exact and applies more generally than just to the specific model. We show that the extended Harris criterion also applies to percolation with long-range-correlated site or bond-occupation probabilities, so that the scaling law should be obeyed by such systems. Results for the percolation properties of the triangular Ising model are in agreement with these predictions.
Using a weak-disorder scheme and real-space renormalization-group techniques, we obtain analytical results for the critical behavior of various q-state Potts models with correlated disordered exchange interactions along d(1) of d spatial dimensions on hierarchical (Migdal-Kadanoff) lattices. Our results indicate qualitative differences between the cases d-d(1)=1 (for which we find non-physical random fixed points, suggesting the existence of non-perturbative fixed distributions) and d-d(1)>1 (for which we do find acceptable perturbative random fixed points), in agreement with previous numerical calculations by Andelman and Aharony [Phys. Rev. B 31, 4305 (1985)]. We also rederive a criterion for relevance of correlated disorder, which generalizes the usual Harris criterion.
This paper is a study of the percolation problem with long-range correlations in the site or bond occupations. An extension of the Harris criterion for the relevance of the correlations is derived for the case that the correlations decay as ${x}^{\ensuremath{-}a}$ for large distances xxx.. For a<da<da<d the correlations are relevant if $a\ensuremath{\nu}\ensuremath{-}2<0$ (where $\ensuremath{\nu}$ is the percolation correlation-length exponent), while for a>da>da>d the correlations are relevant if $d\ensuremath{\nu}\ensuremath{-}2<0$.. Applying this criterion to the behavior that results when the correlations are relevant, we argue that the new behavior will have ${\ensuremath{\nu}}_{\mathrm{long}}=\frac{2}{a}$.. It is shown that the correlated bond percolation problem is equivalent to a qqq-state Potts model with quenched disorder in the limit $q\ensuremath{\rightarrow}1$.. With the use of this result, a renormalization-group study of the problem is presented, expanding in $\ensuremath{\epsilon}=6\ensuremath{-}d$ and in $\ensuremath{\delta}=4\ensuremath{-}a$.. In addition to the normal percolation fixed point, we find a new long-range fixed point. The crossover to this new fixed point follows the extended Harris criterion, and the fixed point has exponents ${\ensuremath{\nu}}_{\mathrm{long}}=\frac{2}{a}$ (as predicted) and ${\ensuremath{\eta}}_{\mathrm{long}}=\frac{1}{11}(\ensuremath{\delta}\ensuremath{-}\ensuremath{\epsilon})$. Finally, several results on the percolation properties of the Ising model at its critical point are shown to be in agreement with the predictions of this paper.