Physical review. B, Condensed matter

Critical behavior of theq-state Potts model with correlated random fields

Bosiljka Tadić, R. Pirc

Published Jul 1, 1985• 2 citations

Abstract

We study the critical behavior of the q-state Potts model in the presence of quenched random fields, which locally prefer ordering of any one of the q states, and are long-range correlated such that [H(0)H(R${)]}_{\mathrm{av}\mathrm{\ensuremath{\sim}}{R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}\ensuremath{\theta})}}$. We find a stable critical behavior controlled by a new random fixed point, and obtain a set of critical exponents within the \ensuremath{\epsilon}\ifmmode \tilde{}\else \~{}\fi{} expansion (\ensuremath{\epsilon}\ifmmode \tilde{}\else \~{}\fi{}=8+\ensuremath{\theta}-d), which cannot be related to those of a pure Potts model of lower dimensionality.

DOI: 10.1103/physrevb.32.350

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Physical review. B, Condensed matter

Critical behavior of theq-state Potts model with correlated random fields

Bosiljka Tadić, R. Pirc

Published Jul 1, 1985• 2 citations

Publisher

Abstract

{)]}_{\mathrm{av}\mathrm{\ensuremath{\sim}}{R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}\ensuremath{\theta})}}

. We find a stable critical behavior controlled by a new random fixed point, and obtain a set of critical exponents within the \ensuremath{\epsilon}\ifmmode \tilde{}\else ~{}\fi{} expansion (\ensuremath{\epsilon}\ifmmode \tilde{}\else ~{}\fi{}=8+\ensuremath{\theta}-d), which cannot be related to those of a pure Potts model of lower dimensionality.

DOI: 10.1103/physrevb.32.350

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Correlated percolation with long-range interactions

Bosiljka Tadić, R. Pirc

We study the critical behavior of the q-state Potts model with long-range interactions decaying asymptotically as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d+\ensuremath{\sigma})}$ in the presence of random- ${T}_{c}$ impurities correlated over large distances such that the correlations fall off as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}a)}$, a<d. We find that the renormalization-group scaling equations have a new fixed point in the appropriate double \ensuremath{\epsilon}, x expansion, where \ensuremath{\epsilon}=3\ensuremath{\sigma}-d and x=a-\ensuremath{\sigma}. This fixed point, however, is never both stable and physical.We study the critical behavior of the q-state Potts model with long-range interactions decaying asymptotically as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d+\ensuremath{\sigma})}$ in the presence of random-${T}_{c}$ impurities correlated over large distances such that the correlations fall off as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}a)}$, a<d. We find that the renormalization-group scaling equations have a new fixed point in the appropriate double \ensuremath{\epsilon}, x expansion, where \ensuremath{\epsilon}=3\ensuremath{\sigma}-d and x=a-\ensuremath{\sigma}. This fixed point, however, is never both stable and physical.

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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 ${[h(0)h(R)]}_{\mathrm{av}}\ensuremath{\sim}{R}^{\ensuremath{-}(d\ensuremath{-}\ensuremath{\theta})}$. 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 ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}^{2}$.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 ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}^{2}$.

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Scale-invariant quenched disorder and its stability criterion at random critical points

David Andelman, A. Nihat Berker

The critical properties of systems with quenched bond disorder are determined from a fixed distribution, under renormalization group, of the random bonds. Full fixed distributions with all moments are obtained numerically by histograms and, to a good approximation, in terms of $\ensuremath{\Gamma}$ distributions. For such systems, the specific-heat exponent $\ensuremath{\alpha}$ does not equal the crossover exponent $\ensuremath{\varphi}$ at random criticality. We derive a new relation between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$, which invokes characteristics of the fixed distribution. The difference between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$ is noted for $n$ -vector models in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive $\ensuremath{\alpha}$ appears to be possible. We develop a general treatment of quenched disorder and illustrate it by calculating specific-heat curves. It is suggested that the critical exponents of the three- and four-state random-bond Potts models in two dimensions are $\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19$.The critical properties of systems with quenched bond disorder are determined from a fixed distribution, under renormalization group, of the random bonds. Full fixed distributions with all moments are obtained numerically by histograms and, to a good approximation, in terms of $\ensuremath{\Gamma}$ distributions. For such systems, the specific-heat exponent $\ensuremath{\alpha}$ does not equal the crossover exponent $\ensuremath{\varphi}$ at random criticality. We derive a new relation between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$, which invokes characteristics of the fixed distribution. The difference between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$ is noted for $n$-vector models in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive $\ensuremath{\alpha}$ appears to be possible. We develop a general treatment of quenched disorder and illustrate it by calculating specific-heat curves. It is suggested that the critical exponents of the three- and four-state random-bond Potts models in two dimensions are $\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19$.

Physical review. B, Condensed matterMar 1, 1984104 citations

{)]}_{\mathrm{av}\mathrm{\ensuremath{\sim}}{R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}\ensuremath{\theta})}}

{T}_{c}

impurities correlated over large distances such that the correlations fall off as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}a)}$, a<d. We find that the renormalization-group scaling equations have a new fixed point in the appropriate double \ensuremath{\epsilon}, x expansion, where \ensuremath{\epsilon}=3\ensuremath{\sigma}-d and x=a-\ensuremath{\sigma}. This fixed point, however, is never both stable and physical.

The problem of critical behavior in systems with correlated random impurities has recently attracted a great deal of attention. For the

m

-component Heisenberg model, renormalization-group calculations using a novel double expansion predict second-order transitions described by a new random fixed point, whereas the usual single $\ensuremath{\epsilon}$ expansion leads to a runaway, often interpreted as a smeared transition. In this paper the corresponding results for the

p

-state Potts model with impurities which are perfectly correlated in certain directions are presented. (For $p\ensuremath{\le}2$ the pure Potts model has a continuous transition;

p=1

corresponds to the percolation problem.) In this case the new random fixed point is, in direct contrast to the

m

-component Heisenberg situation, never both physical and stable. Thus for the Potts model the runaway obtained in the single $\ensuremath{\epsilon}$ expansion persists. For sufficiently strong disorder the runaway is always obtained, even above the upper critical dimension where the Gaussian fixed point is (locally) stable. We speculate that the runaways may be symptomatic of a "smearing" of the phase transition by the strongly correlated impurities.

The effect of quenched impurities on systems undergoing first-order phase transitions is studied within the framework of the

q

-state Potts model. For large

q

a mapping to the random-field Ising model explains the absence of any latent heat in 2D, and suggests that for

d>2

such systems exhibit a tricritical point with an exponent $\ensuremath{\nu}$ related to those of the random-field model by $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\nu}}_{\mathrm{RF}}/(2\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{RF}}\ensuremath{-}{\ensuremath{\beta}}_{\mathrm{RF}})$. In 2D we analyze the model using finite-size scaling and conformal invariance, and find a continuous transition with a ratio $\ensuremath{\beta}/\ensuremath{\nu}$ which varies continuously with

q

, and a weakly varying exponent $\ensuremath{\nu}\ensuremath{\approx}1$. We find strong evidence for the multiscaling of the correlation functions.

\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 ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}^{2}$.

The effect of the long-range correlated random magnetic fields which behave like

〈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 $(m\ensuremath{-}2)\ensuremath{\sigma}\ensuremath{-}d+4=0$ (

m

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.

n

-vector models in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive $\ensuremath{\alpha}$ appears to be possible. We develop a general treatment of quenched disorder and illustrate it by calculating specific-heat curves. It is suggested that the critical exponents of the three- and four-state random-bond Potts models in two dimensions are $\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19$.