Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Europhysics Letters (EPL)

Crossover from Random Exchange to Random Field Critical Behaviour

Amnon Aharony A

Published Jun 15, 1986• 45 citations
Publisher

Abstract

The equality φ = γ between the exponent φ, which describes crossover from zero to finite-random-field critical behaviour, and the order parameter susceptibility exponent, γ, is shown to be replaced by φ > γ when the random exchange dominates. Rough renormalization group estimates yield φ/γ ≃ 1.1 for three-dimensional random Ising models. The implications to experiments on dilute antiferromagnets in uniform fields are discussed. Similarly to spin glasses, the nonlinear susceptibility of the latter is shown to diverge as ( T - T N ) -0.7 .

DOI
10.1209/0295-5075/1/12/002

Related papers

Critical behaviour of the random field Ising model

Jean-Yves Fortin, P. C. W. Holdsworth

We present an analytic real-space renormalization group calculation for the random-field Ising model. We apply the Migdal-Kadanoff approximation for the renormalization of a cubic cell in dimensions d , introducing a new field partitioning scheme which allows us to treat the random-field fluctuations in a coherent manner. Our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension. In three dimensions the magnetization exponent and the Schwartz-Soffer inequality is almost satisfied as an equality. We expand analytically in . Further, we show that and the magnitude of the inequality go to zero exponentially with . We calculate the crossover exponent, from pure to the random-field system and find surprisingly good agreement with experimental values. We find that satisfies the Schwartz-Soffer inequality: , the susceptibility exponent of the pure system. We expand in and find that the magnitude of the inequality varies exponentially in . Finally we find that dimensional reduction is satisfied to first order in , with the reduced dimension .

Journal of Physics A Mathematical and General•Jan 9, 1998•5 citations

Random coupling crossover in Ising ferromagnets

Rajiv R. P. Singh, Michael Ellis Fisher

Scaling and renormalization-group theories of crossover in criticality induced by random interactions are searchingly tested for Ising (n=1) systems. Extrapolation of 15th-order series in two variables for the susceptibility of the \ifmmode\pm\else\textpm\fi{}J simple-cubic Ising model yields a crossover exponent \ensuremath{\varphi}=0.125\ifmmode\pm\else\textpm\fi{}0.025. This agrees well with the specific-heat exponent, \ensuremath{\alpha}\ensuremath{\simeq}0.11. Consistency between free-energy and susceptibility scaling with \ensuremath{\varphi}=\ensuremath{\alpha} is cross-checked and leading universal scaling-function coefficients and nonuniversal scaling field amplitudes are determined.

Physical review. B, Condensed matter•Feb 1, 1988•7 citations

Dimensional crossover in quantal and finite-size Ising models

Ian D. Lawrie

The d-dimensional, quantum-mechanical Ising model in a transverse field Gamma displays a critical line, Gamma = Gamma c (T), which terminates in a multicritical point, ( Gamma ,T)=( Gamma c (0),0), with the exponents of the (d+1)-dimensional, nonquantal Ising model. Analogous behaviour is found for a nonquantal, (d+1)-dimensional n-component model of finite thickness, L in the last dimension, with the replacement Gamma to T,T to 1/L. A field theoretic method is used to study crossover behaviour in the vicinity of the multicritical point, for 3<d<4, where the multicritical exponents are mean-field-like. Multicritical behaviour is found to be governed by a Gaussian-like fixed point of the renormalisation group, yielding the crossover exponent, phi T = 1 / 2 (4-d). Crossover scaling functions are constructed to first order in epsilon =4-d for the longitudinal susceptibility and specific heat.

Journal of Physics C Solid State Physics•Sep 28, 1978•27 citations

Crossover from random-exchange to random-field critical behavior in Ising models

Pasquale Calabrese, Andrea Pelissetto C, Ettore Vicari

We compute the crossover exponent $\ensuremath{\varphi}$ describing the crossover from the random-exchange to the random-field critical behavior in Ising systems. For this purpose, we consider the field-theoretical approach based on the replica method, and perform a six-loop calculation in the framework of a fixed-dimension expansion. The crossover from random-exchange to random-field critical behavior has been observed in dilute anisotropic antiferromagnets, such as ${\mathrm{Fe}}_{x}{\mathrm{Zn}}_{1\ensuremath{-}x}{\mathrm{F}}_{2}$ and ${\mathrm{Mn}}_{x}{\mathrm{Zn}}_{1\ensuremath{-}x}{\mathrm{F}}_{2},$ when applying an external magnetic field. Our result $\ensuremath{\varphi}=1.42(2)$ for the crossover exponent is in good agreement with the available experimental estimates.

arXiv:cond-mat/0305041•Physical review. B, Condensed matter•Sep 24, 2003•8 citations

Hyperscaling and crossover exponents near the percolation threshold

Yuval Gefen, Amnon Aharony A, Yonathan Shapir +1 more

The critical exponents which describe quantities which are measured per unit 'mass' of the infinite cluster near percolation are shown to be shifted by beta p (the exponent describing the probability of belonging to this cluster). The fractal dimensionality of the infinite cluster then replaces the Euclidean one in hyperscaling relations. The crossover exponent for the effects of random fields on dilute Ising models are zero temperature is then shown to be phi h= gamma p+ beta p. Similarly, that for random local concentrations is phi p= alpha p+ beta p.

Journal of Physics C Solid State Physics•Aug 30, 1982•10 citations

Critical phenomena during a dimensional crossover

Denjoe O’Connor, Christopher Rhodes Stephens

The critical behaviour of a system undergoing a dimensional crossover is investigated. A renormalization group equation is obtained that interpolates between different dimensions. The susceptibility of an Ising-like system is treated in some detail, in particular and effective critical exponent for the susceptibility is computed, which for the particular crossover examined interpolates between the susceptibility exponents appropriate for a (4- in )-dimensional and a (3- in )-dimensional Ising model.

Journal of Physics A Mathematical and General•Jan 7, 1992•21 citations
The equality φ = γ between the exponent φ, which describes crossover from zero to finite-random-field critical behaviour, and the order parameter susceptibility exponent, γ, is shown to be replaced by φ > γ when the random exchange dominates. Rough renormalization group estimates yield φ/γ ≃ 1.1 for three-dimensional random Ising models. The implications to experiments on dilute antiferromagnets in uniform fields are discussed. Similarly to spin glasses, the nonlinear susceptibility of the latter is shown to diverge as ( T - T N ) -0.7 .
We present an analytic real-space renormalization group calculation for the random-field Ising model. We apply the Migdal-Kadanoff approximation for the renormalization of a cubic cell in dimensions d , introducing a new field partitioning scheme which allows us to treat the random-field fluctuations in a coherent manner. Our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension. In three dimensions the magnetization exponent and the Schwartz-Soffer inequality is almost satisfied as an equality. We expand analytically in . Further, we show that and the magnitude of the inequality go to zero exponentially with . We calculate the crossover exponent, from pure to the random-field system and find surprisingly good agreement with experimental values. We find that satisfies the Schwartz-Soffer inequality: , the susceptibility exponent of the pure system. We expand in and find that the magnitude of the inequality varies exponentially in . Finally we find that dimensional reduction is satisfied to first order in , with the reduced dimension .
Scaling and renormalization-group theories of crossover in criticality induced by random interactions are searchingly tested for Ising (n=1) systems. Extrapolation of 15th-order series in two variables for the susceptibility of the \ifmmode\pm\else\textpm\fi{}J simple-cubic Ising model yields a crossover exponent \ensuremath{\varphi}=0.125\ifmmode\pm\else\textpm\fi{}0.025. This agrees well with the specific-heat exponent, \ensuremath{\alpha}\ensuremath{\simeq}0.11. Consistency between free-energy and susceptibility scaling with \ensuremath{\varphi}=\ensuremath{\alpha} is cross-checked and leading universal scaling-function coefficients and nonuniversal scaling field amplitudes are determined.
The d-dimensional, quantum-mechanical Ising model in a transverse field Gamma displays a critical line, Gamma = Gamma c (T), which terminates in a multicritical point, ( Gamma ,T)=( Gamma c (0),0), with the exponents of the (d+1)-dimensional, nonquantal Ising model. Analogous behaviour is found for a nonquantal, (d+1)-dimensional n-component model of finite thickness, L in the last dimension, with the replacement Gamma to T,T to 1/L. A field theoretic method is used to study crossover behaviour in the vicinity of the multicritical point, for 3<d<4, where the multicritical exponents are mean-field-like. Multicritical behaviour is found to be governed by a Gaussian-like fixed point of the renormalisation group, yielding the crossover exponent, phi T = 1 / 2 (4-d). Crossover scaling functions are constructed to first order in epsilon =4-d for the longitudinal susceptibility and specific heat.
We compute the crossover exponent $\ensuremath{\varphi}$ describing the crossover from the random-exchange to the random-field critical behavior in Ising systems. For this purpose, we consider the field-theoretical approach based on the replica method, and perform a six-loop calculation in the framework of a fixed-dimension expansion. The crossover from random-exchange to random-field critical behavior has been observed in dilute anisotropic antiferromagnets, such as ${\mathrm{Fe}}_{x}{\mathrm{Zn}}_{1\ensuremath{-}x}{\mathrm{F}}_{2}$ and ${\mathrm{Mn}}_{x}{\mathrm{Zn}}_{1\ensuremath{-}x}{\mathrm{F}}_{2},$ when applying an external magnetic field. Our result $\ensuremath{\varphi}=1.42(2)$ for the crossover exponent is in good agreement with the available experimental estimates.
The critical exponents which describe quantities which are measured per unit 'mass' of the infinite cluster near percolation are shown to be shifted by beta p (the exponent describing the probability of belonging to this cluster). The fractal dimensionality of the infinite cluster then replaces the Euclidean one in hyperscaling relations. The crossover exponent for the effects of random fields on dilute Ising models are zero temperature is then shown to be phi h= gamma p+ beta p. Similarly, that for random local concentrations is phi p= alpha p+ beta p.
The critical behaviour of a system undergoing a dimensional crossover is investigated. A renormalization group equation is obtained that interpolates between different dimensions. The susceptibility of an Ising-like system is treated in some detail, in particular and effective critical exponent for the susceptibility is computed, which for the particular crossover examined interpolates between the susceptibility exponents appropriate for a (4- in )-dimensional and a (3- in )-dimensional Ising model.