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

Journal of Physics C Solid State Physics

Renormalisation group analysis of random n-vector model. II. Crossover behaviour

Prabodh Shukla

Published Aug 28, 1977• 0 citations
Publisher

Abstract

For pt.I see ibid., vol.10, p.2985 (1977). The crossover function for the specific heat of a disordered material is constructed to order epsilon by solving the renormalisation group equations for the m n model (m to O,n>1). From this an effective exponent is obtained which provides a local measure of the degree of singularity of the specific heat in the critical region. It is seen that as the critical region is approached from above the critical temperature the effective exponent behaves initially as if no disorder were present. But as the critical temperature is more closely approached the effective exponent crosses over to the value predicted for the disordered material.

DOI
10.1088/0022-3719/10/16/009

Related papers

Renormalisation group analysis of random n-vector model. I. Smooth cut-off renormalisation group equations

Prabodh Shukla

The fixed points of the exact renormalisation group equations with smooth cut-off are determined in the epsilon -expansion for systems characterised by mn-component vector order-parameters. The critical behaviour of quenched random n-component spin systems is obtained by taking the limit m=O. The critical exponent is obtained to order epsilon 2 , and the other exponents to order epsilon . A derivation of Nelson's trajectory-integral expression for the free energy, appropriate for the case of smooth cut-off renormalisation group equations, is presented.

Journal of Physics C Solid State Physics•Aug 28, 1977•1 citation

Crossover behaviour and effective critical exponents in isotropic and anisotropic Heisenberg systems

Ann Bruce, D. J. Wallace

Renormalisation group methods are used to determine, to second order in epsilon =4-d, the scaling function describing the crossover from Gaussian to Heisenberg behaviour in the susceptibility of an isotropic n-component spin system. The results are used in conjunction with an earlier Feynman graph calculation to obtain an O( epsilon 2 ) representation of the n-to-m-component susceptibility crossover function, and the corresponding effective exponents, for an anisotropic n-component system.

Journal of Physics A Mathematical and General•Jul 1, 1976•74 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

New singularities in the critical behavior of random Ising models at marginal dimensionalities

Amnon Aharony A

Renormalization-group equations are exactly solved for the random Ising model with (i) short-range interaction at $d=4$, and (ii) dipolar interactions at $d=3$. In both cases, the leading singularities of the susceptibility $\ensuremath{\chi}$ and of the specific heat $C$ are found to be $\ensuremath{\chi}\ensuremath{\propto}{t}^{\ensuremath{-}1}\mathrm{exp}[{(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ and $C\ensuremath{\propto}\ensuremath{-}{|\mathrm{ln}t|}^{\frac{1}{2}}\mathrm{exp}[{\ensuremath{-}2(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$. $D$ is a universal constant, equal to 6/53 in case (i) and to 9/[81ln(4/3) + 53] in case (ii). Relations between amplitudes of $C$ and of the correlation length, corrections to the leading singularities, crossover effects from the nonrandom region or from the meanfield region to the asymptotic critical region and possible experiments are also discussed.

Physical review. B, Solid state•Mar 1, 1976•111 citations

The renormalisation of probability distributions in the random field problem

B. Boechat, M. A. Contínentino

The authors study the Ising ferromagnet in a random field following the renormalisation of the distribution of random fields. The disorder in the local field induces fluctuations in the exchange couplings which are taken into account. The method allows a straightforward determination of the exponents associated with the zero-temperature fixed point which governs the critical behaviour along the critical line and in particular the critical slowing down. They find the correlation length exponent v=1.0 which is in very good agreement with numerical calculations. They also obtain the crossover exponent for small random fields.

Journal of Physics Condensed Matter•Jun 11, 1990•9 citations

On the crossover to universal criticality in dilute Ising systems

H. K. Janssen, Klaus Oerding, E Sengespeick

Monte Carlo simulations of the critical behaviour of disordered Ising systems have led to concentration-dependent critical exponents which seem to violate universality. We apply the renormalization group to investigate the crossover effects which cause the observed behaviour. We improve the three-loop expansion of the Callan-Symanzik and Wilson functions by a Pade-Borel approximation and solve the flow equations for various initial points. The exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point.

Journal of Physics A Mathematical and General•Nov 7, 1995•57 citations
For pt.I see ibid., vol.10, p.2985 (1977). The crossover function for the specific heat of a disordered material is constructed to order epsilon by solving the renormalisation group equations for the m n model (m to O,n>1). From this an effective exponent is obtained which provides a local measure of the degree of singularity of the specific heat in the critical region. It is seen that as the critical region is approached from above the critical temperature the effective exponent behaves initially as if no disorder were present. But as the critical temperature is more closely approached the effective exponent crosses over to the value predicted for the disordered material.
The fixed points of the exact renormalisation group equations with smooth cut-off are determined in the epsilon -expansion for systems characterised by mn-component vector order-parameters. The critical behaviour of quenched random n-component spin systems is obtained by taking the limit m=O. The critical exponent is obtained to order epsilon 2 , and the other exponents to order epsilon . A derivation of Nelson's trajectory-integral expression for the free energy, appropriate for the case of smooth cut-off renormalisation group equations, is presented.
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.
Renormalization-group equations are exactly solved for the random Ising model with (i) short-range interaction at d=4d=4d=4, and (ii) dipolar interactions at d=3d=3d=3. In both cases, the leading singularities of the susceptibility $\ensuremath{\chi}$ and of the specific heat CCC are found to be $\ensuremath{\chi}\ensuremath{\propto}{t}^{\ensuremath{-}1}\mathrm{exp}[{(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ and $C\ensuremath{\propto}\ensuremath{-}{|\mathrm{ln}t|}^{\frac{1}{2}}\mathrm{exp}[{\ensuremath{-}2(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$. DDD is a universal constant, equal to 6/53 in case (i) and to 9/[81ln(4/3) + 53] in case (ii). Relations between amplitudes of CCC and of the correlation length, corrections to the leading singularities, crossover effects from the nonrandom region or from the meanfield region to the asymptotic critical region and possible experiments are also discussed.
The authors study the Ising ferromagnet in a random field following the renormalisation of the distribution of random fields. The disorder in the local field induces fluctuations in the exchange couplings which are taken into account. The method allows a straightforward determination of the exponents associated with the zero-temperature fixed point which governs the critical behaviour along the critical line and in particular the critical slowing down. They find the correlation length exponent v=1.0 which is in very good agreement with numerical calculations. They also obtain the crossover exponent for small random fields.
Monte Carlo simulations of the critical behaviour of disordered Ising systems have led to concentration-dependent critical exponents which seem to violate universality. We apply the renormalization group to investigate the crossover effects which cause the observed behaviour. We improve the three-loop expansion of the Callan-Symanzik and Wilson functions by a Pade-Borel approximation and solve the flow equations for various initial points. The exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point.