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 Applied Physics

Dynamical Aspect of Magnetic Critical Phenomena

Kyozi Kawasaki

Published Mar 1, 1970• 16 citations
Publisher

Abstract

Dynamical behaviors of magnetic systems near the critical points in the nonhydrodynamical regime are considered. For this purpose we construct a kinetic equation to describe the time evolution of critical fluctuations that corresponds in a sense to the Boltzmann equation of dilute gases. A graphical technique is developed to deduce the self-consistent equation for time-correlation functions of critical fluctuations from this kinetic equation, and the properties of the time-correlation functions of spin fluctuations for the Heisenberg isotropic and planar ferro- and isotropic antiferromagnets are discussed. The results are then applied to the high-frequency sound attenuation in magnetic systems.

DOI
10.1063/1.1658918

Related papers

Dynamics of Critical Fluctuations. II

Kyozi Kawasaki

The dynamical theory of critical fluctuations put forward in a previous work is rephrased and extended so as to be applicable below the transition points as well. The theory is illustrated for the isotropic Heisenberg spin systems where the critical dynamical variables can be chosen in such a way as to reproduce the correct characteristic times governing the dynamics of critical fluctuations which were obtained by the dynamical scaling laws. In this connection the general forms of various equal time correlations of long wave length fluctuations have been obtained with the help of the scaling law equation of state. The theory is useful as a starting point for devising approximate schemes of evaluating time correlation functions of physical quantities of our interest, where at least the correct scaling properties of the characteristic times of the problem are retained. This is illustrated by deriving a new self-consistent equation for the time-displaced spin auto-correlation function, and the truncated moments for describing the part of the spectral line shape that is associated with critical fluctuations are introduced and discussed.

Progress of Theoretical Physics•Jul 1, 1968•42 citations

Dynamics of Critical Fluctuations. III: A Study of the Dynamical Scaling Law

Kyozi Kawasaki

The nature of our approach to the dynamics of critical fluctuations initiated in earlier works of this series is studied by applying it to the kinetic Ising model and anisotropic Heisenberg ferromagnets, and by including the Hamiltonian density in constructing the critical dynamical variables (CDV). In the isotropic Heisenberg magnets as well as in the anisotropic Heisenberg ferromagnets where the transverse components of spins first become critical (Matsubara-Matsuda model), the dynamics of the CDV can be asymptotically closed within themselves (that is, the energy transfer among the CDV modes dominates that between the CDV modes and other microscopic modes) if the CDV is constructed from spin densities alone. The dynamical scaling law (DSL) then holds among phenomena associated with the spin densities. For other cases studied, the dynamics cannot be closed within the chosen CDV, and then the DSL may fail among the phenomena associated with the CDV. A tentative criterion is given for the CDV where the DSL holds. In particular, the characteristic frequency in the Matsubara-Matsuda model behaves as κ3/2f with κ the inverse correlation range of spin fluctuations, which agrees with that found for the liquid helium near the λ transition by Ferrell et al. using the DSL. However, the thermal conductivities of the magnetic systems studied remain finite at the critical points. Various equal-time correlations that involve certain short wave length spin fluctuations are studied in Appendices.

Progress of Theoretical Physics•Oct 1, 1968•61 citations

Scaling Laws for Dynamics Near the Critical Point

Pierre C. Hohenberg, Bertrand I. Halperin

The static scaling laws may be generalized to dynamic phenomena by making assumptions on the behavior of time-dependent correlation functions near the magnetic critical point. At any temperature different from Tc, the correlation functions are assumed to reflect the ``hydrodynamic'' behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range ξ of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to very small wavenumbers, satisfying the inequality kξ≪1. The dynamic scaling assumptions relate the behavior of the time-dependent correlation functions in the ``hydrodynamic'' (kξ≪1) and ``critical'' (kξ≫1) regions. It is thus possible to predict certain features of the dynamic behavior at Tc, from a knowledge of the behavior away from Tc. Application is made to the spin-correlation function of the isotropic Heisenberg antiferromagnet, which is directly measurable by inelastic neutron scattering. Detailed predictions are presented on the form of the cross section as a function of energy and momentum transfer, and of temperature near Tc.

Journal of Applied Physics•Mar 1, 1970•4 citations

Generalization of Scaling Laws to Dynamical Properties of a System Near its Critical Point

Bertrand I. Halperin, Pierre C. Hohenberg

The Widom-Kadanoff scaling laws are generalized to dynamic phenomena, by making assumptions on the structure of time-dependent correlation functions near ${T}_{c}$. The theory is applied to isotropic Heisenberg antiferromagnet and predictions are made, which can be tested by inelastic neutron-scattering and nmr measurements. The ferromagnet is also briefly discussed.

Physical Review Letters•Sep 18, 1967•259 citations

Critical dynamics of isotropic antiferromagnets using renormalization-group methods: T≥TN

R. Feeedman, Gene F. Mazenko

We study a model equation of motion describing the dynamics of isotropic antiferromagnets near the critical point. We perform a renormalization-group analysis of this equation of motion correct to $O(\ensuremath{\epsilon})$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, finding recursion relations and analyzing their fixed points and their stability. For the one stable fixed point we obtain the dynamical index $z=\frac{d}{2}=2\ensuremath{-}\frac{\ensuremath{\epsilon}}{2}$. We also calculate the response and correlation functions in the scaling region using perturbation theory correct to $O(\ensuremath{\epsilon})$. Our calculation gives analytical forms for the correlation functions for the staggered magnetization and the magnetization. These correlation functions can be written in the dynamical scaling form. The shape function that characterizes the frequency spectrum for the staggered magnetization is Lorentzian in the hydrodynamical regime, but shows fluctuation induced peaks at and near the N\ifmmode\dagger\else\textdagger\fi{}eel temperature. The shape function for the magnetization is essentially Lorentzian for all $x=q\ensuremath{\xi}$ (where $q$ is the wave number and $\ensuremath{\xi}$ is the correlation length) and shows a narrow lydrodynamical width as $x\ensuremath{\rightarrow}0$.

Physical review. B, Solid state•Jun 1, 1976•48 citations

High-Frequency Sound Attenuation and Dispersion in the Critical Region

Kyozi Kawasaki

A general theory is presented for sound attenuation and dispersion near the critical point in the region of frequencies which are much greater than the characteristic frequency of order-parameter dynamics, but are much smaller than the frequency of sound with the wavelength equal to the range of correlation of local order-parameter fluctuations. The method used is the most general version of the mode-mode coupling theory plus the static and dynamic scaling hypotheses and some general properties of time-correlation functions. If we ignore the small critical exponent $\ensuremath{\alpha}$ associated with the heat-capacity singularity, we find that the attenuation behaves as $f{\ensuremath{\epsilon}}^{0}$ and the dispersion (relative sound-velocity change with frequency) as ${f}^{0}{\ensuremath{\epsilon}}^{0}$, where $f$ is the dimensionless frequency and $\ensuremath{\epsilon}$ the dimensionless temperature distance from the critical point.

Physical Review A•Mar 1, 1971•19 citations
Dynamical behaviors of magnetic systems near the critical points in the nonhydrodynamical regime are considered. For this purpose we construct a kinetic equation to describe the time evolution of critical fluctuations that corresponds in a sense to the Boltzmann equation of dilute gases. A graphical technique is developed to deduce the self-consistent equation for time-correlation functions of critical fluctuations from this kinetic equation, and the properties of the time-correlation functions of spin fluctuations for the Heisenberg isotropic and planar ferro- and isotropic antiferromagnets are discussed. The results are then applied to the high-frequency sound attenuation in magnetic systems.
The dynamical theory of critical fluctuations put forward in a previous work is rephrased and extended so as to be applicable below the transition points as well. The theory is illustrated for the isotropic Heisenberg spin systems where the critical dynamical variables can be chosen in such a way as to reproduce the correct characteristic times governing the dynamics of critical fluctuations which were obtained by the dynamical scaling laws. In this connection the general forms of various equal time correlations of long wave length fluctuations have been obtained with the help of the scaling law equation of state. The theory is useful as a starting point for devising approximate schemes of evaluating time correlation functions of physical quantities of our interest, where at least the correct scaling properties of the characteristic times of the problem are retained. This is illustrated by deriving a new self-consistent equation for the time-displaced spin auto-correlation function, and the truncated moments for describing the part of the spectral line shape that is associated with critical fluctuations are introduced and discussed.
The nature of our approach to the dynamics of critical fluctuations initiated in earlier works of this series is studied by applying it to the kinetic Ising model and anisotropic Heisenberg ferromagnets, and by including the Hamiltonian density in constructing the critical dynamical variables (CDV). In the isotropic Heisenberg magnets as well as in the anisotropic Heisenberg ferromagnets where the transverse components of spins first become critical (Matsubara-Matsuda model), the dynamics of the CDV can be asymptotically closed within themselves (that is, the energy transfer among the CDV modes dominates that between the CDV modes and other microscopic modes) if the CDV is constructed from spin densities alone. The dynamical scaling law (DSL) then holds among phenomena associated with the spin densities. For other cases studied, the dynamics cannot be closed within the chosen CDV, and then the DSL may fail among the phenomena associated with the CDV. A tentative criterion is given for the CDV where the DSL holds. In particular, the characteristic frequency in the Matsubara-Matsuda model behaves as κ3/2f with κ the inverse correlation range of spin fluctuations, which agrees with that found for the liquid helium near the λ transition by Ferrell et al. using the DSL. However, the thermal conductivities of the magnetic systems studied remain finite at the critical points. Various equal-time correlations that involve certain short wave length spin fluctuations are studied in Appendices.
The static scaling laws may be generalized to dynamic phenomena by making assumptions on the behavior of time-dependent correlation functions near the magnetic critical point. At any temperature different from Tc, the correlation functions are assumed to reflect the ``hydrodynamic'' behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range ξ of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to very small wavenumbers, satisfying the inequality kξ≪1. The dynamic scaling assumptions relate the behavior of the time-dependent correlation functions in the ``hydrodynamic'' (kξ≪1) and ``critical'' (kξ≫1) regions. It is thus possible to predict certain features of the dynamic behavior at Tc, from a knowledge of the behavior away from Tc. Application is made to the spin-correlation function of the isotropic Heisenberg antiferromagnet, which is directly measurable by inelastic neutron scattering. Detailed predictions are presented on the form of the cross section as a function of energy and momentum transfer, and of temperature near Tc.
The Widom-Kadanoff scaling laws are generalized to dynamic phenomena, by making assumptions on the structure of time-dependent correlation functions near Tc{T}_{c}Tc​. The theory is applied to isotropic Heisenberg antiferromagnet and predictions are made, which can be tested by inelastic neutron-scattering and nmr measurements. The ferromagnet is also briefly discussed.
We study a model equation of motion describing the dynamics of isotropic antiferromagnets near the critical point. We perform a renormalization-group analysis of this equation of motion correct to $O(\ensuremath{\epsilon})$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, finding recursion relations and analyzing their fixed points and their stability. For the one stable fixed point we obtain the dynamical index $z=\frac{d}{2}=2\ensuremath{-}\frac{\ensuremath{\epsilon}}{2}$. We also calculate the response and correlation functions in the scaling region using perturbation theory correct to $O(\ensuremath{\epsilon})$. Our calculation gives analytical forms for the correlation functions for the staggered magnetization and the magnetization. These correlation functions can be written in the dynamical scaling form. The shape function that characterizes the frequency spectrum for the staggered magnetization is Lorentzian in the hydrodynamical regime, but shows fluctuation induced peaks at and near the N\ifmmode\dagger\else\textdagger\fi{}eel temperature. The shape function for the magnetization is essentially Lorentzian for all $x=q\ensuremath{\xi}$ (where qqq is the wave number and $\ensuremath{\xi}$ is the correlation length) and shows a narrow lydrodynamical width as $x\ensuremath{\rightarrow}0$.
A general theory is presented for sound attenuation and dispersion near the critical point in the region of frequencies which are much greater than the characteristic frequency of order-parameter dynamics, but are much smaller than the frequency of sound with the wavelength equal to the range of correlation of local order-parameter fluctuations. The method used is the most general version of the mode-mode coupling theory plus the static and dynamic scaling hypotheses and some general properties of time-correlation functions. If we ignore the small critical exponent $\ensuremath{\alpha}$ associated with the heat-capacity singularity, we find that the attenuation behaves as $f{\ensuremath{\epsilon}}^{0}$ and the dispersion (relative sound-velocity change with frequency) as ${f}^{0}{\ensuremath{\epsilon}}^{0}$, where fff is the dimensionless frequency and $\ensuremath{\epsilon}$ the dimensionless temperature distance from the critical point.