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 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 dynamics of critical fluctuations is treated theoretically by extending the usual macroscopic dynamical variables to include all other dynamical variables that could exhibit the critical slowing-down. The equations of motion for these generalized macroscopic dynamical variables are derived with the help of Mori's general theory of Brownian motion, which involve the “first moment frequency matrix” Ωq and the microscopic damping for these new dynamical variables, where q is the wave vector characterizing macroscopic inhomogeneity. The latter involves only the dynamical variables that perform rapid random motions and is disregarded, or is disposed of by means of a dimensional argument. Then the dynamics of critical fluctuations is completely governed by Ωq which is expressed in terms of equaltime correlations of long wave-length fluctuations only and allows the analysis with the help of the Widom-Kadanoff static scaling laws. The theory is applied to the problem of anomalous spin relaxations in ferro and antiferromagnets in the paramagnetic region, and the inverse characteristic times near the transition points are found to behave asymptotically as κ(5-η)/2f(q/κ) and κ3/2f(q/κ), respectively, where κ is the inverse correlation range of spin fluctuations and η the parameter measuring the deviation of spin pair correlation from the Ornstein-Zernike form. Finally a close analogy is pointed out between the present treatment of critical fluctuations are the theory of hydrodynamic turbulence.
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.
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 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.
"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."
Equipped with the field theory representation of non-linear Langevin equations, the tools of dynamic perturbation theory, and the dynamic renormalization group introduced in Chapters 4 and 5, we are now in the position to revisit models for dynamic critical behavior that entail reversible mode couplings and other conserved hydrodynamic modes. We have already encountered some of these in Section 3.3. In models C and D, respectively, a non-conserved or conserved n -component order parameter is coupled to a conserved scalar field, the energy density. Through a systematic renormalization group analysis, we may critically assess the earlier predictions from scaling theory, and discuss the stability of fixed points characterized by strong dynamic scaling, wherein the order parameter and conserved non-critical mode fluctuate with equal rates, and weak dynamic scaling regimes, where these characteristic time scales differ. Next we investigate isotropic ferromagnets (model J), with the conserved spin density subject to reversible precession in addition to diffusive relaxation. Exploiting rotational invariance, we can now firmly establish the scaling relation z = ( d + 2 − η )/2. Similar symmetry arguments yield a scaling relation for the dynamic exponents associated with the order parameter and the non-critical fields in the O ( n )-symmetric SSS model that encompasses model E for planar ferromagnets and superfluid helium 4 (for n = 2), and model G for isotropic antiferromagnets ( n = 3). There exist competing strong- and weak-scaling fixed points, with the former stable to one-loop order, and characterized by z = d /2 for all slow modes.