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Physical review. B, Solid state

Dynamical Properties of the Spherical Model in the Low-Temperature and Critical Regions

J. M. J. van Leeuwen, J. D. Gunton

Published Jul 1, 1972• 6 citations
Publisher

Abstract

The properties of the time-dependent spin correlation functions are investigated for the spherical model of the classical magnetic system. Equations for the correlation functions are derived by a diagrammatic analysis. The lowest-order approximation is for $T>{T}_{c}$ identical to the equation derived by Resibois and De Leener for the Weiss limit of the Heisenberg model. Attention has been given to the hydrodynamic regime along the line of spontaneous magnetization from $T=0$ up to $T={T}_{c}$, where spin-wave frequencies and damping rates as well as the "diffusive" behavior have been calculated to lowest order in the wave number. The dynamical scaling problem is examined within this diagrammatic analysis. The scaling properties and the equations determining the homogeneous correlation function are established explicitly.

DOI
10.1103/physrevb.6.231

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The properties of the time-dependent spin correlation functions are investigated for the spherical model of the classical magnetic system. Equations for the correlation functions are derived by a diagrammatic analysis. The lowest-order approximation is for T>TcT>{T}_{c}T>Tc​ identical to the equation derived by Resibois and De Leener for the Weiss limit of the Heisenberg model. Attention has been given to the hydrodynamic regime along the line of spontaneous magnetization from T=0T=0T=0 up to T=TcT={T}_{c}T=Tc​, where spin-wave frequencies and damping rates as well as the "diffusive" behavior have been calculated to lowest order in the wave number. The dynamical scaling problem is examined within this diagrammatic analysis. The scaling properties and the equations determining the homogeneous correlation function are established explicitly.
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.
We present an exact solution for the spin-spin correlation function of the dynamical spherical model (the spherical-model equilibrium state together with the classical-Heisenberg-model equations of motion) in one dimension at T=0T=0T=0. The physical picture of the system is clarified, as is the difference between the Heisenberg and dynamical spherical model arising from the presence of the constraint ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}=1$ in the former model. The effect of the higher-order correlations, induced by the constraint, on the dynamics is made explicit. The solution satisfies the dynamical-scaling hypothesis. The mode-coupling approximation, although it gives the correct dynamical exponents, yields a qualitatively incorrect spectral function. It cannot be improved by including higher-order terms in the expansion of the self energy in terms of the number of interactions (the extended-mode-coupling theory), as the higher-order terms yield physically unrealistic solutions. A qualitatively correct approximate solution has been found, that exhibits both the central peak and the spin wave sidebands of the exact solution. The solution cannot be improved by including higher-order terms either. There does not appear to be any way of constructing a series of approximations from the leading terms in a renormalized-perturbation-series expansion that will converge to the exact solution.
The thermodynamic properties of a ddd-dimensional classical vector-spin system in a random magnetic field are calculated explicitly in the spherical-model limit. The system is characterized by significant spin-glass order of the Edwards-Anderson type even when no transition to a state with spontaneous magnetization occurs. This spin-glass order should result in observable effects in the magnetic susceptibility and specific heat. The properties of the random-field spherical model are compared with those reported for the corresponding Ising model. The behavior of the coherence length is identical in both models at their respective lower critical dimensionalities.
We consider a “generalized Heisenberg model”, with a Hamiltonian given by ℋ︁(D) = –JΣij S(D)j, where the quantities Si are isotropically-interacting D-dimensional classical spins and the summation is restricted to nearest-neighbor pairs of sites <i,j>. Thus when D = 1, 2, and 3, this model reduces to the S = 1/2 Ising, plane rotator, and classical Heisenberg models which are useful for describing phase transitions in, respectively, liquid-gas systems, Bose fluids, and magnetic materials. Unfortunately this Hamiltonian is exactly soluble only in the limit D → ∞, in which case the solution corresponds to the spherical model of Berlin and Kac. For finite D, we must consider approximation procedures in order to obtain any useful information. In this paper we develop a diagrammatic representation, valid for arbitrary D, for successive terms in the high-temperature expansions of the free energy and the spin-spin correlation function—from which we can obtain series expansions for all the thermodynamic properties of the system.
A statistical theory of the localized model of ferromagnetism is developed by considering the effective density matrices for a single spin, and two nearest-neighbor spins, respectively. The Ising Model and the Heisenberg model, and even the case of the anisotropic exchange interaction, are treated within a unified mathematical scheme. The result is reduced to that of the Bethe approximation in the case of the Ising model, and the constant coupling approximation in the case of the Heisenberg model. The method is generalized to calculate the static spin-correlation function. It is obtained in Ornstein-Zernike type. In the nonequilibrium state, the expectation value of the spin decays by the homogeneous damping as well as the spin-diffusion process. This relaxation equation is different from that by Van Hove because of the existence of the homogeneous damping. The diffusion constant does not vanish even at the Curie point TC, contrary to Van Hove's result, although the damping constant vanishes at TC. The latter fact causes the critical slowing down of the regression of fluctuation. The solution of this relaxation equation gives the time-dependent spin-correlation function. The method can be applied to the antiferromagnetic case, too.
Spin-dynamics techniques have been used to perform large-scale simulations of the dynamic behavior of the classical Heisenberg antiferromagnet in simple cubic lattices with linear sizes L< 60.L<~60.L< 60. This system is widely recognized as an appropriate model for the magnetic properties of RbMnF3.{\mathrm{RbMnF}}_{3}.RbMnF3​. Time evolutions of spin configurations were determined numerically from coupled equations of motion for individual spins using an algorithm implemented by Krech et al., which is based on fourth-order Suzuki-Trotter decompositions of exponential operators. The dynamic structure factor was calculated from the space- and time-displaced spin-spin correlation function. The crossover from hydrodynamic to critical behavior of the dispersion curve and spin-wave half-width was studied as the temperature was increased towards the critical temperature. The dynamic critical exponent was estimated to be $z=(1.43\ifmmode\pm\else\textpm\fi{}0.03),$ which is slightly lower than the dynamic scaling prediction, but in good agreement with a recent experimental value. Direct, quantitative comparisons of both the dispersion curve and the line shapes obtained from our simulations with very recent experimental results for RbMnF3{\mathrm{RbMnF}}_{3}RbMnF3​ are presented.