Dynamic Critical Phenomena in Magnetic Systems. I

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Abstract
In order to deal with the long-time behavior of the time correlations of spins and take into account the life-time effects of all critical variables involved, we formulate a generalized continued fraction expansion of the time correlation functions. It is shown that, if the correlation length of spin fluctuations κ-1 and the wave-length of external disturbance κ-1 are very long, then the long range correlations of spin fluctuations involved yield the most dominant part in the limit of long times or small frequencies. The asymptotic behavior of the most dominant part is determiend by the equal-time correlations of long wave-length spin fluctuations which are treated with the static scaling laws. It is shown that, in the cas of η=0, η being the parameter measuring the deviation of the spin pair correlation from the Ornstein-Zernike form, the dynamic scaling law proposed by Halperin and Hohenberg holds with the characteristic frequencies of the form κθg(k/κ), where θ=5/2 in ferromagnets and 3/2 in antiferromagnets. In the case of η≠0, however, the dynamic scaling law does not hold. In ferromagnets, this is due to a non-similarity between the longitudinal and transverse components in the ordered phase, and in antiferromagnets, this is due to a non-similarity between the critical slowing-down of the staggered polarization and the kinematical slowing-down of the small wave-number polarization. In ferromagnets in the paramagnetic region, however, there exists a characteristic frequency with θ=(5+η)/2. These results are derived by first using the pair correlation approximation and then removing such an approximation.

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