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Journal of Physics A Mathematical and General

Critical transport anomalies in 4-ϵ and 6-ϵ dimensions

J. D. Gunton, Kyozi Kawasaki

Published Jan 1, 1975• 18 citations
Publisher

Abstract

The mode coupling equations for the time-dependent correlation functions are solved to lowest order in epsilon =d c -d for several models of critical dynamics where the 'conventional' theory is expected for the dimensionality d greater than d c . The results are in agreement with recent renormalization group calculations. New results are obtained for the tricritical behaviour of 3 He- 4 He, as well as for the critical amplitude of various transport coefficients.

DOI
10.1088/0305-4470/8/1/003

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The mode coupling equations for the time-dependent correlation functions are solved to lowest order in epsilon =d c -d for several models of critical dynamics where the 'conventional' theory is expected for the dimensionality d greater than d c . The results are in agreement with recent renormalization group calculations. New results are obtained for the tricritical behaviour of 3 He- 4 He, as well as for the critical amplitude of various transport coefficients.
We discuss renormalization-group and mode-coupling theories of critical dynamics for several models. We show that to lowest order in $\ensuremath{\epsilon}={d}_{c}\ensuremath{-}d$, where for d>dcd>{d}_{c}d>dc​ conventional theory is valid, the renormalizationgroup differential equations are related by a simple transformation to those obtained from mode-coupling theory. Furthermore, the values of the dynamical fixed-point parameters essentially coincide with the critical amplitudes of the diverging transport coefficients, as determined by mode-coupling theory. In addition, we present a correct renormalization-group treatment for dynamical systems with more than one time scale, as for the binary liquid. We stress the fact that there are two distinct classes of critical dynamics, as illustrated by the results for the models studied here. We also discuss the instability of the conventional fixed point (time-dependent Ginzburg-Landau like) against mode-coupling perturbations for d<dcd<{d}_{c}d<dc​, with particular emphasis on a model for superfluid helium.
Recursion relations correct to lowest order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ are obtained for simple models representing the critical dynamics of a binary liquid, a Heisenberg antiferromagnet, superfluid helium, and related systems. The exponents for the divergences of the transport coefficients satisfy the "scaling" relations predicted by mode-mode coupling theory. The viscosity of the binary liquid is predicted to diverge as ${k}^{\ensuremath{-}\frac{\ensuremath{\epsilon}}{19}}$ near d=4d=4d=4, which is a new result.
Employing Wilson's renormalization group scheme, we investigate the critical behaviour of a modified Ginzburg-Landau model with a nonlocal mode-coupling interaction in the quartic term. Carrying out the calculations at one-loop order, we obtain the critical exponents in the leading order of , where ρ is an exponent occurring in the nonlocal interaction term and d is the space dimension. Interestingly, the correlation exponent η is found to be non-zero at one-loop order and the expansion corresponds to an expansion about the tricritical mean-field theory in three dimensions, unlike the conventional theory. The ensuing critical exponents are in good agreement with experimental values for samples close to tricriticality. Our analysis indicates that tricriticality is a feature only in three dimensions.
We study the purely relaxational tricritical dynamics of a non-conserved order parameter (model A) in the upper critical dimension d c =3 and in 3- epsilon dimensions. We are especially interested in the relaxation, starting from a macroscopically prepared initial state with a small correlation length. Using the methods of renormalized field theory we obtain the scaling behaviour of the correlation and response functions and study the nonlinear relaxation of the order parameter M(t). In three dimensions M(t) displays a crossover from the purely logarithmic short-time behaviour M(t) approximately (In(t/t 0 )) -a to a t - 1 4 / power law with logarithmic corrections. For dimensions d<3 we obtain the exponents which govern the tricritical relaxation at lowest non-trivial order in epsilon =3-d. The dynamic scaling exponent z is calculated at second order in epsilon .