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Physical Review Letters

Renormalization-Group Calculations of Divergent Transport Coefficients at Critical Points

Bertrand I. Halperin, Pierre C. Hohenberg, Eric D. Siggia

Published Jun 1, 1974• 136 citations
Publisher

Abstract

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 $d=4$, which is a new result.

DOI
10.1103/physrevlett.32.1289

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Recursion relations correct to lowest order in \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=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−\ensuremathϵ19{k}^{\ensuremath{-}\frac{\ensuremath{\epsilon}}{19}}k\ensuremath−19\ensuremathϵ​ near d=4d=4d=4, which is a new result.
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.
Phenomenological models with "planar-spin" and "antiferromagnetic" dynamics are introduced, and their critical behavior is analyzed using renomalization-group methods. Dynamic scaling is shown to hold for these models to all orders in $\ensuremath{\epsilon}=4\ensuremath{-}d$, and the dynamic exponents are expressed entirely in terms of static exponents, in agreement with earlier phenomenological and mode-coupling theories. The magnitudes of the diverging transport and kinetic coefficients are expressed purely in terms of static properties and of universal constants which are calculated to second order in $\ensuremath{\epsilon}$. Matching conditions are proved between the characteristic frequencies above and below Tc{T}_{c}Tc​, and the corresponding universal amplitude ratios are calculated to second order in $\ensuremath{\epsilon}$. The principal applications are to liquid helium and the Heisenberg antiferromagnet RbMnF3{\mathrm{F}}_{3}F3​, where the experimental exponents and amplitudes both agree reasonably well with theory. In the case of liquid helium ("asymmetric" case with $\ensuremath{\alpha}<0$) the asymptotic critical behavior is somewhat masked by correction terms, due to the slow approach of the spectific heat to its finite value at ${T}_{\ensuremath{\lambda}}$. These correction terms are analyzed in detail, and a proposal is made for extracting the true asymptotic behavior from the data. The effects of other conserved fields such as the mass density and momentum in helium, and the energy density in the magnet, are considered, and shown to be irrelevant for the critical behavior of the order parameter.
We review the progress made in dynamic bulk critical behaviour in equilibrium in the last 25 years since the review of Halperin and Hohenberg. We unify the presentation of the theoretical background by restricting ourselves to the field-theoretic renormalization group method. The main results obtained in the different universality classes are presented. This contains the critical dynamics near the gas–liquid transition in pure fluids (model H), the plait point and consolute point in mixtures (model H � ), the superfluid transition in 4 He (model F) and 4 He– 3 He mixtures (model F � ), the Curie point (model J) and Neel point (model G) in Heisenberg magnets and the superconducting transition. In comparison with experimental results, it became clear that in most cases one has to consider apart from the universal asymptotic critical behaviour also the non-universal effective behaviour. Either because it turned out to be inevitable due to a small dynamical transient exponent inhibiting the system to reach the asymptotics (e.g., at the superfluid transition) or because one is interested in the region further away from the phase transition like in pure fluids and mixtures at their gas–liquid or demixing transition. The calculation of the critical dynamics is adequate in most cases only in two-loop order. We review these results and present the solution to unreasonable features found for some models. Thus, we consider model C where relaxational and diffusive dynamics are coupled and the scaling properties and the limit to a purely relaxational model (model A) have not been understood. In general for models where the order parameter couples to other conserved densities time scale ratios between the kinetic coefficients of the order parameter and the conserved densities play an important role. Their fixed-point values and the approach to the fixed point are changed considerably in two-loop order compared to their values in one-loop order. These considerations are relevant for the explanation of the dynamical critical shape functions of systems such as superfluid helium (model F) and the isotropic antiferromagnet (model G). As far as possible, the comparison of results obtained by the renormalization group theory with numerical simulations has been made.
The Martin-Siggia-Rose field theories associated with the critical dynamics of mode-coupling systems, are renormalized in a way which decouples statics from dynamics, through the minimal-renormalization procedure. This simplifies considerably the computations which are thus carried out to two-loop order for exponent and Wilson functions. We obtain all dynamic transients for helium, O(n)O(n)O(n) symmetric antiferromagnets, and liquid-gas systems. We also discuss the relevance for helium of a fixed point where dynamic scaling is weakly violated.