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Progress of Theoretical Physics

On the Critical Dynamics in Bose Systems and the Time-Dependent Ginzburg-Landau Model

Masuo Suzuki

Published Jan 1, 1975• 9 citations
Publisher

Abstract

An extended expression of the dynamic self-energy of one-particle Green function is obtained up to order 1/m in m-component Bose systems, by means of a complex unperturbed propagator. As applications of it, dynamic critical exponents of the TDGL model as well as of a Bose system are evaluated to order 1/m, for both short-range and long-range interactions. These confirm the previous results of the TDGL model and of the Bose system for short-range interactions. An effective interaction Veff(p, iωn) is obtained to order 1/m, which is useful for discussion on anomalies of transport coefficients near the λ-transition.

DOI
10.1143/ptp.53.97

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An extended expression of the dynamic self-energy of one-particle Green function is obtained up to order 1/m in m-component Bose systems, by means of a complex unperturbed propagator. As applications of it, dynamic critical exponents of the TDGL model as well as of a Bose system are evaluated to order 1/m, for both short-range and long-range interactions. These confirm the previous results of the TDGL model and of the Bose system for short-range interactions. An effective interaction Veff(p, iωn) is obtained to order 1/m, which is useful for discussion on anomalies of transport coefficients near the λ-transition.
We study the dynamic critical exponent from effective and microscopic theories. We employ a simple TDGL model, or model A in the classification of Hohenberg and Halperin, as an effective theory and the imaginary time formalism of the finite-temperature filed theory as a microscopic theory. Taking an O(N) scalar model as an example and carrying out the 1/N expansion up to the NLO in the 1PI and 2PI effective actions, we compare the low-energy and low-momentum behavior of the response function in the effective theory and of the retarded Green's function in the microscopic theory. At the NLO of the 1PI 1/N expansion the low-energy and low-momentum behavior of the two-point function is very much different in the microscopic and effective theories: in the field theory it is dominated by the propagating mode while in model A it is dominated by the diffusive mode. Also, in the microscopic theory the dynamic critical exponent, z, depends on whether the kinematics is relativistic or nonrelativistic. In contrast, at the NLO of the 2PI 1/N expansion the microscopic and effective theories are equivalent. They satisfy exactly the same Kadanoff-Baym equation. Also, whether the kinematics is relativistic or nonrelativistic in the microscopic theory becomes irrelevant. This implies that the diffusive mode with z = 2 + O(1/N) is dominant at low energies and momenta even in the microscopic theory at the NLO of the 2PI 1/N expansion, though we do not explicitly solve the Kadanoff-Baym equation. We also try to improve the calculation of the dynamic critical exponent of model A by incorporating the static 2PI NLO correlations. The obtained critical exponent is slightly smaller than the previous result and its N dependence is also milder than the previous one.
The theoretic renormalization-group approach is applied to the study of short-time dynamics of the d-dimensional n-component spin systems with long-range interactions r-(d+σ) and quenched disorder which has long-range correlations r-(d-ρ). Asymptotic scaling laws are obtained in a frame of double expansions in ∊=2σ-d and ρ with ρ of the order ∊. The static exponents are obtained exactly to all the order. The initial slip exponents θ′ for the order parameter and θ for the response function, as well as the dynamic exponent z, are calculated upto the first order in ∊. In d=2σ, in contrast to the unique logarithmic decay in the long-time regime which does not depend on σ, ρ, n and the disorder, we find rich scaling structures including logarithmic and exponential-logarithmic scalings in the short-time regime. Non-universal critical scalings of Ising systems are also discussed for d=2σ.
Renormalization techniques of field theory are used to prove (order by order to all orders) dynamical scaling laws on a Ginzburg-Landau stochastic model studied by Halperin, Hohenberg, and Ma. The dynamical exponent is calculated to order ${\ensuremath{\epsilon}}^{3}$ and so is the new exponent ${\ensuremath{\omega}}_{b}$, which governs the vanishing of the imaginary part of the (renormalized) kinetic coefficient, and appears among the corrections to scaling. Difficulties of previous calculations taking a microscopic approach to the critical dynamics of a Bose system are commented upon.
The renormalization-group method for studying critical phenomena is generalized to a class of dynamical systems---the time-dependent Ginzburg-Landau models. The effects of conservation laws on the critical dynamics are investigated through the study of models with different conservation properties for the energy and the space integral of the order parameter. Dynamic critical exponents near four dimensions ($d\ensuremath{\approx}4$) are obtained from recursion relations, analogous to those of Wilson and Fisher. The physical significance of the time-dependent Ginzburg-Landau models is explored and the applicability of the results to experiments on the NMR linewidth of FeF2{\mathrm{F}}_{2}F2​ is discussed.
We investigate the critical dynamics of the time-dependent Landau-Ginzburg model with nonconserved n-component order parameter (model A) in the presence of long-range correlated quenched impurities. We use a special kind of long-range correlations $g(|r\ensuremath{-}{r}^{\ensuremath{'}}|)\ensuremath{\sim}|r\ensuremath{-}{r}^{\ensuremath{'}}{|}^{\ensuremath{-}a},$ $a=4\ensuremath{-}\ensuremath{\delta},$ previously introduced by Weinrib and Halperin. The parameter $\ensuremath{\delta}\ensuremath{\sim}\ensuremath{\epsilon},$ where $\ensuremath{\epsilon}=4\ensuremath{-}d$ is the usual small parameter within the Wilson-Fisher renormalization group theory. Using a double expansion in $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ we calculate the critical exponent zzz up to second order on the small parameters. We show that quenched impurities of this kind affect the critical dynamics already in first order of $\ensuremath{\epsilon}$ and $\ensuremath{\delta},$ leading to a relevant correction for the mean field value of the exponent z.z.z.