Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Solid state

Field-theoretic techniques and critical dynamics. I. Ginzburg-Landau stochastic models without energy conservation

C. De Dominicis, E. Brézin, Jean Zinn‐Justin

Published Dec 1, 1975• 131 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.12.4945

Related papers

Renormalization-group methods for critical dynamics: I. Recursion relations and effects of energy conservation

Bertrand I. Halperin, Pierre C. Hohenberg, Shang-keng Ma

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 Fe${\mathrm{F}}_{2}$ is discussed.

Physical review. B, Solid state•Jul 1, 1974•550 citations

The renormalization group

James Jeffrey Binney, N J Dowrick, Andrew J. Fisher +1 more

Abstract We are part of the way towards understanding the Landau-Ginzburg model in the critical regime. We have been able to calculate two critical exponents, one of which describes the system at T = Tc and the other at T ≠ Tc. If we are prepared to accept the scaling laws for which we found experimental evidence in Chapter 1, and theoretical evidence in Chapter 5, we can stop there; all four remaining critical exponents follow. However, this is rather unsatisfactory. In Chapter 5 we saw that the scaling relations can be deduced from the properties of the critical fixed point that is reached by repeatedly reblocking and rescaling the system; it would be good to provide a similarly general derivation in the context of the Landau-Ginzburg model.

•Jun 11, 1992•4 citations

The Theory of Critical Phenomena

James Jeffrey Binney, N J Dowrick, Andrew J. Fisher +1 more

Abstract The successful calculation of critical exponents for continuous phase transitions is one of the main achievements of theoretical physics over the last quarter-century. This was achieved through the use of scaling and field-theoretic techniques which have since become standard equipment in many areas of physics, especially quantum field theory. This book provides a thorough introduction to these techniques. Continuous phase transitions are introduced, then the necessary statistical mechanics is summarized, followed by standard models, some exact solutions and techniques for numerical simulations. The real-space renormalization group and mean-field theory are then explained and illustrated. The final chapters cover the Landau-Ginzburg model, from physical motivation, through diagrammatic perturbation theory and renormalization to the renormalization group and the calculation of critical exponents above and below the critical temperature.

•Jun 11, 1992•1,224 citations

Dynamic scaling

Uwe Claus Täuber

In the preceding chapter, we have introduced several levels for the mathematical description of stochastic dynamics. We now use the kinetic Ising models introduced in Section 2.3.3 to formulate the dynamic scaling hypothesis which appropriately generalizes the homogeneity property of the static correlation function in the vicinity of a critical point, as established in Chapter 1. The dynamic critical exponent z is defined to characterize both the critical dispersion and the basic phenomenon of critical slowing-down. As a next step, and building on the results of Section 2.4, a continuum effective theory for the mesoscopic order parameter density, basically the dynamical analog to the Ginzburg–Landau approach, is constructed in terms of a non-linear Langevin equation. The distinction between dissipative and diffusive dynamics for the purely relaxational kinetics of either a non-conserved or conserved order parameter field, respectively, defines the universality classes A and B. Following the analysis of these models in the Gaussian approximation, they also serve to outline the construction of a dynamical perturbation theory for non-linear stochastic differential equations through direct iteration. In general, however, the order parameter alone does not suffice to fully capture the critical dynamics near a second-order phase transition. Additional hydrodynamic modes originating from conservation laws need to be accounted for as well. The simplest such situation is entailed in the relaxational models C and D, which encompass the static coupling of the order parameter to the energy density. Further scenarios emerge through reversible non-linear mode couplings in the Langevin equations of motion.

Cambridge University Press eBooks•Mar 6, 2014•0 citations

Critical dynamics: a field-theoretical approach

R. Folk, G. Moser

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.

Journal of Physics A Mathematical and General•May 31, 2006•150 citations

Dynamics of a coupled-mode system: Explicit analysis at order ε2

K. K. Murata

The renormalization-group equations for critical behavior in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions are generalized to the discussion of dynamic exponents at $T={T}_{c}$ and then applied to a situation recently considered by Hohenberg, Halperin, and Ma (HHM); namely, one in which an $n$-component primary field $\ensuremath{\psi}$ is coupled to a secondary field $\ensuremath{\phi}$ via a ${\ensuremath{\lambda}}_{0}(\ensuremath{\psi}\ensuremath{\psi})\ensuremath{\phi}$ coupling. In this work their analysis of relaxational fields is explicitly considered at order ${\ensuremath{\epsilon}}^{2}$. For $n<4$, we find the dynamic exponents differ in one case at $O({\ensuremath{\epsilon}}^{2})$ from that given by HHM. In contrast to HHM, the dynamic scaling postulate is also preserved by the second-order fixed points in all instances and detailed analysis is given. The fixed-point values of $R$, a dimensionless ratio of relaxation rates for the $\ensuremath{\phi}$ and $\ensuremath{\psi}$ fields, in certain cases, is found to go as ${R}^{*}\ensuremath{\sim}\frac{1}{\ensuremath{\epsilon}}$ which is a new result. This large value where it occurs is found essential to understanding dynamic scaling. Analysis of the case of a propagating conserved $\ensuremath{\phi}$ field also is given. It is found that its dynamic critical properties can be characterized by its damping term alone.

Physical review. B, Solid state•Mar 1, 1976•24 citations
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 \ensuremathϵ3{\ensuremath{\epsilon}}^{3}\ensuremathϵ3 and so is the new exponent \ensuremathωb{\ensuremath{\omega}}_{b}\ensuremathωb​, 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.
Abstract We are part of the way towards understanding the Landau-Ginzburg model in the critical regime. We have been able to calculate two critical exponents, one of which describes the system at T = Tc and the other at T ≠ Tc. If we are prepared to accept the scaling laws for which we found experimental evidence in Chapter 1, and theoretical evidence in Chapter 5, we can stop there; all four remaining critical exponents follow. However, this is rather unsatisfactory. In Chapter 5 we saw that the scaling relations can be deduced from the properties of the critical fixed point that is reached by repeatedly reblocking and rescaling the system; it would be good to provide a similarly general derivation in the context of the Landau-Ginzburg model.
Abstract The successful calculation of critical exponents for continuous phase transitions is one of the main achievements of theoretical physics over the last quarter-century. This was achieved through the use of scaling and field-theoretic techniques which have since become standard equipment in many areas of physics, especially quantum field theory. This book provides a thorough introduction to these techniques. Continuous phase transitions are introduced, then the necessary statistical mechanics is summarized, followed by standard models, some exact solutions and techniques for numerical simulations. The real-space renormalization group and mean-field theory are then explained and illustrated. The final chapters cover the Landau-Ginzburg model, from physical motivation, through diagrammatic perturbation theory and renormalization to the renormalization group and the calculation of critical exponents above and below the critical temperature.
In the preceding chapter, we have introduced several levels for the mathematical description of stochastic dynamics. We now use the kinetic Ising models introduced in Section 2.3.3 to formulate the dynamic scaling hypothesis which appropriately generalizes the homogeneity property of the static correlation function in the vicinity of a critical point, as established in Chapter 1. The dynamic critical exponent z is defined to characterize both the critical dispersion and the basic phenomenon of critical slowing-down. As a next step, and building on the results of Section 2.4, a continuum effective theory for the mesoscopic order parameter density, basically the dynamical analog to the Ginzburg–Landau approach, is constructed in terms of a non-linear Langevin equation. The distinction between dissipative and diffusive dynamics for the purely relaxational kinetics of either a non-conserved or conserved order parameter field, respectively, defines the universality classes A and B. Following the analysis of these models in the Gaussian approximation, they also serve to outline the construction of a dynamical perturbation theory for non-linear stochastic differential equations through direct iteration. In general, however, the order parameter alone does not suffice to fully capture the critical dynamics near a second-order phase transition. Additional hydrodynamic modes originating from conservation laws need to be accounted for as well. The simplest such situation is entailed in the relaxational models C and D, which encompass the static coupling of the order parameter to the energy density. Further scenarios emerge through reversible non-linear mode couplings in the Langevin equations of motion.
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 renormalization-group equations for critical behavior in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions are generalized to the discussion of dynamic exponents at T=TcT={T}_{c}T=Tc​ and then applied to a situation recently considered by Hohenberg, Halperin, and Ma (HHM); namely, one in which an nnn-component primary field $\ensuremath{\psi}$ is coupled to a secondary field $\ensuremath{\phi}$ via a ${\ensuremath{\lambda}}_{0}(\ensuremath{\psi}\ensuremath{\psi})\ensuremath{\phi}$ coupling. In this work their analysis of relaxational fields is explicitly considered at order ${\ensuremath{\epsilon}}^{2}$. For n<4n<4n<4, we find the dynamic exponents differ in one case at $O({\ensuremath{\epsilon}}^{2})$ from that given by HHM. In contrast to HHM, the dynamic scaling postulate is also preserved by the second-order fixed points in all instances and detailed analysis is given. The fixed-point values of RRR, a dimensionless ratio of relaxation rates for the $\ensuremath{\phi}$ and $\ensuremath{\psi}$ fields, in certain cases, is found to go as ${R}^{*}\ensuremath{\sim}\frac{1}{\ensuremath{\epsilon}}$ which is a new result. This large value where it occurs is found essential to understanding dynamic scaling. Analysis of the case of a propagating conserved $\ensuremath{\phi}$ field also is given. It is found that its dynamic critical properties can be characterized by its damping term alone.