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, Condensed matter

Renormalization-group flow equations of modelF

V. Dohm

Published Aug 1, 1991• 65 citations
Publisher

Abstract

The dynamic renormalization-group flow equations for model F of Halperin, Hohenberg, and Siggia [Phys. Rev. B 13, 1299 (1976)] are calculated by means of renormalized field theory within the minimal-renormalization scheme up to two-loop order. These equations are combined with the Borel-resummation results for the static renormalization-group functions computed by Schloms and Dohm. The corresponding static fixed point destabilizes the dynamic-scaling fixed point in two-loop order. The nonuniversal initial values of the static and dynamic flow equations are identified for the \ensuremath{\lambda} transition of $^{4}\mathrm{He}$ at various pressures. Predictions are made for the bulk thermal conductivity very close to ${\mathit{T}}_{\ensuremath{\lambda}}$ where the departures from dynamic scaling should be observable. Effective static and dynamic parameters are computed that can be applied to other critical phenomena above and below the \ensuremath{\lambda} line of $^{4}\mathrm{He}$.

DOI
10.1103/physrevb.44.2697

Related papers

Model F in two-loop order and the thermal conductivity near the superfluid transition of He4

V. Dohm

A nonlinear renormalization-group (RG) analysis of the thermal conductivity data near the $\ensuremath{\lambda}$ line of $^{4}\mathrm{He}$ is performed on the basis of the corrected dynamic RG flow equations of model $F$ in two-loop order. Our analysis eliminates recent uncertainties in the regime $(T\ensuremath{-}{T}_{\ensuremath{\lambda}})∕{T}_{\ensuremath{\lambda}}\ensuremath{\lesssim}{10}^{\ensuremath{-}6}$, which is most sensitive to the effect of the weak-scaling fixed point. This effect is found to be stronger than described previously. We also improve the Borel summation of the amplitude of the specific heat.

Physical Review B•Mar 13, 2006•11 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

Field theory of bicritical and tetracritical points. IV. Critical dynamics including reversible terms

R. Folk, Yurij V. Holovatch, G. Moser

This article concludes a series of papers [Folk, Holovatch, and Moser, Phys. Rev. E 78, 041124 (2008); 78, 041125 (2008); 79, 031109 (2009)] where the tools of the field theoretical renormalization group were employed to explain and quantitatively describe different types of static and dynamic behavior in the vicinity of multicritical points. Here we give the complete two-loop calculation and analysis of the dynamic renormalization-group flow equations at the multicritical point in anisotropic antiferromagnets in an external magnetic field. We find that the time scales of the order parameters characterizing the parallel and perpendicular ordering with respect to the external field scale in the same way. This holds independent whether the Heisenberg fixed point or the biconical fixed point in statics is the stable one. The nonasymptotic analysis of the dynamic flow equations shows that due to cancellation effects the critical behavior is described, in distances from the critical point accessible to experiments, by the critical behavior qualitatively found in one-loop order. Although one may conclude from the effective dynamic exponents (taking almost their one-loop values) that weak scaling for the order parameter components is valid, the flow of the time-scale ratios is quite different, and they do not reach their asymptotic values.

arXiv:1106.2026•Physical Review E•Feb 24, 2012•3 citations

Critical dynamics in a real-time formulation of the functional renormalization group

Johannes V. Roth, Lorenz von Smekal

A bstract We present first calculations of critical spectral functions of the relaxational Models A, B, and C in the Halperin-Hohenberg classification using a real-time formulation of the functional renormalization group (FRG). We revisit the prediction by Son and Stephanov that the linear coupling of a conserved density to the non-conserved order parameter of Model A gives rise to critical Model-B dynamics. We formulate both 1-loop and 2-loop self-consistent expansion schemes in the 1PI vertex functions as truncations of the effective average action suitable for real-time applications, and analyze in detail how the different critical dynamics are properly incorporated in the framework of the FRG on the closed-time path. We present results for the corresponding critical spectral functions, extract the dynamic critical exponents for Models A, B, and C, in two and three spatial dimensions, respectively, and compare the resulting values with recent results from the literature.

Journal of High Energy Physics•Oct 11, 2023•24 citations

Critical dynamics in a real-time formulation of the functional renormalization group

Johannes V. Roth, Lorenz von Smekal

We present first calculations of critical spectral functions of the relaxational Models A, B, and C in the Halperin-Hohenberg classification using a real-time formulation of the functional renormalization group (FRG). We revisit the prediction by Son and Stephanov that the linear coupling of a conserved density to the non-conserved order parameter of Model A gives rise to critical Model-B dynamics. We formulate both 1-loop and 2-loop self-consistent expansion schemes in the 1PI vertex functions as truncations of the effective average action suitable for real-time applications, and analyze in detail how the different critical dynamics are properly incorporated in the framework of the FRG on the closed-time path. We present results for the corresponding critical spectral functions, extract the dynamic critical exponents for Models A, B, and C, in two and three spatial dimensions, respectively, and compare the resulting values with recent results from the literature.

arXiv:2303.11817•Mar 21, 2023•1 citation

Renormalization-Group Functions and Nonuniversal Critical Behaviour

R. Schloms, V. Dohm

The method of Borel resummation is used to obtain an accurate representation of renormalization-group functions of the n -vector model within the minimal renormalization scheme. This permits to relate the nonuniversal critical behaviour of different quantities well away from criticality. The results can be used to estimate the range of applicability of the n -vector model. Application to the λ-transition of 4 He explains the measured effective exponent of the superfluid density along the λ-line.

Europhysics Letters (EPL)•Feb 15, 1987•67 citations
The dynamic renormalization-group flow equations for model F of Halperin, Hohenberg, and Siggia [Phys. Rev. B 13, 1299 (1976)] are calculated by means of renormalized field theory within the minimal-renormalization scheme up to two-loop order. These equations are combined with the Borel-resummation results for the static renormalization-group functions computed by Schloms and Dohm. The corresponding static fixed point destabilizes the dynamic-scaling fixed point in two-loop order. The nonuniversal initial values of the static and dynamic flow equations are identified for the \ensuremath{\lambda} transition of 4He^{4}\mathrm{He}4He at various pressures. Predictions are made for the bulk thermal conductivity very close to T\ensuremathλ{\mathit{T}}_{\ensuremath{\lambda}}T\ensuremathλ​ where the departures from dynamic scaling should be observable. Effective static and dynamic parameters are computed that can be applied to other critical phenomena above and below the \ensuremath{\lambda} line of 4He^{4}\mathrm{He}4He.
A nonlinear renormalization-group (RG) analysis of the thermal conductivity data near the $\ensuremath{\lambda}$ line of 4He^{4}\mathrm{He}4He is performed on the basis of the corrected dynamic RG flow equations of model FFF in two-loop order. Our analysis eliminates recent uncertainties in the regime $(T\ensuremath{-}{T}_{\ensuremath{\lambda}})∕{T}_{\ensuremath{\lambda}}\ensuremath{\lesssim}{10}^{\ensuremath{-}6}$, which is most sensitive to the effect of the weak-scaling fixed point. This effect is found to be stronger than described previously. We also improve the Borel summation of the amplitude of the specific heat.
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.
This article concludes a series of papers [Folk, Holovatch, and Moser, Phys. Rev. E 78, 041124 (2008); 78, 041125 (2008); 79, 031109 (2009)] where the tools of the field theoretical renormalization group were employed to explain and quantitatively describe different types of static and dynamic behavior in the vicinity of multicritical points. Here we give the complete two-loop calculation and analysis of the dynamic renormalization-group flow equations at the multicritical point in anisotropic antiferromagnets in an external magnetic field. We find that the time scales of the order parameters characterizing the parallel and perpendicular ordering with respect to the external field scale in the same way. This holds independent whether the Heisenberg fixed point or the biconical fixed point in statics is the stable one. The nonasymptotic analysis of the dynamic flow equations shows that due to cancellation effects the critical behavior is described, in distances from the critical point accessible to experiments, by the critical behavior qualitatively found in one-loop order. Although one may conclude from the effective dynamic exponents (taking almost their one-loop values) that weak scaling for the order parameter components is valid, the flow of the time-scale ratios is quite different, and they do not reach their asymptotic values.
A bstract We present first calculations of critical spectral functions of the relaxational Models A, B, and C in the Halperin-Hohenberg classification using a real-time formulation of the functional renormalization group (FRG). We revisit the prediction by Son and Stephanov that the linear coupling of a conserved density to the non-conserved order parameter of Model A gives rise to critical Model-B dynamics. We formulate both 1-loop and 2-loop self-consistent expansion schemes in the 1PI vertex functions as truncations of the effective average action suitable for real-time applications, and analyze in detail how the different critical dynamics are properly incorporated in the framework of the FRG on the closed-time path. We present results for the corresponding critical spectral functions, extract the dynamic critical exponents for Models A, B, and C, in two and three spatial dimensions, respectively, and compare the resulting values with recent results from the literature.
We present first calculations of critical spectral functions of the relaxational Models A, B, and C in the Halperin-Hohenberg classification using a real-time formulation of the functional renormalization group (FRG). We revisit the prediction by Son and Stephanov that the linear coupling of a conserved density to the non-conserved order parameter of Model A gives rise to critical Model-B dynamics. We formulate both 1-loop and 2-loop self-consistent expansion schemes in the 1PI vertex functions as truncations of the effective average action suitable for real-time applications, and analyze in detail how the different critical dynamics are properly incorporated in the framework of the FRG on the closed-time path. We present results for the corresponding critical spectral functions, extract the dynamic critical exponents for Models A, B, and C, in two and three spatial dimensions, respectively, and compare the resulting values with recent results from the literature.
The method of Borel resummation is used to obtain an accurate representation of renormalization-group functions of the n -vector model within the minimal renormalization scheme. This permits to relate the nonuniversal critical behaviour of different quantities well away from criticality. The results can be used to estimate the range of applicability of the n -vector model. Application to the λ-transition of 4 He explains the measured effective exponent of the superfluid density along the λ-line.