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International Journal of Modern Physics B

Two-Loop RG Calculation of Sound Attenuation and Dispersion Near the Liquid-Gas Critical Point

L. Ts. Adzhemyan, Александр Николаевич Васильев, A. V. Serdukov

Published May 30, 1998• 2 citations
Publisher

Abstract

We calculate the singular part of the sound attenuation and dispersion near the liquid-gas critical point on the critical isochore above T c . It is shown that the corresponding scaling function is related to the correlation function <ψ2ψ2> of two composite operators of the Halperin-Hohenberg–Siggia model H.1 Using the renormalization group (RG) and ∊-expansion we obtain this scaling function in second order in ∊.

DOI
10.1142/s0217979298000685

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We calculate the singular part of the sound attenuation and dispersion near the liquid-gas critical point on the critical isochore above T c . It is shown that the corresponding scaling function is related to the correlation function <ψ2ψ2> of two composite operators of the Halperin-Hohenberg–Siggia model H.1 Using the renormalization group (RG) and ∊-expansion we obtain this scaling function in second order in ∊.
We employ a novel real-time formulation of the functional renormalization group (FRG) to compute universal scaling functions of the thermal diffusivity and the shear viscosity in the vicinity of the liquid-gas critical point, i.e., for the dynamic universality class of Model H from the Halperin-Hohenberg classification. We map out the universal dependence of the transport coefficients on temperature, external magnetic field, and wavenumber, and provide a detailed comparison with the Kawasaki approximation, which is here obtained from a perturbative one-loop approximation to our real-time FRG flow. In contrast to the Kawasaki approximation, the non-perturbative scaling functions from the full real-time FRG flow show a mild dependence on the thermodynamic path towards the critical point. We further compare our FRG results for the universal wavenumber and temperature dependence of the thermal diffusivity with experimental data from critical fluids.
A general theory is presented for sound attenuation and dispersion near the critical point in the region of frequencies which are much greater than the characteristic frequency of order-parameter dynamics, but are much smaller than the frequency of sound with the wavelength equal to the range of correlation of local order-parameter fluctuations. The method used is the most general version of the mode-mode coupling theory plus the static and dynamic scaling hypotheses and some general properties of time-correlation functions. If we ignore the small critical exponent $\ensuremath{\alpha}$ associated with the heat-capacity singularity, we find that the attenuation behaves as $f{\ensuremath{\epsilon}}^{0}$ and the dispersion (relative sound-velocity change with frequency) as ${f}^{0}{\ensuremath{\epsilon}}^{0}$, where fff is the dimensionless frequency and $\ensuremath{\epsilon}$ the dimensionless temperature distance from the critical point.
We describe numerical simulations of the critical dynamics near the superfluid phase transition. The calculations are based on an implementation of a stochastic hydrodynamic theory known as model F in the classification of Hohenberg and Halperin. This theory is expected to describe dynamic scaling near the lambda transition in liquid 4^44He, Bose-Einstein condensation in ultracold atomic gases, and the superfluid transition in the unitary Fermi gas. Our simulation is based on a Metropolis algorithm previously applied to the critical endpoint of the liquid-gas phase transition in ordinary fluids. In the model E truncation of model F we obtain the expected dynamical exponent z≃3/2z\simeq 3/2z≃3/2. We observe the emergence of a propagating second sound mode at the phase transition. The second sound diffusivity DsD_sDs​ is consistent with the scaling relation Ds∼ξxκD_s\sim ξ^{x_κ}Ds​∼ξxκ​, where ξξξ is the correlation length and xκ=1/2x_κ=1/2xκ​=1/2.
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 ${\mathit{T}}_{\ensuremath{\lambda}}$ 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 4He^{4}\mathrm{He}4He.
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.