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Physical Review Letters

Critical Thermal Conductivity of He4 for T>~Tλ

Guenter Ahlers, Pierre C. Hohenberg, Avinoam Kornblit

Published Feb 16, 1981• 26 citations
Publisher

Abstract

Nonlinear renormalization-group recursion relations are used to show that the thermal conductivity of $^{4}\mathrm{He}$ has a crossover between a weak-coupling regime for ${t}_{c}\ensuremath{\ll}t<1$ and a scaling regime for $t\ensuremath{\ll}{t}_{c}$, where $t=\frac{(T\ensuremath{-}{T}_{\ensuremath{\lambda}})}{{T}_{\ensuremath{\lambda}}}$ and ${t}_{c}\ensuremath{\approx}{10}^{\ensuremath{-}3}$. The smallness of ${t}_{c}$---an essential feature---is explained by a small bare dynamic coupling constant. The asymmetric spin model ($F$) provides a quantitative fit to experiment, whereas the symmetric model ($E$) is only semiquantitative. The border-line dimension ${d}^{*}$, below which dynamic scaling breaks down, is shown to satisfy ${d}^{*}\ensuremath{\lesssim}2.54$.

DOI
10.1103/physrevlett.46.493

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Nonlinear renormalization-group recursion relations are used to show that the thermal conductivity of 4He^{4}\mathrm{He}4He has a crossover between a weak-coupling regime for tc\ensuremath≪t<1{t}_{c}\ensuremath{\ll}t<1tc​\ensuremath≪t<1 and a scaling regime for t\ensuremath≪tct\ensuremath{\ll}{t}_{c}t\ensuremath≪tc​, where t=(T\ensuremath−T\ensuremathλ)T\ensuremathλt=\frac{(T\ensuremath{-}{T}_{\ensuremath{\lambda}})}{{T}_{\ensuremath{\lambda}}}t=T\ensuremathλ​(T\ensuremath−T\ensuremathλ​)​ and tc\ensuremath≈10\ensuremath−3{t}_{c}\ensuremath{\approx}{10}^{\ensuremath{-}3}tc​\ensuremath≈10\ensuremath−3. The smallness of tc{t}_{c}tc​---an essential feature---is explained by a small bare dynamic coupling constant. The asymmetric spin model (FFF) provides a quantitative fit to experiment, whereas the symmetric model (EEE) is only semiquantitative. The border-line dimension d∗{d}^{*}d∗, below which dynamic scaling breaks down, is shown to satisfy d∗\ensuremath≲2.54{d}^{*}\ensuremath{\lesssim}2.54d∗\ensuremath≲2.54.
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.
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.
We establish new scaling properties for the universality class of Model C, which describes relaxational critical dynamics of a nonconserved order parameter coupled to a conserved scalar density. We find an anomalous diffusion phase, which satisfies weak dynamic scaling while the conserved density diffuses only asymptotically. The properties of the phase diagram for the dynamic critical behavior include a significantly extended weak scaling region, together with a strong and a decoupled scaling regime. These calculations are done directly in $2\ensuremath{\le}d\ensuremath{\le}4$ space dimensions within the framework of the nonperturbative functional renormalization group. The scaling exponents characterizing the different phases are determined along with subleading indices featuring the stability properties.
By expressing the free energy as a line integral along a renormalization-group trajectory, a technique for calculating the crossover scaling functions which describe tricritical systems, anisotropic spin systems, etc., is developed. This formalism is applied to the model recursion relations of Riedel and Wegner, which simulate crossover behavior. A simple mechanism for the breakdown of dimensionality dependent hyperscaling relationships emerges from the analysis. The specific-heat crossover scaling function describing crossovers from Gaussian to Heisenberg critical behavior is constructed to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$.
We calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ in the paramagnetic regime of quantum Heisenberg ferromagnets for temperatures TTT close to the critical temperature Tc{T}_{c}Tc​ using our recently developed functional renormalization group approach to quantum spin systems. In d=3d=3d=3 dimensions, we find that for small momenta $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ the dynamic structure factor assumes the scaling form $S(\mathbit{k},\ensuremath{\omega})=(\ensuremath{\tau}TG(\mathbit{k})/\ensuremath{\pi})\mathrm{\ensuremath{\Phi}}(k\ensuremath{\xi},\ensuremath{\omega}\ensuremath{\tau})$, where $G(\mathbit{k})$ is the static spin-spin correlation function, $\ensuremath{\xi}$ is the correlation length, and the characteristic timescale $\ensuremath{\tau}$ is proportional to ${\ensuremath{\xi}}^{5/2}$. We explicitly calculate the dynamic scaling function $\mathrm{\ensuremath{\Phi}}(x,y)$ and find satisfactory agreement with neutron scattering experiments probing the critical spin dynamics in EuO and EuS. Precisely at the critical point where $\ensuremath{\xi}=\ensuremath{\infty}$, our result for the dynamic structure factor can be written as $S(\mathbit{k},\ensuremath{\omega})={(\ensuremath{\pi}{\ensuremath{\omega}}_{k})}^{\ensuremath{-}1}{T}_{c}G(\mathbit{k}){\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\omega}/{\ensuremath{\omega}}_{k})$, where ${\ensuremath{\omega}}_{k}\ensuremath{\propto}{k}^{5/2}$. We find that ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ vanishes as ${\ensuremath{\nu}}^{\ensuremath{-}13/5}$ for large $\ensuremath{\nu}$, and as ${\ensuremath{\nu}}^{3/5}$ for small $\ensuremath{\nu}$. While the large-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ is consistent with calculations based on mode-coupling theory and with perturbative renormalization group calculations to second order in $\ensuremath{\epsilon}=6\ensuremath{-}d$, our result for small frequencies disagrees with previous calculations. We argue that until now, neither experiments nor numerical simulations have been sufficiently accurate to determine the low-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$. We also calculate the low-temperature behavior of $S(\mathbit{k},\ensuremath{\omega})$ in one- and two-dimensional ferromagnets and find that it satisfies dynamic scaling with exponent z=2z=2z=2 and exhibits a pseudogap for small frequencies.
The field-theoretic renormalization-group formalism is used to calculate the light scattering spectrum of 4He^{4}\mathrm{He}4He for $T>~{T}_{\ensuremath{\lambda}}$ self-consistently to second order in the loop expansion. It is shown that owing to the finite wave vector of the experiments the system never comes close to its fixed point (even for $T={T}_{\ensuremath{\lambda}}$), and transients must be included consistently to explain the data. Previous explanations of the absence of temperature dependence of the linewidth, in terms of the effect of the weak-scaling fixed point, are shown to be incorrect.