Physical Review B

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

V. Dohm

Published Mar 13, 2006• 11 citations

Abstract

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.

DOI: 10.1103/physrevb.73.092503

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Physical Review B

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

V. Dohm

Published Mar 13, 2006• 11 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevb.73.092503

Related papers

Renormalization-group flow equations of modelF

V. Dohm

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 $^{4}\mathrm{He}$ .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 $^{4}\mathrm{He}$.

Physical review. B, Condensed matterAug 1, 199165 citations

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Guenter Ahlers, Pierre C. Hohenberg, Avinoam Kornblit

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 ${d}^{*}\ensuremath{\lesssim}2.54$.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 ${d}^{*}\ensuremath{\lesssim}2.54$.

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The Renormalization group method (RG) is applied to the investigation of the E model of critical dynamics, which describes the transition from the normal to the superfluid phase in He4. The technique “Sector decomposition” with R’ operation is used for the calculation of the Feynman diagrams. The RG functions, critical exponents and critical dynamical exponent z, which determines the growth of the relaxation time near the critical point, have been calculated in the two-loop approximation in the framework of ε-expansion. The relevance of a fixed point for helium, where the dynamic scaling is weakly violated, is briefly discussed.The Renormalization group method (RG) is applied to the investigation of the E model of critical dynamics, which describes the transition from the normal to the superfluid phase in He4. The technique “Sector decomposition” with R’ operation is used for the calculation of the Feynman diagrams. The RG functions, critical exponents and critical dynamical exponent z, which determines the growth of the relaxation time near the critical point, have been calculated in the two-loop approximation in the framework of ε-expansion. The relevance of a fixed point for helium, where the dynamic scaling is weakly violated, is briefly discussed.

EPJ Web of ConferencesJan 1, 201613 citations

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Pierre C. Hohenberg, Sarben Sarkar

The field-theoretic renormalization-group formalism is used to calculate the light scattering spectrum of $^{4}\mathrm{He}$ 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.The field-theoretic renormalization-group formalism is used to calculate the light scattering spectrum of $^{4}\mathrm{He}$ 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.

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We present a detailed study of thermal transport in the disordered Fermi liquid with short-range interactions. At temperatures smaller than the impurity scattering rate, i.e., in the diffusive regime, thermal conductivity acquires nonanalytic quantum corrections. When these quantum corrections become large at low temperatures, the calculation of thermal conductivity demands a theoretical approach that treats disorder and interactions on an equal footing. In this paper, we develop such an approach by merging Luttinger's idea of using gravitational potentials for the analysis of thermal phenomena with a renormalization group calculation based on the Keldysh nonlinear sigma model. The gravitational potentials are introduced in the action as auxiliary sources that couple to the heat density. These sources are a convenient tool for generating expressions for the heat density and its correlation function from the partition function. Already in the absence of the gravitational potentials, the nonlinear sigma model contains several temperature-dependent renormalization group charges. When the gravitational potentials are introduced into the model, they acquire an independent renormalization group flow. We show that this flow preserves the phenomenological form of the correlation function, reflecting its relation to the specific heat and the constraints imposed by energy conservation. The main result of our analysis is that the Wiedemann-Franz law holds down to the lowest temperatures even in the presence of disorder and interactions and despite the quantum corrections that arise for both the electric and thermal conductivities.We present a detailed study of thermal transport in the disordered Fermi liquid with short-range interactions. At temperatures smaller than the impurity scattering rate, i.e., in the diffusive regime, thermal conductivity acquires nonanalytic quantum corrections. When these quantum corrections become large at low temperatures, the calculation of thermal conductivity demands a theoretical approach that treats disorder and interactions on an equal footing. In this paper, we develop such an approach by merging Luttinger's idea of using gravitational potentials for the analysis of thermal phenomena with a renormalization group calculation based on the Keldysh nonlinear sigma model. The gravitational potentials are introduced in the action as auxiliary sources that couple to the heat density. These sources are a convenient tool for generating expressions for the heat density and its correlation function from the partition function. Already in the absence of the gravitational potentials, the nonlinear sigma model contains several temperature-dependent renormalization group charges. When the gravitational potentials are introduced into the model, they acquire an independent renormalization group flow. We show that this flow preserves the phenomenological form of the correlation function, reflecting its relation to the specific heat and the constraints imposed by energy conservation. The main result of our analysis is that the Wiedemann-Franz law holds down to the lowest temperatures even in the presence of disorder and interactions and despite the quantum corrections that arise for both the electric and thermal conductivities.

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Georg Schwiete, Alexander M. Finkel’stein

We study thermal transport in a disordered Fermi liquid. We use gravitational potentials as sources for finding the heat density and its correlation function. Thermal conductivity is obtained from the dynamical part of the correlation function; it is thereby fully determined by diffusion modes, which are renormalized by the interaction. For a comprehensive study of thermal transport, we extend the Renormalization Group (RG) analysis by including the gravitational potentials into the renormalization scheme. We demonstrate explicitly the connection between the specific heat and the static part of the heat density correlation function, $\chi_{kk}^{st}=-Tc$ . In the developed scheme this fundamental relation is closely connected with a fixed point in the multi-parametric RG-flow of the gravitational potentials.We study thermal transport in a disordered Fermi liquid. We use gravitational potentials as sources for finding the heat density and its correlation function. Thermal conductivity is obtained from the dynamical part of the correlation function; it is thereby fully determined by diffusion modes, which are renormalized by the interaction. For a comprehensive study of thermal transport, we extend the Renormalization Group (RG) analysis by including the gravitational potentials into the renormalization scheme. We demonstrate explicitly the connection between the specific heat and the static part of the heat density correlation function, $\chi_{kk}^{st}=-Tc$. In the developed scheme this fundamental relation is closely connected with a fixed point in the multi-parametric RG-flow of the gravitational potentials.

Dec 30, 20130 citations

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}

^{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}

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 ${d}^{*}\ensuremath{\lesssim}2.54$.

The field-theoretic renormalization-group formalism is used to calculate the light scattering spectrum of

^{4}\mathrm{He}

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.

\chi_{kk}^{st}=-Tc

. In the developed scheme this fundamental relation is closely connected with a fixed point in the multi-parametric RG-flow of the gravitational potentials.