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Physical review. B, Solid state

Renormalization-group proof of Josephson's relation and ε expansion for the helicity modulus (superfluid density)

Joseph Rudnick, David Jasnow

Published Sep 1, 1977• 42 citations
Publisher

Abstract

We show that Josephson's relation, $\ensuremath{\upsilon}=2\ensuremath{\beta}\ensuremath{-}\ensuremath{\eta}\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$, connecting the critical exponent $\ensuremath{\upsilon}$, describing the vanishing of the helicity modulus (superfluid density), to other thermodynamic and correlation exponents is an exact scaling law within the renormalization-group framework. We use the standard isotropic $n$-vector model with continuous spins; when $d<4$ hyperscaling presumably holds, and we have $\ensuremath{\upsilon}=(d\ensuremath{-}2)\ensuremath{\nu}$. Secondly, we derive the first two leading terms of the helicity modulus in an $\ensuremath{\epsilon}=4\ensuremath{-}d$ expansion; this allows second-order evaluation of universal amplitude ratios associated with the superfluid transition.

DOI
10.1103/physrevb.16.2032

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Two-scale-factor universality and the renormalization group

Pierre C. Hohenberg, Amnon Aharony A, Bertrand I. Halperin +1 more

The hypothesis of two-scale-factor universality, originally proposed by Stauffer, Ferer, and Wortis, is shown to follow from the renormalization-group approach, for systems close to their critical point. Values of the universal ratios involving correlation length and specific-heat amplitudes are obtained from the $\ensuremath{\epsilon}$ expansion, for Ising, $X\ensuremath{-}Y$, and Heisenberg models. In the latter two cases the correlation function has a power-law behavior at large distances below ${T}_{c}$, and the (transverse) correlation length is defined in terms of the stiffness constant ${\ensuremath{\rho}}_{s}$. Experimental values of the correlation lengths and amplitude ratios are determined for superfluid $^{4}\mathrm{He}$, which is $X\ensuremath{-}Y$-like, and for the Heisenberg antiferromagnet ${\mathrm{RbMnF}}_{3}$. Comparisons are made between the values of the amplitude ratios coming from $\ensuremath{\epsilon}$ expansions, series, and experiments.

Physical review. B, Solid state•Apr 1, 1976•278 citations
We show that Josephson's relation, \ensuremathυ=2\ensuremathβ\ensuremath−\ensuremathη\ensuremathν=2\ensuremath−\ensuremathα\ensuremath−2\ensuremathν\ensuremath{\upsilon}=2\ensuremath{\beta}\ensuremath{-}\ensuremath{\eta}\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}\ensuremathυ=2\ensuremathβ\ensuremath−\ensuremathη\ensuremathν=2\ensuremath−\ensuremathα\ensuremath−2\ensuremathν, connecting the critical exponent \ensuremathυ\ensuremath{\upsilon}\ensuremathυ, describing the vanishing of the helicity modulus (superfluid density), to other thermodynamic and correlation exponents is an exact scaling law within the renormalization-group framework. We use the standard isotropic nnn-vector model with continuous spins; when d<4d<4d<4 hyperscaling presumably holds, and we have \ensuremathυ=(d\ensuremath−2)\ensuremathν\ensuremath{\upsilon}=(d\ensuremath{-}2)\ensuremath{\nu}\ensuremathυ=(d\ensuremath−2)\ensuremathν. Secondly, we derive the first two leading terms of the helicity modulus in an \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d expansion; this allows second-order evaluation of universal amplitude ratios associated with the superfluid transition.
The ordered state of a ddd-dimensional isotropic system with an nnn-vector ($n\ensuremath{\ge}2$) order parameter is considered. By the imposition of suitable boundary conditions it is shown how to define explicitly a helicity modulus $\ensuremath{\Upsilon}(T)$ which measures the free-energy increment associated with "twisting" the direction of the order parameter. For a Bose system the superfluid density is seen to be ${\ensuremath{\rho}}_{s}(T)={(\frac{m}{\ensuremath{\hbar}})}^{2}\ensuremath{\Upsilon}(T)$. A critical exponent vvv is defined by $\ensuremath{\Upsilon}(T)\ensuremath{\sim}{|T\ensuremath{-}{T}_{c}|}^{v}$ as $T\ensuremath{\rightarrow}{T}_{c}$; for an ideal Bose gas and spherical model ($n\ensuremath{\rightarrow}\ensuremath{\infty}$), v=1v=1v=1 is an exact result for all d>2d>2d>2. The difficulties of defining a correlation length in the ordered phase are discussed. A full scaling theory of the correlations avoids these problems and may be linked to a phenomenological hydrodynamic approach, to clarify and rederive Josephson's relation $v=2\ensuremath{\beta}\ensuremath{-}\ensuremath{\eta}\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$. This reduces to $v=(d\ensuremath{-}2)\ensuremath{\nu}$ (used by some authors with d=3d=3d=3), only if one accepts ddd-dependent, "hyperscaling" relations such as $d\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$; however, both these latter relations fail for the ideal Bose gas when d>4d>4d>4. An alternative derivation of the formula $v=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$ is based on the scaling theory for systems with a large but finite dimension.
The field-theoretical renormalization-group techniques previously developed for the description of the mestastable phases of isotropic, homogeneous, ddd-dimensional systems, which (close to their critical temperature) can be described in terms of the Landau-Ginzburg-Wilson functional integral with a two-component order parameter, and applied to obtain the scaling form of the helicity modulus, defined as a measure of the free-energy increment associated with a metastable, helical twisting of the order parameter. The general scaling form of this quantity is determined even in the presence of an external helical field, for arbitrary helix pitch and temperature. In the case of vanishing external field, this quantity represents the superfluid density and is a function of the temperature and the superfluid velocity (which corresponds to the inverse of the helix pitch). In the limit of small velocities this expression exhibits the well-known Josephson asymptotic form ${\ensuremath{\rho}}_{s}\ensuremath{\sim}{({T}_{c}\ensuremath{-}T)}^{\ensuremath{\nu}(d\ensuremath{-}2)}={({T}_{c}\ensuremath{-}T)}^{2\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}\ensuremath{\eta}}$. An explicit $\ensuremath{\epsilon}=4\ensuremath{-}d$ expansion for the helicity modulus is presented in the case of arbitrary external field to the one-loop approximation.
A general derivation of the two-scale-factor universality is given using the renormalized ${\ensuremath{\varphi}}^{4}$ theory. As a consequence ten universal relations among critical amplitudes are obtained, any other universal relations being a combination of them. The situation is shown to be the same as for the scaling laws for critical exponents. The calculations up to order ${\ensuremath{\epsilon}}^{2}$ for systems with continuous symmetry are performed for some of these relations such as the specific-heat amplitudes ratio and ${R}_{\ensuremath{\xi}}^{+}$, which relates the amplitudes of the correlation length and of the specific heat, etc. Comparison with experiments and series results and, in particular, the discussion of the superfluid helium and RbMnF3{\mathrm{F}}_{3}F3​ cases, are repeated with our improved numerical results.
We present a field-theoretic renormalization-group treatment of the critical behavior of the cphi4{\mathit{cphi}}^{4}cphi4 model within the minimal-subtraction scheme at fixed dimension d4. We show that correlation functions above and below Tc{\mathit{T}}_{\mathit{c}}Tc​ can be calculated within this renormalization scheme without using the \ensuremath{\epsilon}=4-d expansion. This is demonstrated for the specific heat, for the order parameter, and for the superfluid density (stiffness constant). Various asymptotic amplitudes and correction amplitudes are determined, and some of their universal ratios are calculated in three dimensions. Our result for ${\mathit{a}}_{\mathit{c}}^{\mathrm{\ensuremath{-}}}$/${\mathit{a}}_{\mathrm{\ensuremath{\rho}}\mathit{s}}$ eliminates a previous error of the \ensuremath{\epsilon} expansion regarding the sign of this ratio for d=3 and n=2.
Using the Landau Hamiltonian for the description of second-order phase transitions, we give a proof of scaling for any continuous number of dimensions below four. The proof is based on a summation of diagrams having a power-law divergence and standard renormalization-group methods. The proof is constructive in that it leads to an unambiguous calculation for the critical exponents $\ensuremath{\eta}$ and $\ensuremath{\gamma}$. We present in this paper a detailed discussion of the proof; we also compare our method with the $\ensuremath{\epsilon}$ expansion leading to an interesting aspect of that theory: We find that the contribution to the critical exponents of order $\ensuremath{\epsilon}$ can be gotten without any calculation of diagrams. In this paper we have only made a lowest-order calculation in three dimensions. To this order we are of course unable to locate the relevant fixed point, but it leads to a relation between $\ensuremath{\eta}$ and $\ensuremath{\gamma}$---also to lowest order---which is such that if $\ensuremath{\gamma}$ is fixed to be 1.25, then $\ensuremath{\eta}$ turns out to be 0.12.
The hypothesis of two-scale-factor universality, originally proposed by Stauffer, Ferer, and Wortis, is shown to follow from the renormalization-group approach, for systems close to their critical point. Values of the universal ratios involving correlation length and specific-heat amplitudes are obtained from the $\ensuremath{\epsilon}$ expansion, for Ising, $X\ensuremath{-}Y$, and Heisenberg models. In the latter two cases the correlation function has a power-law behavior at large distances below Tc{T}_{c}Tc​, and the (transverse) correlation length is defined in terms of the stiffness constant ${\ensuremath{\rho}}_{s}$. Experimental values of the correlation lengths and amplitude ratios are determined for superfluid 4He^{4}\mathrm{He}4He, which is $X\ensuremath{-}Y$-like, and for the Heisenberg antiferromagnet RbMnF3{\mathrm{RbMnF}}_{3}RbMnF3​. Comparisons are made between the values of the amplitude ratios coming from $\ensuremath{\epsilon}$ expansions, series, and experiments.