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Physical review. B, Solid state

Two-scale-factor universality and the renormalization group

Pierre C. Hohenberg, Amnon Aharony A, Bertrand I. Halperin, Eric D. Siggia

Published Apr 1, 1976• 278 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.13.2986

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Renormalized coupling constants and related amplitude ratios for Ising systems

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Using modern estimates and exact values of critical points and exponents for Ising systems, existing series expansions for various lattices are analyzed to estimate the universal amplitude ratio ${\mathit{R}}_{0}$\ensuremath{\equiv}(${\mathit{C}}_{4}^{+}$${)}^{2}$/${\mathit{C}}^{+}$${\mathit{C}}_{6}^{+}$, where ${\mathit{C}}^{\ifmmode\pm\else\textpm\fi{}}$ are the amplitudes of divergence of the susceptibility \ensuremath{\chi}, above and below ${\mathit{T}}_{\mathit{c}}$, while ${\mathit{C}}_{\mathit{k}}^{\ifmmode\pm\else\textpm\fi{}}$ are, similarly, the amplitudes for (${\mathrm{\ensuremath{\partial}}}^{\mathit{k}\mathrm{\ensuremath{-}}2}$\ensuremath{\chi}/\ensuremath{\partial}${\mathit{h}}^{\mathit{k}\mathrm{\ensuremath{-}}2}$${)}_{\mathit{h}=0}$; we obtain ${\mathit{R}}_{0}$=0.1582\ifmmode\pm\else\textpm\fi{}0.0002 for d=2 dimensions and 0.1275\ifmmode\pm\else\textpm\fi{}0.0003 for d=3. Similarly, we find the universal ratio ${\mathit{C}}_{4}^{+}$/${\mathit{C}}_{4}^{\mathrm{\ensuremath{-}}}$=-9.0\ifmmode\pm\else\textpm\fi{}0.3 and -121.5\ifmmode\pm\else\textpm\fi{}0.5 for d=3 and 2, respectively. On using existing estimates for ${\mathit{C}}^{+}$, for the second-moment correlation-length amplitude ${\mathit{f}}_{1}^{+}$, and for the universal ratios ${\mathit{C}}^{+}$/${\mathit{C}}^{\mathrm{\ensuremath{-}}}$ and ${\mathit{f}}_{1}^{+}$/${\mathit{f}}_{1}^{\mathrm{\ensuremath{-}}}$, we estimate the renormalized coupling constants to be ${\mathit{g}}_{+}^{\mathrm{*}}$=${\mathit{G}}_{1}^{+}$\ensuremath{\equiv}-${\mathit{C}}_{4}^{+}$/(${\mathit{C}}^{+}$${)}^{2}$(${\mathit{f}}_{1}^{+}$a${)}^{\mathit{d}}$/${\mathit{v}}_{0}$=24.45\ifmmode\pm\else\textpm\fi{}0. 15 for d=3 and 14.700\ifmmode\pm\else\textpm\fi{}0.017 for d=2. Beneath ${\mathit{T}}_{\mathit{c}}$ we find ${\mathit{G}}_{1}^{\mathrm{\ensuremath{-}}}$=-501\ifmmode\pm\else\textpm\fi{}60, -1768\ifmmode\pm\else\textpm\fi{}80 and -${\mathit{C}}_{3}^{\mathrm{\ensuremath{-}}}$B/(${\mathit{C}}^{\mathrm{\ensuremath{-}}}$${)}^{2}$=6.${4}_{7}$\ifmmode\pm\else\textpm\fi{}0.2, 33.${0}_{6}$\ifmmode\pm\else\textpm\fi{}0.1, where B is the spontaneous-magnetization amplitude, and thence ${\mathit{g}}_{\mathrm{\ensuremath{-}}}^{\mathrm{*}}$\ensuremath{\simeq}85 and 23 for d=3 and 2, respectively. \textcopyright{} 1996 The American Physical Society.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 1, 1996•62 citations
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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion, for Ising, X\ensuremath−YX\ensuremath{-}YX\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 \ensuremathρs{\ensuremath{\rho}}_{s}\ensuremathρs​. Experimental values of the correlation lengths and amplitude ratios are determined for superfluid 4He^{4}\mathrm{He}4He, which is X\ensuremath−YX\ensuremath{-}YX\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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansions, series, and experiments.
A universality hypothesis of Stauffer, Ferer, and Wortis, concerning a relation between the scale factors of the correlation length and of the magnetic field, is proven to order ${\ensuremath{\epsilon}}^{2}$ with $\ensuremath{\epsilon}=4\ensuremath{-}d$. Agreement of explicit estimates of the universal quantities with high-TTT data is discussed.
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.
Abstract In Chapter 15, the scaling behaviour of correlation functions at criticality, T = Tc, has been derived. This chapter is devoted to the critical domain, where the correlation length is large with respect to the microscopic scale, but finite. In dimensions d < 4, above Tc, the property of strong scaling is derived: in the critical domain above Tc, all correlation functions, after rescaling, can be expressed in terms of universal correlation functions, in which the scale of distance is provided by the correlation length. However, because the correlation length is singular at Tc, in this formalism, the critical temperature cannot be crossed. Alternatively, one can expand correlation functions in formal power series of the deviation (T −Tc) from the critical temperature, in presence of a magnetic field. The sum of the expansion satisfies renormalization group (RG) equations also valid for T < Tc and in a magnetic field, from which follow scaling properties in the whole critical domain. The universal two-point function can be expanded when T approaches Tc, using the short-distance expansion (SDE). A few terms of the ϵ expansion (ϵ is the deviation from dimension 4) of a few universal quantities are reported. Calculations at fixed dimension and summation of perturbative expansions are described. The conformal bootstrap based on the SDE and conformal invariance at the infrared (IR) fixed point provides an alternative method to determine critical exponents.
A universality hypothesis relates the amplitude of the singular parts of the coherence length and the specific heat (or surface tension). For the spin-\textonehalf{} Ising model it is exact in two dimensions and numerically accurate to within 1% in three. It is consistent with measurements on the "Ising-like" systems CO2{\mathrm{O}}_{2}O2​, Xe, and $\ensuremath{\beta}$-brass and the "Heisenberg-like" systems RbMnF3{\mathrm{F}}_{3}F3​ and EuS to within experimental uncertainties (\ensuremath{\sim} 20%). It provides a sensitive and experimentally convenient indicator of symmetry ("universality") class.
Using modern estimates and exact values of critical points and exponents for Ising systems, existing series expansions for various lattices are analyzed to estimate the universal amplitude ratio R0{\mathit{R}}_{0}R0​\ensuremath{\equiv}(C4+{\mathit{C}}_{4}^{+}C4+​)2{)}^{2})2/C+{\mathit{C}}^{+}C+C6+{\mathit{C}}_{6}^{+}C6+​, where ${\mathit{C}}^{\ifmmode\pm\else\textpm\fi{}}$ are the amplitudes of divergence of the susceptibility \ensuremath{\chi}, above and below Tc{\mathit{T}}_{\mathit{c}}Tc​, while ${\mathit{C}}_{\mathit{k}}^{\ifmmode\pm\else\textpm\fi{}}$ are, similarly, the amplitudes for (${\mathrm{\ensuremath{\partial}}}^{\mathit{k}\mathrm{\ensuremath{-}}2}$\ensuremath{\chi}/\ensuremath{\partial}${\mathit{h}}^{\mathit{k}\mathrm{\ensuremath{-}}2}$)h=0{)}_{\mathit{h}=0})h=0​; we obtain R0{\mathit{R}}_{0}R0​=0.1582\ifmmode\pm\else\textpm\fi{}0.0002 for d=2 dimensions and 0.1275\ifmmode\pm\else\textpm\fi{}0.0003 for d=3. Similarly, we find the universal ratio C4+{\mathit{C}}_{4}^{+}C4+​/${\mathit{C}}_{4}^{\mathrm{\ensuremath{-}}}$=-9.0\ifmmode\pm\else\textpm\fi{}0.3 and -121.5\ifmmode\pm\else\textpm\fi{}0.5 for d=3 and 2, respectively. On using existing estimates for C+{\mathit{C}}^{+}C+, for the second-moment correlation-length amplitude f1+{\mathit{f}}_{1}^{+}f1+​, and for the universal ratios C+{\mathit{C}}^{+}C+/${\mathit{C}}^{\mathrm{\ensuremath{-}}}$ and f1+{\mathit{f}}_{1}^{+}f1+​/${\mathit{f}}_{1}^{\mathrm{\ensuremath{-}}}$, we estimate the renormalized coupling constants to be g+∗{\mathit{g}}_{+}^{\mathrm{*}}g+∗​=G1+{\mathit{G}}_{1}^{+}G1+​\ensuremath{\equiv}-C4+{\mathit{C}}_{4}^{+}C4+​/(C+{\mathit{C}}^{+}C+)2{)}^{2})2(f1+{\mathit{f}}_{1}^{+}f1+​a)d{)}^{\mathit{d}})d/v0{\mathit{v}}_{0}v0​=24.45\ifmmode\pm\else\textpm\fi{}0. 15 for d=3 and 14.700\ifmmode\pm\else\textpm\fi{}0.017 for d=2. Beneath Tc{\mathit{T}}_{\mathit{c}}Tc​ we find ${\mathit{G}}_{1}^{\mathrm{\ensuremath{-}}}$=-501\ifmmode\pm\else\textpm\fi{}60, -1768\ifmmode\pm\else\textpm\fi{}80 and -${\mathit{C}}_{3}^{\mathrm{\ensuremath{-}}}$B/(${\mathit{C}}^{\mathrm{\ensuremath{-}}}$)2{)}^{2})2=6.47{4}_{7}47​\ifmmode\pm\else\textpm\fi{}0.2, 33.06{0}_{6}06​\ifmmode\pm\else\textpm\fi{}0.1, where B is the spontaneous-magnetization amplitude, and thence ${\mathit{g}}_{\mathrm{\ensuremath{-}}}^{\mathrm{*}}$\ensuremath{\simeq}85 and 23 for d=3 and 2, respectively. \textcopyright{} 1996 The American Physical Society.