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Progress of Theoretical Physics

1/n Expansion Up to Order 1/n2. IV: Critical Amplitude Ratio R

M. Oku, O. Yutaka

Published Feb 1, 1979• 5 citations
Publisher

Abstract

As one of the series of papers on 1/n expansion of the classical n-vector model up to O(1/n2), the universal amplitude ratio Rχ=1"DB8-1 is investigated up to O(1/n2) in the present paper. The 1/n2 term for Rχ is calculated numerically for d=3. In case of the critical exponent β, γ or ν, the result of the second-order calculation is not satisfactory for small values of n. On the contrary, the situation is much better for Rχ. The numerical value obtained is Rχ=1.20 for n=3 at d=3, which is almost equal to that of high-temperature expansion, Rχ=1.23.

DOI
10.1143/ptp.61.443

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1/n Expansion Up to Order 1/n2. III: Critical Exponents and for d=3

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1/n Expansion Up to Order 1/n2. II: Critical Exponent for d=3

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Critical Exponents from Other Expansions

It is useful to compare the results obtained so far with other approaches to the critical exponents. One is a similar field-theoretic approach based on perturbation expansions of φ 4-theories. But instead of working in D = 4−ε dimensions and continuing the results to ε = 1 to obtain critical exponents in the physical dimension D = 3, perturbation expansions can be derived directly in three dimensions, and studied in the regime of small masses. Fundamentally, this approach is less satisfactory than the previous one since there is no small parameter analogous to ε which permits us to prove the existence of a scaling limit at least for small ε. Another disadvantage is that it is impossible to calculate the expansion coefficients analytically, except for one- and two loop-diagrams. In one respect the three-dimensional approach has, however, an advantage: the power series have long been available up to six loops for ω, η, and ν, and there exist recent results up to seven loops for the critical exponents η and ν. Although numerical expansions are not as esthetic as the exact expansions derived in this book, their increased order leads to a higher accuracy in the critical exponents, as we shall see below in Section 20.4. Another approach is based on high-temperature expansions for lattice models of the O(N)symmetric classical Heisenberg model with the energy of Eq. (1.52). The critical exponents will be derived from such expansions in Section 20.8. Finally we take advantage of the fact that the O(N)-symmetric classical Heisenberg model is exactly solvable in the limit of large N, where it reduces to the spherical model [recall the remark after Eq. (1.69)]. It is therefore possible to expand the φ 4-theory around this limit in powers of 1/N. The results will be compared with those from the perturbation expansions in

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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.

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Critical Exponent η up to 1/n2for the Three-Dimensional System with Short-Range Interaction

Ryuzo Abe

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Progress of Theoretical Physics•Jun 1, 1973•43 citations
As one of the series of papers on 1/n expansion of the classical n-vector model up to O(1/n2), the universal amplitude ratio Rχ=1"DB8-1 is investigated up to O(1/n2) in the present paper. The 1/n2 term for Rχ is calculated numerically for d=3. In case of the critical exponent β, γ or ν, the result of the second-order calculation is not satisfactory for small values of n. On the contrary, the situation is much better for Rχ. The numerical value obtained is Rχ=1.20 for n=3 at d=3, which is almost equal to that of high-temperature expansion, Rχ=1.23.
It is useful to compare the results obtained so far with other approaches to the critical exponents. One is a similar field-theoretic approach based on perturbation expansions of φ 4-theories. But instead of working in D = 4−ε dimensions and continuing the results to ε = 1 to obtain critical exponents in the physical dimension D = 3, perturbation expansions can be derived directly in three dimensions, and studied in the regime of small masses. Fundamentally, this approach is less satisfactory than the previous one since there is no small parameter analogous to ε which permits us to prove the existence of a scaling limit at least for small ε. Another disadvantage is that it is impossible to calculate the expansion coefficients analytically, except for one- and two loop-diagrams. In one respect the three-dimensional approach has, however, an advantage: the power series have long been available up to six loops for ω, η, and ν, and there exist recent results up to seven loops for the critical exponents η and ν. Although numerical expansions are not as esthetic as the exact expansions derived in this book, their increased order leads to a higher accuracy in the critical exponents, as we shall see below in Section 20.4. Another approach is based on high-temperature expansions for lattice models of the O(N)symmetric classical Heisenberg model with the energy of Eq. (1.52). The critical exponents will be derived from such expansions in Section 20.8. Finally we take advantage of the fact that the O(N)-symmetric classical Heisenberg model is exactly solvable in the limit of large N, where it reduces to the spherical model [recall the remark after Eq. (1.69)]. It is therefore possible to expand the φ 4-theory around this limit in powers of 1/N. The results will be compared with those from the perturbation expansions in
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.
High-temperature expansions for the susceptibility and the second correlation moment of the classical N-vector model [also known as the O(N) symmetric Heisenberg classical spin model] on the square lattice are extended from order ${\mathrm{\ensuremath{\beta}}}^{14}$ to ${\mathrm{\ensuremath{\beta}}}^{21}$ for arbitrary N. For the second field derivative of the susceptibility the series expansion is extended from order ${\mathrm{\ensuremath{\beta}}}^{14}$ to ${\mathrm{\ensuremath{\beta}}}^{17}$. For -2\ensuremath{\le}N2, a numerical analysis of the series is performed in order to compare the critical exponents \ensuremath{\gamma}(N), \ensuremath{\nu}(N), and \ensuremath{\Delta}(N) to exact (though nonrigorous) formulas and to compute the ``dimensionless four point coupling constant'' g${\mathrm{^}}_{\mathit{r}}$(N). For N>2, we present a study of the analyticity properties of \ensuremath{\chi}, \ensuremath{\xi}, etc. in the complex \ensuremath{\beta} plane and describe a method to estimate the parameters which characterize their low-temperature behaviors. We compare our series estimates to the predictions of the perturbative renormalization group theory, to exact (but nonrigorous or conjectured) formulas and to the results of the 1/N expansion, always finding a good agreement. \textcopyright{} 1996 The American Physical Society.