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Physical Review Letters

Ising-Model Critical Indices in Three Dimensions from the Callan-Symanzik Equation

George Allen Baker Jr., Bernie G. Nickel, Melville S. Green, Daniel I. Meiron

Published Jun 7, 1976• 316 citations
Publisher

Abstract

The coefficients in the Callan-Symanzik equations for a three-dimensional, continuous spin Ising model with an $\mathrm{exp}(\ensuremath{-}A{s}^{4}+B{s}^{2})$ spin-weight factor are expanded in the dimensionless, renormalized coupling constant. These series are summed by the Pad\'e-Borel method to yield the critical indices $\ensuremath{\gamma}=1.241\ifmmode\pm\else\textpm\fi{}0.002$, $\ensuremath{\eta}=0.02\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\nu}=0.63\ifmmode\pm\else\textpm\fi{}0.01$, and ${\ensuremath{\Delta}}_{1}=0.49\ifmmode\pm\else\textpm\fi{}0.01$.

DOI
10.1103/physrevlett.36.1351

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The coefficients in the Callan-Symanzik equations for a three-dimensional, continuous spin Ising model with an exp(\ensuremath−As4+Bs2)\mathrm{exp}(\ensuremath{-}A{s}^{4}+B{s}^{2})exp(\ensuremath−As4+Bs2) spin-weight factor are expanded in the dimensionless, renormalized coupling constant. These series are summed by the Pad'e-Borel method to yield the critical indices \ensuremathγ=1.241\ifmmode±\else\textpm\fi0.002\ensuremath{\gamma}=1.241\ifmmode\pm\else\textpm\fi{}0.002\ensuremathγ=1.241\ifmmode±\else\textpm\fi0.002, \ensuremathη=0.02\ifmmode±\else\textpm\fi0.02\ensuremath{\eta}=0.02\ifmmode\pm\else\textpm\fi{}0.02\ensuremathη=0.02\ifmmode±\else\textpm\fi0.02, \ensuremathν=0.63\ifmmode±\else\textpm\fi0.01\ensuremath{\nu}=0.63\ifmmode\pm\else\textpm\fi{}0.01\ensuremathν=0.63\ifmmode±\else\textpm\fi0.01, and \ensuremathΔ1=0.49\ifmmode±\else\textpm\fi0.01{\ensuremath{\Delta}}_{1}=0.49\ifmmode\pm\else\textpm\fi{}0.01\ensuremathΔ1​=0.49\ifmmode±\else\textpm\fi0.01.
High-temperature series expansions are derived for ($\frac{{\ensuremath{\partial}}^{2}\ensuremath{\chi}}{\ensuremath{\partial}{H}^{2}}$) for arbitrary dimension. These series and others already available are analyzed to determine if hyperscaling holds. Hyperscaling is found to hold for d=2d=2d=2 but fails in three and four dimensions as $2\ensuremath{\Delta}\ensuremath{-}d\ensuremath{\nu}\ensuremath{-}\ensuremath{\gamma}=\ensuremath{-}0.028\ifmmode\pm\else\textpm\fi{}0.003 (d=3)$, $\ensuremath{-}0.302\ifmmode\pm\else\textpm\fi{}0.038 (d=4)$. The triviality of hyper-strong-coupling, Euclidean, boson, ${:{\ensuremath{\varphi}}^{4}:}_{3,4}$ field theory follows. The expected location of the zero of the $\ensuremath{\beta}$ function in the Callan-Symanzik equation is computed in two and three dimensions.
Four-loop expansions of the coefficients of the Callan-Symanzik equation for 3D and 2D dilute Ising models are calculated. Fixed-point coordinates and critical exponents are estimated by two methods of summation of double series: a generalisation of the Pade-Borel approximation and the first confluent form of the in algorithm of Wynn. Summation of the double series for the 2D dilute Ising model gives exponents very close to exponents of the pure Ising model, in accordance with the exact solution. The two methods are also applied to summation of single-variable series of pure Ising and polymer models, both in 3D and 2D cases. Both methods are shown to provide close agreement between present estimates and results obtained earlier either exactly in conformal invariant theories (2D) or numerically with high accuracy (3D). An additional test is provided by estimation of critical exponents of the 2D O(n) model for n=-1. The methods tested this way are used to calculate critical exponents of the 3D dilute Ising model. The values obtained are consistent with recent experimental results.
A scaling-field representation of Wilson's exact renormalization-group equation is derived and used for the approximate calculation of critical exponents for the continuous-spin Ising model in three dimensions. The truncation of the hierarchy of scaling-field equations that retains only the four most relevant scaling fields yields the critical exponents $\ensuremath{\nu}\ensuremath{\approx}0.60$, $\ensuremath{\eta}\ensuremath{\approx}0.06$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\approx}\ensuremath{-}0.47$ (where ${\ensuremath{\Delta}}_{1}$ denotes Wortis's correction-to-scaling exponent).
The renormalization-group (RG) functions for the three-dimensional n-vector cubic model are calculated in the five-loop approximation. High-precision numerical estimates for the asymptotic critical exponents of the three-dimensional impure Ising systems are extracted from the five-loop RG series by means of the Pad\'e-Borel-Leroy resummation under n=0.n=0.n=0. These exponents are found to be $\ensuremath{\gamma}=1.325\ifmmode\pm\else\textpm\fi{}0.003,$ $\ensuremath{\eta}=0.025\ifmmode\pm\else\textpm\fi{}0.01,$ $\ensuremath{\nu}=0.671\ifmmode\pm\else\textpm\fi{}0.005,$ $\ensuremath{\alpha}=\ensuremath{-}0.0125\ifmmode\pm\else\textpm\fi{}0.008,$ and $\ensuremath{\beta}=0.344\ifmmode\pm\else\textpm\fi{}0.006.$ For the correction-to-scaling exponent, the less accurate estimate $\ensuremath{\omega}=0.32\ifmmode\pm\else\textpm\fi{}0.06$ is obtained.