This is the third article in a sequence in which we reexamine the calculation of the critical exponents of the N-vector model from field theory. In the first article we have applied summation methods to the ε-expansion for which new terms have been calculated (up to order ε5). In the second article we have incorporated in the summation method, in the case of Ising-like systems (N = 1), the knowledge of the exact values of exponents in d = 2 dimensions. Since exact values are now also known in the case of polymers (N = 0), we apply here the same method to this case. Moreover, since for the exponent η for N = 1 the results coming from the ε-expansion and perturbation series at fixed dimension are slightly different, we have recalculated the exponents in the latter case using the variant of the summation method developed for the ε-expansion. The results do not change significantly showing both the reliability and the limits of the method.
A numerical technique, termed the scaling-field method, is developed for solving by successive approximation Wilson's exact renormalization-group equation for critical phenomena in three-dimensional spin systems. The approach uses the scaling-field representation of the Wilson equation derived by Riedel, Golner, and Newman. A procedure is proposed for generating in a nonperturbative and unbiased fashion sequences of successively larger truncations to the infinite hierarchy of scaling-field equations. A "principle of balance" is introduced and used to provide a self-consistency criterion. The approach is then applied to the isotropic
N-vector model. Truncations to order 13 (10, when
N=1) scaling-field equations yield the leading critical exponents, $\ensuremath{\nu}$ and $\ensuremath{\eta}$, and several of the correction-to-scaling exponents, ${\ensuremath{\Delta}}_{m}$, to high precision. Results for
N=0,1,2,and3 are tabulated. For the Ising case (
N=1), the estimates $\ensuremath{\nu}=0.626\ifmmode\pm\else\textpm\fi{}0.009$, $\ensuremath{\eta}=0.040\ifmmode\pm\else\textpm\fi{}0.007$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\equiv}{\ensuremath{\Delta}}_{400}=0.54\ifmmode\pm\else\textpm\fi{}0.05$ are in good agreement with recent high-temperature-series results, though exhibiting larger confidence limits at the present level of approximation. For the first time, estimates are obtained for the second and third correction-to-scaling exponents. For example, for the Ising model the second "even" and first "odd" correction-to-scaling exponents are ${\ensuremath{\Delta}}_{422}=1.67\ifmmode\pm\else\textpm\fi{}0.11$ and ${\ensuremath{\Delta}}_{500}=1.5\ifmmode\pm\else\textpm\fi{}0.3$, respectively. Extensions necessary to improve the accuracy of the calculation are discussed, while applications of the approach to anisotropic
N-vector models are described elsewhere. Finally, the scaling-field method is compared with other techniques for the high-precision calculation of critical phenomena in three dimensions, i.e., high-temperature-series, Monte Carlo renormalization-group, and field-theoretic perturbation expansions.
This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\ensuremath{\eta}$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\ensuremath{\eta}$. They are intended to demonstrate, qualitatively, how nonzero values for $\ensuremath{\eta}$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$. The equations are solved both by $\ensuremath{\epsilon}$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\ensuremath{\epsilon}}^{2}$ we obtain $\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}$. Numerical results in three dimension are $\ensuremath{\eta}=0.058$, $\ensuremath{\nu}=0.588$, and $\ensuremath{\gamma}=1.14$. The relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$ is confirmed.
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. 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.
The critical exponent $\ensuremath{\eta}$ for three-dimensional systems with an
n-vector order parameter is evaluated in the framework of the pseudo-$\ensuremath{\epsilon}$ expansion approach. The pseudo-$\ensuremath{\epsilon}$ expansion ($\ensuremath{\tau}$ series) for $\ensuremath{\eta}$ found up to the ${\ensuremath{\tau}}^{7}$ term for
n = 0, 1, 2, 3 and within the ${\ensuremath{\tau}}^{6}$ order for general
n is shown to have a structure that is rather favorable for getting numerical estimates. The use of Pad\'e approximants and direct summation of the $\ensuremath{\tau}$ series result in iteration procedures rapidly converging to the asymptotic values that are very close to the most reliable numerical estimates of $\ensuremath{\eta}$ known today. The origin of such an efficiency is discussed and shown to lie in the general properties of the pseudo-$\ensuremath{\epsilon}$ expansion machinery interfering with some peculiarities of the renormalization group expansion of $\ensuremath{\eta}$.
A careful Monte Carlo investigation of the phase transition very close to the critical point (T -> Tc, H -> 0) in relatively large d = 3, s = 1/2 Ising lattices did produce critical exponents beta = 0.3126(4) =~ 5/16, delta^{-1} = 0.1997(4) =~ 1/5 and gamma_{3D} = 1.253(4) =~ 5/4. Our results indicate that, within experimental error, they are given by simple fractions corresponding to the linear interpolations between the respective two-dimensional (Onsager) and four-dimensional (mean field) critical exponents. An analysis of our inverse susceptibility data chi^{-1}(T) vs. /T - Tc/ shows that these data lead to a value of gamma_{3D} compatible with gamma' = gamma and Tc = 4.51152(12), while gamma values obtained recently by high and low temperature series expansions and renormalization group methods are not.