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).
A modification of the Wilson-Kogut smooth-cutoff formulation of the renormalization group is developed. The resulting hierarchy of equations is truncated at four levels, with the renormalization-group parameters being approximated by the leading terms of a mometum-space Taylor series. The equations are then solved for
d=3,
N=1 systems. A line of fixed points with associated $\ensuremath{\eta}\ensuremath{\cong}0.045$ is obtained. The largest two eigenvalues of the near-fixed-point hierarchy are ${y}_{1}\ensuremath{\cong}1.6$ and ${y}_{2}\ensuremath{\cong}\ensuremath{-}1.6$.
A scaling‐field representation of Wilson’s exact renormalization‐group equation is derived and used to calculate critical exponents for the continuous‐spin Ising model in three dimensions. Here the results for two successive truncations of the infinite hierarchy of scaling‐field equations are presented. The truncation that retains only the two most relevant scaling fields yields the critical exponents ν?0.55 and η=0; the truncation retaining the four most relevant fields yields the improved values ν?0.60 and η?0.06. Approximations to the full hierarchy of scaling‐field equations introduce a weak dependence of the critical exponents on redundant parameters in the renormalization‐group Hamiltonian. That question is also discussed.
For the n-vector model of the general type a renormalization-group (RG) scheme in momentum space is developed. Using this scheme we obtain a generalized exact equation of the renormalization group. The main idea of this approach is not to allow the zero-order critical Hamiltonian to move away from the critical surface under the influence of the RG transformation. The equation found differs from the known ones in that all redundant operators are excluded. On the basis of the developed approach we argue that critical exponents are functionals of some function introducing the cutoff into the phase-transition theory.
A renormalization-group scheme is developed for the three-dimensional
O(2N)-symmetric Ginzburg-Landau-Wilson model, which is consistent with the use of a
1/N expansion as a systematic method of approximation. It is motivated by an application to the critical properties of superconductors, reported in a separate paper. Within this scheme, the infrared stable fixed point controlling critical behavior appears at
z=0, where $z={\ensuremath{\lambda}}^{\ensuremath{-}1}$ is the inverse of the quartic coupling constant, and an efficient renormalization procedure consists of the minimal subtraction of ultraviolet divergences at
z=0. This scheme is implemented at next-to-leading order, and the standard results for critical exponents calculated by other means are recovered. An apparently novel result of this nonperturbative method of approximation is that corrections to scaling (or confluent singularities) do not, as in perturbative analyses, appear as simple power series in the variable ${y=zt}^{\ensuremath{\omega}\ensuremath{\nu}}.$ At least in three dimensions, the power series are modified by powers of
lny.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.