Critical exponents by the scaling-field method: The isotropicN-vector model in three dimensions

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, and, and several of the correction-to-scaling exponents, ₌, to high precision. Results for N=0, 1, 2, and 3 are tabulated. For the Ising case (N=1), the estimates =0. 6260. 009, =0. 0400. 007, and ₁₄₀₀=0. 540. 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 ₄₂₂=1. 670. 11 and ₅₀₀=1. 50. 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.