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 threedimensional 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 selfconsistency criterion. The approach is then applied to the isotropic ¹ectormodel. Truncations to order 13 (10, when %=1) scaling-field equations yield the leading critical exponents, v and q, and several of the correction-to-scaling exponents, 6, to high precision. Results for N =0, 1, 2, and 3 are tabulated. For the Ising case (%=1), the estimates v=0. 626+0.009, g=0.040+0.007, and 6& =-5400 --0.54+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 6422 --1.67+0. 11 and 5&00 --1.5+0.3, respectively. Extensions necessary to improve the accuracy of the calculation are discussed, while applications of the approach to anisotropic S-vector models are described else-where. 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.

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Critical properties of models defined by continuous-spin Landau Hamiltonians of cubic symmetry are calculated as functions of spatial dimensionality, 2. 8 &d &4, and number of spin components, E. The investigation employs the scaling-field method developed by Golner and Riedel for Wilson's exact momentum-space renormalization-group equation.Fixed points studied include the isotropic and decoupled Ising ( -2&% & 00 ), the faceand corner-ordered cubic (1 &N & tx) ), and, via the replica method for )V~0, the quenched random Ising fixed point. Variations of X and d are used to link the results to exact results or results from other calculational methods, such as e expansions near two and four dimensions. This establishes the consistency of the calculation for three dimensions. Specifically, truncated sets involving seven (twelve) scaling-field equations are derived for the cubic ¹ectormodel. A stable random Ising fixed point is found and shown to be distinct from the cubic fixed point and to connect, as a function of d, with the Khmelnitskii e' fixed point. At d =3, the short truncation yields a=0. 11 for the pure Ising and a= -0.09 for the random Ising fixed point. A search for a random tricritical fixed point was inconclusive. For the E-component cubic model, the spin dimensionality X"at which the isotropic and cubic fixed points change stability, is determined as a function of d. The results support E, )3 for three dimensions.

We have extended the original damped-shifted force (DSF) electrostatic kernel and have been able to derive three new electrostatic potentials for higher-order multipoles that are based on truncated Taylor expansions around the cutoff radius. These include a shifted potential (SP) that generalizes the Wolf method for point multipoles, and Taylor-shifted force (TSF) and gradient-shifted force (GSF) potentials that are both generalizations of DSF electrostatics for multipoles. We find that each of the distinct orientational contributions requires a separate radial function to ensure that pairwise energies, forces, and torques all vanish at the cutoff radius. In this paper, we present energy, force, and torque expressions for the new models, and compare these real-space interaction models to exact results for ordered arrays of multipoles. We find that the GSF and SP methods converge rapidly to the correct lattice energies for ordered dipolar and quadrupolar arrays, while the TSF is too severe an approximation to provide accurate convergence to lattice energies. Because real-space methods can be made to scale linearly with system size, SP and GSF are attractive options for large Monte Carlo and molecular dynamics simulations, respectively.

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