We propose a simple tractable method which extracts the long-range terms of the coordinate-space renormalization group in the critical domain. The critical indices and temperatures of the two- and three-dimensional Ising models are calculated to an accuracy of few percent.
The equivalence of the critical behavior of the d-dimensional Ising model at a finite temperature to that of a d- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.
Discoveries in physics often hinge on a disagreement between theory and experiment of a percent or less. Because of such a discrepancy, many workers in statistical mechanics have been concerned about the complete validity of the scaling and renormalization-group theories of critical phenomena. For the three-dimensional Ising model, the experimentally derived values for the critical exponents in fluids have agreed to three significant figures with those predicted by the renormalization group. However, the predictions disagreed by a couple of percent with exponents calculated by series expansions, and the claimed accuracy of these expansion estimates was much less than a percent.