Renormalization·Group Transformation of Three-Dimensional Ising Models

Ising models on the square and simple cubic lattices are studied by a slightly modified version of the Niemeijer-van Leeuwen renormalization-group transformation. For the simple cubic lattice, we find the transition temperature K,=0.226 with critical exponents a =-0.420 and &=5.04 from the third order cumulant expansion. Ising models are also treated by the two-, three- and four-step decimation transformations among the simple cubic, body-centered cubic and face-centered cubic lattices. It is found from the three-step trans-formations that three systems have the same critical exponents. In contrast to the results from the cumulant expansion method, this method yields the positive specific heat exponent except one case. These results are compared with the known series values. Niemeijer and van Leeuwen (N v L) n proposed two methods of directly applying Wilson's renormalization-group theory21 to the Ising model, the cumulant and cluster expansion methods. Recently many authors3l~7) have proven these methods to be very useful to evaluate critical exponents as well as critical temperatures for a two