A New Approach to Real Space Renormalization Group Treatment of Ising Model for Square and Simple Cubic Lattice

Abstract

Real Space Renormalization Group (RSRG) treatment of Ising model for square and simple cubic lattice is investigated and critical coupling strengths of these lattices are obtained. The mathematical complications, which appear inevitable in the decimated partition function due to Block-spin transformation, is treated with a relevant approximation. The approximation is based on the approximate equivalence of $\ln(1+f(K,\{\sigma_{n.n}\})) \simeq f(K,\{\sigma_{n.n}\})$ for small $f(K,\{\sigma_{n.n}\})$, here $K$ is the nearest neighbor coupling strength and $\{\sigma_{n.n}\}$ is the nearest neighbor spins degrees of freedom around a central spin. The values of the critical coupling strengths are obtained as $0.4830$ for square lattice and $0.2225$ for simple cubic (SC) lattice. The corresponding critical exponents values $\alpha$ and $\nu$ are also calculated within very acceptable agreement with those values obtained from numerical works.