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

TL;DR

This paper introduces an approximate real space renormalization group method for the Ising model on square and cubic lattices, estimating critical couplings and exponents with good agreement to numerical results.

Contribution

It presents a novel approximation approach to handle mathematical complexities in RSRG for the Ising model on square and cubic lattices, providing critical parameters.

Findings

Critical coupling for square lattice: 0.4830

Critical coupling for cubic lattice: 0.2225

Critical exponents match numerical results well

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.