VERIFICATION OF THE THREE-CLUSTER MOLECULAR-FIELD RENORMALIZATION GROUP APPROACH FOR THE ISING MODEL

Abstract

A transfer matrix approach has been worked out to test the predictions of the three-step molecular-field renormalization group (MFRG) for square Ising clusters with linear size u p to L = 11. The convergence of the finite-size critical couplings and the critical exponents towards the exact values is shown. The molecular field approximation has become a valuable tool for a qualitative understanding of the phase diagrams of three-dimensional systems and for obtaining quantitative estimates of non-singular thermodynamic properties. Its most serious drawback is the prediction of classical critical exponents. A successful attempt to improve this deficiency (1-3) is the idea of MFRG which will be illustrated for the ferromagnetic nearest-neighbour Ising model on the square lattice, inasmuch as for this case the exact results are known. Consider three square clusters consisting of N = L 2 , N' = L' 2 and N'' = L'' 2 sites, where L, Land L'' denote the linear size of a given cluster, with L > L' > L''. In the spirit of the mean-field theory, the border spins of the clusters are subject to the symmetry-breaking (effective) field b, band b'', respectively. For the cluster of the size L, the Hamiltonian in the presence of magnetic field can be written as where K stands for the coupling βJ and the parameters h and b denote the external and the effective field, respectively. The first sum runs over the nearest-neighbour pairs, the second over all the cluster sites and the third over the edge spins only, so that the factor nj counts the number of effective fields (nj = 1 except for the corner spins where nj = 2). In the standard MFA, the self-consistency condition for the magnetization per site