Physica A 183(1992) 563-573 ~ 1 [~ North-Holland Renormalization group study of the two-dimensional Ising model with crossing bonds J.A. Plascak Departamento de Fisica- ICEx, Universidade Federal de Minas Gerais, C.P. 702, 30161 Belo Horizonte, MG, Brazil Received 17 July 1991 The simple quadratic Ising model with nearest- and next-nearest-neighbor interactions is investigated by means of the mean field renormalization group approach. By conveniently choosing the size of the blocks one obtains the expected phase diagram exhibiting ferromag- netic, antiferromagnetic, paramagnetic and superantiferromagnetic phases. The critical lines which separate the paramagnetic phase from the ordered ones have the expected asymptotic behavior for large values of the nearest-neighbor coupling. I. Introduction The reduced Hamiltonian of the two-dimensional Ising model with crossing bonds can be written as H = K~ ~ o-~o-j + K 2 ~'~ o-~o-j, o-~ = ---1, (1) (nn) (nnn) where a factor -/3 = -1/kT is included in the Hamiltonian and in K 1 and K:. The sums run over nearest- and next-nearest-neighbor pairs of spins, respec- tively. Although being a simple generalization of the square Ising model, it has not been solved exactly and the Onsager's solution [1] applies only to the case K 2 = 0 and to the case K 1 =0, when one has two decoupled square Ising models. However, this model has been investigated by several methods including mean field theory [2], series expansion [3, 4], renormalization group [5], Monte Carlo [6, 7], Monte Carlo renormalization [8], finite-size scaling [9, 10], and interface method [11], among others. It has been shown that the phase diagram in the (K1, K2) plane is symmetric for K~--~-K~ and presents ferromagnetic (F), antiferromagnetic (AF), superantiferromagnetic (SAF) and paramagnetic (P) phases. The renormalization group procedure [5] suggests 0378-4371/92/$05.00 © 1992-Elsevier Science Publishers B.V. All rights reserved