Equivalence of the O(n) vector ferromagnetic and antiferromagnetic models

The effective-field renormalization group (EFRG) approach is used to find the Néel temperature (TN) of the O(n) vector model with antiferromagnetic (AF) interaction. The EFRG method is illustrated by employing approximations in which clusters with one (N′=1) and two (N=2) spins are used. The critical temperature TN is obtained as a function of component (n) and coordination (z) numbers. For all values of n and z we show that TN=Tc, where Tc is the Curie temperature for the ferromagnetic (F) case. As a comparison, the results of the quantum Heisenberg model (n=3) with F and AF interactions are also presented, and we find that TN>Tc, which is different from the classical result Tc=TN.