CHINESE JOURNAL OF PHYSICS, vol.66, pp.415-421, 2020 (Journal Indexed in SCI)
In this work, the order parameter and the two-site correlation functions are expressed properly using the decimation transformation process in the presence of an external field so that their applications lead to some significant physical results. Indeed, their applications produce or reproduce some relevant and important results which were included in cumbersome mathematics in the previous studies, if not in a form impossible to understand. The average magnetization or the order parameter < sigma> is expressed as < sigma(0,i )> =< tanh [ kappa( sigma(1,i) + sigma(2,i)+ ...+sigma(z,i)) + H]>. Here, kappa is the coupling strength, z is the number of nearest neighbors. sigma(0,i )denotes the central spin at the i(th) site, while sigma(l,i), l = 1, 2, ..., z are the nearest neighbor spins around the central spin. H is the normalized external magnetic field. We show that the application of this relation to the 1D Ising model reproduces readily the previously obtained exact results in the absence of an external field. Furthermore, the three-site correlation functions of square and honeycomb lattices of the form < sigma(1)sigma(2)sigma(3)> are analytically obtained. One finds that the three-site correlation functions are equal to f(kappa) < sigma > . Here f(kappa) depends on the lattice types and is an analytic function of coupling constant. This result indicates that the critical properties of three-site correlation functions of those lattices are the same as the corresponding order parameters < sigma > of those lattices. This will mean that the uniqueness of the average magnetization as an order parameter is questionable. The application of the two site correlation function relation obtained in this paper produces a relevant exact relation between four site correlation function and two site correlation functions for square lattice. It also produces a relation between correlation functions of honeycomb lattice. Making use of this relation leads to the exact calculation of the constant A appearing in the proposed correlation function relation in the scaling theory. It is also indicated that the correlation length xi(kappa) can be obtained exactly in the realm of scaling theory from the obtained correlation function relation for the honeycomb lattice. In addition, the average magnetization relation obtained in this paper leads easily to the result of the conventional mean field critical coupling strength which, as a first approximation, is equal to kappa(c) = 1/z. It is also shown that systematic improvements of the values of the critical coupling strengths are possible with mean field treatments in this picture.