Relevant alternative analytic average magnetization calculation method for the square and the honeycomb Ising lattices

Abstract

In this work, the order parameter or average magnetization expressions are obtained for the square and the honeycomb lattices based on recently obtained magnetization relation,Where, κ is the coupling strength and z is the number of nearest neighbors. σ 0,i denotes the central spin at the i th site while σ l,i , l = 1, 2, . . . , z, are the nearest neighbor spins around the central spin. In our investigation, inevitably we have to make a conjecture about the three site correlation function appearing in the obtained relation of this paper. The conjectured form of the the three spin correlation function is given by the relation, < σ 1 σ 2 σ 3 >= a < σ > +(1 − a) < σ > (1+β −1 ) , here β denotes the critical exponent for the average magnetization and a is positive real number less than one. The relevance of this conjecture is based on fundamental physical reasoning. In addition, it is tested and investigated by comparing the obtained relations of this paper with the previously obtained exact results for the square and honeycomb lattices. It is seen that the agreements of the obtained average magnetization relations with those of the previously obtained exact results are unprecedentedly perfect.

Relevant spontaneous magnetization relations for the triangular and the cubic lattice Ising model

“…where we keep only numerical indices for notational simplicity. Apparently, this final equation is very similar to the previously used ones [19] except for extra more spin terms which create eventually some mathematical complication as expected. But, the treatment is going to be the same.…”

“…To investigate and test the validity and relevance of the method which is going to be used in this paper, we start by considering its application on the triangular Ising lattice. The treatment is going to be very similar to the procedure applied in the calculation of to the honeycomb and the square lattice cases studied in [19]. To start the calculation, we need to rewrite the equation introduced in the above introduction once more as,…”

“…When this relation is applied to a particular lattice, it produces inevitably odd spin correlation functions such as the three spins and five spins correlation functions. At this point it is important to mention that its applications to the honeycomb and the square lattices are the simplest in that only three spins correlation function appear in the final equation [19], making the mathematical treatment of the honeycomb and the square lattices are less complicated. The spontaneous magnetization relations have also been investigated and obtained [19].…”

“…At this point it is important to mention that its applications to the honeycomb and the square lattices are the simplest in that only three spins correlation function appear in the final equation [19], making the mathematical treatment of the honeycomb and the square lattices are less complicated. The spontaneous magnetization relations have also been investigated and obtained [19]. In the derivations of the spontaneous magnetization relation of those lattices, we had to use a conjectured mathematical form for the corresponding three spin correlation functions, given as < σ 1 σ 2 σ 3 >= a < σ > +(1 − a) < σ >.…”

“…It is going to be our main objective to obtain an analytical spontaneous magnetization expression for the cubic lattice This paper is organized as follows. In the next section, the average magnetization of the the triangular lattice is going to be calculated with an alternative analytic method based on the previously derived average magnetization relation by us [19]. The obtained expression is also going to be compared by the already available exact result derived by Potts.…”