On the Solution of One-Dimensional Ising Models

In this paper it is shown that the thermodynamic limit of the partition function of the statistical models under consideration on a one-dimensional lattice with an arbitrary finite number of interacting neighbors is expressed in terms of the principal eigenvalue of a matrix of finite size. The high sparseness of these matrices for any number of interactions makes it possible to perform an effective numerical analysis of the macro characteristics of these models.


Introduction
The data of modern studies of the magnetic properties of monoatomic chains [1] [2] raise the question of choosing a model for describing these phenomena and how to solve it.Here we study the problems of solving translationally invariant models with a binary interaction of spins located at the nodes of a one-dimensional lattice.

The Partition Function
be the binary variables (spins) that take two values of 1 ± , and , , M N M N < are some natural numbers.We consider a statistical model on a one-dimensional lattice, with nodes numbered by natural numbers 1, , N M +  , whose partition function has the form ( ) ( ) ( ) { } ( ) Here summation over ( ) , , , N σ σ σ σ =  means summation over all states of the spins, , 1, , are some real parameters of the interspin interaction, H is the parameter of interaction with the external field, and the boundary We are interested in the free energy ( ) ( ) F M , which for a specific value of M is calculated by the formula For 1 M = , the Formula (1) gives the partition function of the one-dimensional Ising model with the interaction between nearest spins, which has an exact solution.The free energy of this model is calculated through the principal eigenvalue (PEV) (positive single and maximum modulo eigenvalues) of a matrix of size , , and , x y are integers.Consider the sum where the summation is over the sets of binary variables , , , σ σ σ σ =  and the boundary condition , , , M σ σ σ  , we get the partition function (1), on the other hand, the sum ( ) ( ) M can be written in the form In this case, we can represent the set of values of each value ( ) the set of values of elements of some matrix ( ) .
The matrix ( ) A M is called the transfer matrix, the free energy ( ) ( ) Next, the transfer matrix ( ) A M will be expressed in the form convenient for applications.To do this, we present some information from the theory of in-

Indexed Matrices
The theory of indexed matrices for lattice models is presented in the monograph [2].We define single-index matrices, permutation operators for indices, and formulas that will be needed later.
The operator of permutations of indices is the operator permuting the canonical basis of the Euclidean space (Here J is a positive integer and T is a transpose sign) in the following order: × in this basis will be denoted by the symbol J P .Let q be a number matrix of size 2 2 × , then we denote by [ ] × whose blocks are the matrix q.The in- q is a block-diagonal matrix whose blocks are the matrix obtained by replacing the identity matrices ( ) ( ) It follows from the block structure of indexed matrices that the matrices q and w commute with different indices , and from the definition of indexed matrices it follows that We introduce the following notation for basis matrices of size Then any matrix B of size 2 2

J J
× can be written in the form

,
, , where , β σ σ -number of coefficients and the following equalities hold The trace of the matrix B is calculated as follows , .
i j i j i j i j i j

P e e f g g f h h i j
They have the following properties J i i j i j j i j i m j m i j J J q P P q P P P P P P P P in particular, the equalities J m j m J j j J j J j J P P P P q P P q = = (12) are satisfied.
We note that it follows from (8) that the matrix B can be written in the form where all the indexed matrices have the size 2 2 J J × .But, proceeding from the block structure of indexed matrices, matrix B can be given the form , , , , where the elements of a matrix of size 2 2 × should be understood as matrices of size ( ) ( ) . Thus, the equations ( 12), useful for our further calculations, in this notation take the form j j j j j j j q e g f h e g f h q = where all the indexed matrices have the size ( ) ( )

Transfer Matrix
For a given value of M, the specific form of the transfer matrix ( ) A M is de- termined by the correspondence between the sets of binary variables and the natural numbers that number the elements of the matrix.Consider matrix, which is defined by formula , g µ µ according to (3) has the form ( ) ( ) ( ) and all indexed matrices have the size 2 2

M M
× .Let us show that this matrix is the transfer matrix of the model (1).To do this, we calculate the partition function by the Formula (4).In these calculations, we use sets of binary variables ( ) The expression for Q can be expressed through the product of matrices of size 2 2 × with elements from indexed matrices.Let us write it in more detail and denote by the symbols u and v the vectors [ ]

∏
We transpose this polynomial from indexed matrices by transposing the indexed matrices without changing their previous places in the polynomial, which is possible due to their commutation (6).Then we obtain the following expres- ∏ where the symbol ( ) i r K denotes a matrix that has the form Using the commutation relations (13), we write the expression for T Q in the form and, using the commutation relations and Formulas ( 6), ( 7), (11), we get In the same way, we obtain the following expression for

Some Questions of Numerical Model Analysis
The resulting transfer matrix has a structure that makes it possible to use the well-known method of power iterations of the matrix very effectively to calculate its PEV.The effectiveness of this method is due to the extreme thinness of the transfer matrix and the existence of the PEV.For any M, the matrix of the permutation operator of indices has only one nonzero element equal to one in each Yu.N. Kharchenko Yu. N. Kharchenko DOI: 10.4236/jamp.2018.65082961 Journal of Applied Mathematics and Physics

2 ×
with each element of the matrix q, multiplied by the value of this element.For concrete matrices of size 2 , we use the notation ( ) ( ){ }, , , * * * * , where we write row elements in parentheses.
addition to the operator J P , we also introduce permutation operators whose matrices [ ]

[
Hence we obtain the required expression for the transfer matrix ( )