The data of modern studies of the magnetic properties of monoatomic chains   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.
2. The Partition Function
Let , , be the binary variables (spins) that take two values of , and are some natural numbers. We consider a statistical model on a one-dimensional lattice, with nodes numbered by natural numbers , whose partition function has the form
Here summation over means summation over all states of the spins, are some real parameters of the interspin interaction, H is the parameter of interaction with the external field, and the boundary condition is satisfied. We are interested in the free energy , which for a specific value of M is calculated by the formula
For , 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   (Ising solution). We show that the free energy of a model with an arbitrary value of M is calculated through the PEV of a matrix of size (this eigenvalue exists for finite values of M by the Perron-Frobenius theorem). We call such models one-dimensional Ising models.
Let be a set of binary variables. We define the value of by the formula
where are the Kronecker symbols, , , and are integers. Consider the sum , which has the form
where the summation is over the sets of binary variables and the boundary condition . Performing the summation over the sets , we get the partition function (1), on the other hand, the sum can be written in the form
In this case, we can represent the set of values of each value as the set of values of elements of some matrix of size enumerated by the values of the sets . Hence we obtain
The matrix is called the transfer matrix, the free energy (2) is expressed in terms of the PE transfer-matrix by the formula   .
Next, the transfer matrix will be expressed in the form convenient for applications. To do this, we present some information from the theory of indexed matrices.
3. Indexed Matrices
The theory of indexed matrices for lattice models is presented in the monograph  . 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: and . Its matrix in this basis will be denoted by the symbol .
Let q be a number matrix of size , then we denote by a block-diagonal matrix of size whose blocks are the matrix q. The indexed matrix is defined by the formula . The matrix is a block-diagonal matrix whose blocks are the matrix obtained by replacing the identity matrices with each element of the matrix q, multiplied by the value of this element.
For concrete matrices of size , we use the notation , where we write row elements in parentheses.
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 can be written in the form
where ―number of coefficients and the following equalities hold
The trace of the matrix B is calculated as follows
In addition to the operator , we also introduce permutation operators whose matrices have the form
They have the following properties
in particular, the equalities
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 . But, proceeding from the block structure of indexed matrices, matrix B can be given the form
where the elements of a matrix of size should be understood as matrices of size . Thus, the equations (12), useful for our further calculations, in this notation take the form
where all the indexed matrices have the size .
4. Transfer Matrix
For a given value of M, the specific form of the transfer matrix is determined 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
where according to (3) has the form
and all indexed matrices have the size . 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 similar to . First, we compute , for this we write in the form
Taking into account the commutation relations (6) and the equalities (9), we write this product in the form
Continuing this process of multiplication by , we obtain
We multiply this expression for by the matrix , using the sets of binary variables and
As a result, we obtain the following expression for
Calculating the trace of the right side of this equation by the Formula (10), we obtain the partition function (1) in the form (4).
Now we compute the transfer matrix (14). The sum over all values of the set of the transfer matrix (14) can be written in the form , where
The expression for Q can be expressed through the product of matrices of size with elements from indexed matrices. Let us write it in more detail
We now introduce the following measurands
and denote by the symbols u and v the vectors , . Then Q takes the form
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 expression for
where the symbol denotes a matrix that has the form
Using the commutation relations (13), we write the expression for in the form
We introduce new notation
and, using the commutation relations and Formulas (6), (7), (11), we get
In the same way, we obtain the following expression for
Then for we obtain the formula
where the matrix has the form
Hence we obtain the required expression for the transfer matrix of the one-dimensional Ising model (1)
5. 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 row and each column, and the matrix that faces the matrix of the permutation operator of the indices is block-diagonal with blocks of size . On the other hand, the M-th power of the transfer matrix is the matrix with positive elements, and therefore Perron-Frobenius theorem  is fulfilled for it, according to which its PEV exists and depends analytically on the parameters of the matrix . The existence of the PEV is critical for the application of the method, and the sparseness of the transfer matrix makes it possible to construct an economical, fast-acting algorithm for calculating the PEV  .
This kind of matrix was used to calculate the macro characteristics of two-dimensional Ising-type models with the interaction between nearest neighbors   . There they were called the root transfer matrices (RTM) and the free energy of these two-dimensional models with RTM was determined by the formula
where is the RTM of the matrix .
There are at least three cases in which the transfer matrix (15) in form exactly coincides with the RTM of the corresponding multidimensional models with the interaction between the nearest neighbors. The difference is that in one-dimensional models the parameter M determines, in a certain sense, the distance between the interacting spins, and the parameter M in multidimensional models is equal to the number of nodes of the two-dimensional lattice along the horizontal, while the number of nodes along the vertical is infinite. Let us consider these cases.
1) We choose the following values for the parameters in the transfer matrix (15): , , . In this case, the transfer matrix of this model takes the form
This transfer matrix coincides, up to the notation, with the RTM of the two-dimensional Ising model on a square lattice with the interaction between nearest neighbors (  , p. 128).
2) Let the parameters take the following values , , , . Then the transfer matrix of this model takes the form
and it, to within a similarity transformation, coincides with the RTM of the two-dimensional Ising model on a triangular lattice  .
3) For the parameters , which take the values , , , , , , , the transfer matrix of the one-dimensional model takes the form
This transfer matrix for M = RN coincides with the RTM of the three-dimensional Ising model on a cubic lattice with the interaction between nearest neighbors (  , p. 147).
Neither of these models exists for an exact solution with H not equal to zero. In such cases, the necessary information on the behavior of macro-characteristics of the model can give a numerical analysis. And the links between one-dimensional and multidimensional models can be used to solve problems in studies of various models   . In the iterative method, the rate of convergence to the limit is significantly influenced by the value of the PEV ratio to another higher absolute value of its own value. Calculations showed  that this ratio reaches its minimum value near points that in the limit as M tends to infinity they tend to singular points of the model. Thus, the study of one-dimensional models can give an idea of a pattern of the arrangement of singular lines in the corresponding multidimensional models. It should be noted that the study of one-dimensional models is interesting also for and since their free energy is determined by the PEV of a matrix of finite size, their thermodynamic properties can in principle be investigated by numerical methods with the required accuracy.
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