The method of reduction of transfer matrix for modulated systems

We present a new method of calculation of partition function for the layered systems with the arbitrary spin-modulated structure in the linear cluster approximation. The thermodynamic description of the system in question is based on the Bogolyubov variational principle (inequality). The transfer matrix technique is used to determine the partition function, finally the free energy of the system, in terms of its largest eigenvalue. However, the compositional modulation introduces different types of transfer matrices related to different pure components of the system as well as the interface regions between them. The reduction of transfer matrices related to high-spin components obtained by a partial summation of the partition function gives us a simplified expression for the free energy in characteristic form already known for a low-spin component. In particular, we study two periodic magnetic superstructures ABAB with a strong perpendicular anisotropy, spin SA = 1⁄2 and the large spin value SB = 1 or SB = 3/2. In each case, the method presented leads to a simple renormalized expression for the free energy of anisotropic homogeneous structure with only spins S = 1⁄2. Next, as a numerical result interesting discontinuous thermal transition between new stable ordered phases is obtained.