Static Fluctuation of Spins in Systems with Competing Anisotropies

Spin structure and phase transition of mixtures with competing anisotropies are investigated by a theory which takes static fluctuation of thermal averages of spins into consideration. It is found that random off-diagonal interaction between neighboring spins induces orientational distribution of spins and affect the second order phase transition. Random mixtures of two kinds of spins with orthogonal anisotropies have three distinct ordered phases. For a mixture composed of Aand B-spins which prefer to aline to the zand *axes, respectively, the z component of the spin has the long range order (LRO) in the high concentration range of A-spin and the 5 components has LRO in the low concentration range. In the intermediate concentration range both component achieve LRO and a phase called the oblique phase appears. (We denote these phases as [z] , [XI and [xz] phases in this report.) One of characteristic features of the mixture is that the thermal average of each spin points to various directions depending on its local environment in [xz] phase [l, 21. On the other hand the spin structure of 1x1 or [z] phases is rather simple. All spins point to a; or zaxes. This situation changes drastically if there exists random off-diagonal interaction which couples the a; and zcomponents of neighboring spins. Such interaction is expected to appear generally, even if its magnitude is small, in random mixtures since the bulk crystal symmetry is destroyed locally [3]. In this work I investigate how the off-diagonal interaction affects the spin structure of the ordered phases and the phase transition paying main attention to the orientational distribution of spins. For this purpose I consider a ferromagnetic mixture of Aand B-spins whose magnitudes are one. Its Hamiltonian, If = Ho + H', is given as spins fluctuates in its direction. I have developed a theory where the fluctuation of the directions of the thermal averages of spins are taken into consideratibn. The effective field acting on a spin is obtained by replacing each neighboring spin operator by SP, 1 or SP, 2 (P = A or B) with probability pp or 1 pp, while in the ordinary mean field theory it is replaced by the thermal average of P-spin, (Sp). Two vectors SP, 1 and Sp, 2 are determined by introducing two pararneters SP, I, t and SP, 2, t as SP, I = SP, n n + S ~ , I, tt, SP, 2 = SP, n n + S ~ , 2, t t , (3) where n and t are unit vectors parallel and perpendicular to (SP) and Sp, , is the magnitude of (Sp) . The parameters Sp, 1, t and Sp, 2, t represent the degree of the static fluctuation of P-spins and obtained as the averages of the component of (SP ( E ) ) , the thermal average of P-spin placed in a local environment designated by E, perpendicular to n under the condition that (Sp (E)) -t are positive and negative, respectively. They satisfy a equation that ppSp, 1, t + (1 pp) SP, 2, = 0. In this theory local environment of a spin is classified into 462 types for the simple cubic lattice. The parameters (Sp) , Sp, 1, t and Sp, 2, 2 are determined self-consistently in the framework of a single site approximation. The phase diagrams for r = 0.1 and Jxz / J = 0,0.1, 0.2 are shown in figure 1. The transition temperature (2) H' = -2 J F (5';s; + si"sja) ,