Studies in the Theory of Cooperative Phenomena.

In this dissertation, two distinct problems have been addressed; spin dynamics of the anisotropic and/or alternating XY chain in a field, and the ANNNI model using a variation of Kikuchi's method. The eigenvalues and eigenvectors for the anisotropic alternating XY chain at T = 0 have been obtained via the Jordan-Wigner transformation to free fermions, as well as an expression for <S('2)(t)S('2)(0)>, the time-dependent spin correlation in the z-direction. For the pure alternating and pure anisotropic cases, the excitation spectra, consisting of two-parameter continua, and the dynamic structure factor have been obtained, the latter as a sum of integrals. Explicit evaluation of these integrals has been possible in general for the alternating case, and in the anisotropic case in the transverse Ising limit and for the special field h = SQRT.(1-(gamma)('2)), for which the system has an ordered ground state. The magnetization has also been found in these cases. For the anisotropic chain in the special field, the nth frequency moments of S('22)(q,(omega)) have been explicitly evaluated for n = -1, 0, 1, 2, 3. The odd moments, n > 0, agree with a set of sum rules developed by Muller. Limitations in the applicability of spin-wave theory, even in the ordered state, are indicated. In the study of the ANNNI model, a variant of Kikuchi's cluster-variation method is described which is capable of handling systems for which many clusters are important. This method is strictly numerical. Ordering is described in terms of normal modes of the order parameters necessary for a particular Kikuchi entropy. The method immediately gives the free energy, entropy, specific heat, and order parameters as functions of temperature. A phase diagram for the 3-D ANNNI model which interpolates between simple mean-field and low-temperature expansion results is obtained. Also, some information is obtained for the 2-D ANNNI model using three different approximations for the entropy. Some peculiar behaviors are encountered for this model. The existence of an incommensurate phase may be indicated.