Critical properties of many-component systems

Critical properties are discussed for systems with order parameters given by n vectors S/sub a/, each with m components. The Hamiltonian has an arbitrary symmetry for each vector separately, but there is a particular kind of coupling between them. It is shown that there is an integral representation for the partition function which reduces n to an explicit parameter in an averaged partition function for the m-component model. This leads to a simple discussion of properties of the system as a function of n. In particular, it is possible to give a coherent derivation of several known and new results without the aid of perturbation theory or the renormalization-group method. It is shown that, in certain special cases, the exponents are Gaussian when n is a negative even integer and that n = 0 corresponds to the excluded-volume problem. The general case is shown to reduce to an arbitrary m-component model which is random when n = 0 and constrained when n ..-->.. infinity. A direct derivation of the large-n limit is given and leads to a variety of exactly solvable models. Expressions for the order n/sup -1/ correction are obtained in terms of correlation functions. This expansion is validmore » at all temperatures and for any order of transition, so that it is particularly suitable for considering tricritical phenomena. (auth)« less