Critical properties of many-component systems

Critical properties are discussed for systems with order parameters given by $n$ vectors ${\overrightarrow{\mathrm{S}}}_{\alpha }$, 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 $n=0\mathrm{}$ corresponds to the excluded-volume problem. The general case is shown to reduce to an arbitrary $m$-component model which is random when $n=0\mathrm{}$ and constrained when $n\to \infty$. A direct derivation of the large-$n$ limit is given and leads to a variety of exactly solvable models. Expressions for the order $n^{-1\mathrm{}}$ correction are obtained in terms of correlation functions. This expansion is valid at all temperatures and for any order of transition, so that it is particularly suitable for considering tricritical phenomena.