Generalized Scaling Hypothesis in Multicomponent Systems. I. Classification of Critical Points by Order and Scaling at Tricritical Points

The goal of this work is to provide an analysis of spaces of critical points for multicomponent systems. First, we propose the geometric concept of order $\mathcal{O}$ for critical points; we distinguish it from a previous definition of a "multicritical" point. Specifically, we may define the intersection of spaces of critical points of order $\mathcal{O}$ to be a space of critical points of order ($\mathcal{O}+1\mathrm{}$). Ordinary critical points are defined to be of order $\mathcal{O}=2\mathrm{}$, so that the tricritical points introduced by Griffiths are of order $\mathcal{O}=3\mathrm{}$. We discuss more general examples of critical spaces of order $\mathcal{O}=3\mathrm{}$ which are known for a wide variety of systems; we also propose several examples of models of magnetic systems showing critical points of order $\mathcal{O}=4\mathrm{}$—i.e., systems having intersecting lines of tricritical points. The analysis of critical and coexistence spaces also provides a new form of the Gibbs phase rule suitable for complex magnetic models. Next we define—for the critical points of order $\mathcal{O}$ of which examples have been given—special directions in terms of which to make a scaling hypothesis. We give the hypothesis for simple systems and then for tricritical points, and then, in a subsequent paper, part II, the special directions are used to make a scaling hypothesis at spaces of critical points of any order. Certain predictions (e.g., scaling laws and "single-power" scaling functions) follow in a simple and straightforward fashion. We consider the scaling hypothesis at a critical space of order $\mathcal{O}$ in terms of a group of transformations. We can define a set of invariants of the group. It is possible, for $\mathcal{O}\ge 3$, to make a second scaling hypothesis for the space of order $\mathcal{O}-1\mathrm{}$ using certain of these invariants as independent variables. This is advantageous because certain "double-power" scaling functions then follow directly; these predict that for $\mathcal{O}=3\mathrm{}$, experimental data collapse from a volume onto a line. This prediction is to be contrasted with ordinary scaling functions, which predict that data collapse by only a single dimension (e.g., from a volume onto a surface or from a surface onto a line).