We make the scaling hypothesis at a space of critical points of arbitrary order, using the geometric classification of critical points by order proposed in a preceding paper, part I. It follows from the definition of order that there are regions in which several scaling hypotheses are simultaneously valid, one at each space of critical points of order $\mathcal{O}\ensuremath{\ge}2$ which is connected to the point of highest order. The successive hypotheses are conveniently framed in terms of invariants of the groups of transformations about the points of higher order. This procedure facilitates the formation of multiple-power scaling functions. We suggest that the regions of validity of later hypotheses are bounded by hypersurfaces which scale and which are therefore most easily visualized in the spaces of invariants. A general critical point of order 4 is treated as a preliminary example. We then apply the hypothesis in detail to the critical point of order 4 in a specific example: the Ising model with variable interplanar interaction. This is done in two ways: the first shows specifically the relationship to the case of a metamagnet and to scaling with a parameter for change of lattice dimension; the second explicitly demonstrates that the connectivity between the various surfaces of ordinary (
O=2) critical points is the same as for the edges of a tetrahedron. As a consequence of this scaling hypothesis, we can make predictions about the shapes of the coexistence and critical surfaces meeting at the critical point of order 4.
The first phase diagram explicitly exhibiting intersecting lines of tricritical points is presented. Their point of intersection is a critical point of order 4. The system is a simple assembly of ferromagnetic Ising planes coupled with an arbitrary interplanar interaction, with Hamiltonian $\mathcal{H}=\ensuremath{-}J\left(\ensuremath{\Sigma}\stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{s}_{i}{s}_{j}+\mathcal{R}\ensuremath{\Sigma}\stackrel{z}{〈\mathrm{ij}〉}{s}_{i}{s}_{j}\right)\ensuremath{-}\ensuremath{\mu}H\ensuremath{\Sigma}\stackrel{}{i}{s}_{i}\ensuremath{-}\ensuremath{\mu}{H}^{\ensuremath{'}}{(\ensuremath{-}1)}^{\ensuremath{\eta}}\ensuremath{\Sigma}\stackrel{}{i}{s}_{i},$ where
J>0, ${s}_{i}=\ifmmode\pm\else\textpm\fi{}1$, and $\ensuremath{\mu}$ is the magnetic moment per spin. The first sum is over nearestneighbor (nn) spins in an $x\ensuremath{-}y$ plane, the second sum over nn spins coupled along the
z direction,
H is the magnitude of a uniform magnetic field, and ${H}^{\ensuremath{'}}$ is a staggered magnetic field which acts oppositely on adjacent planes of constant
z ($\ensuremath{\eta}=0$ on even planes, +1 on odd planes). The Hamiltonian is invariant under $\mathcal{R}\ensuremath{\rightarrow}\ensuremath{-}\mathcal{R}$, $H\ensuremath{\rightarrow}{H}^{\ensuremath{'}}$, and ${s}_{i}\ensuremath{\rightarrow}{(\ensuremath{-}1)}^{\ensuremath{\eta}}{s}_{i}$, so that the Gibbs potential is also invariant, $G(T, H, {H}^{\ensuremath{'}}, \mathcal{R})=G(T, {H}^{\ensuremath{'}}, H, \ensuremath{-}\mathcal{R})$. Using this symmetry, we make a scaling hypothesis about the special point $\ensuremath{\tau}[\ensuremath{\equiv}T\ensuremath{-}{T}_{c} (R=0)]=H={H}^{\ensuremath{'}}=R=0$, namely, that the Gibbs potential obeys the functional equation $G({\ensuremath{\lambda}}^{{a}_{\ensuremath{\tau}}}\ensuremath{\tau}, {\ensuremath{\lambda}}^{\mathrm{aH}}H, {\ensuremath{\lambda}}^{a{H}^{\ensuremath{'}}}{H}^{\ensuremath{'}}, {\ensuremath{\lambda}}^{a\mathcal{R}}\mathcal{R})=\ensuremath{\lambda}G(\ensuremath{\tau}, H, {H}^{\ensuremath{'}}, \mathcal{R})$; the four scaling powers are found to be ${a}_{\ensuremath{\tau}}=\frac{1}{2}$,
aH=aH=1615,
aR=87.
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
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
O to be a space of critical points of order (
O+1). Ordinary critical points are defined to be of order
O=2, so that the tricritical points introduced by Griffiths are of order
O=3. We discuss more general examples of critical spaces of order
O=3 which are known for a wide variety of systems; we also propose several examples of models of magnetic systems showing critical points of order
O=4---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
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
O in terms of a group of transformations. We can define a set of invariants of the group. It is possible, for $\mathcal{O}\ensuremath{\ge}3$, to make a second scaling hypothesis for the space of order $\mathcal{O}\ensuremath{-}1$ using certain of these invariants as independent variables. This is advantageous because certain "double-power" scaling functions then follow directly; these predict that for
O=3, 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).
In this chapter, we investigate the concept of scaling, in particular the scaling hypothesis that has proved so important in the development of the modern theory of critical phenomena. Scaling refers to the way in which properties such as the susceptibility behave in the critical region as they approach infinity (or zero). In this discussion, it is important to be specific about the thermodynamic field variables used to represent the system as well as the direction in which the critical point is approached. A useful schematic in this regard is shown in figure 9.1, which shows phase coexistence diagrams for both an Ising-type magnet and a typical fluid. The fields for the magnet are (H, T ) while for the fluid they are (μ, T ). We note immediately from this figure that the magnet’s coexistence curve is coincident with the H = 0 axis and symmetric about the M = 0 axis. However, these features do not reflect in the corresponding diagrams for the fluid, with important implications for a scaling analysis in this system, which we take up later in this chapter. We illustrate the basic critical scaling laws using the Ising (magnet) system; however, the fact that similar scaling laws also apply to fluids is an important consequence of critical-point universality, a topic we have often alluded to in this text and which is discussed in more detail in chapter 10 dealing with the renormalization-group method.
The conventional renormalization-group analysis of the tricritical behavior of a metamagnet is completed by a description of the first-order transition below the tricritical temperature, and the conditions for the discontinuity in the magnetization are explicitly verified in an Ising spin model. The singularity structure of the crossover scaling function is derived from relations between the scaling fields for the critical and tricritical fixed points. Numerical calculations illustrating various aspects of the theory are given for a square-lattice Ising model in a four-cell cluster approximation.