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Physical review. B, Solid state

Generalized Scaling Hypothesis in Multicomponent Systems. II. Scaling Hypothesis at Critical Points of Arbitrary Order

Alex Hankey, T. S. Chang, Eugene Stanley

Published Aug 1, 1973• 27 citations
Publisher

Abstract

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 ($\mathcal{O}=2$) 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.

DOI
10.1103/physrevb.8.1178

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

T. S. Chang, Alex Hankey, Eugene Stanley

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 ($\mathcal{O}+1$). Ordinary critical points are defined to be of order $\mathcal{O}=2$, so that the tricritical points introduced by Griffiths are of order $\mathcal{O}=3$. We discuss more general examples of critical spaces of order $\mathcal{O}=3$ which are known for a wide variety of systems; we also propose several examples of models of magnetic systems showing critical points of order $\mathcal{O}=4$---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 $\mathcal{O}$ 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 $\mathcal{O}\ensuremath{\ge}3$, 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 $\mathcal{O}=3$, 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).

Physical review. B, Solid state•Jul 1, 1973•80 citations

Scaling Hypothesis for a System Exhibiting a Critical Point of Order Four: Ising Planes with Variable Interplanar Interactions

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A scaling theory for critical points of higher order is reviewed and applied to a point of intersection of tricritical lines in a specific example: the Ising model with variable interplanar interactions. As a consequence of the scaling hypothesis, we can make predictions about the shapes of the coexistence and critical surfaces meeting at this critical point of order 4 where four phases become identical simultaneously.

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At tricritical points thermodynamic functions scale with respect to three independent field variables. Such points have been called critical points of the third order. We present, in this paper, a general scaling hypothesis for such critical points and demonstrate how the concept of invariant spaces leads naturally to the prediction of data collapsing from volumes to lines in terms of “double‐power scaling functions”.

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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.

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Scaling transformations and the XY-model

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Condensed media show a large variety of critical phenomena, ranging from critical opalescence at the end point of the liquid–gas coexistence curve, the Curie transition of ferromagnetic materials, to the superfluid transition of helium, the behaviour of solutions of polymers, the conductivity of random media, … To this list should be added systems with local symmetries, such as those suggested by particle physics in order to understand quark confinement. Surprisingly all these phenomena can be classified into a few universality classes, characterized by specific large distance behaviour with the same critical exponents. In this chapter, we sketch the methods and ideas of the renormalization procedure. We illustrate the concepts with simple approximations in the language of classical spin models in the first section, treating in more detail the XY -model as an example in the second section. Scaling laws. Real space renormalization Homogeneity and scale invariance The discussion given in previous chapters suggests that, close to a continuous transition, critical systems exhibit universal properties. Correlations at large distance are not sensitive to the details of microscopic interactions. Their behaviour is described by a specific dimensional analysis governed by some essential characteristics of the system, such as the dimension of space, the nature of the order parameter and the underlying symmetries. The mean field approximation gave a first idea of a simple critical behaviour. Fluctuations have only a quantitative effect in dimension greater than the upper critical one.

Cambridge University Press eBooks•Sep 29, 1989•0 citations
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 O\ensuremath≥2\mathcal{O}\ensuremath{\ge}2O\ensuremath≥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\mathcal{O}=2O=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 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\mathcal{O}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\mathcal{O}O to be a space of critical points of order (O+1\mathcal{O}+1O+1). Ordinary critical points are defined to be of order O=2\mathcal{O}=2O=2, so that the tricritical points introduced by Griffiths are of order O=3\mathcal{O}=3O=3. We discuss more general examples of critical spaces of order O=3\mathcal{O}=3O=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\mathcal{O}=4O=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\mathcal{O}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\mathcal{O}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\mathcal{O}=3O=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.
Critical points of higher order can exist in complex magnetic and fluid systems. By definition, at a critical point of order O\mathcal{O}O, O\mathcal{O}O phases become identical simultaneously. Here the Wilson renormalization-group method is generalized from ordinary critical points (O=2\mathcal{O}=2O=2) and Gaussian tricritical points (O=3\mathcal{O}=3O=3) to critical points of arbitrary order O\mathcal{O}O. An expansion scheme in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)\ensuremath{-}d}$ is proposed. The nontrivial fixed points and the critical exponents for O=3,4\mathcal{O}=3,4O=3,4 are calculated to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$. We present the relevant scaling fields and densities for O=3\mathcal{O}=3O=3, and, in an appendix, justify the validity of the approximate recursion relation to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$.
Condensed media show a large variety of critical phenomena, ranging from critical opalescence at the end point of the liquid–gas coexistence curve, the Curie transition of ferromagnetic materials, to the superfluid transition of helium, the behaviour of solutions of polymers, the conductivity of random media, … To this list should be added systems with local symmetries, such as those suggested by particle physics in order to understand quark confinement. Surprisingly all these phenomena can be classified into a few universality classes, characterized by specific large distance behaviour with the same critical exponents. In this chapter, we sketch the methods and ideas of the renormalization procedure. We illustrate the concepts with simple approximations in the language of classical spin models in the first section, treating in more detail the XY -model as an example in the second section. Scaling laws. Real space renormalization Homogeneity and scale invariance The discussion given in previous chapters suggests that, close to a continuous transition, critical systems exhibit universal properties. Correlations at large distance are not sensitive to the details of microscopic interactions. Their behaviour is described by a specific dimensional analysis governed by some essential characteristics of the system, such as the dimension of space, the nature of the order parameter and the underlying symmetries. The mean field approximation gave a first idea of a simple critical behaviour. Fluctuations have only a quantitative effect in dimension greater than the upper critical one.