An argument is presented for a certain asymptotic form for the correlation functions of two local dynamical variables when the system is near its critical point and in the one phase region. If
A(r) and
B(r) are local dynamical variables and $\ensuremath{\omega}$ stands for the variables specifying the thermodynamic state, it is found that for $\ensuremath{\omega}$ near the critical point ${\ensuremath{\omega}}_{c}$ and for
r large enough, $〈A(\mathbf{r})B(0)〉\ensuremath{-}〈A(\mathbf{r})〉〈B(0)〉\ensuremath{\simeq}\frac{{d}_{A}(\ensuremath{\omega}){d}_{B}(\ensuremath{\omega}){e}^{\ensuremath{-}\ensuremath{\kappa}(\ensuremath{\omega})}}{{r}^{b(\ensuremath{\omega})}}$, where $\ensuremath{\kappa}(\ensuremath{\omega})$ and $b(\ensuremath{\omega})$ are the same for a large class of
A and
B and $\ensuremath{\kappa}(\ensuremath{\omega})\ensuremath{\rightarrow}0$, $b(\ensuremath{\omega})\ensuremath{\rightarrow}b\ensuremath{\ge}0 \mathrm{as} \ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}$. For the Ising model of any dimension in zero field and below the critical temperature, the additional assumption of a single correlation length implies the relationship ${\ensuremath{\gamma}}^{\ensuremath{'}}+2\ensuremath{\beta}=2\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}}$, which is often conjectured for the critical exponents.
Introduction We noted in the previous chapter that the singular behavior of thermodynamic functions at a critical point (the termination of a coexistence line) can be characterized by a set of critical exponents {α, β, γ, …}. Experimental observations indicate that these exponents are quite universal , i.e. independent of the material under investigation, and to some extent, of the nature of the phase transition. For example, the vanishing of the coexistence boundary in the condensation of CO 2 has the same singular behavior as that of the phase separation of protein solutions into dilute and dense components. This universality of behavior needs to be explained. We also noted that the divergence of the response functions, as well as direct observations of fluctuations via scattering studies, indicate that fluctuations have long wavelengths in the vicinity of the critical point, and are correlated over distances ξ ≫ a , where a is a typical interparticle spacing. Such correlated fluctuations involve many particles and a coarse-graining approach, in the spirit of the theory of elasticity, may be appropriate to their description. Here we shall construct such a statistical field theory . We shall frame the discussion in the language of a magnetic system whose symmetries are more transparent, although the results are of more general applicability. Consider a material such as iron, which is experimentally observed to be ferromagnetic below a Curie temperature T c , as in Fig. 1.4.
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 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).
The fact noted by Schofield that the specific heat at constant volume, albeit singular, is less singular than the specific heat at constant pressure implies an asymptotic relation between the two-, three-, and four-body distribution functions. The experimental background of this fact is discussed and a fluctuation formula which expresses
CV in terms of two-, three-, and four-body correlation functions is derived. A heuristic explanation of the asymptotic properties of the distribution functions is given based on the fact that local fluctuations proportional to the critical eigenvector are overwhelmingly the most probable near the critical point. It is shown that if
CV is indeed infinite there exists a second critical eigenvector linearly independent of the first. Some consequences of the existence of two critical eigenvectors are discussed, and a form for the short-range behavior of the distribution functions near the critical point is conjectured.
There are two independent critical exponents that describe the behavior of systems near their critical point. However, at the critical point only the exponent η, which describes the decay of the correlation function, is usually discussed. We emphasize that there is a second independent exponent η′ that describes the decay of the fourth-order correlation function. The exponent η′ is related to the exponents determining the behavior of thermodynamic functions near criticality via a fluctuation-response equation for the specific heat. We also discuss a scaling law for η′.
Abstract This chapter is devoted to a brief review of general properties of phase transitions in macroscopic physics and, in particular in lattice models. Some of these lattice models actually appear as lattice regularizations of Euclidean (imaginary time) quantum physics theory (QFT). Most of the transitions considered in this work have the following character: spins on the lattice, or macroscopic particles in the continuum, interact through short-range forces, assumed, for simplicity, to decay exponentially. For simple systems, it is possible to find a local observable, called order parameter, whose expectation values depend on the phase in the several phase region, for example, the spin in ferromagnetic systems. In the disordered phase, the connected two-point function decreases exponentially at large distance, at a rate characterized by the correlation length (the inverse of the smallest physical mass in particle physics). In continuous transitions, the correlation length diverges at the critical temperature. Within the mean-field approximation (consistent with Landau's theory of critical phenomena), it can be shown that the singular behaviour of thermodynamic quantities at the critical temperature is universal. These properties can also be reproduced by calculating correlation functions with a perturbed Gaussian measure. It is then shown that the leading corrections to the mean-field approximation, in Ising-like systems, diverge at the critical temperature for dimensions smaller than or equal to
4.