3.1. The Models

The critical behavior of the different universality classes can be derived from a range of simple model systems. I briefly describe the Ising model and its extension to the n-vector model, which defines a broad range of models classified according to their values for two variables. This model is crucial to understanding the real-space RG and is abstracted to provide the basis for the field-theoretic RG. Microphysical models are not defined for multiple members of the same universality class; rather, a representative model is used for each class.

Niss (Reference Niss2005) describes the early history of the Lenz-Ising model.Footnote ⁶ The Ising model was specifically designed to represent the physical characteristics of magnetic systems rather than the broader range of systems that display critical phenomena. According to the model, such systems are composed of an array of interacting micromagnets that have a discrete range of orientations. This latter assumption arose out of a combination of empirical data and considerations from early quantum mechanics.

In modern formulations, the Ising model consists of a D-dimensional cubic lattice with {e_i} basis vectors with sites labeled $k=(k_1{e}_1,\dots ,k_D{e}_D)$. At each site there is a spin variable ${\sigma }_k\in \{\mathrm{-1},1\}$, although in extensions to this model the spin variable can take a greater range of values. A Hamiltonian is defined

(1)$\begin{array}{}& H=-J\sum _{k,k+\mu }{\sigma }_k{\sigma }_{k+\mu }-B\sum _k{\sigma }_k\mathrm{.}\end{array}$

The coupling constant J takes a positive value and is assumed to be independent of all variables other than the system volume. The Ising model interaction is generally defined over nearest or next-nearest neighbors; thus, μ is a lattice vector that takes any vector to the relevant neighbor in the positive direction; B is an external magnetic field.

The Hamiltonian of a system corresponds to the energy of the system in a particular configuration; thus, we see (as is confirmed empirically) that the Ising Hamiltonian will take a lower value when the spins are aligned and a higher value when spins are disordered. The ferromagnetic-paramagnetic transition can be defined over this lattice as the transition from the spin configuration with all spins aligned to that where there is no general correlation between the spin directions. This transition will take place at the critical temperature (T_c). No analytic derivation of critical behavior for any three-dimensional model has been achieved.Footnote ⁷

Behaviors characteristic of systems approaching T_c are termed ‘critical phenomena’, and it is with respect to the power laws that describe such behavior that universality can be observed. Current mathematical procedures to describe such behavior involve the RG, described below. The n-vector model generalizes the Ising model to various universality classes. Stanley (Reference Stanley1999, S361) notes: “empirically, one finds that all systems in nature belong to one of a comparatively small number of such universality classes.”

The n-vector model includes spins that can take on a continuum of states:

(2)$\begin{array}{}& H(d,n)=-J\sum _{k,k+\mu }{\sigma }_k\cdot {\sigma }_{k+\mu }-B\sum _k{\sigma }_k\mathrm{.}\end{array}$

Here, the spin ${\sigma }_k=({\sigma }_{k,1},{\sigma }_{k,2},\dots ,{\sigma }_{k,n})$ is an n-dimensional unit vector. The two parameters that determine the universality class are the system dimensionality d (which will determine the set of nearest neighbors) and the spin dimensionality n. The standard, three-dimensional Ising model corresponds to $H$(3, 1), as it describes three-dimensional magnets that, in the ferromagnetic phase, will have spins either aligned or anti-aligned with a single axis.Footnote ⁸

I now turn to a discussion of the RG derivation of critical exponents. A full exposition would require more space than we have here, but I sketch the procedure below.Footnote ⁹ RG transformations are constructed to preserve thermodynamical properties of the system of interest while increasing the mean size of correlations. Thus, for example, the RG transformations take a ferromagnetic system toward the critical point (where the order parameter fluctuates wildly).

3.2. Field-Theoretic and Real-Space Renormalization

The different RG approaches posit competing methods for deriving critical exponents:

Field-Theoretic RG

The Hamiltonian (eq. [5]) considered in this case is more abstract (technically it is a functional of the order parameter [OP])—I outline one way of deriving it below. The calculation of this Hamiltonian for real systems involves integration over a range of scales and energies. The highest energy (smallest scale) cutoff (denoted Λ) corresponds to the impossibility of fluctuations on a scale smaller than the distance between the particles in the physical system. The RG transformation in this case involves decreasing the cutoff, thus increasing the minimum scale of fluctuations considered. This procedure is analogous to increasing the lattice size and will similarly generate a flow through parameter space designed to maintain the Hamiltonian form and qualitative properties of the system in question.

Both approaches to the RG can be formalized as follows. The RG transformation $R$ transforms a set of (coupling) parameters {K} to another set {K′} such that $ℛ\left\{K\right\}=\left\{K^{\prime }\right\}$. The set of parameters that corresponds to a fixed point is {K*}, defined such that $ℛ\left\{K^{*}\right\}=\left\{K^{*}\right\}$.Footnote ¹¹ If we assume that $R$ is differentiable at the fixed point, this leads us to a version of the RG equations:

(3)$\begin{array}{}& \begin{array}{c}{K^{\prime }}_a-K_a^{*}\sim \sum _b{T}_{\mathrm{ab}}(K_b-K_b^{*}),\\ \text{where}{T}_{\mathrm{ab}}={\frac{\partial K_a^{\prime }}{\partial K_b}|}_{K=K^{*}}\mathrm{.}\end{array}\end{array}$

There are now two more steps before we can define relevance and irrelevance. First, we define the eigenvalues of the matrix T_ab as {λⁱ} and its left eigenvectors as {eⁱ}. Scaling variables ($u_i\equiv {\sum }_a{e}_a^i(K_a-K_a^{*})$) are linear combinations of the deviations from the fixed points. By construction, these scaling variables will transform multiplicatively near the fixed point such that ${u^{\prime }}_i={\lambda }^i{u}_i$. The second (trivial) step is to redefine the eigenvalues as ${\lambda }^i=b^{y_i}$, where b is the renormalization rescaling factor and y_i are known as the RG eigenvalues.

If $y_i>0$, then u_i is relevant; if $y_i<0$, u_i is irrelevant; and if $y_i=0$, u_i is marginally relevant. The relevant scaling variables will increase in magnitude after repeated RG transformations, while the irrelevant scaling variables will tend to zero after multiple iterations. (The behavior of the marginal scaling variables requires more analysis to determine.) Thus, given the Hamiltonian of one of our models, one can define an RG transformation that will allow one to (i) classify certain of the coupling parameters of the system in question as (ir)relevant to its behavior near the fixed point and (ii) extract the critical exponents from the scaling behavior near the fixed point. Up to this point the description is generic. Note that in the real-space approach the coupling parameters to the Ising-type Hamiltonians are marked as relevant or irrelevant, while in the field-theoretic approach it is the operators—functions of the order parameter—that are so labeled.

The real-space RG depends on the application of a blocking transformation: a standard example is depicted in section 4.2, although almost any blocking transformation would do equally well. It is required that the Hamiltonian form is stable across these transformations. Since the Hamiltonians are not renormalizable this involves the application of a transformation and subsequent truncation of the Hamiltonian.Footnote ¹²

The field-theoretic RG approach derives the critical exponents using diagrammatic perturbation theory—I do not have space to elaborate this here. The Hamiltonian in this context is macroscopic and depends on the OP (ɸ) that, in the Ising model context, is proportional to a sum of the spins in a small region of volume δV at x: $ɸ\left(x\right)\propto (1/\delta V){\sum }_{i\in \delta V}{\sigma }_i$.Footnote ¹³ We require that $a\ll \delta V\ll l$, where a is the physical lattice spacing and l is the dominant statistical length (often the correlation length). One can approach the Hamiltonian’s construction from the Ising model as follows (see Klein, Gould, and Tobochnik Reference Klein, Gould and Tobochnik2012).Footnote ¹⁴

Start with the Ising model (eq. [1]); then postulate a form for the Helmholtz free energy $F$(ɸ) of a system in contact with a heat bath. The terms in equation (4) correspond (a) to the interaction of the coarse-grained Ising spins with an external magnetic field, (b) the interactions between the coarse-grained spins that depend only on the distance between blocks, and (c) an approximation of the entropy (using Stirling’s approximation): $F=U-TS$.

(4)$\begin{array}{}& \begin{array}{c}ℱ\left(ɸ\right)=-\stackrel{\left(a\right)}{\overbrace{B\int ɸ\left(x\right)dx}}-\stackrel{\left(b\right)}{\overbrace{\frac{1}{2}\int \int J\left(\right|x-y\left|\right)ɸ\left(x\right)ɸ\left(y\right)dxdy}}\\ -k_{B}T\underset{c}{\underbrace{\left(\int [1+ɸ(x\left)\right]\ln (1+ɸ(x\left)\right)dx+\int [1-ɸ(x\left)\right]\ln (1-ɸ(x\left)\right)dx\right)\mathrm{.}}}\end{array}\end{array}$

After only a few steps, and a generalization to dimension d, one ends up with the Landau-Ginzburg-Wilson (LGW) Hamiltonian:

(5)$\begin{array}{}& H=\int d^{d}x\left[\frac{1}{2}{\zeta }^2|\nabla ɸ(x){|}^2+\frac{1}{2}\theta |ɸ\left(x\right){|}^2+\frac{1}{4!}\eta \left|ɸ\right(x){|}^4\right]\mathrm{.}\end{array}$

Note, however, that the LGW Hamiltonian is not the Ising model effective Hamiltonian. This latter object is more complicated; however, it is demonstrated in Binney et al. (Reference Binney1992, app. K)—and is plausible given its derivation—that equation (5) is a good approximation to a truncated form of the Ising Hamiltonian near the critical point.

The construction of equation (5) is quite different from equations (1) and (2). It builds on these models but abstracts from them. More details can be found in (e.g.) Fisher (Reference Fisher1974). There he demonstrates the field-theoretic methods that allow one to derive expressions for the critical exponents as functions of d and n. See equation (6) for the first few terms of the exponent α; this will give a value for various universality classes. This derivation depends on the functional integration of the LGW Hamiltonian over all functions ɸ(x).

(6)$\begin{array}{}& \alpha =\frac{4-n}{2(n+8)}(4-d)+\frac{{(n+2)}^2(n+28)}{4{(n+8)}^3}{(4-d)}^2+\dots \mathrm{.}\end{array}$

Crucially, it can be shown that the addition of certain terms to the LGW Hamiltonian will lead to irrelevant contributions; these do not affect the values for critical exponents describing the approach to a given fixed point. In Binney et al. (Reference Binney1992, chap. 14), the criteria for relevance and irrelevance are derived. An operator O_p is relevant if $p-d(p-2)/2>0$ and irrelevant if $p-d(p-2)/2<0$, where d is the dimension of the system under investigation and p is a function of the exponent of ɸ.Footnote ¹⁵

This serves to establish that for the LGW Hamiltonian, for $d=3$, any O_p with $p>6$ will be irrelevant at the fixed point.Footnote ¹⁶ This is an important result for the discussion in the remainder of this article. Its generality depends on the justification for the applicability of the LGW Hamiltonian to various models. As we will see in what follows, this will depend in part on the OP assigned to each member of each universality class.

The theory behind this result is relatively involved, but the idea is simple: the LGW Hamiltonian is renormalizable. This means that applying an RG transformation to the Hamiltonian will not add terms that cannot be absorbed into the parameters ζ, θ, η in equation (5). At the fixed point—which describes the location of the critical phase transition—renormalizable Hamiltonians are scale invariant: they are unaffected by RG transformations.Footnote ¹⁷ Thus, at the fixed point, the only elements that contribute to behavior are those in the renormalizable Hamiltonian; all irrelevant terms that may be added to that Hamiltonian will not contribute there. By contrast, the Hamiltonians employed in the real-space approach are not renormalizable, and the description of their behavior near the critical point relies on the imposition of scale invariance by truncating the Hamiltonian after each iteration of the RG transformation; as such, criteria for relevance and irrelevance of additions to those Hamiltonians cannot be specified in such generality.

The next section will explore the extent to which each RG approach can be considered to explain the universality of critical phenomena.

4.1. Field-Theoretic RG

We have sound theoretical reasons to think that the LGW Hamiltonian represents a wide range of physical systems. The physical analysis behind this claim is the renormalizability of the LGW Hamiltonian and the demonstration that certain classes of operators are irrelevant, as discussed above. Binney et al. (Reference Binney1992, 366) express this as follows: “to the accuracy of our calculation we have shown that any three-dimensional physical system whose Hamiltonian can be written as an even functional of a one-component scalar field should have the same critical behavior as the Landau-Ginzburg model.” Thus, we need to show, for each system of interest, that its order parameter (OP) can be written as a one-component scalar field. Paying close attention to the OP of each system will also ground the various assignations of systems to different universality classes. The OP accounts for the symmetry group (i.e., the n of the n-vector model) and the dimensionality. Defining the OP for a condensed matter system is not a straightforward process; it depends subtly on the kind of phase transition the systems undergoes and on which macroscopic features change at such a phase transition. Footnote 13 provides some examples of OPs.

The central operators of the LGW Hamiltonian together with the OPs represent the common aspects of the various systems in the same class at the critical point. Such systems are distinguished, at most, by operators that are, by RG arguments, irrelevant to the behavior at the critical point. Thus, the common aspects are sufficient for the systems’ exhibiting universal behavior, and we have a common-type explanation. Furthermore, this, in principle, justifies Abs_E: the various systems are represented by Hamiltonians distinguished by irrelevant operators, and the flows of the distinct systems converge at the fixed point.

One might have the following worry: once the OP has been specified, common representation is assured for the different systems in the same class. As such, specification of the OP might be said to do all the explanatory work.

I do not think that this argument goes through. I claimed in section 2 that Com_E explanations must identify common features and show that they are sufficient for common behavior.Footnote ²⁰ The OP together with the LGW Hamiltonian represent the common features of the different systems, but the field-theoretic RG framework is required to demonstrate that such operators are sufficient for the universal behavior and that all other aspects of the descriptions are irrelevant at the critical point. Part of this explanation does lie in the matching of OPs to systems, but it also relies on the fact that all the nonrenormalizable operators, which otherwise represent the differences between systems’ behavior away from the critical point, are irrelevant at the critical point. This is nontrivial; in some contexts the irrelevance of certain details may be obvious, and thus may remain implicit, but where it is not obvious demonstrations of irrelevance play an important role in the explanation. It is also worth noting that, even were this point not granted, the OP does not play a role in the real-space approach, and, as such, it is only through the field-theoretic approach that we are able to identify the common features of universally behaving systems.

A further concern with this explanation rests on the observation that the OP and LGW Hamiltonian correspond to large-scale features of the systems of interest; thus, our explanation is not tied in detail to systems’ microscopic heterogeneities. So long as we have reason to believe that our model represents the different systems that exhibit universality, the explanation need not include all the details of each system; it has long been acknowledged that good explanations may abstract from underlying details (see, e.g., Woodward Reference Woodward2003).Footnote ²¹ We do have good reasons to link field-theoretic models to the physical systems of interest; chiefly, a microscopic justification of the choice of OP, discussion of irrelevant operators (continued below), and the empirical observation of approximate scale symmetry in physical systems at and near the critical point. Further justification akin to that provided by the derivation of the LGW Hamiltonian from the Ising model (see sec. 3.2) may provide a bottom-up account of critical phenomena, although this claim is subject to worries about the infinite idealizations invoked (see n. 11). I assume that unreduced explanations can be shown to be adequate before such questions of reduction are settled.

That we cannot, in practice, write down the irrelevant operators may be worrying. It may be thought that evidence is scant for the claim that such operators indeed represent the heterogeneities that distinguish physical systems away from the critical point.

I hope partially to alleviate such worries by briefly considering crossover theory.Footnote ²² The theoretical description of crossover tells us that in some cases we may derive a correspondence between certain operators and the details of physical systems. By showing that certain operators may represent the details that distinguish systems away from the critical point, this ought to bolster the analysis of the field-theoretic RG explanation presented just above.

Systems undergoing crossover display critical behavior characteristic of some universality class as they approach T_c, but under repeated iterations of the RG transformations (read: as the temperature moves closer to T_c) they deviate from that behavior and cross over to a different universality class. For example, a system near the Heisenberg fixed point may have an additional relevant operator; we might thus define a Heisenberg-type ($n=3$) Hamiltonian including operators for isotropic and anisotropic couplings. It turns out that a system so described will cross over to Ising-type behavior; for further details, see Fisher (Reference Fisher1974) and Cardy (Reference Cardy1996).

Crossover theory is empirically successful, and such successes are predicated on deriving a relationship between operators and the details of physical systems. Although the operators for which such a correspondence can be shown are not irrelevant—these are relevant or marginally relevant operators—such correspondences help to establish that operators may play the required role in the field-theoretic RG explanation of universality.

For most instances of universality, we have yet to discover irrelevant operators that are physically interpreted as representing those features that distinguish multiple members of the same class. The phenomenon of crossover does suggest that such differences can be modeled. This in turn justifies the claim that the field-theoretic approach explains the universality of critical phenomena: it identifies shared features in our systems of interest (represented by the LGW Hamiltonian) sufficient to predict their display of the critical exponents. The expanded Hamiltonians with the irrelevant operators, together with the flow induced by the RG, may be depicted as in figure 1 and thus explain universality.

Is universality thus explained? I claimed above that one way to explain universality is by constructing a map between convergent flows and real physical systems (akin to Abs_E). However, as noted above, no map can be explicitly constructed in this case since we do not know how to write down the irrelevant operators for the various systems of interest. Thus, an explanation of type Abs_E with the necessary link to Com_Q may be found only in principle; in practice Com_E goes through.

4.2. Real-Space RG

The real-space RG may be understood by appeal to simple diagrams like that in figure 2. It is thus unfortunate that, as I argue in this section, the explanation provided by the real-space RG is inadequate.

Figure 2. Single application of the real-space RG where a block of spins is replaced by a single larger spin. From Fisher (Reference Fisher1998); reproduced with permission of the American Physical Society via Copyright Clearance Center.

The real-space approach allows for the derivation of critical exponents consistent with empirical observation for various models. Furthermore, we have an account of relevance and irrelevance and the claim that “in general, for fixed points describing second-order critical points, there are two relevant parameters: the temperature and the field conjugate to the order parameter (for the magnet it is the magnetic field)” (Cheung Reference Cheung2011, 51). Why is this explanation of universality not sufficient?

In section 2 I categorized a few options for how universality may be explained. I claimed that Spec_E was insufficient but Com_E or a supplemented Abs_E could do the job. I think that neither latter option is live in the real-space case. This is because the mathematical model employed does not have the tools to represent systems other than the archetypal system for each universality class; the problem is that the Ising model is designed specifically for magnetic systems. In what follows, I explore a number of ways it might be adapted to represent other members of its class: Kadanoff’s proposal and the lattice-gas analogy. I conclude that none of these adaptations allows for a general RG explanation of universality: insofar as the correct critical exponents are predicted, this is done for individual systems and a common-type explanation is unavailable.

While the field-theoretic approach makes use of a Hamiltonian derived from the Ising model, the Hamiltonian used in that approach is renormalizable. As such, it includes a scale-invariant core at the critical point that represents a range of different systems. That is how Com_E is achieved: by showing what is in common and the general demonstration that all possible distinguishing features are irrelevant.

The real-space approach does not make use of a renormalizable Hamiltonian, nor does it have a formalism that establishes generally that the Hamiltonians apply across a wide range of systems. Likewise, there seems to be little hope that a picture like that in figure 1 can be shown to correspond to distinct systems in the same universality class having convergent flows. The reason is that the real-space approach relies on Ising-type models, which do not represent the other systems in the same class—the lattice-gas analogy provides a possible counterargument, discussed below. To claim that Ising models in fact represent all the different systems would be to assume universality rather than to explain it. While one may be able to demonstrate that each system in the same universality class shares certain features (e.g., symmetry and dimensionality), the real-space approach does not provide the resources to demonstrate that such commonality leads to common behavior in general: we do not have an argument available on the real-space approach that common symmetry and dimensionality is sufficient for universality.Footnote ²³ Thus, commonality sufficient for common behavior cannot be adduced for a common-type explanation, nor can an Abs_E-Com_Q link be established. It is only by recourse to the field-theoretic approach that a full explanation of universality is available. Three responses to these claims ought to be considered.

First, it could be noted that irrelevant couplings are discussed in the real-space context, and we know that only a few relevant couplings determine the critical exponents; thus, perhaps these relevant couplings provide the Com_E explanation. But the model is still tied to the details of the system it was created to represent. Thus, the irrelevant couplings are those aspects of that system that will not affect its critical exponents. To show that some aspects of a given system are irrelevant to its behavior in a given context is quite different from showing that all systems with the same relevant properties (and with different properties otherwise) will display the behavior. The former, system-relative claim is established by the real-space RG, but the latter more general claim is not. The success of the field-theoretic RG explanation is due to the fact that we can categorize operators as relevant or irrelevant quite generally. In the real-space RG, potential couplings are only categorized for a given model.

Second, since the blocking transformation is tailored specifically to the behavior of systems at criticality, the transformation itself might represent the common features for Com_E. The transformation is constructed so as to mirror the self-similarity of such systems, and its application to systems at criticality is thus justified. This would mean that the real-space RG explains universality by appeal to the fact that all these systems have some commonality (i.e., their self-similarity) that justifies the use of these techniques to derive their critical exponents.

The problem with this claim is that the real-space RG approach does not simply derive the exponents from the blocking transformations. In fact, such exponents are derived by applying the blocking transformation and then truncating the Hamiltonian so that it will retain its original form. As such, the original Hamiltonian significantly determines the application of the real-space RG and the exponents derived. It is thus not quite right to claim that self-similarity is a common feature sufficient for universality. Moreover, if one considers the phenomenon of crossover, it is clear that this depends specifically on the terms in the Hamiltonian of each system. I raise this here as evidence that the initial Hamiltonian is crucial to the real-space derivation and explanation. As such, the appeal to a common blocking RG transformation cannot provide a common-type explanation.

Third, for the remainder of this section I consider a pair of specific elaborations of the real-space RG approach that provide limited explanations of universality. The first demonstrates that critical exponents derived on the real-space approach do not depend on certain couplings in the Ising model. The second—the lattice-gas analogy—relates liquid gas to magnetic models. Neither, I argue, provides a general account of the range of classes and systems that behave universally. Such general accounts are currently unavailable.

Batterman (Reference Batterman2017b) highlights an argument found in Kadanoff (Reference Kadanoff and Green1971) to the effect that one can introduce a parameter λ into the free energy function for the Ising model, and it can be demonstrated that the critical exponents do not depend on the value of this parameter. In Kadanoff’s example this parameter corresponds to the ratio of the couplings for nearest-neighbor to next-nearest-neighbor models. As such, we may be assured of a particular kind of generality of the Ising model representation. The independence of the critical exponents from such parameters is, however, insufficient to establish the requisite generality for the real-space approach. If a similar argument were available for the variation between a liquid-gas system and a magnetic system, then this explanation would be far more convincing; such an argument has not yet been presented.

The lattice-gas analogy exemplifies a possible mapping that may provide reasons to view the Ising model as representing liquid-gas systems in addition to magnetic systems. If this succeeds it would justify a common-type explanation for the limited case at hand. However, it would not justify claims to an RG explanation of universality because the mapping is not one sourced in the relevance and irrelevance criteria of the RG. Rather, it would provide a distinct explanation of universality. Furthermore, it is not generalizable to other members of the various universality classes. In this respect the field-theoretic approach outdoes the real-space approach, even with the lattice-gas analogy.

The lattice-gas model is summarized as follows:

Consider the Hamiltonian

$\begin{array}{}& H=\mathrm{-4}J\sum _{〈\mathrm{ij}〉}{\rho }_i{\rho }_j-\mu \sum _i{\rho }_i,\end{array}$

where ${\rho }_i=0$, 1 depending if the site is empty or occupied, and μ is the chemical potential. If we define ${\sigma }_i=2{\rho }_i-1$, we reobtain the Ising-model Hamiltonian with $B=2\mathrm{qJ}+\mu /2$, where q is the coordination number of the lattice. Thus, for $\mu =\mathrm{-4}qJ$, there is an equivalent transition separating the gas phase for $T>T_c$ from a liquid phase for $T<T_c$.

(Pelissetto and Vicari Reference Pelissetto and Vicari2002, 554)

This mapping is clear enough but merely shifts the burden of justification. As Pelissetto and Vicari acknowledge, “The lattice gas is a crude approximation of a real fluid” (Reference Pelissetto and Vicari2002, 554). Their justification for this approximation is empirical: “Nonetheless, the universality of the behavior around a continuous phase-transition point implies that certain quantities, e.g., critical exponents … are identical in a real fluid and in a lattice gas, and hence in the Ising model” (554). The model is provided the following rationale in the context of its original presentation: “Theoretically speaking, by making the lattice constant smaller and smaller one could obtain successively better approximations to the partition function of the real gas” (Lee and Yang Reference Lee and Yang1952, 412). Although gases may often be modeled as continuum gases, this is itself an idealization that requires a physical justification. Furthermore, the problem with the application of the Ising model to a physical gas is not that the Ising model is discretized—we expect gases to contain finitely many particles. Rather one should be concerned that the molecules have more degrees of freedom available to them than the components of uniaxial magnets. This is not to suggest that idealized models are intrinsically problematic; rather, I am skeptical that this physical justification of the analogy is sufficient to explain universality.

If we were to accept this justification of the lattice-gas model, further questions would be raised, for magnets and liquid-gas systems do not display the same behavior away from the critical point. It is precisely because the systems behave so differently much of the time that universality is startling. Thus, even if the lattice-gas analogy gave a good account of liquid-gas systems at criticality, additional details are needed to explain the limited applicability away from criticality.

Do we have an explanation why these different systems undergo similar behavior near the critical point? It turns out, and this is surprising and interesting, that uniaxial magnets and fluids have some behavior that is approximately described by the same model, namely, the Ising model. But this result is a consequence of careful mapping between the systems; it was not an RG result. The RG was used for the derivation of the critical exponents from the models, not in the justification of the applicability of the models to various physical systems. The lattice-gas analogy provides an account of what is in common between systems with diverse microphysics by mapping the Ising model to a liquid-gas model. However, no generalized real-space RG explanation is available that tells us why all the members of the same class have identical behavior.

On the field-theoretic approach we were able to identify common features and demonstrate their sufficiency for common behavior—that is how Com_E was achieved. On the real-space approach, insofar as we can derive the same critical exponents for different systems, we still have no characterization of their common features that would serve to demystify universality. I do not purport to rule out real-space explanations in principle: if the physics were sufficiently developed to allow real-space derivations for each member of each model, and one could use these to adduce common features, then universality would likely be explained. Alternatively, a Kadanoff-type approach might be extended for a more generalized account of universality. In section 2 I argued that Spec_E and Com_E explanations were to be distinguished: the real-space RG currently only provides Spec_E for certain models but is insufficient for Com_E.

On neither approach to the RG do we have a fully worked out bottom-up explanation of universality. Nonetheless, I claim that the field-theoretic approach is adequate just because we have a generalized account of universality on that approach, which provides conditions that, if satisfied, predict common behavior across a class of systems.