The $\epsilon$-Expansion: Perturbation Around the Upper Critical Dimension

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Hello! In our last session, we jumped right into the action and calculated the critical exponent $ν$ to first order in $ϵ = 4 - d$ . While that calculation showcased the power of the renormalization group (RG), it's crucial to understand the foundation upon which it's built.

Today, we will take a step back to explore the core logic of the $ϵ$ -expansion. Our goal is to answer the question: Why is this a valid, controlled, and powerful method for studying critical phenomena? We will see how this ingenious idea, pioneered by Kenneth Wilson and Michael Fisher, turns a seemingly intractable strong-coupling problem into a solvable perturbative one by treating the dimensionality of space itself as a variable.

1. The Need for a New Approach: The Limits of Mean-Field Theory

As you know from your background and our earlier discussions, mean-field theories provide a first, qualitative picture of phase transitions. However, they neglect fluctuations, which become dominant near a critical point. The accuracy of mean-field theory turns out to be highly dependent on the spatial dimension, $d$ .

Statistical Field Theory - DAMTP

To set the stage, let's first formalize the concept of mean-field theory's dimensional dependence. The following excerpt from David Tong's lecture notes on Statistical Field Theory clearly summarizes this idea and introduces the concept of an 'upper critical dimension'.

Please read section 1.2.3 'Validity of Mean Field Theory' (pages 16-17). Pay close attention to the bullet points that describe how the theory's validity changes with dimension d. Then, glance at the table of critical exponents on page 18, which starkly contrasts the mean-field predictions with the more accurate values for d=2 and d=3.

As the notes explain, there is a key concept:

The Upper Critical Dimension ( $d_{c}$ ): For a given system, this is the dimension at and above which mean-field theory gives the correct critical exponents.

An intuitive way to understand this is to consider the effect of fluctuations. In higher dimensions, a spin has many more neighbors, and the fluctuations from any single neighbor have a smaller effect on its overall state. The spin experiences something closer to the average magnetization, which is the core assumption of mean-field theory. For the $ϕ^{4}$ theory, which models systems like the Ising model, the upper critical dimension is $d_{c} = 4$ .

2. The Renormalization Group View of $d_{c} = 4$

The RG framework provides a precise way to understand why $d = 4$ is special. Recall the simplified RG flow equation (the beta function) for the quartic coupling $u$ under a change of length scale $l = ln b$ :

\frac{d u}{d l} = β (u) \approx (4 - d) u - C u^{2}

Here, $C$ is a positive constant that comes from integrating out the high-momentum modes in a one-loop calculation. Let's analyze the behavior of this equation near the Gaussian Fixed Point (GFP) at $u = 0$ :

For $d > 4$ : The term $(4 - d)$ is negative. If we start with a small positive $u$ , $\frac{d u}{d l}$ is negative, driving the coupling $u \to 0$ as we zoom out (increase $l$ ). The interaction is an irrelevant operator. The physics is governed by the non-interacting GFP, and mean-field theory is exact.
For $d = 4$ : The linear term vanishes. The flow equation becomes $\frac{d u}{d l} = - C u^{2}$ . The coupling still flows to zero, but much more slowly. The interaction is marginal. This is the borderline case.
For $d < 4$ : The term $(4 - d)$ is positive. For a small positive $u$ , $\frac{d u}{d l}$ is now positive, driving the coupling away from the GFP. The interaction is a relevant operator. The GFP is unstable.

This last point is the crux of the problem. For the physically interesting cases of $d = 2$ and $d = 3$ , the RG flow tells us that even a small interaction will grow, driving the system away from the simple Gaussian theory. Our simple perturbative approach fails.

3. The Wilson-Fisher Insight: Finding a New Fixed Point

The failure of the GFP for $d < 4$ seems like a dead end. However, Kenneth Wilson and Michael Fisher had a revolutionary idea. Instead of giving up, they asked: If the flow runs away from the GFP, where does it go? Perhaps it flows towards a new, non-trivial fixed point at some $u^{*} \neq = 0$ .

How can we find this new fixed point? Let's treat the dimension $d$ not as a fixed integer, but as a continuous parameter. Specifically, let's consider a dimension infinitesimally close to 4:

d = 4 - ϵ

where $ϵ$ is a small, positive parameter. Substituting this into our beta function gives:

β (u) = ϵ u - C u^{2}

Now, we search for fixed points by setting $β (u^{*}) = 0$ :

u^{*} (ϵ - C u^{*}) = 0

This equation has two solutions:

$u^{*} = 0$ : The unstable Gaussian Fixed Point.
$u^{*} = ϵ / C$ : A new, non-trivial fixed point! This is the Wilson-Fisher Fixed Point (WFFP).

This is a profound result. By moving the dimension slightly below 4, a new interacting fixed point emerges.

4. A Controlled Perturbative Expansion

The most important feature of this new fixed point is that its location, $u^{*} = ϵ / C$ , is small when $ϵ$ is small.

This is the central logic of the $ϵ$ -expansion:
Because the interacting fixed point is located at a small value of the coupling $u^{*} \sim O (ϵ)$ , we can use $u^{*}$ as a small parameter to study the properties of this fixed point. This means we can calculate physical quantities, like critical exponents, as a power series in $ϵ$ .

We have successfully turned a non-perturbative problem (a flow to strong coupling) into a controlled, perturbative one, where the small expansion parameter is $ϵ = 4 - d$ .

[PDF] Lecture 12: ε-expansion and Wilson-Fisher fixed point

Let's see this logic applied to the full flow equations. The following lecture notes by Naoki Kawashima provide a concise mathematical argument for why u* is of order ε and the related coupling t* is of order ε².

Please review slide 5, titled 'ε-expansion'. Focus on the flow equations for t and u, the definition ε = 4-d, and the crucial argument in the parentheses: 'By considering the order in ε, we see t*=O(ε²) and u*=O(ε).'

Test your understanding!

Why is the $ϵ$ -expansion considered a "controlled" expansion, whereas a naive expansion in the coupling $u$ for a system with $d = 3$ is not?

Show answer

A naive expansion in $u$ for $d = 3$ is uncontrolled because the RG flow shows that $u$ is a relevant parameter, meaning it grows larger and larger as we approach the critical point (zoom out to large length scales). You cannot do a perturbative expansion in a parameter that becomes large.

The $ϵ$ -expansion is "controlled" because it identifies a new fixed point, the WFFP, where the coupling u* is fixed at a small value proportional to $ϵ$ . The expansion is performed around this stable fixed point, and the expansion parameter $ϵ$ is assumed to be small, ensuring the corrections are also small and the series converges (or is at least a valid asymptotic expansion).

5. Visualizing the Logic

We can visualize this concept in two ways. First, by plotting the beta function $β (u)$ for a small, positive $ϵ$ :

Beta Function and Fixed Points in Epsilon Expansion — This graph shows the beta function `β(u)` versus the coupling `u` for a small, positive ε. There are two fixed points where `β(u)=0`: the unstable Gaussian fixed point at `u=0` and the stable Wilson-Fisher fixed point at `u* > 0`. The arrows indicate the direction of the RG flow, away from the unstable point (`u<u*` flows right, `u>u*` flows left) and towards the stable one.

Second, we can see the flow in the larger parameter space. The WFFP acts as an attractor for the RG flow on the critical surface.

Renormalization Group Flow Diagram — This RG flow diagram illustrates the situation in the parameter space. For `d < 4` (ε > 0), the Gaussian fixed point (GFP) at the origin is unstable for perturbations along the `g` (or `u`) direction. The flow lines on the critical surface are driven away from the GFP and towards the stable Wilson-Fisher fixed point (WFFP), which correctly describes the critical behavior of the system.

This flow towards a common fixed point is the RG explanation for universality: systems with very different microscopic details (different starting points in the parameter space) will flow to the same fixed point and thus exhibit the same critical exponents.

Given your QFT background, you might see a parallel with dimensional regularization. In both techniques, the dimension $d$ is treated as a continuous variable to manage problematic behavior (divergences in QFT, strong coupling here). The $ϵ$ -expansion takes this a step further by using the analytic continuation in $d$ to uncover new, non-perturbative physics.

Conclusion

In this lesson, we have unpacked the theoretical justification for the $ϵ$ -expansion, the method we used in our previous lesson to calculate $ν$ .

Key Takeaways:

Mean-field theory is only exact at or above the upper critical dimension, which is $d_{c} = 4$ for $ϕ^{4}$ theory.
From an RG perspective, the $ϕ^{4}$ interaction is relevant for $d < 4$ , making the non-interacting Gaussian fixed point unstable.
By setting $d = 4 - ϵ$ , Wilson and Fisher showed that a new, stable, and interacting Wilson-Fisher fixed point appears at a coupling $u^{*} \sim O (ϵ)$ .
The smallness of $u^{*}$ for small $ϵ$ allows for a controlled perturbative expansion in $ϵ$ to calculate critical exponents and other universal quantities.

Preview of the Next Lesson:

Now that we have a solid conceptual understanding of why the $ϵ$ -expansion works, we are well-equipped to apply it further. In the next lesson, we will calculate the remaining key critical exponents, $η$ and $γ$ , to first order in $ϵ$ . This will require us to analyze how the scaling of the field $ϕ$ itself is modified at the Wilson-Fisher fixed point.