Classifying Operators by Relevance to Fixed Points

The Physical Meaning of Relevance

Let's begin with the core physical idea. When we perform an RG step, we are "zooming out" and looking at our system at a larger length scale, or equivalently, at a lower energy scale. The question we want to answer is: how does the strength of a particular interaction term in our Hamiltonian change as we zoom out?

• An interaction that becomes stronger as we flow to lower energies is called relevant. It has a dominant effect on the long-distance physics.
• An interaction that becomes weaker is called irrelevant. Its effects are washed out at large scales, and it only modifies the short-distance details.
• An interaction whose strength does not change (at least at the simplest level of analysis) is called marginal. Its fate requires a more delicate investigation.

The following video provides an excellent conceptual overview of this idea, framing it in the context of identifying which interactions are important for low-energy physics.

This picture of flowing coupling constants is key. A perturbation to a Hamiltonian at a fixed point is generated by adding a new operator, multiplied by a small coupling constant. The RG flow then acts on this coupling. If the flow drives the coupling away from zero, the operator is relevant. If it drives it towards zero, the operator is irrelevant.

The following reading from David Tong's lecture notes provides a crisp definition of these concepts and explains how they relate to the structure of the "theory space" we navigate with the RG.

From Eigenvalues to Scaling Dimensions

In the previous lesson, we saw that the stability of a fixed point is governed by the eigenvalues of the linearized RG transformation matrix. Let's connect this to our new classification.

Recall that for a discrete RG step with length rescaling factor $b$, a small perturbation $\delta g$ from a fixed point transforms as $\delta g^{\prime }\approx \lambda \delta g$.

• Relevant: $|\lambda |>1$. The perturbation grows.
• Irrelevant: $|\lambda |<1$. The perturbation shrinks.
• Marginal: $|\lambda |=1$. The perturbation's fate is decided by higher-order terms.

It is conventional and extremely useful to parameterize the eigenvalue as $\lambda =b^y$. The exponent $y$ is called the scaling dimension of the coupling $g$. The classification becomes:

• Relevant if $y>0$.
• Irrelevant if $y<0$.
• Marginal if $y=0$.

The scaling dimension $y$ quantifies how fast a coupling grows or shrinks with scale. The video below makes this connection between eigenvalues and scaling dimensions explicit.

Dimensional Analysis and Operator Classification

While finding the eigenvalues of the full RG transformation can be difficult, we can often determine the classification of an operator near a simple fixed point using dimensional analysis. This approach is particularly insightful around the Gaussian fixed point (the free theory), where interactions are turned off.

Consider an interaction term in the free energy (or action) in $d$ spatial dimensions of the form:

Here, ${\mathcal{O}}_n$ is an operator (like ${\varphi }^n$) and $g_n$ is its coupling constant. For the action $S_{int}$ to be dimensionless (as it appears in $e^{-S}$), the coupling $g_n$ must have dimensions that compensate for $[d^{d}x]$ and $[{\mathcal{O}}_n]$.

Under an RG transformation where lengths are rescaled $x\to x/b$, the coupling transforms to maintain this balance. This "engineering dimension" of the coupling often determines its relevance. The key insight is that an operator's scaling dimension, which determines its relevance, is directly related to its mass dimension from this simple analysis, at least near the Gaussian fixed point.

Let's see how this works for our familiar ${\varphi }^4$ theory. The field $\varphi$ has mass dimension $(d-2)/2$.

• The mass term is $\int d^{d}xr{\varphi }^2$. The operator ${\varphi }^2$ has dimension $d-2$. The coupling $r$ must have dimension 2. This corresponds to a positive scaling dimension $y_r=2$, so the mass operator ${\varphi }^2$ is always relevant.
• The interaction term is $\int d^{d}xg{\varphi }^4$. The operator ${\varphi }^4$ has dimension $2(d-2)$. The coupling $g$ must have dimension $4-d$. This corresponds to a scaling dimension $y_g=4-d$.

This simple result is incredibly powerful. The relevance of the ${\varphi }^4$ operator depends on the dimension of space!

• If $d>4$, then $y_g<0$. The operator is irrelevant. The interaction weakens at long distances, and the theory flows towards the free Gaussian fixed point.
• If $d<4$, then $y_g>0$. The operator is relevant. The interaction strengthens, driving the system away from the Gaussian fixed point towards an interacting one (the Wilson-Fisher fixed point).
• If $d=4$, then $y_g=0$. The operator is marginal. Its relevance is not determined by simple scaling and depends on quantum corrections, which lead to a slow, logarithmic running of the coupling.

The following text formalizes this dimensional analysis argument and presents a useful table of examples.

A Note on Marginal and Irrelevant Operators

It's tempting to think that irrelevant operators can be completely ignored. This is often true, but not always.

1. Marginal Operators: When an operator is marginal by dimensional analysis (like ${\varphi }^4$ in $d=4$), its fate is decided by the sign of its beta function, $\beta (g)=\frac{dg}{d(\ln L)}$. If quantum corrections lead to $\beta (g)>0$, the coupling slowly grows, and the operator is called marginally relevant. If $\beta (g)<0$, it's marginally irrelevant. For ${\varphi }^4$ theory, it turns out to be marginally irrelevant, meaning the theory becomes trivial (free) at the lowest energies.
2. Dangerously Irrelevant Operators: Sometimes an operator can be irrelevant (its coupling flows to zero), but it still has a crucial impact on physical quantities, often because its coupling appears in the denominator of an expression. Such operators are called dangerously irrelevant. This is a beautiful example of the subtlety involved in the "craft" of theoretical physics, showing that you can't always discard terms just because they are small.

Conclusion

In this lesson, we have attached deep physical meaning to the stability analysis of RG fixed points. By classifying operators based on how their couplings behave under the RG flow, we have developed a powerful predictive tool.

Key Takeaways:

• Operators are classified based on their behavior under RG flow towards the infrared (long distances). Relevant operators grow, irrelevant operators shrink, and marginal operators are borderline cases.
• This classification is determined by the scaling dimension of the operator's coupling, which in turn is related to the eigenvalues of the linearized RG transformation.
• Near the Gaussian fixed point, an operator's relevance can often be determined by simple dimensional analysis. For ${\varphi }^4$ theory, the dimension $d=4$ is special: it is the upper critical dimension above which the interaction is irrelevant.
• Irrelevant does not always mean "ignorable." Dangerously irrelevant operators can still have a profound effect on physical predictions.

The classification of operators is the key that unlocks the mystery of universality. The fact that, near a critical point, there are typically only a few relevant operators and infinitely many irrelevant ones explains why the long-distance physics is insensitive to most of the microscopic details of a system. In our next lesson, we will explore this profound consequence in detail.