Operators in Momentum-Space RG

Hello! Let's dive into our next lesson on the Renormalization Group.

In our last session, we took a major step by identifying the Gaussian and Wilson-Fisher fixed points from the RG flow equations for ${\varphi }^4$ theory. We saw that for dimensions $d<4$, the interacting Wilson-Fisher fixed point (WFFP) is the stable one (in the sense of being an attractor on the critical surface), while the non-interacting Gaussian fixed point (GFP) is unstable. This hinted that the WFFP is the key to understanding critical phenomena.

The terms 'relevant' and 'irrelevant' came up when we discussed the stability of these fixed points in the directions of the ${\mu }^2$ and $g$ couplings. Today, we will formalize and generalize this crucial idea. We will move beyond just ${\varphi }^2$ and ${\varphi }^4$ to consider a theory with an infinite number of possible interaction terms. Our goal is to explain the concept of relevant, irrelevant, and marginal operators in the context of momentum-space RG. This classification scheme is the heart of the RG and the theoretical foundation for the concept of universality.

The Big Picture: RG Flow in Theory Space

Imagine a general free energy functional, not just with ${\varphi }^2$ and ${\varphi }^4$ terms, but with every possible term consistent with the symmetries of the system (e.g., ${\varphi }^6$, $(\nabla \varphi {)}^2{\varphi }^2$, $({\nabla }^2\varphi {)}^2$, etc.).

$F[\varphi ]=\int d^{d}x\left[\frac{1}{2}(\nabla \varphi {)}^2+{\sum }_i{g}_i{O}_i(\varphi ,\nabla \varphi ,...)\right]$

Here, the $O_i$ are the various interaction terms, which in the language of field theory are called operators, and the $g_i$ are their corresponding coupling constants. The state of our theory is defined by a point in an infinite-dimensional parameter space—the "theory space"—spanned by all possible couplings $\{g_i\}$.

The RG transformation induces a flow in this space. As we integrate out short-distance details and rescale, the couplings $\{g_i\}$ evolve. A fixed point is a special point $\{g_i^{\ast }\}$ where this flow stops, corresponding to a scale-invariant theory.

A Formal Definition via Linearization

As the reading suggests, we can classify the operators $O_i$ by analyzing the RG flow in the immediate vicinity of a fixed point ${\vec{g}}^{\ast }$. Let's make this mathematically precise.

Suppose we perturb the theory by moving slightly away from the fixed point: $\vec{g}(0)={\vec{g}}^{\ast }+\delta \vec{g}$. The beta functions, which describe the flow, can be linearized:

$\frac{d{g}_i}{dl}={\beta }_i(\vec{g})⟹\frac{d(\delta g_i)}{dl}={\sum }_j{\frac{\partial {\beta }_i}{\partial g_j}|}_{{\vec{g}}^{\ast }}\delta g_j={\sum }_j{M}_{ij}\delta g_j$

This is a system of linear differential equations for the perturbations, $\frac{d(\delta \vec{g})}{dl}=M\delta \vec{g}$, where $M$ is the stability matrix we encountered last lesson, but now potentially infinite-dimensional.

The key step is to find a basis of operators (which are often linear combinations of simple operators like ${\varphi }^n$) that are eigenvectors of this transformation. In this basis of "eigenoperators," the flow equations decouple. For the coupling $u_k$ of an eigenoperator ${\mathcal{O}}_k$, the flow equation becomes:

$\frac{d{u}_k}{dl}={\lambda }_k{u}_k$

where ${\lambda }_k$ is the corresponding eigenvalue. The solution is simply $u_k(l)=u_k(0)e^{{\lambda }_{k}l}$, where $l=\ln b$ is the logarithmic length scale factor. This leads to the fundamental classification:

• Relevant Operator: If ${\lambda }_k>0$, the coupling $u_k$ grows exponentially as we flow to larger scales ($l\to \infty$). A small perturbation by this operator is amplified and drastically alters the long-distance physics. The system is driven away from the fixed point.
• Irrelevant Operator: If ${\lambda }_k<0$, the coupling $u_k$ decays to zero. The effect of this operator vanishes at large scales. The RG flow is attracted towards the fixed point along this direction.
• Marginal Operator: If ${\lambda }_k=0$, the coupling does not change at the linear level. We must examine higher-order terms in the beta function to determine if the operator is marginally relevant (coupling eventually grows), marginally irrelevant (coupling eventually shrinks), or exactly marginal (coupling can be varied continuously, defining a line or surface of fixed points).

The Physical Picture: Scaling Dimensions

The eigenvalues ${\lambda }_k$ have a deep physical meaning related to the scaling dimension of the operator. Recall that the RG procedure involves rescaling space $x^{\prime }=x/b$ and the field ${\varphi }^{\prime }(x^{\prime })=b^{{\Delta }_{\varphi }}\varphi (x)$, where ${\Delta }_{\varphi }$ is the scaling dimension of the field itself.

An operator ${\mathcal{O}}_k$ is said to have a scaling dimension ${\Delta }_k$ if it transforms as ${\mathcal{O}}_k^{\prime }(x^{\prime })=b^{{\Delta }_k}{\mathcal{O}}_k(x)$. Now, consider its contribution to the free energy, $\int d^{d}x{u}_k{\mathcal{O}}_k$. For the free energy to have the same form after rescaling, the new coupling $u_k^{\prime }$ must compensate for the scaling of $d^{d}x$ and ${\mathcal{O}}_k$:

$u_k^{\prime }\int d^d{x}^{\prime }{\mathcal{O}}_k^{\prime }(x^{\prime })=u_k\int d^{d}x{\mathcal{O}}_k(x)$
$u_k^{\prime }\int b^{-d}{d}^{d}x{b}^{{\Delta }_k}{\mathcal{O}}_k(x)=u_k\int d^{d}x{\mathcal{O}}_k(x)$

This implies $u_k^{\prime }=b^{d-{\Delta }_k}{u}_k$. Writing $b=e^l$, we have $u_k(l)=u_k(0)e^{(d-{\Delta }_k)l}$. Comparing this with our previous result $u_k(l)=u_k(0)e^{{\lambda }_{k}l}$, we find the crucial link:

${\lambda }_k=d-{\Delta }_k$

This gives us an intuitive way to classify operators via "power counting":

Example: Operators at the Gaussian Fixed Point

Let's apply this powerful formalism to the Gaussian Fixed Point, where the theory is free. In a free theory, the scaling dimensions are simply the "engineering dimensions" determined by dimensional analysis. The action must be dimensionless. From the kinetic term $\int d^{d}x(\nabla \varphi {)}^2$, we deduce:
$[\nabla ]=1$, $[d^{d}x]=-d$, so to make the integral dimensionless, we must have $[(\nabla \varphi {)}^2]=d$. This gives the scaling dimension of the field $\varphi$ itself:

${\Delta }_{\varphi }=\frac{d-2}{2}$

Now we can find the dimension of any operator $\mathcal{O}={\varphi }^n$: ${\Delta }_{{\varphi }^n}=n{\Delta }_{\varphi }=n\frac{d-2}{2}$.

Let's check the operators we know:

1. Mass operator ${\varphi }^2$:

   • Scaling dimension: ${\Delta }_{{\varphi }^2}=2\frac{d-2}{2}=d-2$.
   • Eigenvalue: ${\lambda }_2=d-{\Delta }_{{\varphi }^2}=d-(d-2)=2$.
   • Since ${\lambda }_2=2>0$, the mass operator is relevant for all $d$. This confirms our finding from last lesson that the ${\mu }^2$ direction is always relevant.

2. Interaction operator ${\varphi }^4$:

   • Scaling dimension: ${\Delta }_{{\varphi }^4}=4\frac{d-2}{2}=2d-4$.
   • Eigenvalue: ${\lambda }_4=d-{\Delta }_{{\varphi }^4}=d-(2d-4)=4-d$.
   • If $d<4$, ${\lambda }_4>0$, so ${\varphi }^4$ is relevant.
   • If $d>4$, ${\lambda }_4<0$, so ${\varphi }^4$ is irrelevant.
   • If $d=4$, ${\lambda }_4=0$, so ${\varphi }^4$ is marginal.

This perfectly reproduces the stability analysis from the previous lesson, but within a more general and powerful framework.

For the marginal case $d=4$, we must go beyond the linear approximation. As we saw in the derivation of the beta function, loop corrections give $\frac{dg}{dl}\approx -B{g}^2$. This negative sign means the flow is towards $g=0$, making the ${\varphi }^4$ operator marginally irrelevant at the GFP.

Universality Explained

The classification of operators as relevant, irrelevant, or marginal provides a profound explanation for universality.

Imagine starting with two very different microscopic models—say, the Ising model on a lattice and a real fluid near its critical point. In our language, these two systems start at two very different points in the infinite-dimensional theory space. They have different couplings for all sorts of complicated operators.

However, if they are in the same universality class, they flow towards the same fixed point (e.g., the Wilson-Fisher fixed point).

• To get to the critical point, we only need to tune the few relevant couplings (like temperature) to place our system on the critical surface that flows into the fixed point.
• The couplings of all irrelevant operators, which encode the non-universal microscopic details, will shrink to zero under the RG flow.
• Therefore, at long distances (deep in the infrared), both systems are described by the same fixed-point theory, and thus exhibit the same critical exponents and scaling functions.

The set of all theories that flow to the same fixed point is called its basin of attraction. Universality is the statement that microscopic details are irrelevant.

Conclusion

This lesson has formalized the concepts that underpin our understanding of RG flows and universality.

Key Takeaways:

• Operators are classified based on the behavior of their couplings under RG flow near a fixed point. This is determined by the sign of the eigenvalues $\lambda$ of the linearized flow matrix.
• Relevant operators ($\lambda >0$) have couplings that grow, driving the system away from the fixed point. They correspond to the parameters we must tune to observe criticality.
• Irrelevant operators ($\lambda <0$) have couplings that decay, meaning their effects vanish at long distances. They represent the non-universal microscopic details.
• Marginal operators ($\lambda =0$) require higher-order analysis to determine their behavior.
• This classification is directly related to the operator's scaling dimension $\Delta$ via the relation $\lambda =d-\Delta$.
• This framework provides the mathematical foundation for universality: systems in the same basin of attraction flow to the same fixed point, where the physics is governed only by the few relevant operators.

Preview of the Next Lesson:

We now have a complete conceptual framework. We know that for $d<4$, the WFFP governs criticality, and we know how to classify the operators that define its properties. It's time to put this machinery to work. In the next lesson, we will perform our first beyond-mean-field calculation: we will calculate the critical exponent $\nu$ at the Wilson-Fisher fixed point to first order in the $ϵ$-expansion.