Gaussian and Wilson-Fisher Fixed Points

Hello! Welcome to our next lesson on the Renormalization Group.

In our last session, we successfully derived the one-loop RG flow equations for ${\varphi }^4$ theory. This was a significant step, turning the abstract RG framework into a concrete set of differential equations that describe how the parameters of our theory—the mass-squared ${\mu }^2$ and the coupling $g$—evolve as we zoom out to larger length scales. We ended with the exciting discovery that these equations predict a new, non-trivial fixed point for dimensions $d<4$: the Wilson-Fisher fixed point.

Today, we will build directly on that foundation. Our goal is to identify the Gaussian fixed point and the Wilson-Fisher fixed point from the flow equations and, more importantly, to understand their roles by analyzing their stability. We will see precisely why the simple Gaussian fixed point is insufficient for describing critical phenomena in realistic dimensions and why the interacting Wilson-Fisher fixed point is the key to universality.

Recap: The RG Flow Equations

Let's recall the flow equations we derived for the couplings $g$ and ${\mu }^2$, where $l=\ln b$ is the flow parameter and $ϵ=4-d$:

$$\frac{dg}{dl}={\beta }_g(g)=ϵg-B{g}^2$$ $$\frac{d{\mu }^2}{dl}={\beta }_{{\mu }^2}({\mu }^2,g)=2{\mu }^2+Ag$$

Here, $A$ and $B$ are positive constants that arise from the loop integrals. A fixed point of this flow is a point $({\mu }^{2\ast },g^{\ast })$ in the parameter space where the couplings cease to evolve: ${\beta }_g=0$ and ${\beta }_{{\mu }^2}=0$. Physically, a fixed point corresponds to a scale-invariant theory, which is precisely the situation at a critical point.

RG Flow, Fixed Points, and Stability

Before we solve these specific equations, it's useful to solidify our intuition about the general structure of RG flows. The core idea is that we can classify fixed points by analyzing the flow in their immediate vicinity. Does a small perturbation grow, driving the system away from the fixed point, or does it shrink, pulling the system back towards it?

As the video explained, the stability of a fixed point is determined by the eigenvalues of the linearized flow matrix. A direction with a positive eigenvalue is called relevant, as perturbations in that direction grow and drive the system away from the fixed point. A direction with a negative eigenvalue is irrelevant, as perturbations decay, and the flow is attracted towards the fixed point. An unstable fixed point (with one or more relevant directions) is repulsive, while a stable fixed point acts as an attractor for the RG flow, defining the universal long-distance physics.

1. The Gaussian Fixed Point (GFP)

The most obvious fixed point of our flow equations is the Gaussian Fixed Point (GFP), which occurs at ${\mu }^2=0$ and $g=0$. This represents the free, non-interacting theory. Let's analyze its stability.

To make the stability analysis explicit, we linearize our flow equations around the fixed point $({\mu }^2,g)=(0,0)$. Let $\delta {\mu }^2$ and $\delta g$ be small deviations:

$$\frac{d(\delta g)}{dl}\approx ϵ(\delta g)$$ $$\frac{d(\delta {\mu }^2)}{dl}\approx 2(\delta {\mu }^2)+A(\delta g)$$

In matrix form, this is:

$$\frac{d}{dl}\left(\begin{array}{c}\delta {\mu }^2\\ \delta g\end{array}\right)=\left(\begin{array}{cc}2& A\\ 0& ϵ\end{array}\right)\left(\begin{array}{c}\delta {\mu }^2\\ \delta g\end{array}\right)$$

The matrix $Y_{GFP}=\left(\begin{array}{cc}2& A\\ 0& ϵ\end{array}\right)$ is the stability matrix at the GFP. Its eigenvalues are simply the diagonal entries:

• ${\lambda }_t=2$
• ${\lambda }_g=ϵ=4-d$

Interpretation:

• The eigenvalue ${\lambda }_t=2$ is positive, meaning the ${\mu }^2$ direction is always relevant. This makes physical sense: ${\mu }^2\propto (T-T_c)$, and we know we must precisely tune the temperature to reach a critical point.
• The stability in the coupling direction depends on the dimension $d$:
  • For $d>4$: $ϵ<0$, so ${\lambda }_g<0$. The coupling $g$ is irrelevant. Any small interaction will die away under RG flow. The system flows to the free theory, and mean-field theory becomes exact.
  • For $d<4$: $ϵ>0$, so ${\lambda }_g>0$. The coupling $g$ is relevant. Any small interaction is amplified by the RG flow, driving the system away from the free theory. The GFP is unstable.
  • For $d=4$: $ϵ=0$, so ${\lambda }_g=0$. The coupling is marginal. The linear term vanishes, and we must look at the next term in the beta function: ${\beta }_g=-B{g}^2$. This term is negative, so it drives the flow towards $g=0$. The coupling is marginally irrelevant, and mean-field theory holds with logarithmic corrections.

This analysis is profound: it mathematically explains why mean-field theory fails for physical dimensions $d=2$ and $d=3$. The existence of a relevant coupling ($g$) means the non-interacting GFP cannot describe the critical point; the system is driven to a new, interacting fixed point.

2. The Wilson-Fisher Fixed Point (WFFP)

Since the GFP is unstable for $d<4$, we must search for the other fixed point we identified last lesson. Setting ${\beta }_g=ϵg-B{g}^2=0$ gives a non-trivial solution for $g^{\ast }>0$ when $ϵ>0$:

$g^{\ast }=\frac{ϵ}{B}$

Plugging this into the ${\beta }_{{\mu }^2}=0$ equation gives the fixed point value for ${\mu }^2$:

$2{\mu }^{2\ast }+A{g}^{\ast }=0⟹{\mu }^{2\ast }=-\frac{A{g}^{\ast }}{2}=-\frac{Aϵ}{2B}$

This is the Wilson-Fisher Fixed Point (WFFP). It exists only for $d<4$, and it has a non-zero interaction strength $g^{\ast }$.

Let's perform the linearization around $({\mu }^{2\ast },g^{\ast })$. The stability matrix $Y_{WFFP}$ is the Jacobian of the beta functions evaluated at the fixed point:

$$Y_{WFFP}={\left(\begin{array}{cc}\frac{\partial {\beta }_{{\mu }^2}}{\partial {\mu }^2}& \frac{\partial {\beta }_{{\mu }^2}}{\partial g}\\ \frac{\partial {\beta }_g}{\partial {\mu }^2}& \frac{\partial {\beta }_g}{\partial g}\end{array}\right)}_{({\mu }^{2\ast },g^{\ast })}=\left(\begin{array}{cc}2& A\\ 0& ϵ-2B{g}^{\ast }\end{array}\right)$$

Substituting $g^{\ast }=ϵ/B$, the lower-right element becomes $ϵ-2B(ϵ/B)=-ϵ$.

$$Y_{WFFP}=\left(\begin{array}{cc}2& A\\ 0& -ϵ\end{array}\right)$$

The eigenvalues are:

• ${\lambda }_1=2$
• ${\lambda }_2=-ϵ$

Interpretation:
For $d<4$, we have $ϵ>0$. Therefore, the WFFP has one positive eigenvalue (${\lambda }_1=2$) and one negative eigenvalue (${\lambda }_2=-ϵ$). This means the WFFP is a saddle point.

• It has one relevant direction. To flow into the fixed point, a physical system must be tuned along this direction (by changing temperature) to land on the critical surface—the set of points that are attracted to the WFFP.
• It has one irrelevant direction. If a system is on the critical surface, any small perturbation along this irrelevant direction will decay, and the system will inevitably flow to the WFFP.

This is the key result. For $d<4$, the GFP is unstable and repels the flow, while the WFFP is an attractor (a stable fixed point for flows on the critical surface). Therefore, the long-distance, universal properties of the critical point are described not by the free GFP, but by the interacting WFFP.

Visualizing the Flow

The following diagram illustrates the RG flow in the $({\mu }^2,g)$ parameter space for $d<4$, summarizing our entire analysis.

Conclusion

In this lesson, we formally identified the two principal fixed points of ${\varphi }^4$ theory and analyzed their stability.

Key Takeaways:

• Fixed points of the RG flow correspond to scale-invariant theories and represent possible critical points.
• The Gaussian Fixed Point ($g^{\ast }=0$) is stable for $d>4$ but unstable for $d<4$. This explains why mean-field theory works in high dimensions but fails in physical dimensions.
• The Wilson-Fisher Fixed Point ($g^{\ast }\propto (4-d)$) exists for $d<4$. It is an interacting fixed point that is attractive (on the critical surface), making it the one that governs universal critical behavior in dimensions 2 and 3.
• The stability analysis, which classifies directions as relevant or irrelevant based on the eigenvalues of the linearized flow, is the central tool for understanding which fixed point controls the physics and defines the universality class.

Preview of the Next Lesson:

We have now identified the correct, interacting fixed point that describes criticality. The next logical step is to use its properties to compute physical observables. In our next lesson, we will calculate the critical exponent $\nu$ at the Wilson-Fisher fixed point to the first order in the $ϵ$-expansion. This will be our first calculation of a critical exponent beyond mean-field theory.