Wilson-Fisher Fixed Point: Critical Exponent ν to First Order in ε

Hello! Let's get straight to our goal for this lesson.

In our previous discussions, we established that for dimensions $d<4$, the interacting Wilson-Fisher fixed point (WFFP), not the non-interacting Gaussian fixed point, governs the physics of critical phenomena. We also developed a powerful formal framework for classifying operators as relevant, irrelevant, or marginal based on the eigenvalues of the linearized RG flow.

Today, we will put this machinery into practice to achieve a landmark result: we will calculate the critical exponent $\nu$ at the Wilson-Fisher fixed point to first order in $ϵ=4-d$. This calculation represents the first correction to the mean-field results and demonstrates the power of the RG approach to compute non-trivial physical observables.

The Strategy

Our path to calculating $\nu$ involves three main steps:

1. Locate the Wilson-Fisher Fixed Point (WFFP): We'll start with the RG flow equations for the mass-squared coupling $t$ and the interaction coupling $u$ that we introduced in a previous lesson. By setting the beta functions to zero, we will solve for the fixed-point coordinates $(t^{\ast },u^{\ast })$ to leading order in $ϵ$.
2. Linearize the Flow: We will then analyze the RG flow in the vicinity of this fixed point by computing the stability matrix $Y_{ij}=\frac{\partial {\beta }_i}{\partial g_j}$ evaluated at $(t^{\ast },u^{\ast })$.
3. Extract the Exponent $\nu$: The eigenvalues of this matrix correspond to the scaling exponents $y_k$. The critical exponent $\nu$ is related to the eigenvalue $y_t$ of the temperature-like relevant operator by the relation $\nu =1/y_t$. We will calculate $y_t$ to first order in $ϵ$ and from it, find our desired expression for $\nu$.

Step 1: Finding the Wilson-Fisher Fixed Point

Let's start with the one-loop RG flow equations for the couplings $t$ (related to $T-T_c$) and $u$ (the ${\varphi }^4$ interaction strength), which we can take from our previous work. These equations describe how the couplings change under a change in length scale $l$.

As the notes show, the flow equations (beta functions) are:

$${\beta }_t(t,u)=\frac{dt}{dl}=2t-24tu-96u^2$$ $${\beta }_u(t,u)=\frac{du}{dl}=ϵu-16tu-72u^2$$

To find the fixed point $(t^{\ast },u^{\ast })$, we set both beta functions to zero. We are looking for the non-trivial solution where $u^{\ast }\ne 0$. As discussed in the reading, a simple power-counting argument suggests that for the terms to balance, we must have $u^{\ast }$ be of order $ϵ$, and $t^{\ast }$ of order ${ϵ}^2$. This insight is key to solving the system perturbatively.

From the second equation, ${\beta }_u=0$:

$$ϵu^{\ast }-16t^{\ast }{u}^{\ast }-72(u^{\ast }{)}^2=0$$

Dividing by $u^{\ast }$ (since we want the non-trivial fixed point) gives:

$$ϵ-16t^{\ast }-72u^{\ast }=0$$

Since we know $t^{\ast }\sim O({ϵ}^2)$, we can neglect the $16t^{\ast }$ term to find $u^{\ast }$ to leading order in $ϵ$:

$$ϵ-72u^{\ast }\approx 0⟹u^{\ast }=\frac{ϵ}{72}+O({ϵ}^2)$$

Now, we substitute this result into the first equation, ${\beta }_t=0$:

$$2t^{\ast }-24t^{\ast }{u}^{\ast }-96(u^{\ast }{)}^2=0$$

We are solving for $t^{\ast }$, which we expect to be $O({ϵ}^2)$. The term $24t^{\ast }{u}^{\ast }$ is of order ${ϵ}^3$, so we can neglect it to find $t^{\ast }$ to leading order:

$$2t^{\ast }-96(u^{\ast }{)}^2\approx 0⟹2t^{\ast }\approx 96{\left(\frac{ϵ}{72}\right)}^2=96\frac{{ϵ}^2}{5184}=\frac{{ϵ}^2}{54}$$ $$t^{\ast }=\frac{{ϵ}^2}{108}+O({ϵ}^3)$$

So, we have successfully located the Wilson-Fisher fixed point to leading order:

$$(t^{\ast },u^{\ast })=\left(\frac{{ϵ}^2}{108},\frac{ϵ}{72}\right)$$

This fixed point exists and has a small positive interaction strength ($u^{\ast }>0$) for any $ϵ>0$, i.e., for any dimension $d<4$.

Step 2: Linearizing the Flow

Now we analyze the flow near the WFFP. We need to compute the stability matrix $Y$ whose elements are $Y_{ij}=\frac{\partial {\beta }_i}{\partial g_j}$ evaluated at $(t^{\ast },u^{\ast })$.

$$Y={\left(\begin{array}{cc}\frac{\partial {\beta }_t}{\partial t}& \frac{\partial {\beta }_t}{\partial u}\\ \frac{\partial {\beta }_u}{\partial t}& \frac{\partial {\beta }_u}{\partial u}\end{array}\right)}_{(t^{\ast },u^{\ast })}$$

First, we calculate the partial derivatives of the beta functions:

• $\frac{\partial {\beta }_t}{\partial t}=2-24u$
• $\frac{\partial {\beta }_t}{\partial u}=-24t-192u$
• $\frac{\partial {\beta }_u}{\partial t}=-16u$
• $\frac{\partial {\beta }_u}{\partial u}=ϵ-16t-144u$

Now, we evaluate these at the fixed point, keeping only terms up to order $ϵ$:

• $Y_{11}=2-24u^{\ast }=2-24(\frac{ϵ}{72})=2-\frac{ϵ}{3}$
• $Y_{12}=-24t^{\ast }-192u^{\ast }=-24(\frac{{ϵ}^2}{108})-192(\frac{ϵ}{72})=O({ϵ}^2)-\frac{8}{3}ϵ\approx -\frac{8}{3}ϵ$
• $Y_{21}=-16u^{\ast }=-16(\frac{ϵ}{72})=-\frac{2}{9}ϵ$
• $Y_{22}=ϵ-16t^{\ast }-144u^{\ast }=ϵ-16(\frac{{ϵ}^2}{108})-144(\frac{ϵ}{72})=ϵ-O({ϵ}^2)-2ϵ\approx -ϵ$

So, the linearized flow matrix is:

$$Y\approx \left(\begin{array}{cc}2-\frac{ϵ}{3}& -\frac{8}{3}ϵ\\ -\frac{2}{9}ϵ& -ϵ\end{array}\right)$$

Step 3: Finding the Exponent $\nu$

The eigenvalues of this matrix $Y$ are the scaling exponents $y_k$ that we are looking for. Since we only need the eigenvalues to order $ϵ$, we can neglect the off-diagonal terms, which are themselves of order $ϵ$. The eigenvalues of a triangular matrix are its diagonal entries, and this approximation holds for our matrix to the order we are interested in.

The eigenvalues are therefore:

$$y_t=Y_{11}=2-\frac{ϵ}{3}+O({ϵ}^2)$$ $$y_u=Y_{22}=-ϵ+O({ϵ}^2)$$

As expected, $y_t$ is positive (relevant operator) and $y_u$ is negative (irrelevant operator). This confirms that the WFFP is stable on the critical surface.

Finally, we connect this to the critical exponent $\nu$. The correlation length $\xi$ diverges near the critical temperature as $\xi \sim |T-T_c{|}^{-\nu }$. In our field theory, the temperature-like variable is $t$, so $\xi \sim t^{-\nu }$. Under an RG transformation that rescales lengths by $b$, the correlation length transforms as $\xi \to \xi /b$. The variable $t$ flows according to its eigenvalue $y_t$, so $t\to t^{\prime }=b^{y_t}t$. For the scaling law to hold, we must have:

$${\xi }^{\prime }\sim (t^{\prime }{)}^{-\nu }⟹\frac{\xi }{b}\sim (b^{y_t}t{)}^{-\nu }=b^{-y_t\nu }{t}^{-\nu }=b^{-y_t\nu }\xi$$

Comparing the powers of $b$ gives $-1=-y_t\nu$, which leads to the fundamental relation:

$$\nu =\frac{1}{y_t}$$

Substituting our result for $y_t$:

$$\nu =\frac{1}{2-\frac{ϵ}{3}}=\frac{1}{2}{\left(1-\frac{ϵ}{6}\right)}^{-1}$$

Using the Taylor expansion $(1-x{)}^{-1}\approx 1+x$ for small $x$, we get:

$$\nu \approx \frac{1}{2}\left(1+\frac{ϵ}{6}\right)=\frac{1}{2}+\frac{ϵ}{12}$$

This is our final result for $\nu$ to first order in $ϵ=4-d$.

Test your understanding!

For the 3D Ising model, $d=3$, which means $ϵ=4-3=1$. What value does our first-order expansion give for $\nu$? How does it compare to the mean-field value? Highly precise numerical simulations give $\nu \approx 0.630$. What does this comparison tell you about the $ϵ$-expansion?

Show answer

• Mean-field value: ${\nu }_{MF}=1/2=0.5$.
• First-order $ϵ$-expansion for $d=3$: $\nu =\frac{1}{2}+\frac{1}{12}\approx 0.5+0.0833=0.5833$.
• Numerical value: ${\nu }_{num}\approx 0.630$.

The first-order correction significantly improves the prediction, moving it from the mean-field value of 0.5 up to ~0.583, much closer to the true value. This shows that even though the expansion parameter $ϵ=1$ is not infinitesimally small, the $ϵ$-expansion provides a surprisingly effective and systematically improvable framework for calculating critical exponents. Higher-order calculations in $ϵ$ yield results that are in remarkable agreement with experiments and simulations.

Conclusion

In this lesson, we have successfully executed a complete RG calculation, moving from the abstract flow equations to a concrete, non-trivial prediction for a critical exponent.

Key Takeaways:

• The Wilson-Fisher fixed point is located at $(t^{\ast },u^{\ast })=(\frac{{ϵ}^2}{108},\frac{ϵ}{72})$ to leading order in $ϵ=4-d$.
• By linearizing the RG flow around this fixed point, we found the relevant eigenvalue associated with the temperature-like perturbation to be $y_t=2-\frac{ϵ}{3}$.
• Using the relation $\nu =1/y_t$, we derived the critical exponent $\nu =\frac{1}{2}+\frac{ϵ}{12}$.
• This result provides the first-order correction to the mean-field value $\nu =1/2$ and gives a much better estimate for physical systems in three dimensions.

Preview of the Next Lesson:

We have now calculated one of the key critical exponents. But what about the others, like $\eta$ and $\gamma$? To find $\eta$, we need to consider the scaling of the field $\varphi$ itself, which we have so far assumed to be canonical. In the next lesson, we will see that interactions at the WFFP introduce a correction to the field's scaling, known as the anomalous dimension $\eta$. We will calculate the critical exponents $\eta$ and $\gamma$ to first order in $ϵ$.