Critical Exponents η and γ to First Order in ε

Hello! Welcome to our next lesson.

In our previous session, we established the core logic of the $ϵ$-expansion. We saw that for dimensions $d=4-ϵ$, the unstable Gaussian fixed point gives way to a new, stable Wilson-Fisher fixed point (WFFP). Crucially, this fixed point occurs at a weak coupling strength, $u^{\ast }\sim O(ϵ)$, which provides us with a small, controllable parameter to perform perturbative calculations.

Today, we will put this powerful machinery to work. Our goal is to calculate two of the most important critical exponents, $\eta$ and $\gamma$, to the first order in $ϵ$. This will not only give us concrete, non-mean-field results but also deepen our understanding of how the renormalization group connects microscopic interactions to macroscopic critical behavior.

1. Scaling Relations and Anomalous Dimensions

Before we begin the calculations, let's recall the physical meaning of $\eta$ and $\gamma$ and how they relate to the underlying field theory.

• The exponent $\gamma$ describes the divergence of the magnetic susceptibility $\chi$ as the temperature $T$ approaches the critical temperature $T_c$. With $t=(T-T_c)/T_c$, we have $\chi \sim |t{|}^{-\gamma }$.
• The exponent $\eta$ characterizes the decay of the two-point correlation function $G(x)=⟨\varphi (x)\varphi (0)⟩$ precisely at the critical point ($t=0$). Simple dimensional analysis suggests $G(x)\sim 1/|x{|}^{d-2}$, but interactions modify this: $G(x)\sim 1/|x{|}^{d-2+\eta }$. The term $\eta$ is thus called the anomalous dimension.

From a field theory perspective, these exponents are not independent fundamental constants but are determined by the properties of the RG fixed point. In particular, they can be expressed in terms of just two quantities: the anomalous dimensions of the field $\varphi$ and the operator ${\varphi }^2$.

As you've just read, the key relations are:

• $\alpha =2-d\nu$ (Hyperscaling/Josephson relation)
• $\beta =\frac{1}{2}\nu (d-2+\eta )$
• $\gamma =\nu (2-\eta )$ (Fisher's identity)
• $\delta =\frac{d+2-\eta }{d-2+\eta }$

Our path forward is clear: if we can calculate $\eta$ and $\nu$ in the $ϵ$-expansion, we can find $\gamma$. Let's start with $\eta$.

2. The Anomalous Dimension $\eta$ at Order $ϵ$

The anomalous dimension $\eta$ corrects the scaling dimension of the field $\varphi$. In the continuum RG framework you're familiar with from QFT, this correction is captured by the Callan-Symanzik function ${\gamma }_{\varphi }(u)$, which describes the scaling of the field itself. At the fixed point $u^{\ast }$, the relationship is exact:

The function ${\gamma }_{\varphi }(u)$ is determined by the field renormalization constant $Z_{\varphi }$, which in turn comes from the momentum-dependent part of the one-particle irreducible (1PI) self-energy, $\Sigma (p^2)$. Specifically, $Z_{\varphi }$ is chosen to cancel the divergence in the derivative of the self-energy: $\frac{\partial \Sigma }{\partial p^2}$.

Let's consider the one-loop contribution to the self-energy in ${\varphi }^4$ theory. The corresponding Feynman diagram is:

A one-loop Feynman diagram for the self-energy in φ⁴ theory. A propagator line enters from the left, forms a loop, and exits to the right. This diagram represents the leading-order correction to the propagator.

The value of this loop is given by the integral:

Notice a crucial feature of this integral: it has no dependence on the external momentum $p$. The integral evaluates to a constant (which is divergent and needs to be regularized). Because it's a constant, its derivative with respect to $p^2$ is zero:

This means that at the one-loop level, there is no correction to the kinetic term $\frac{1}{2}(\partial \varphi {)}^2$. Consequently, the field renormalization $Z_{\varphi }=1$ to this order, and the Callan-Symanzik function ${\gamma }_{\varphi }(u)$ is zero.

Since $\eta =2{\gamma }_{\varphi }(u^{\ast })$ and ${\gamma }_{\varphi }(u)$ is zero to order $u$, and we know $u^{\ast }\sim O(ϵ)$, it follows that ${\gamma }_{\varphi }(u^{\ast })=0$ to order $ϵ$. Therefore:

To first order in $ϵ$, the anomalous dimension is zero. The first non-zero contribution to $\eta$ famously arises at the two-loop level and is found to be $\eta =\frac{{ϵ}^2}{54}$. For our purposes, we can confidently set $\eta =0$.

3. Calculating the Susceptibility Exponent $\gamma$

With $\eta =0$, Fisher's identity simplifies to $\gamma =2\nu$. In our previous lesson, we saw that the RG eigenvalue associated with the temperature-like perturbation away from the WFFP was $y_t=2-\frac{ϵ}{3}$. This gave us the exponent $\nu$:

Using our simplified scaling relation, we can immediately find $\gamma$:

This gives the correct result and shows the power of the scaling relations.

However, it is more rigorous and instructive to derive this directly from the RG formalism, just as we did for $\nu$. The following resource lays out the complete theoretical framework for deriving all critical exponents from the anomalous dimensions of the relevant operators at the WFFP.

Let's synthesize what we've learned. The reading establishes the following key relations at the Wilson-Fisher fixed point (${\lambda }^{\ast }$ or $u^{\ast }$):

• $\eta =2{\gamma }_{\varphi }(u^{\ast })$
• $\nu =\frac{1}{2-{\gamma }_{{\varphi }^2}(u^{\ast })}$
• $\gamma =\frac{2(1-{\gamma }_{\varphi }(u^{\ast }))}{2-{\gamma }_{{\varphi }^2}(u^{\ast })}$

Here, ${\gamma }_{{\varphi }^2}(u^{\ast })$ is the anomalous dimension of the composite operator ${\varphi }^2$, which corresponds to the mass/temperature term in the Hamiltonian. It is this quantity that determines the eigenvalue $y_t$. The eigenvalue of the mass term under the RG flow is $y_t=2-{\gamma }_{{\varphi }^2}(u^{\ast })$.

From our previous calculation of $\nu$, we have $y_t=2-ϵ/3$. This allows us to identify the anomalous dimension of ${\varphi }^2$:

Now we have everything we need. Plugging our results for the anomalous dimensions into the expression for $\gamma$:

• ${\gamma }_{\varphi }(u^{\ast })=\eta /2=0$ (to order $ϵ$)
• ${\gamma }_{{\varphi }^2}(u^{\ast })=ϵ/3$

We get:

Expanding this for small $ϵ$, we find:

This confirms our earlier, simpler derivation and provides a solid grounding for the result within the full RG framework.

Conclusion

In this lesson, we have successfully calculated two key critical exponents using the $ϵ$-expansion. This exercise showcases the concrete predictive power of the renormalization group.

Key Takeaways:

• The critical exponent $\eta$ is directly related to the anomalous dimension of the field, $\eta =2{\gamma }_{\varphi }(u^{\ast })$.
• At one-loop order in ${\varphi }^4$ theory, the self-energy is momentum-independent, leading to ${\gamma }_{\varphi }(u^{\ast })=0$ and thus $\eta =0+O({ϵ}^2)$.
• The critical exponent $\gamma$ can be calculated from the scaling relation $\gamma =\nu (2-\eta )$ or directly from the RG formalism using $\gamma =\frac{2(1-{\gamma }_{\varphi }(u^{\ast }))}{2-{\gamma }_{{\varphi }^2}(u^{\ast })}$.
• Both methods yield the same result to first order in $ϵ$: $\gamma =1+\frac{ϵ}{6}+O({ϵ}^2)$.

These results, while derived for an infinitesimal $ϵ$, provide remarkably good approximations for the physical dimension $d=3$ (where $ϵ=1$), giving $\gamma \approx 1.17$. This is a significant improvement over the mean-field value $\gamma =1$ and is impressively close to the best numerical estimates of $\gamma \approx 1.24$.

Preview of the Next Lesson:

We have now seen how the $ϵ$-expansion works and used it to calculate several exponents. Next, we will compare the predictions of the $ϵ$-expansion for $d=3$ with the precise numerical and experimental results for the 3D Ising model. This will allow us to assess the strengths and limitations of this perturbative approach and will set the stage for discussing the concept of crossover phenomena—how a system's behavior can change from being described by one fixed point to another as a parameter is varied.