Scaling Laws and Dimensionality: Mean-Field vs. 2D Ising Exponents

Hello! Welcome to the fourth lesson in our module on Scaling and Universality.

In the previous lesson, we took a significant step by deriving the four major scaling relations—Rushbrooke, Widom, Fisher, and Josephson—from the scaling hypothesis. We established that these relations connect the six critical exponents ($\alpha ,\beta ,\gamma ,\delta ,\nu ,\eta$), implying that only two are independent.

Today, our goal is to put these powerful relations to the test. We will perform a crucial verification exercise using the two primary theoretical models we have studied: mean-field theory and the exact solution of the 2D Ising model. This process is more than just a mathematical check; it will reveal deep physical differences between these models, particularly concerning the role of fluctuations and the anomalous dimension exponent, $\eta$.

The Cast of Characters: Critical Exponents

Before we begin our verification, let's assemble the data we need. The critical exponents for mean-field theory and the 2D Ising model have been derived or stated in previous modules. The table below consolidates them for easy reference. You can find these values discussed across several of our resources, such as Table 1 in the notes from the University of Edinburgh or the summaries in the videos by Prof. Kardar and Prof. Thijssen.

Our task is to check if these two distinct sets of numbers satisfy the scaling relations we derived.

Verification for Mean-Field Theory

Let's begin with mean-field theory, which you'll recall is an approximation that neglects fluctuations. We will test each of the four relations.

1. Rushbrooke Identity: $\alpha +2\beta +\gamma =2$
   Plugging in the MFT values:
   $0+2\left(\frac{1}{2}\right)+1=1+1=2$
   The identity holds.

2. Widom Identity: $\gamma =\beta (\delta -1)$
   Plugging in the MFT values:
   $1=\frac{1}{2}(3-1)=\frac{1}{2}(2)=1$
   The identity holds.

3. Fisher Identity: $\gamma =(2-\eta )\nu$
   Plugging in the MFT values:
   $1=(2-0)\left(\frac{1}{2}\right)=2\cdot \frac{1}{2}=1$
   This also holds. Note that the MFT value $\eta =0$ is crucial here. This is consistent with the fact that the mean-field correlation function has the simple Ornstein-Zernike form $G(r)\sim 1/r^{d-2}$ at criticality, which by definition corresponds to $\eta =0$.

4. Josephson (Hyperscaling) Identity: $d\nu =2-\alpha$
   This identity explicitly involves the spatial dimension $d$. Let's plug in the MFT values:
   $d\left(\frac{1}{2}\right)=2-0⟹\frac{d}{2}=2⟹d=4$
   This is a remarkable result. The hyperscaling relation is only satisfied by the mean-field exponents at the very specific dimension $d=4$. This dimension is known as the upper critical dimension ($d_c$) for this universality class. For any dimension $d>4$, where mean-field theory is expected to be correct, the hyperscaling relation is violated. This failure is a fundamental signature of mean-field theory and is tied to its neglect of fluctuations.

Verification for the 2D Ising Model

Now let's turn to the 2D Ising model, an exactly solved system where fluctuations are fully accounted for. Here, we must use $d=2$.

1. Rushbrooke Identity: $\alpha +2\beta +\gamma =2$
   $0+2\left(\frac{1}{8}\right)+\frac{7}{4}=\frac{1}{4}+\frac{7}{4}=\frac{8}{4}=2$
   It holds.

2. Widom Identity: $\gamma =\beta (\delta -1)$
   $\frac{7}{4}=\frac{1}{8}(15-1)=\frac{14}{8}=\frac{7}{4}$
   It holds.

3. Fisher Identity: $\gamma =(2-\eta )\nu$
   This is the most interesting check. Let's first see what happens if we incorrectly assume $\eta =0$, as in mean-field theory. The relation would predict $\gamma =2\nu$.
   $\frac{7}{4}\ne 2(1)=2$
   The identity fails. This demonstrates that a zero anomalous dimension is inconsistent with the other exact exponents. Now, let's use the correct value, $\eta =1/4$.
   $\frac{7}{4}=\left(2-\frac{1}{4}\right)\cdot 1=\frac{7}{4}$
   The identity holds perfectly. This is a central result: the scaling relations are only satisfied for the 2D Ising model if the anomalous dimension $\eta$ is non-zero. This directly links the internal consistency of the theory to the presence of non-trivial correlations that alter the decay of $G(r)$ at criticality.

4. Josephson (Hyperscaling) Identity: $d\nu =2-\alpha$
   Using $d=2$:
   $2(1)=2-0⟹2=2$
   Hyperscaling holds exactly for the 2D Ising model.

The following video segment provides an excellent recap of the Fisher and Josephson relations and explicitly performs these checks for the 2D Ising model.

Discussion: What It All Means

This verification exercise reveals the deep physical distinctions between mean-field and fluctuation-dominated systems.

• The Meaning of $\eta \ne 0$: The non-zero value $\eta =1/4$ in the 2D Ising model is a direct quantitative measure of how much critical fluctuations alter the spatial correlations from the simple mean-field picture. This is not a small correction; it fundamentally changes the nature of the critical state.

• Hyperscaling and Dimensionality: The fact that hyperscaling holds for the 2D Ising model but fails for MFT above $d=4$ is a profound insight. It tells us that the simple geometric picture underlying hyperscaling—that the singular free energy density is set by the packing of correlation-sized "blobs" ($f_{sing}\sim {\xi }^{-d}$)—is only valid in low dimensions where fluctuations are strong. In high dimensions ($d>4$), spins have so many neighbors that fluctuations are suppressed, the system behaves in a mean-field-like manner, and this simple geometric argument breaks down.

The following video provides an intuitive physical argument for why mean-field theory becomes more reliable in higher dimensions.

Conclusion

Key Takeaways:

• The critical exponents of mean-field theory satisfy the thermodynamic scaling relations (Rushbrooke, Widom, Fisher with $\eta =0$). However, the hyperscaling relation ($d\nu =2-\alpha$) is only satisfied at the upper critical dimension $d_c=4$.
• The exact exponents of the 2D Ising model satisfy all four scaling relations.
• Crucially, verifying the Fisher identity for the 2D Ising model requires a non-zero anomalous dimension ($\eta =1/4$), providing a quantitative link between the consistency of the scaling theory and the non-trivial nature of critical correlations.
• The validity of hyperscaling is a key differentiator: it holds for fluctuation-dominated systems like the 2D Ising model (below $d_c$) but fails for mean-field theory (above $d_c$).

Preview of the Next Lesson:

We have now seen two very different sets of critical exponents that each form a consistent family under the scaling relations. This naturally leads us to the concept of universality classes. In the next lesson, we will formalize this idea, exploring which physical properties of a system—like spatial dimensionality and the symmetry of the order parameter—determine the universality class it belongs to, and thus dictate its critical exponents.