Scaling Hypothesis for Free Energy and Correlation Functions

The Scaling Hypothesis for Free Energy

The core idea of the scaling hypothesis, pioneered by Widom and Kadanoff, is that the singular part of the free energy, $f_s$, does not depend on the reduced temperature $t=(T-T_c)/T_c$ and external field $h$ in an arbitrary way. Instead, it is a generalized homogeneous function of these two variables. This reflects the physical intuition that near the critical point, the system loses its intrinsic scales of energy and length, and its behavior is governed by the ratio of the energy scales associated with $t$ and $h$.

To see how this functional form might arise, let's watch a segment from a lecture by Prof. Mehran Kardar. He begins by re-examining the mean-field (saddle-point) free energy and shows that it can be written in a specific homogeneous form, which he then elevates to a general hypothesis.

As motivated in the video, the scaling hypothesis for the singular part of the free energy density is:

where:

• $A$ and $B$ are non-universal amplitudes.
• The prefactor $|t{|}^{2-\alpha }$ is chosen to correctly recover the specific heat singularity, $C_h=-\frac{{\partial }^{2}f}{\partial T^2}\propto |t{|}^{-\alpha }$, in the limit $h\to 0$.
• $\Delta$ is a new exponent called the gap exponent. It governs the crossover between temperature-dominated ($h/|t{|}^{\Delta }\ll 1$) and field-dominated ($h/|t{|}^{\Delta }\gg 1$) behavior.
• $g_{\pm }(x)$ is a universal scaling function, with the subscript denoting the sign of $t$ (i.e., above or below $T_c$). Analyticity arguments require that $g_{+}$ and $g_{-}$ are related, so we often drop the subscript.

This single assumption is the cornerstone of scaling theory. Let's now explore its consequences.

Thermodynamic Scaling Relations

If the scaling hypothesis is true, we can derive all thermodynamic exponents from just two fundamental exponents (e.g., $\alpha$ and $\Delta$). The derivation involves taking derivatives of the scaling form of $f_s$ and matching the results to the definitions of the other exponents.

For a rigorous walkthrough of these derivations, please study the following text.

Let's summarize the key steps and results from that reading:

1. Magnetization ($\beta ,\delta$): The magnetization is $m=-(\partial f_s/\partial h{)}_T$. Applying this to the scaling form gives:

   $$m(t,h)\sim |t{|}^{2-\alpha -\Delta }{g}_{\pm }^{\prime }\left(\frac{h}{|t{|}^{\Delta }}\right)$$
   • For $t<0,h\to 0$: The argument of $g^{\prime }$ goes to zero, so $g^{\prime }(0)$ is a constant. We recover $m\sim (-t{)}^{\beta }$, which implies the relation:
     $\beta =2-\alpha -\Delta$
   • For $t=0$: The result must be independent of $t$. For this to hold, the function $g_{\pm }^{\prime }(x)$ must behave like $x^p$ for large $x$, where $p\Delta =2-\alpha -\Delta$. This leads to $m(0,h)\sim h^{(2-\alpha -\Delta )/\Delta }$. Comparing with $m\sim h^{1/\delta }$ gives:
     $\delta =\frac{\Delta }{2-\alpha -\Delta }=\frac{\Delta }{\beta }$

2. Susceptibility ($\gamma$): The susceptibility is $\chi =(\partial m/\partial h{)}_T$. Differentiating again yields:

   $$\chi (t,h)\sim |t{|}^{2-\alpha -2\Delta }{g}_{\pm }^{\prime \prime }\left(\frac{h}{|t{|}^{\Delta }}\right)$$
   • For $h\to 0$: The argument of $g^{\prime \prime }$ is zero. We recover $\chi \sim |t{|}^{-\gamma }$, which implies:
     $\gamma =2\Delta -(2-\alpha )$

Exponent Identities

These relationships can be rearranged to eliminate the unknown exponents $\alpha$ and $\Delta$, resulting in identities that relate the directly measurable exponents $\alpha ,\beta ,\gamma ,\delta$.

• Rushbrooke Identity: By substituting the expressions for $\beta$ and $\gamma$ into $\alpha +2\beta +\gamma$, we find:

  $$\alpha +2(2-\alpha -\Delta )+(2\Delta -2+\alpha )=\alpha +4-2\alpha -2\Delta +2\Delta -2+\alpha =2$$

  $\alpha +2\beta +\gamma =2$

• Widom Identity: From the relation $\delta =\Delta /\beta$, we have $\beta \delta =\Delta$. Substituting $\Delta$ into the expression for $\gamma$ gives $\gamma =2\beta \delta -2+\alpha$. Rearranging and using the Rushbrooke identity gives a simpler form:
  $\gamma =\beta (\delta -1)$

These identities are powerful predictions. They state that if you experimentally measure any three of these four exponents, the fourth is automatically determined. The remarkable agreement of these relations with experimental data is strong evidence for the validity of the scaling hypothesis.

Let's watch Prof. Kardar derive these same identities and check them against experimental data.

Hyperscaling: Connecting to Correlations

So far, we have only discussed bulk thermodynamic exponents. The scaling hypothesis can be extended to include the correlation length $\xi$ and its exponent $\nu$. The extension rests on a crucial physical assumption:

Near the critical point, the diverging correlation length $\xi$ is the only relevant length scale that determines the singular behavior of the system.

This means the singular part of the free energy density, $f_s$, which has units of (energy/volume), must be proportional to $k_{B}T/{\xi }^d$, where $d$ is the spatial dimension. Each "correlation volume" ${\xi }^d$ contributes roughly one unit of thermal energy, $k_{B}T$, to the free energy.

Since we also assume that $\xi$ itself is a homogeneous function, $\xi (t,h)\sim |t{|}^{-\nu }{g}_{\xi }(h/|t{|}^{\Delta })$, we can write:

Comparing this with our original postulate, $f_s\sim |t{|}^{2-\alpha }$, we arrive at the Josephson identity:

This is a hyperscaling relation because it explicitly involves the dimensionality $d$. It provides a direct link between a microscopic quantity ($\xi$, via $\nu$) and a macroscopic one ($C_h$, via $\alpha$).

Finally, we can relate the correlation function exponent $\eta$ by considering the fluctuation-dissipation theorem, which connects susceptibility to the integral of the correlation function: $\chi \sim \int d^d\mathbf{r}G(\mathbf{r})$.

As shown in the video, approximating the integral up to the correlation length $\xi$, we get $\chi \sim {\xi }^{2-\eta }$.
Since $\chi \sim |t{|}^{-\gamma }$ and $\xi \sim |t{|}^{-\nu }$, we find the Fisher identity:

With these two additional relations, all six critical exponents are now linked, leaving only two as independent. For instance, knowing $\nu$ and $\eta$ allows one to calculate all others.

Experimental Test: Data Collapse

The scaling hypothesis makes a stunning and testable prediction. Since the scaling function $g(x)$ is universal, we can rewrite the scaling form for magnetization, $m(t,h)=|t{|}^{\beta }{g}_m(h/|t{|}^{\Delta })$, as:

This equation implies that if we measure magnetization curves $m(h)$ at various temperatures $T$ (and thus various $t$) near $T_c$, and then plot the rescaled data $m/|t{|}^{\beta }$ against the rescaled field $h/|t{|}^{\Delta }$, all the data points should "collapse" onto a single, universal curve described by $g_m(x)$.

This phenomenon of data collapse is one of the most powerful confirmations of the scaling hypothesis. Below is an image from a 2024 research paper studying a magnetic material.

Let's analyze plot (a) in detail. The y-axis is $M|ϵ{|}^{-\beta }$ and the x-axis is $H|ϵ{|}^{-(\beta +\gamma )}$. Here, $M$ is the magnetization, $H$ is the field, and $ϵ$ is the reduced temperature $t$. Notice that the exponent on the x-axis is $\beta +\gamma$. Using our scaling relations: $\Delta =\beta \delta =\beta (\frac{\gamma }{\beta }+1)=\gamma +\beta$. So this is exactly a plot of the rescaled variable $m/|t{|}^{\beta }$ versus $h/|t{|}^{\Delta }$.

The different colored data points represent measurements taken at different fixed temperatures around $T_c$. When plotted this way, they all fall onto two universal branches: one for temperatures below $T_c$ (top branch) and one for temperatures above $T_c$ (bottom branch). This successful data collapse provides compelling evidence that the singular part of the free energy is indeed a homogeneous function of the form predicted by the scaling hypothesis.

Conclusion

In this lesson, we have moved from a disparate collection of critical exponents to a unified picture governed by a single, elegant idea. The scaling hypothesis, while phenomenological, provides a powerful framework for understanding the physics of critical points.

Key Takeaways:

• The scaling hypothesis postulates that the singular part of the free energy $f_s(t,h)$ is a generalized homogeneous function of the reduced temperature $t$ and external field $h$.
• The canonical form is $f_s(t,h)\approx |t{|}^{2-\alpha }g(h/|t{|}^{\Delta })$, depending on two independent exponents, conventionally chosen as $\alpha$ and $\Delta$.
• From this single assumption, we derived a set of scaling relations connecting the six main critical exponents:
  • Thermodynamic: $\alpha +2\beta +\gamma =2$ (Rushbrooke), $\gamma =\beta (\delta -1)$ (Widom).
  • Hyperscaling: $d\nu =2-\alpha$ (Josephson), $\gamma =(2-\eta )\nu$ (Fisher).
• These relations reduce the number of independent exponents from six to two.
• The scaling hypothesis predicts the phenomenon of data collapse, where experimental data taken at different temperatures near $T_c$ fall onto a single universal curve when appropriately rescaled.

The scaling hypothesis tells us how things scale, but it doesn't tell us why they scale this way, nor does it provide a method to calculate the two independent exponents from first principles. To address these deeper questions, we need to develop a more microscopic theory. In the next lesson, we will begin this journey by studying Landau-Ginzburg theory, which incorporates spatial fluctuations of the order parameter. This will lay the groundwork for understanding when mean-field theory fails and ultimately motivate the need for the Renormalization Group—the theory that finally provides a rigorous foundation for scaling and universality.