Scaling Hypothesis in Critical Phenomena

Hello! Welcome back to our study of scaling and universality.

In our last lesson, we formally defined the six standard critical exponents ($\alpha ,\beta ,\gamma ,\delta ,\nu ,\eta$) that provide a quantitative "fingerprint" for a universality class. We noted that the experimental and numerical values for these exponents are not independent, suggesting a deeper underlying structure. Mean-field theory failed to capture these values correctly, but it hinted that thermodynamic quantities are interrelated.

Today, we will formalize this deep connection by introducing the scaling hypothesis. This is a powerful phenomenological idea that assumes the singular parts of thermodynamic functions near a critical point possess a specific mathematical form known as a generalized homogeneous function. This single, elegant assumption will be the key to unlocking the relationships between all the critical exponents. Our goal is to state and understand the scaling hypothesis for the two most fundamental quantities: the singular part of the free energy and the correlation function.

The Scaling Hypothesis for the Singular Free Energy

The failure of mean-field theory suggests we need a new starting point. Instead of trying to calculate the free energy from a specific Hamiltonian, what if we just postulate its mathematical structure near the critical point, guided by the idea of universality? This is the essence of the scaling hypothesis.

Let's begin by watching a portion of a lecture where this idea is developed. The video starts by examining the structure of the free energy in the (failed) saddle-point approximation and then boldly generalizes it into a new hypothesis.

As you saw in the video, the central idea is that the physics near a critical point, as a function of the reduced temperature $t=(T-T_c)/T_c$ and the ordering field $h$, doesn't depend on $t$ and $h$ independently. Instead, it depends on a specific combination of them.

To formalize this, let's turn to a concise written resource.

Let's break down the key statement from the reading. The scaling hypothesis for the singular part of the free energy density, $f_{sing}$, states that it takes the form of a generalized homogeneous function:

Let's dissect this equation:

• The prefactor $|t{|}^{2-\alpha }$ is chosen deliberately. Since the specific heat $C\propto -{\partial }^{2}f/\partial t^2$, this choice ensures that when we set $h=0$ and differentiate twice, we recover the defining power law for the specific heat, $C\sim |t{|}^{-\alpha }$.
• The function $g_{\pm }(x)$ is a universal scaling function, with $g_{+}$ for $t>0$ and $g_{-}$ for $t<0$. "Universal" means this function is the same for all systems in the same universality class.
• The argument of the function, $x=h/|t{|}^{\Delta }$, is the crucial scaling variable.
• $\Delta$ is a new exponent called the gap exponent. It determines the relative importance of the field and temperature as we approach the critical point.

This hypothesis is a profound statement. It claims that the complex, non-analytic behavior of the free energy near $T_c$ collapses into a dependence on a single variable, $x$. All the complexity is encoded in the shape of the one-dimensional function $g(x)$.

This structure immediately implies that the critical exponents cannot be independent. For example, by taking the derivative of $f_{sing}$ with respect to $h$ to find the magnetization $m$, we get:
$m(t,h)=-\frac{\partial f_{sing}}{\partial h}=-|t{|}^{2-\alpha }\frac{1}{|t{|}^{\Delta }}{g}_{\pm }^{\prime }\left(\frac{h}{|t{|}^{\Delta }}\right)=-|t{|}^{2-\alpha -\Delta }{g}_{\pm }^{\prime }\left(\frac{h}{|t{|}^{\Delta }}\right)$
To find the spontaneous magnetization (exponent $\beta$), we take the limit $h\to 0$. Assuming $g_{\pm }^{\prime }(0)$ is a non-zero constant, we find $m(t,0)\sim |t{|}^{2-\alpha -\Delta }$. But we defined $m(t,0)\sim |t{|}^{\beta }$, so we must have:
$\beta =2-\alpha -\Delta$
This is our first scaling relation! It's a direct consequence of the scaling hypothesis. In the next lesson, we will derive the full set of these relations.

The Power of Data Collapse

A direct and visually compelling consequence of the scaling hypothesis is the phenomenon of data collapse. The hypothesis implies that if we measure a quantity like magnetization $m$ at various temperatures $t$ and fields $h$, we can make all the data fall onto a single, universal curve.

Rearranging our expression for $m(t,h)$, we can write:
$\frac{m(t,h)}{|t{|}^{\beta }}=-g_{\pm }^{\prime }\left(\frac{h}{|t{|}^{\alpha +\beta }}\right)$
(using $\Delta =2-\alpha -\beta$). This equation predicts that if we plot the rescaled magnetization $m/|t{|}^{\beta }$ against the rescaled field $h/|t{|}^{\alpha +\beta }$, all our data points, regardless of the original $T$ and $H$, will collapse onto the single curve given by $-g^{\prime }(x)$. This has been confirmed with stunning precision in countless experiments and simulations.

The Scaling Hypothesis for Correlations

The first hypothesis successfully linked the thermodynamic exponents $\alpha ,\beta ,\gamma ,\delta$. But what about the exponents $\nu$ and $\eta$ that describe spatial correlations? This requires a second, related hypothesis about the correlation length $\xi$ and the correlation function $G(r)$.

The physical intuition is that at criticality, the divergence of the correlation length $\xi$ is the ultimate origin of all singularities. $\xi$ becomes the only relevant length scale in the problem, and all singular thermodynamic properties must be governed by it.

Let's explore this idea with the next section of the lecture notes.

This leads us to two more statements of the scaling hypothesis.

1. Scaling of the Correlation Length

In direct analogy to the free energy, we assume the correlation length is also a generalized homogeneous function of $t$ and $h$:

Here, $g_{\xi }$ is another universal scaling function. The prefactor $|t{|}^{-\nu }$ is chosen to match the definition of the exponent $\nu$ at $h=0$. Notice that the scaling variable $h/|t{|}^{\Delta }$ is exactly the same, with the same gap exponent $\Delta$. This is crucial and reflects the fact that there is a single underlying scaling structure.

The physical argument presented in the notes (and which you may recognize as related to Kadanoff's block-spin picture that we'll see later) is that the number of effectively independent degrees of freedom in a system of volume $L^d$ is $(L/\xi {)}^d$. Since the singular part of the free energy density should be proportional to this density of degrees of freedom, we have:
$f_{sing}\propto \frac{1}{{\xi }^d}$
Combining this with our scaling forms for $f_{sing}$ and $\xi$ gives a profound connection, which we will formalize into the Josephson identity in the next lesson.

2. Scaling of the Correlation Function

Finally, we can state a scaling form for the two-point correlation function $G(r,t,h)$. For simplicity, let's consider the case $h=0$. The hypothesis is:

where $\mathcal{G}(x)$ is yet another universal scaling function.

This form elegantly contains the behavior in the two most important limits:

• At the critical point ($t\to 0,\xi \to \infty$): For any fixed distance $r$, the argument $r/\xi \to 0$. If we assume $\mathcal{G}(0)$ is a constant, we recover the definition of the exponent $\eta$: $G(r,0)\propto r^{-(d-2+\eta )}$.
• Away from the critical point ($t\ne 0$) for large distances ($r\gg \xi$): The argument $r/\xi$ is large. In this regime, correlations should decay exponentially. This means the scaling function $\mathcal{G}(x)$ must fall off very rapidly (e.g., like $e^{-x}$) for large $x$, ensuring that $G(r,t)$ decays exponentially for $r\gg \xi$.

This scaling form for the correlation function completes the picture, providing a hypothesis that incorporates all six critical exponents into a single coherent framework.

Conclusion

Key Takeaways:

• The scaling hypothesis postulates that the singular part of thermodynamic functions and correlation functions near a critical point are generalized homogeneous functions.
• The scaling form for the free energy density is $f_{sing}(t,h)=|t{|}^{2-\alpha }g(h/|t{|}^{\Delta })$. It depends on a universal scaling function $g(x)$ and a single new "gap" exponent $\Delta$.
• The scaling form for the correlation function (at $h=0$) is $G(r,t)=r^{-(d-2+\eta )}\mathcal{G}(r/\xi (t))$, where the correlation length $\xi (t)\sim |t{|}^{-\nu }$.
• These hypotheses are not derived from first principles (at this stage) but are phenomenological assumptions justified by their ability to explain experimental observations like data collapse and to relate the critical exponents to each other.

Preview of the Next Lesson:

We have laid the groundwork and stated the powerful hypotheses that govern critical phenomena. In the next lesson, we will put them to work. We will take these scaling forms as our starting point and systematically derive the famous scaling relations—Rushbrooke, Widom, Fisher, and Josephson—that prove that only two of the six critical exponents are independent.