Landau Theory and Mean-Field Critical Exponents

Introduction

In our last lesson, we developed the powerful "recipe" for constructing the Landau free energy, a phenomenological tool that captures the essence of a phase transition by focusing on symmetry and the behavior of an order parameter. We saw that for a continuous transition with an up-down ( $M \to - M$ ) symmetry, the free energy density near the critical point takes the universal form $f (M) = f_{0} + \frac{1}{2} a (T) M^{2} + \frac{1}{4} c M^{4}$ , where the coefficient $a (T)$ drives the transition by changing sign at $T_{c}$ .

Having constructed this function, our objective now is to use it to make concrete, quantitative predictions. This lesson is dedicated to demonstrating that by simply minimizing this free energy, we can derive the equilibrium value of the order parameter and calculate the behavior of key thermodynamic quantities like specific heat and susceptibility. This process will allow us to extract the set of critical exponents that characterize the transition. As you will see, this procedure—known as Landau mean-field theory—reproduces a universal set of exponents, providing a foundational benchmark for understanding critical phenomena.

Before we begin the calculations, let's briefly review the physical meaning of the critical exponents we aim to find.

2. Lec 1 (continued); The Landau-Ginzburg Approach Part 1

This segment from Prof. Mehran Kardar's lecture provides a concise physical definition of the critical exponents β, γ, α, and δ in the context of a ferromagnetic transition.

Please watch the portion from this segment. Focus on associating each exponent with a specific physical quantity's singular behavior near T_c: β (beta): How the spontaneous magnetization (order parameter) grows below T_c. δ (delta): How magnetization depends on the external field exactly at T_c. γ (gamma): How the susceptibility (the response to a small field) diverges as T_c is approached. α (alpha): How the specific heat diverges at T_c.

Equilibrium and Spontaneous Symmetry Breaking

The equilibrium state of the system corresponds to the configuration that minimizes the free energy. For a uniform system, we simply need to find the value of the order parameter $M$ that minimizes our Landau potential:

f (M, H = 0) = f_{0} + \frac{1}{2} a_{0} (T - T_{c}) M^{2} + \frac{1}{4} c M^{4}

where we have used the linear approximation $a (T) \approx a_{0} (T - T_{c})$ with $a_{0} > 0$ , and we require $c > 0$ for stability.

To find the minima, we set the first derivative with respect to $M$ to zero:

\frac{\partial f}{\partial M} = a_{0} (T - T_{c}) M + c M^{3} = M [a_{0} (T - T_{c}) + c M^{2}] = 0

This equation has two types of solutions:

$M = 0$
$M^{2} = - \frac{a _{0}}{c} (T - T_{c}) = \frac{a _{0}}{c} (T_{c} - T)$

To determine which of these corresponds to a stable equilibrium (a minimum), we examine the sign of the second derivative, $\frac{\partial ^{2} f}{\partial M ^{2}} = a_{0} (T - T_{c}) + 3 c M^{2}$ :

For $T > T_{c}$ : The coefficient $a (T) = a_{0} (T - T_{c})$ is positive. The only real solution is $M = 0$ . At this point, $\frac{\partial ^{2} f}{\partial M ^{2}} ∣_{M = 0} = a (T) > 0$ , indicating a stable minimum. The system is in its disordered, high-symmetry phase, as expected.
For $T < T_{c}$ : The coefficient $a (T)$ is negative. We now have three real solutions: $M = 0$ and $M_{0} = \pm \frac{a _{0}}{c} (T_{c} - T)$ .
- At $M = 0$ , $\frac{\partial ^{2} f}{\partial M ^{2}} ∣_{M = 0} = a (T) < 0$ , so this is now an unstable maximum.
- At $M = M_{0}$ , $\frac{\partial ^{2} f}{\partial M ^{2}} ∣_{M = M_{0}} = a (T) + 3 c (- \frac{a ( T )}{c}) = - 2 a (T) > 0$ . These two non-zero solutions are the degenerate, stable minima.

The system spontaneously chooses one of these two equivalent states, breaking the $M \to - M$ symmetry and developing a non-zero order parameter $M_{0}$ . This elegant result, arising from a simple polynomial minimization, is the mathematical embodiment of spontaneous symmetry breaking.

The Landau free energy $F - F_0$ versus the order parameter $\eta$ (equivalent to $M$). Above $T_c$ (red curve), there is a single minimum at $\eta=0$. Below $T_c$ (blue curve), the $\eta=0$ state becomes unstable, and two new, degenerate minima appear at non-zero values $\pm\eta_0$.

Deriving the Mean-Field Critical Exponents

With the equilibrium behavior of $M$ established, we can now systematically derive the critical exponents. The following document and video provide clear, step-by-step derivations which we will follow.

[PDF] Phase Transitions and Collective Phenomena - TCM Group

This section of the lecture notes from the University of Cambridge provides a concise and rigorous derivation of all four primary mean-field exponents directly from the Landau free energy.

Please read the subsections within section 2.2, "Landau Mean-Field Theory": Magnetisation: This derives the exponent β. Heat Capacity: This derives the exponent α. Susceptibility: This derives the exponent γ. Equation of State: This derives the exponent δ. As you read, focus on how each calculation follows a logical sequence: Write down the relevant physical quantity (e.g., magnetization, specific heat). Relate it to the free energy or the order parameter. Substitute the equilibrium solutions we just found. Analyze the behavior as t = (T-T_c)/T_c \to 0 to extract the exponent.

Let's summarize the results from this reading, explicitly showing the key steps.

1. The Order Parameter Exponent: β

As found above, for $T < T_{c}$ , the spontaneous magnetization $M_{0}$ is:
$M_{0} (T) = \frac{a _{0}}{c} (T_{c} - T)$
Near the critical point, we can express this in terms of the reduced temperature $t = (T - T_{c}) / T_{c}$ , where $T_{c} - T = - T_{c} t$ .
$M_{0} (t) = - \frac{a _{0} T _{c}}{c} \cdot (- t)^{1/2}$
The general definition is $M_{0} \propto (- t)^{β}$ . By comparison, we find:
$β = \frac{1}{2}$

2. The Specific Heat Exponent: α

The specific heat involves the second derivative of the free energy. First, we find the value of the free energy at equilibrium, $f_{e q} (T)$ , by substituting the equilibrium $M$ values back into the expression for $f$ .

For $T > T_{c}$ : $M_{0} = 0$ , so $f_{e q} (T) = f_{0}$ .
For $T < T_{c}$ : $M_{0}^{2} = - a / c$ , so
$f_{e q} (T) = f_{0} + \frac{1}{2} a (- \frac{a}{c}) + \frac{1}{4} c (- \frac{a}{c})^{2} = f_{0} - \frac{a ^{2}}{4 c} = f_{0} - \frac{a _{0}^{2}}{4 c} (T - T_{c})^{2}$
The singular part of the specific heat per unit volume is $C_{s in g} = - T \frac{\partial ^{2} f _{e q}}{\partial T ^{2}}$ .
For $T > T_{c}$ : $C_{s in g} = 0$ .
For $T < T_{c}$ : $\frac{\partial ^{2} f _{e q}}{\partial T ^{2}} = - \frac{a _{0}^{2}}{2 c}$ . Thus, $C_{s in g} = T \frac{a _{0}^{2}}{2 c}$ .

At $T = T_{c}$ , the specific heat exhibits a finite jump of $Δ C = T_{c} \frac{a _{0}^{2}}{2 c}$ . There is no divergence. Since the definition is $C_{s in g} \propto ∣ t ∣^{- α}$ , a finite value or jump corresponds to:
$α = 0$

3. The Susceptibility Exponent: γ

To find the susceptibility, $χ = (\partial M / \partial H)_{H = 0}$ , we must include the external field $H$ in the free energy: $g (M, H) = f (M) - M H$ . The equilibrium condition becomes:
$\frac{\partial g}{\partial M} = a M + c M^{3} - H = 0$
Differentiating this entire equation with respect to $H$ gives:
$a \frac{\partial M}{\partial H} + 3 c M^{2} \frac{\partial M}{\partial H} - 1 = 0 ⟹ χ = \frac{\partial M}{\partial H} = \frac{1}{a + 3 c M ^{2}}$
Now we evaluate this at $H = 0$ , using our equilibrium solutions for $M$ :

For $T > T_{c}$ : $M_{0} = 0$ .
$χ = \frac{1}{a} = \frac{1}{a _{0} ( T - T _{c} )} \propto t^{- 1}$
For $T < T_{c}$ : $M_{0}^{2} = - a / c$ .
$χ = \frac{1}{a + 3 c ( - a / c )} = \frac{1}{- 2 a} = \frac{1}{2 a _{0} ( T _{c} - T )} \propto (- t)^{- 1}$
In both cases, the exponent is 1. Since $χ \propto ∣ t ∣^{- γ}$ , we have:
$γ = 1$

4. The Critical Isotherm Exponent: δ

Finally, we analyze the relationship between $M$ and $H$ exactly at the critical temperature, $T = T_{c}$ . Here, $a (T_{c}) = 0$ . The equilibrium equation simplifies significantly:
$c M^{3} - H = 0 ⟹ H = c M^{3}$
The definition is $H \propto M^{δ}$ at $t = 0$ . By comparison, we immediately see:
$δ = 3$

The following video provides an excellent summary and re-derivation of these four exponents, which can help solidify your understanding.

Landau Ginzburg theory of Phase Transitions

This video offers a clear, pedagogical walkthrough of the derivation of the mean-field exponents from the Landau free energy expansion.

Watch the segment from this derivation. It covers the calculation of β, α, γ, and δ in a manner that closely mirrors the steps we just outlined.

The Mean-Field Nature of Landau Theory

The set of exponents we have derived, ${α = 0, β = 1/2, γ = 1, δ = 3}$ , are known as the classical or mean-field exponents. It is crucial to understand why Landau theory is fundamentally a mean-field theory.

The approximation lies in our starting point. By writing the free energy as a function of a single, spatially uniform order parameter, $M$ , we have implicitly averaged over all the microscopic degrees of freedom and, more importantly, we have completely ignored any spatial fluctuations of the order parameter itself. The theory describes the behavior of a system where every microscopic constituent feels the same "mean field" generated by all other constituents, rather than its specific, fluctuating local environment.

The document below provides a succinct summary of these exponents and makes the connection between the phenomenological Landau theory and the exactly solved infinite-range Ising model, which is the canonical example of a system where mean-field theory is exact.

1. Exact Solution of the Infinite Range Ising Model

This resource explicitly connects the results of Landau theory to mean-field theory.

First, look at the table in section 2 and the subsequent summary of the mean-field results. Find the table labeled critical exponents (it is on page 6 of the PDF) and the list of results given in equation (33). This confirms the set of exponents we just calculated. Next, read section 3, titled Landau Theory. This part is key. It argues that the critical behavior depended only on the first few terms of the free energy expansion. It then introduces the Landau functional (initially including spatial gradients, which we've ignored so far) and shows that minimizing its spatially uniform part reproduces the same free energy form and thus the same critical behavior as the microscopic mean-field model.

The neglect of fluctuations is the central limitation of Landau theory. Near the critical point, as we saw in the previous module, fluctuations become correlated over all length scales, and they can dramatically alter the critical exponents. The question of when these fluctuations become important is the subject of the Ginzburg criterion, which we will address in the next lesson.

Conclusion

In this lesson, we have taken the crucial step from constructing the Landau free energy to using it for prediction. By applying the fundamental principle of minimizing the free energy, we have successfully described spontaneous symmetry breaking and derived the full set of classical mean-field critical exponents.

Key Takeaways:

Minimizing the Landau free energy $f (M) = \frac{1}{2} a (T) M^{2} + \frac{1}{4} c M^{4}$ naturally explains the emergence of a non-zero order parameter below $T_{c}$ .
This minimization procedure allows for the direct calculation of thermodynamic observables near the critical point.
The theory predicts a universal set of mean-field critical exponents:
- $α = 0$ (a finite jump in specific heat)
- $β = 1/2$ (order parameter grows as $T_{c} - T$ )
- $γ = 1$ (susceptibility diverges as $∣ T - T_{c} ∣^{- 1}$ )
- $δ = 3$ (at $T_{c}$ , $H \propto M^{3}$ )
Landau theory is a mean-field theory because it ignores spatial fluctuations of the order parameter, effectively assuming each part of the system experiences an average, uniform environment.

We have now used Landau's original, simpler formulation. However, the true power of the framework comes from including spatial variations in the order parameter. In our next lesson, we will extend our model to the Landau-Ginzburg free energy functional, which will allow us to study correlations and, most importantly, to derive the Ginzburg criterion—a quantitative measure for the breakdown of mean-field theory itself.

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