Landau Free Energy Expansion: Symmetry & Analyticity

Welcome back to our study of foundational research methods in physics. In the last lesson, we took a "bottom-up" approach to phase transitions, starting from a microscopic Hamiltonian and using mean-field theory to derive a self-consistency equation for the order parameter. This powerful technique gave us a concrete way to calculate properties like the critical temperature. However, it relied on a specific model (the Ising model) and a specific, and not always justified, approximation that neglects fluctuations.

Today, we pivot to a complementary and profoundly influential "top-down" methodology: Landau theory. Instead of building from microscopic details, we will construct a theory for the order parameter directly, based only on the most general and robust principles available to a physicist: symmetry and analyticity. This phenomenological approach elegantly bypasses the messy microscopic details, providing deep insights into why different physical systems, from magnets to liquid-gas mixtures, can exhibit identical behavior near their phase transitions—a phenomenon known as universality.

Your goal for this lesson is to master the core craft of this approach: to construct a Landau free energy expansion for a given physical system by applying arguments of symmetry and analyticity.

From Microscopic Justification to Phenomenological Postulate

The central object in Landau theory is an effective free energy, $F$ , expressed as a function of the order parameter, $ψ$ . This is not the true thermodynamic free energy of the entire system, but a "coarse-grained" potential whose minimum gives the most probable value of the order parameter in thermal equilibrium.

Where does this function come from? In our last lesson, we saw a hint. The mean-field approximation for the Ising model yielded a free energy per site, $f (m)$ , that was a complicated function of the magnetization $m$ . What if we are near a phase transition where the magnetization is small? We can Taylor expand this function.

Let's start by examining this connection, which provides a concrete bridge from our previous work on mean-field theory to the more abstract framework of Landau theory.

Statistical Field Theory

This section from David Tong's notes on Statistical Field Theory provides a perfect motivation for the Landau expansion. It starts with the mean-field free energy for the Ising model and expands it for a small order parameter.

Please read the section titled "Landau's approach focuses on situations where the order parameter is small". Pay close attention to Equation (1.12), which shows the explicit Taylor series of the mean-field free energy. You can see it naturally produces a polynomial in the order parameter m.

The key insight of Landau was to realize that we don't need the microscopic mean-field theory to arrive at such a polynomial expansion. We can postulate its existence based on general principles. This leap from a derived result to a fundamental starting point is a classic move in theoretical physics.

The Recipe for Constructing a Landau Free Energy

The construction of a Landau free energy follows a clear, systematic procedure. We can think of it as a four-step recipe.

Lectures on Landau Theory of Phase Transitions

This resource by Peter D. Olmsted provides a concise summary of the methodology for building a Landau theory.

Please read Section 3.1, "The Recipe". This section outlines the four fundamental steps that we will follow.

Let's break down these steps, focusing on the most critical one: the construction of the free energy functional based on symmetry.

Define an Order Parameter ( $ψ$ ): As we learned previously, this is a quantity that is zero in the disordered (high-symmetry) phase and non-zero in the ordered (low-symmetry) phase.
Assume an Analytic Functional Form: This is the core assumption of analyticity. We assume that near the transition, where $ψ$ is small, the free energy density $f$ can be written as a power series (a Taylor expansion) in $ψ$ and its spatial derivatives.
Apply Symmetries: The form of this expansion is not arbitrary. It is powerfully constrained by the symmetries of the system. The free energy must be invariant under all symmetry operations of the high-temperature (disordered) phase.
Assume Temperature Dependence: The coefficients of the expansion are, in general, functions of temperature $T$ and pressure $P$ . The simplest and most crucial assumption is that only the coefficient of the quadratic term, $ψ^{2}$ , changes sign at the critical temperature $T_{c}$ , while other coefficients are well-behaved.

Step 3 in Action: Symmetry as the Architect

Let's see how symmetry dictates the form of the free energy for a few key examples.

Example 1: The Ising Magnet (Scalar Order Parameter)

The order parameter is the scalar magnetization, $m$ . The high-temperature paramagnetic phase has no preferred direction of magnetization. In zero external magnetic field, the Hamiltonian is invariant under a global spin flip, $S_{i} \to - S_{i}$ . Consequently, the order parameter transforms as $m \to - m$ . The free energy must be invariant under this transformation.

Therefore, $f (m)$ must be an even function of $m$ . An expansion of $f (m)$ can only contain even powers of $m$ .

f (m) = f_{0} + \frac{a}{2} m^{2} + \frac{b}{4} m^{4} + \frac{c}{6} m^{6} + \dots

Odd powers like $m$ and $m^{3}$ are forbidden by this $Z_{2}$ symmetry. The term $f_{0}$ is an $m$ -independent background, and we typically truncate the series at the $m^{4}$ or $m^{6}$ term, arguing that higher-order terms are negligible for small $m$ .

Example 2: The Heisenberg Ferromagnet (Vector Order Parameter)

Here, the order parameter is a vector, $M = (M_{x}, M_{y}, M_{z})$ . The high-temperature phase is isotropic; it is invariant under any rotation in 3D space (the $SO (3)$ group). The free energy, $f (M)$ , must therefore be a rotational scalar.

How do we construct scalars from a vector $M$ ? The only way is to use dot products. The possible terms are powers of the magnitude squared, $M \cdot M = ∣ M ∣^{2}$ .

f (M) = f_{0} + \frac{a}{2} (M \cdot M) + \frac{b}{4} (M \cdot M)^{2} + \dots

Notice that a term like $∣ M ∣^{3}$ is non-analytic at $M = 0$ and is excluded by our analyticity assumption. This shows how different symmetries lead to different functional forms, even though the underlying physics is similar.

For a deeper look at these examples, you can refer back to the Olmsted notes you just read. Section 4.1 covers the Ising magnet (LINK, section 2), and Section 4.2 covers the Heisenberg ferromagnet (LINK, section 3).

Beyond Simple Polynomials: Landau-Ginzburg Theory

So far, we have assumed the order parameter is uniform in space. This is the essence of Landau's original theory, which is, at its heart, a mean-field theory. However, near a critical point, fluctuations become crucial, and the order parameter can vary from point to point. Ginzburg extended Landau's ideas to incorporate these spatial variations, leading to Landau-Ginzburg theory.

The key steps are:

Promote the order parameter to a field: The order parameter $ψ$ becomes a local field $ψ (x)$ , defined via a coarse-graining procedure (averaging microscopic variables over small but macroscopic regions).
Make the free energy a functional: The total free energy $F$ is now a functional of the field $ψ (x)$ , denoted $F [ψ (x)]$ .
Assume locality: The total free energy is the integral over all space of a local free energy density, $f$ . This density at a point $x$ can depend on the field $ψ (x)$ and its derivatives ( $\nabla ψ$ , $\nabla^{2} ψ$ , etc.) at that same point.

F [ψ (x)] = \int d^{d} x f (ψ (x), \nabla ψ (x), \dots)

We then apply our symmetry principles to the free energy density $f$ . For an isotropic system like the Heisenberg model, rotational invariance forbids terms linear in the gradient, like $\nabla M$ . The lowest-order term that couples neighboring regions and penalizes sharp changes in the order parameter must be quadratic in the gradient, such as $(\nabla M_{x})^{2} + (\nabla M_{y})^{2} + (\nabla M_{z})^{2}$ .

The following video lecture by Prof. Mehran Kardar provides a masterful walkthrough of this construction. It synthesizes all the principles we've discussed—the order parameter, analyticity, symmetry, and spatial variations—into a coherent framework.

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

This clip from an MIT OpenCourseWare lecture on Statistical Field Theory explains the construction of the Landau-Ginzburg free energy from first principles.

Please watch from 1:02:59 to 1:23:23. Focus on how Prof. Kardar systematically builds the free energy functional. He argues for: The choice of the magnetization field M(\mathbf{r}) as the relevant degree of freedom. The representation of the total free energy as an integral of a local density. The expansion of this density in powers of the field M and its gradients. The use of rotational and inversion symmetries to determine the allowed terms (M^2, M^4, (\nabla M)^2, etc.).

By the end of the video segment, you will have seen the derivation of the canonical Landau-Ginzburg free energy for a system with a scalar order parameter with up-down ( $m \to - m$ ) symmetry:

F [m (x)] = \int d^{d} x [f_{0} + \frac{a}{2} m^{2} (x) + \frac{b}{4} m^{4} (x) + \frac{κ}{2} (\nabla m (x))^{2} - hm (x)]

Here, $h$ is the external field that couples linearly to the order parameter. Each term has a physical meaning: the $m^{2}$ and $m^{4}$ terms form the local potential, the $(\nabla m)^{2}$ term represents the energy cost of spatial variations (the "stiffness"), and the $hm$ term is the energy of alignment with an external field.

To solidify your understanding of this construction process from a textual perspective, the following resource lays out the same arguments with exceptional clarity.

Statistical Field Theory

This text from David Tong's notes is a perfect complement to the video. It formalizes the concepts of coarse-graining and the free energy functional, and then lists the physical constraints used to construct it.

First, read section 1.3, "Landau-Ginzburg Theory", which introduces the concept of the local order parameter m(x) and the free energy functional F[m(x)]. Then, read section 1.3.1, "The Landau-Ginzburg Free Energy". This section articulates the key constraints—Locality, Invariance, Symmetry, Analyticity—and uses them to write down the general form of the free energy in Equation (1.28).

Conclusion

Today, we have learned the powerful methodology of Landau theory, a cornerstone of modern condensed matter physics. Instead of wrestling with microscopic Hamiltonians, we have seen how to construct an effective theory for the order parameter based on the universal and robust principles of symmetry and analyticity. This "top-down" approach provides a framework that is independent of specific microscopic details, which is the key to understanding universal behavior near phase transitions.

Key Takeaways:

Landau theory provides a phenomenological description of phase transitions by postulating a free energy functional, $F [ψ]$ , as a power series expansion of an order parameter field, $ψ (x)$ .
The construction of this functional is not arbitrary but is governed by a clear set of principles:
- Analyticity: Justifies the power series expansion for small $ψ$ .
- Symmetry: Forbids terms in the expansion that are not invariant under the symmetry operations of the high-temperature phase.
- Locality: Allows the total free energy to be written as an integral of a local density that depends on $ψ$ and its derivatives at a single point.
This procedure leads to the canonical Landau-Ginzburg free energy, which includes a local potential (e.g., $a m^{2} + b m^{4}$ ) and a gradient term ( $κ (\nabla m)^{2}$ ) that penalizes spatial inhomogeneities.

We have successfully constructed our model. The next logical step is to use it. By minimizing this free energy, we can predict the equilibrium value of the order parameter and explore the nature of the phase transition. In our next lesson, we will do just that, analyzing the minima of the Landau potential to distinguish between continuous (second-order) and discontinuous (first-order) phase transitions, a fundamental classification in the study of critical phenomena.

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