Constructing the Landau-Ginzburg Free Energy with Spatial Gradients

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The freely available resources this lesson will direct you to.

Hello! Welcome back to our study of fluctuations and correlations near critical points.

In our last lesson, we uncovered a profound connection: the fluctuation-dissipation theorem, which relates the macroscopic susceptibility $χ$ to the integral of the microscopic correlation function $G_{c} (r)$ . This explained why a diverging correlation length $ξ$ leads to a diverging susceptibility. However, our discussion of Landau theory so far has been based on a spatially uniform order parameter, which is fundamentally unable to describe the very spatial correlations we've been studying.

Today, we will remedy this. Our learning goal is to construct the Landau-Ginzburg free energy functional by including a spatial gradient term to penalize inhomogeneities. This is a pivotal step that elevates Landau theory from a simple model of uniform phases into a powerful field theory capable of describing fluctuations, interfaces, and the rich spatial textures of systems near criticality.

1. From a Uniform Parameter to a Fluctuating Field

Our previous analyses, both in the mean-field theory of the Ising model and in the basic Landau theory, treated the order parameter $m$ as a single number describing the entire system. This is a good approximation deep within an ordered phase, but it breaks down near a critical point where fluctuations at all length scales become important. We saw this vividly in the phenomenon of critical opalescence, where density fluctuations create a cloudy appearance.

To capture this, we must allow the order parameter to vary in space. We replace the scalar $m$ with an order parameter field $m (r)$ .

It's crucial to understand what this field represents. It is not the microscopic value at a point (e.g., a single spin). Instead, it's a coarse-grained or mesoscopic average. Imagine dividing your system into small blocks. Each block is large enough to contain many microscopic constituents (atoms, spins), but small enough compared to the correlation length $ξ$ . The value of $m (r)$ is the average order parameter within the block centered at position $r$ .

The following short video provides an excellent, clear visualization of this coarse-graining concept.

Landau Ginzburg theory of Phase Transitions

To begin, watch this brief explanation from a lecture by Jos Thijssen. It clearly introduces the idea of promoting the order parameter to a spatially-dependent field, m(r), and defines it as a 'mesoscopic average'.

Watch the segment from 13:10 to 15:12. Focus on the conceptual leap from a uniform magnetization M to a spatially varying field m(r) and the definition of the mesoscopic blocks.

This coarse-graining procedure effectively "integrates out" the short-wavelength fluctuations within each block, leaving us with an effective theory for the smooth, long-wavelength fluctuations that dominate near the critical point.

2. Constructing the Functional: Symmetry and Stiffness

Now, how do we write down the free energy for a given configuration of this field, $m (r)$ ? We are looking for a functional, $F [m (r)]$ , which takes the entire function $m (r)$ as input and returns a single number—the total free energy.

Our guiding principles will be the same as in the original Landau theory:

Analyticity: The functional can be expanded in a series of powers of $m$ and its derivatives.
Symmetry: The functional must be invariant under the symmetry operations of the high-temperature phase (e.g., $m \to - m$ for the Ising model, and spatial rotations/translations for an isotropic system).

The following reading guides you through the logical construction of this functional.

Chapter 4 - The Ginzburg-Landau Theory

This document from the University of Mainz lays out the argument for constructing the Ginzburg-Landau functional with perfect clarity. It starts with a naive (and flawed) approach and then introduces the crucial new term based on physical reasoning.

Please read section 4.3.1, titled 'Ansatz'. It's very short, less than half a page. Pay close attention to the 'Problematic' aspect of the naive ansatz and how the 'New Ansatz' solves it by penalizing spatial variations.

Let's summarize and elaborate on the key steps from that reading.

Step 1: The Local Contribution

The most straightforward starting point is to assume the total free energy is simply the integral of the local Landau free energy density over the whole volume. For a simple $m^{4}$ theory (like the Ising model symmetry), this would be:

F_{local} [m (r)] = \int d^{d} r (\frac{1}{2} a_{0} m (r)^{2} + \frac{1}{4} b_{0} m (r)^{4})

(We'll ignore the constant term and the external field for now.)

As the text points out, this is problematic. This functional has no "stiffness". The free energy at point $r$ depends only on $m (r)$ , not on its neighbors. This would allow for infinitely sharp, energetically free variations in the order parameter, which is unphysical. Creating a domain wall, where $m (r)$ changes from $+ m_{0}$ to $- m_{0}$ , must have an energy cost associated with the interface.

Step 2: The Gradient Term

To introduce stiffness, we need a term that couples the order parameter in neighboring regions. In the spirit of a Taylor-like expansion, we look for the lowest-order term involving spatial derivatives of $m (r)$ that respects the system's symmetries.

For an isotropic system (the same in all directions), the term must be a scalar.
For a system with $m \to - m$ symmetry, the term must be even in $m$ .

Consider the possibilities:

$\nabla m$ : This is a vector, not a scalar. Ruled out for an isotropic system.
$Δ m = \nabla^{2} m$ : This is a scalar, but it's not guaranteed to be positive, so it doesn't always penalize fluctuations. More importantly, it can be converted to a surface term via integration by parts, which is not what we want for a bulk energy cost.
$(\nabla m)^{2} = (\partial_{x} m)^{2} + (\partial_{y} m)^{2} + (\partial_{z} m)^{2}$ : This is a scalar and is always non-negative. It is zero for a uniform $m$ and positive for any spatial variation. It is the lowest-order term that correctly penalizes inhomogeneities.

Therefore, we add a "gradient-squared" term to our functional. The complete Landau-Ginzburg free energy functional (for a single-component order parameter, in zero field) is:

F [m (r)] = \int d^{d} r [\frac{1}{2} a_{0} m^{2} + \frac{1}{4} b_{0} m^{4} + \frac{1}{2} κ (\nabla m)^{2}]

The new parameter $κ$ is a positive constant that represents the stiffness of the order parameter field and is related to the energy cost of forming an interface.

Test your understanding!

Imagine a system without $m \to - m$ symmetry, such as a phase transition involving a polar molecule that can align. In such a system, would a term like $m (\nabla m)^{2}$ be allowed in the Landau-Ginzburg expansion by symmetry? What about a term like $m^{2} (\nabla m)^{2}$ ?

Show answer

$m (\nabla m)^{2}$ : This term is odd in $m$ (since it contains one power of $m$ ). If the system does not have $m \to - m$ symmetry, odd powers of $m$ are generally allowed. The term is also a scalar. So yes, this term would be allowed by symmetry.
$m^{2} (\nabla m)^{2}$ : This term is even in $m$ and is a scalar. It would be allowed in any system, including one with $m \to - m$ symmetry. It represents a higher-order coupling between the magnitude of the order parameter and its spatial variation. In our minimal expansion, we typically only keep the lowest-order gradient term, $(\nabla m)^{2}$ .

3. The Landau-Ginzburg Functional as an Effective Action

The Landau-Ginzburg functional is one of the most important concepts in the theory of phase transitions. It's best understood as an effective Hamiltonian or action for the long-wavelength modes of the system.

Given your background in QFT, this analogy should be particularly clear. The probability of observing a particular field configuration $m (r)$ is given by a Boltzmann weight:

P [m (r)] \propto e^{- βF [m (r)]}

The total partition function $Z$ is found by summing (integrating) over all possible configurations of the field, a procedure known as a functional integral or path integral:

Z = \int D m (r) e^{- βF [m (r)]}

This is directly analogous to the path integral formulation of quantum mechanics, where one sums over all paths weighted by $e^{i S /ℏ}$ . Here, the LG functional $F [m (r)]$ plays the role of the Euclidean action. This powerful formalism allows us to use all the tools of quantum field theory to study the statistical mechanics of critical phenomena.

The following comprehensive lecture from Prof. Mehran Kardar will summarize and reinforce these ideas, placing them in the broader context of statistical field theory.

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

Now, let's watch this segment from an MIT OpenCourseWare lecture. Prof. Kardar provides a masterful overview of the entire construction process, emphasizing the roles of coarse-graining and symmetry. He explicitly connects this formalism to concepts from field theory, which will resonate with your background.

Please watch from 1:02:59 to 1:24:11. This is a longer segment, but it synthesizes everything we've discussed. Focus on how he justifies the expansion in powers of m and its gradients, the crucial role of symmetry, and the final interpretation of the partition function as a functional integral over all configurations of the order parameter field.

Conclusion

Key Takeaways:

To describe spatial correlations and inhomogeneities, we promote the order parameter $m$ to a coarse-grained field $m (r)$ .
The Landau-Ginzburg free energy is a functional $F [m (r)]$ that assigns an energy to each possible field configuration.
It is constructed as an expansion in powers of $m (r)$ and its gradients, constrained by the symmetries of the system.
The crucial new ingredient is the gradient-squared term, $\frac{1}{2} κ (\nabla m)^{2}$ , which introduces an energy cost for spatial variations, giving the order parameter field "stiffness".
The full functional acts as an effective Hamiltonian or Euclidean action for the order parameter field, and the partition function is a functional integral over all field configurations. The final form is:
$F [m (r)] = \int d^{d} r [f_{0} + \frac{1}{2} a_{0} m^{2} + \frac{1}{4} b_{0} m^{4} - hm + \frac{1}{2} κ (\nabla m)^{2} + \dots]$

Preview of the Next Lesson:

We have successfully constructed our primary tool: the Landau-Ginzburg functional. Now, we must put it to work. In the next lesson, we will use this functional to analyze the behavior of small fluctuations around the mean-field solution. By doing so, we will derive the Ornstein-Zernike form for the correlation function and obtain an explicit expression for the correlation length $ξ$ in terms of the Landau-Ginzburg parameters $a_{0}$ and $κ$ .