From Landau-Ginzburg to Free Energy: Exploring Spatial Variations

Introduction

In our previous lesson, we explored the superfluid transition in Helium-4 by extending Landau theory to a complex order parameter, $ψ$ . This allowed us to describe the onset of the ordered phase through spontaneous breaking of a continuous U(1) symmetry. However, our model was built on a critical simplification: we assumed the order parameter $ψ$ was uniform throughout the system. This assumption prevents us from describing a vast range of crucial physical phenomena, such as the flow of supercurrents, the structure of interfaces between different phases, or the existence of topological defects like vortices.

Today, we will remove this limitation. Our objective is to construct the Landau-Ginzburg free energy functional by introducing spatial variations of the order parameter. We will allow the order parameter to be a field, $ψ (r)$ , that can change from point to point. This generalization transforms the Landau free energy from a simple function into a functional—a function of a function—and elevates the theory into a powerful framework for studying inhomogeneous systems and fluctuations, which are the essence of critical phenomena.

These plots from a numerical simulation illustrate a scenario our previous theory could not handle: an order parameter, $|\psi|$, that is highly non-uniform in space. It is zero almost everywhere, except in two localized, vortex-like structures. Our goal is to build a theoretical framework that can describe such configurations and determine their energy.

From a Uniform Parameter to a Field

The central idea of Landau-Ginzburg theory is to promote the order parameter from a single value to a continuous field. For a magnet, this would be the magnetization field $m (r)$ ; for a superfluid, it's the macroscopic wavefunction $ψ (r)$ .

It's crucial to understand what this field represents. It is not the state of a single microscopic constituent (like an individual spin). Instead, it's a coarse-grained average over a mesoscopic volume—a region small enough to be treated as a "point" $r$ on a macroscopic scale, but large enough to contain a great number of microscopic degrees of freedom. This averaging process is what allows us to treat $m (r)$ or $ψ (r)$ as a smooth, continuous variable.

The following videos provide excellent explanations of this coarse-graining procedure and the motivation for introducing an order parameter field.

Landau Ginzburg theory of Phase Transitions

This clip from a lecture by Jos Thijssen clearly explains the concept of a mesoscopic average that defines the order parameter field.

Please watch the segment from the beginning of this clip. Note the distinction between the order parameter field M(\mathbf{r}) and the individual microscopic spins.

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

In this excerpt from an MIT OpenCourseWare lecture, Prof. Mehran Kardar provides a detailed motivation for the Landau-Ginzburg approach, defining the order parameter field and discussing the generality of the model.

Watch the segment from this part of the lecture. Pay attention to how the single theory can describe different physical systems by varying the spatial dimension D and the number of order parameter components n. For our superfluid example, we have D=3 and n=2 (real and imaginary parts of \psi).

Because the order parameter is now a field, the total free energy of the system, $F$ , depends on the entire spatial configuration of this field. It is a functional, which we denote by $F [ψ (r)]$ . The standard approach is to express this functional as an integral of a local free energy density over the entire volume of the system:

F [ψ (r)] = \int f_{L G} (r) d^{d} r

Our task is to determine the form of this local density, $f_{L G}$ .

Constructing the Landau-Ginzburg Functional

The free energy density $f_{L G}$ must depend on the order parameter field $ψ (r)$ and its spatial derivatives. We construct it by extending the polynomial expansion from Landau theory, guided by the same principles of symmetry and analyticity.

Local Energy: First, the density must include the original Landau terms, which describe the energy cost of having a non-zero order parameter. For our complex order parameter, this is the familiar "sombrero" potential:
$f_{l oc a l} (ψ (r)) = a (T) ∣ ψ (r) ∣^{2} + \frac{b}{2} ∣ ψ (r) ∣^{4}$
Gradient Energy: A new contribution must arise from the spatial variations themselves. A non-uniform order parameter implies, for example, that neighboring spins in a magnet are not perfectly aligned, which costs exchange energy. In a superfluid, a gradient of the phase corresponds to a kinetic energy of flow. Therefore, we must add a term that penalizes gradients. This term must be:
- Lowest order: We consider the simplest possible gradient terms.
- Rotationally invariant: In an isotropic system, the energy should not depend on the direction of the gradient, only its magnitude. This suggests terms like $(\nabla ψ) \cdot (\nabla ψ)$ .
- Symmetry compliant: The term must respect the symmetries of the order parameter. For our U(1) symmetric superfluid, it must be invariant under the transformation $ψ \to ψ e^{i ϕ}$ .

The simplest term that satisfies all these conditions is $∣\nabla ψ ∣^{2} = \nabla ψ^{*} \cdot \nabla ψ$ . Higher-order terms like $(\nabla^{2} ψ)^{2}$ or $∣\nabla ψ ∣^{4}$ are less relevant for describing long-wavelength variations near the critical point.

Combining these parts, the Landau-Ginzburg free energy functional for a complex order parameter $ψ (r)$ in the presence of a conjugate external field $h (r)$ is:

F [ψ] = \int d^{d} r [a (T) ∣ ψ ∣^{2} + \frac{b}{2} ∣ ψ ∣^{4} + g ∣\nabla ψ ∣^{2} - (h^{*} ψ + h ψ^{*})]

The new coefficient $g$ must be positive ( $g > 0$ ) to ensure that the uniform state is energetically favored and the system is stable against short-wavelength fluctuations. It represents the "stiffness" of the order parameter against spatial variations.

The following resources provide concise and rigorous derivations of this functional.

[PDF] Chapter 4 - The Ginzburg-Landau Theory - staff.uni-mainz.de

This text from a course at the University of Mainz provides a clear and formal construction of the Landau-Ginzburg functional. It is written for a scalar order parameter m, but the logic is identical.

Please read the section "Ansatz" on the first page. Note how the author justifies the need for a new term to penalize spatial variations and introduces the gradient term based on a lowest-order expansion that respects the required symmetries.

[PDF] Chapter 2 Ginzburg-Landau Phenomenology

This chapter from a textbook by Ben Simons at Cambridge University also presents the construction based on fundamental symmetries.

Read the introduction in Section 2.1, which discusses coarse-graining and the fundamental symmetries of locality and symmetry. Then, see how these principles lead directly to the Ginzburg-Landau Hamiltonian in Eq. (2.1). Note that \beta \mathcal{H} is used here instead of F, but the structure is the same. The parameter K corresponds to our g.

The Physical Meaning of the Functional

Now that we have constructed the functional, it is important to reflect on its physical interpretation. This is a point of subtlety where your mathematical background will be particularly helpful. The Landau-Ginzburg functional $F [ψ]$ is not, strictly speaking, the thermodynamic free energy of a system with a fixed field configuration $ψ (r)$ .

Instead, it is best understood as an effective Hamiltonian or a coarse-grained free energy. When we integrated out the microscopic degrees of freedom (e.g., the positions of all individual helium atoms) to obtain the smooth field $ψ (r)$ , we implicitly included their entropic contributions into the phenomenological coefficients $a$ , $b$ , and $g$ .

The full statistical mechanics of the system is recovered by considering all possible fluctuations of the order parameter field. The partition function $Z$ is given by a functional integral over all possible configurations of $ψ (r)$ , weighted by the Boltzmann factor of the Landau-Ginzburg functional:

Z = \int D ψ (r) e^{- F [ψ (r)] / k_{B} T}

This expression is formally identical to the path integral formulation of quantum field theory, which you are encountering in your QFT course. The Landau-Ginzburg functional plays the role of the classical action.

The mean-field theory we have been using corresponds to the simplest possible approximation of this integral: the saddle-point approximation. In this approximation, we assume the integral is completely dominated by the single field configuration $ψ_{0} (r)$ that minimizes the functional $F [ψ]$ . All other configurations (the fluctuations) are ignored. Finding this minimum involves the calculus of variations, a topic we will explore in the next lesson.

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

Prof. Kardar's lecture provides a deep insight into this interpretation, connecting the construction to the central limit theorem and the functional integral.

Please watch the segment from this section. The key points are: The justification for an analytic expansion in \psi comes from the coarse-graining process, which involves summing many random variables, invoking the central limit theorem (CLT justification). The full partition function is expressed as a functional integral over all field configurations, weighted by the effective Hamiltonian (functional integral).

Conclusion

In this lesson, we have taken a major step forward by generalizing Landau theory to describe spatially inhomogeneous systems. By allowing the order parameter to become a field $ψ (r)$ , we moved from a simple free energy function to a free energy functional.

Key Takeaways:

The Landau-Ginzburg theory describes systems using a coarse-grained order parameter field, such as $ψ (r)$ .
To account for the energy cost of non-uniform states (like domain walls or currents), a gradient term ( $g ∣\nabla ψ ∣^{2}$ ) is added to the Landau free energy density.
The resulting Landau-Ginzburg functional, $F [ψ] = \int (a ∣ ψ ∣^{2} + \frac{b}{2} ∣ ψ ∣^{4} + g ∣\nabla ψ ∣^{2}) d^{d} r$ , is an effective Hamiltonian for the order parameter field.
The complete statistical description of the system involves a functional integral over all possible field configurations, connecting statistical mechanics to the path integral formalism of field theory.

Our work so far has been to construct the functional. The next step is to use it. In the upcoming lesson, we will apply the calculus of variations to minimize $F [ψ]$ , which will yield the Euler-Lagrange equation for the order parameter field. Solving this equation will allow us to find the profile of an interface and, most importantly, to derive the Ornstein-Zernike form of the correlation function and extract the behavior of the correlation length, $ξ$ . This will be the key to understanding fluctuations and the limits of mean-field theory itself.

Recommended resources

Introduction

From a Uniform Parameter to a Field

Constructing the Landau-Ginzburg Functional

The Physical Meaning of the Functional

Conclusion