Landau Free Energy Expansion from Symmetry

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome to the fifth module of our course.

In the previous module, we concluded by examining the limitations of mean-field theory. We saw that its failure to predict correct critical exponents stems from its core approximation: replacing local, fluctuating interactions with a static, average field. This neglect of fluctuations is particularly problematic near a critical point where correlations become long-ranged.

This brings us to a pivotal question: how can we build a more general theory of phase transitions? The answer lies in the Landau theory, a powerful phenomenological framework that is the subject of this module. Instead of starting from a microscopic Hamiltonian, Landau's approach begins with the most general principles that must govern any phase transition, chief among them being symmetry.

Today's lesson will focus on the very foundation of this approach. Our goal is to construct the Landau free energy expansion based on symmetry arguments for a given phase transition. We will learn how to identify the correct order parameter and use the symmetries of the system to determine the allowed terms in the free energy expansion. This method is remarkably powerful, often allowing us to predict the nature of a phase transition without knowing any microscopic details.

1. The Philosophy of Landau Theory

Landau's approach is a "top-down" or phenomenological one. It rests on a few key ideas:

The Order Parameter: We characterize the transition by an order parameter, which we'll denote by $η$ . You're already familiar with this concept from previous lessons (magnetization $m$ for a ferromagnet, density difference $ρ_{l} - ρ_{g}$ for a fluid). In general, the order parameter is a quantity that is zero in the high-temperature, more symmetric phase (the disordered phase) and non-zero in the low-temperature, less symmetric phase (the ordered phase). The change in the order parameter reflects the spontaneous symmetry breaking that occurs at the transition.
The Free Energy Expansion: Landau postulated that near a phase transition, the appropriate thermodynamic potential (e.g., the Helmholtz or Gibbs free energy) can be expanded as a power series in the order parameter $η$ . This is justified for a continuous (second-order) transition, where $η$ can be made arbitrarily small by approaching the critical temperature $T_{c}$ .
Symmetry Constraints: The crucial insight is that the free energy is a scalar quantity and must be invariant under all symmetry operations of the high-temperature (disordered) phase. This fundamental requirement severely restricts the mathematical form of the power series expansion.

Let's begin by exploring the general framework that connects symmetry, the order parameter, and the free energy.

Landau Theory From Symmetry

This video from Jonathon Riddell's channel provides a somewhat abstract but powerful introduction to how symmetry guides the construction of Landau theory. It frames the transition in terms of a change in a general 'density' field and its associated symmetry group.

Watch the segments from 00:31 to 04:30 and 19:41 to 21:03. The video uses 'rho' (ρ) for the density/symmetry and 'eta' (η) or 'psi' (ψ) for the order parameter. Focus on the core idea: the state of the system is described by a field (ρ), a phase transition involves a change in this field (ρ = ρ₀ + δρ), and this change can be characterized by an order parameter that quantifies the deviation from the high-symmetry state.

2. Constructing the Expansion: The Role of Invariance

Let's now apply this philosophy to construct the free energy, which we'll call $F (T, η)$ . We expand it in a power series around $η = 0$ :

F (T, η) = F_{0} (T) + a_{1} (T) η + a_{2} (T) η^{2} + a_{3} (T) η^{3} + a_{4} (T) η^{4} + \dots

Now, we use symmetry to eliminate terms.

The Linear Term ( $a_{1} η$ ): The free energy must have an extremum at $η = 0$ in the symmetric phase, so we must have $\frac{\partial F}{\partial η} ∣_{η = 0} = 0$ . This immediately implies that $a_{1} = 0$ . The only exception would be if an external field that couples linearly to the order parameter is present, which we will consider in a later lesson.
Symmetry under Inversion: Consider a system like the Ising model. The disordered (paramagnetic) phase is symmetric under time-reversal, which means flipping all spins. This operation corresponds to changing the sign of the magnetization, $m \to - m$ . Therefore, the free energy must be invariant under this transformation: $F (m) = F (- m)$ . For this to be true for our power series, all coefficients of odd powers of $η$ must be zero.

This leaves us with the canonical form of the Landau free energy for a system with this "Ising-like" up/down symmetry:

F (T, η) = F_{0} (T) + A (T) η^{2} + B (T) η^{4} + O (η^{6})

where we have renamed $a_{2} \to A$ and $a_{4} \to B$ . For stability, we need $B (T) > 0$ near the transition. We truncated the series at the fourth order because it is the minimal form needed to describe a phase transition. This form should look familiar: it's precisely what you get if you expand the mean-field free energy (e.g., from the Bragg-Williams approximation) for small magnetization.

To see this connection explicitly and solidify the symmetry argument, please read the first section of the following notes.

Chapter 4 - The Ginzburg-Landau Theory

These lecture notes from the University of Mainz begin by deriving the Landau expansion from the specific Bragg-Williams mean-field theory, and then generalize the result by showing it can be obtained directly from symmetry considerations.

Please read section 4.1.1, 'Ising symmetry'. Notice how the expansion of the mean-field free energy naturally yields only even powers of m, and how the author then states that this form is generally required for any system with the m -> -m symmetry.

3. The Landau Rule: Cubic Terms and First-Order Transitions

What happens if the system's high-temperature phase does not possess a symmetry that forces odd-powered terms to vanish? In this case, a cubic term, $C (T) η^{3}$ , is generally allowed in the expansion.

Its presence has a profound consequence, which is often called the Landau rule:

If the Landau expansion contains a cubic term, the phase transition will be first-order.

The absence of a cubic term is thus a necessary condition for a transition to be continuous (second-order).

To understand why this is the case, please read the following material.

[PDF] Lecture 3. Symmetry change at a phase transition. Landau theory.

This resource from MIT OpenCourseWare provides a rigorous discussion of how symmetry dictates the type of phase transition. It introduces the key concept of 'cubic invariants' and their role.

Please read the section titled '2. Transition type deduced from symmetry'. Focus on the central statement: if cubic invariants are allowed, the transition is type I (first-order). Follow the contrasting examples given: the ferromagnetic transition, where time-reversal symmetry forbids a cubic term, and crystallization, where no such symmetry exists and the transition is first-order.

As explained in the notes, the presence of a non-zero cubic term creates an asymmetry in the free energy landscape. This leads to a discontinuous jump in the order parameter from $η = 0$ to a finite value, which is the definition of a first-order transition. We will analyze the mechanics of this in more detail in a future lesson on $ϕ^{6}$ theory.

The notes from Uni Mainz (LINK), in section 4.1.2 which you can review, also provide a clear graphical illustration of how a cubic term leads to a first-order transition.

4. Applications to Multi-Component Order Parameters

The power of Landau's symmetry-based approach truly shines when we consider more complex systems where the order parameter is not a simple scalar. Your background in condensed matter physics makes these examples particularly relevant. The principle remains the same: the free energy must be a scalar quantity, formed by combinations of the order parameter components that are invariant under the symmetry group of the disordered phase.

Let's look at a few examples, all of which are covered excellently in the Mainz lecture notes.

Chapter 4 - The Ginzburg-Landau Theory

This resource demonstrates how to apply the symmetry principle to construct Landau expansions for several important physical systems with vector or tensor order parameters.

Read section 4.2, 'Landau theory in systems with multicomponent order parameter'. Pay close attention to the following examples: Heisenberg Model (4.2.1): The order parameter is a 3D vector \vec{m}. The disordered phase has full rotational O(3) symmetry. Why must the free energy depend only on scalar quantities like \vec{m}^2 = m_x^2+m_y^2+m_z^2? Cubic Anisotropy (4.2.2): The crystal lattice reduces the symmetry from O(3) to the cubic group. This allows a new fourth-order invariant, m_x^4 + m_y^4 + m_z^4. See how this is incorporated into the expansion. Liquid Crystals (4.2.5): The order parameter is a tensor, Q_{\alphaeta}. The free energy is expanded in terms of rotational invariants like Tr(Q^2) and Tr(Q^3). Notice the presence of the cubic term and what it immediately implies about the isotropic-nematic transition.

These examples demonstrate the versatility of the method. By simply listing the symmetries of the high-temperature phase and finding the corresponding polynomial invariants of the order parameter, we can write down the general form of the free energy and make powerful predictions. The prediction that the isotropic-nematic liquid crystal transition is first-order, based on the existence of the Tr(Q³) invariant, is a classic success of Landau theory.

Test your understanding!

Consider a 2D system on a square lattice. The order parameter is a two-component vector $m = (m_{x}, m_{y})$ . The high-temperature phase has the symmetries of a square. In addition to the rotationally invariant term $(m^{2})^{2} = (m_{x}^{2} + m_{y}^{2})^{2}$ , what is another fourth-order term allowed by the square symmetry that is not allowed by full rotational symmetry? (Hint: Think about what polynomials in $m_{x}$ and $m_{y}$ are left unchanged by 90-degree rotations and reflections across the x/y axes).

Show answer

A 90-degree rotation transforms $(m_{x}, m_{y}) \to (- m_{y}, m_{x})$ . A term like $m_{x}^{4}$ is not invariant under this transformation. However, the combination $m_{x}^{4} + m_{y}^{4}$ is invariant: $m_{x}^{4} + m_{y}^{4} \to (- m_{y})^{4} + m_{x}^{4} = m_{y}^{4} + m_{x}^{4}$ . This term is also invariant under reflections. Therefore, a term proportional to $m_{x}^{4} + m_{y}^{4}$ is allowed in the Landau expansion. The full fourth-order part would be $B_{1} (m_{x}^{2} + m_{y}^{2})^{2} + B_{2} (m_{x}^{4} + m_{y}^{4})$ . The relative values of $B_{1}$ and $B_{2}$ would determine whether the spins prefer to align along the axes or the diagonals.

Conclusion

Key Takeaways:

Landau theory is a phenomenological framework that describes phase transitions by expanding the free energy in powers of an order parameter.
The expansion is not arbitrary; its form is dictated by the symmetry of the system's high-temperature (disordered) phase. The free energy must be invariant under all symmetry operations of that phase.
For systems with an "Ising-like" symmetry ( $η \to - η$ ), odd powers of the order parameter are forbidden, leading to the standard $F \approx A η^{2} + B η^{4}$ form.
Landau's Rule: If the system's symmetry allows a cubic term ( $η^{3}$ ), the phase transition is generally first-order. The absence of a cubic term is a necessary condition for a continuous transition.
This principle extends to multi-component order parameters (vectors, tensors), where the free energy is constructed from polynomial invariants (e.g., $m^{2}$ , Tr( $Q^{3}$ )) under the relevant symmetry group (e.g., O(3), cubic group).

Preview of the Next Lesson:

We have now learned how to construct the Landau free energy. In our next lesson, we will put it to use. We will analyze the simplest form, the $ϕ^{4}$ theory ( $F = A η^{2} + B η^{4}$ ), to describe a continuous phase transition in detail. We will derive the temperature dependence of the order parameter and see how this powerful, general theory recovers familiar results.