Landau Free Energy: A Symmetry-Constrained Expansion

The Landau-Ginzburg Approach: A Physicist's Guess

The core idea of Landau theory is to write down an effective free energy, not by a rigorous microscopic derivation, but by a physically motivated construction. We focus on the most important degree of freedom—the order parameter field—and write down the simplest possible form for its free energy that is consistent with the symmetries of the problem.

Prof. Mehran Kardar provides an excellent overview of this philosophy. He justifies the approach by drawing an analogy to the central limit theorem and emphasizes the role of locality and symmetry in constraining the form of the effective Hamiltonian.

The Landau Recipe

As motivated by the video, Landau's approach can be distilled into a four-step procedure. This recipe allows us to systematically construct the free energy for a vast range of phase transitions.

The lecture notes by Peter D. Olmsted provide a clear and concise summary of this recipe.

Let's now apply this recipe to several key physical systems.

Example 1: The Ising Magnet (Continuous Transition)

The simplest, yet most illustrative, example is a system with Ising symmetry, like a uniaxial ferromagnet or the liquid-gas transition.

• Step 1: Order Parameter. We choose the scalar magnetization, $M$. In the paramagnetic phase ($T>T_c$), $M=0$. In the ferromagnetic phase ($T<T_c$), a spontaneous magnetization $M\ne 0$ appears.

• Step 2 & 3: Symmetry and Expansion. The crucial symmetry is that the physics is unchanged if all spins are flipped. In the absence of an external magnetic field, this means the free energy density, $f(M)$, must be invariant under the transformation $M\to -M$. Therefore, $f(M)$ must be an even function of $M$. The most general polynomial expansion containing only even powers is:

  $$f(M)=f_0(T)+\frac{1}{2}a(T)M^2+\frac{1}{4}c(T)M^4+O(M^6)$$

  We truncate the series at the fourth order. The coefficient $c$ must be positive ($c>0$) to ensure the free energy is bounded from below for large $M$, guaranteeing thermodynamic stability.

• Step 4: Temperature Dependence. We make the standard Landau assumption that only the coefficient of the quadratic term is sensitive to the proximity to the transition. We expand $a(T)$ near $T_c$:

  $$a(T)\approx a_0(T-T_c)$$

  where $a_0>0$. This ensures that for $T>T_c$, $a>0$, and the minimum of $f(M)$ is at $M=0$. For $T<T_c$, $a<0$, and the $M^2$ term favors a non-zero magnetization.

This construction is detailed in the Olmsted notes.

Example 2: First-Order Transitions and Cubic Invariants

What happens if the symmetry of the system does allow for odd-powered terms? The presence of a cubic term fundamentally changes the nature of the transition.

Consider the isotropic-nematic transition in liquid crystals. The molecules are rod-like and have head-tail symmetry ($\hat{\nu }\equiv -\hat{\nu }$). The ordered nematic phase is characterized by a collective alignment, while the isotropic phase has no preferred direction.

• Order Parameter: A simple scalar like $⟨\cos \theta ⟩$ is insufficient. The proper order parameter is a traceless, symmetric tensor:

  $$Q_{\alpha \beta }=S\left(n_{\alpha }{n}_{\beta }-\frac{1}{3}{\delta }_{\alpha \beta }\right)$$

  where $\hat{\mathbf{n}}$ is the director (average alignment axis) and $S$ is the scalar degree of order. $Q_{\alpha \beta }$ is zero in the isotropic phase and non-zero in the nematic phase.

• Symmetry and Expansion: The free energy must be a scalar and invariant under rotations. This means it can only be built from scalar invariants of the tensor $Q$. The possible low-order invariants are $Tr(Q^2)$ and $Tr(Q^3)$. Unlike the Ising case where the $M\to -M$ symmetry forbade odd powers, here the symmetry allows for a cubic term. The expansion is:

  $$f(Q)=f_0+\frac{1}{2}A(T)\text{Tr}(Q^2)-\frac{1}{3}B\text{Tr}(Q^3)+\frac{1}{4}C(\text{Tr}(Q^2){)}^2$$

  The cubic term, $B\text{Tr}(Q^3)$, is permitted and, as it turns out, is responsible for the transition being first-order.

The presence of this cubic term leads to a discontinuous jump in the order parameter at the transition temperature, which is characteristic of a first-order transition.

Visualizing the Transitions

The difference between continuous (second-order) and first-order transitions is beautifully captured by plotting the shape of the Landau free energy.

• Second-order (Continuous): The blue curve shows the potential for an even-powered expansion like $f=\frac{1}{4}c{M}^4$ at $T=T_c$ (where $a=0$). The minimum is uniquely at $M=0$. Just below $T_c$, two new minima will emerge and move continuously away from zero. This corresponds to the smooth onset of magnetization shown in the inset.

• First-order: The red curve shows a potential with a cubic term. At the transition temperature, there are three coexisting minima with the same free energy: one at $M=0$ (the disordered phase) and two at $M\ne 0$ (the ordered phase). This degeneracy allows the system to jump discontinuously from the $M=0$ state to an $M\ne 0$ state, corresponding to the discontinuous jump in the order parameter shown in the inset.

Other Types of Order Parameters

The power of Landau theory is its generality. The same principles apply to systems with more complex order parameters.

• Vector Order Parameter (Heisenberg Model): For a 3D magnet where the spin can point in any direction, the order parameter is a vector $\mathbf{M}$. Rotational symmetry dictates that the free energy can only depend on the scalar invariant $\mathbf{M}\cdot \mathbf{M}=M^2$. This leads to an expansion in even powers of $M$, similar to the Ising model:
  $f(\mathbf{M})=f_0+\frac{1}{2}a(\mathbf{M}\cdot \mathbf{M})+\frac{1}{4}c(\mathbf{M}\cdot \mathbf{M}{)}^2+\dots$
  You can read about this in section 4.2 of the Olmsted notes.

• Complex Order Parameter (Crystals, Superfluids): In systems where a periodic density wave appears, like in the formation of a crystal, the order parameter is the complex Fourier amplitude of the density modulation, $\psi ={\rho }_q$. Translational invariance of the disordered phase requires that the free energy cannot depend on the phase of $\psi$. It must depend only on the modulus, $|\psi {|}^2={\psi }^{\ast }\psi$. This again leads to an even-powered expansion:
  $f(\psi )=f_0+\frac{1}{2}a|\psi {|}^2+\frac{1}{4}b|\psi {|}^4+\dots$
  This is also discussed in section 4.4.1 of the Olmsted notes. This same form, with $\psi$ being the complex wavefunction of the condensate, is used to describe the superfluid transition.

Conclusion

In this lesson, we have established the systematic procedure for constructing the Landau free energy. This phenomenological approach powerfully distills the physics of a phase transition into a simple polynomial, with the allowed terms dictated entirely by the symmetries of the system.

Key Takeaways:

• Landau theory provides an effective free energy as a polynomial expansion in an order parameter, $\psi$.
• The order parameter is a quantity that is zero in the high-symmetry (disordered) phase and non-zero in the low-symmetry (ordered) phase.
• Symmetry is paramount: The free energy must be invariant under all symmetry operations of the disordered phase. This principle determines which powers of $\psi$ are allowed in the expansion.
• For systems with an inversion-like symmetry ($\psi \to -\psi$), only even powers appear, typically leading to a continuous (second-order) transition.
• If the symmetry allows for a cubic invariant, the transition is typically first-order, characterized by a discontinuous jump in the order parameter.
• The transition is driven by the coefficient of the quadratic term, $a(T)$, which is assumed to change sign at $T_c$.

We have now learned how to write down the free energy. In the next lesson, we will put it to use. By minimizing this free energy, we will calculate the behavior of thermodynamic quantities near the critical point and derive the famous mean-field critical exponents.