Landau Free Energy and Phase Transition Nature

Continuous (Second-Order) Transitions: The Symmetric Case

Let's begin with the canonical model we derived for a system with up-down symmetry ($\psi \to -\psi$), such as an Ising ferromagnet. Ignoring spatial variations for now, the Landau free energy density is:

We established that for the free energy to be bounded from below for large $\psi$, we must have $b>0$. The crucial temperature dependence is captured by the coefficient $a$, which we model as $a(T)=a_0(T-T_c)$ with $a_0>0$. This ensures $a>0$ for $T>T_c$ (the disordered phase) and $a<0$ for $T<T_c$ (the ordered phase).

The equilibrium value of the order parameter, ${\psi }_{eq}$, is found by minimizing $f(\psi )$:

This equation has solutions:

1. $\psi =0$
2. ${\psi }^2=-a/b$

Let's analyze the stability of these solutions:

• For $T>T_c$ ($a>0$): The only real solution is $\psi =0$. The second derivative, $\frac{{\partial }^{2}f}{\partial {\psi }^2}=a+3b{\psi }^2$, is positive at $\psi =0$ (since $a>0$), confirming it is a minimum. The system is in the disordered phase.
• For $T<T_c$ ($a<0$): We now have three real solutions: $\psi =0$ and $\psi =\pm \sqrt{-a/b}$.
  • At $\psi =0$, the second derivative is $\frac{{\partial }^{2}f}{\partial {\psi }^2}=a<0$, so this is now a local maximum. The disordered state is unstable.
  • At $\psi =\pm \sqrt{-a/b}$, the second derivative is $\frac{{\partial }^{2}f}{\partial {\psi }^2}=a+3b(-a/b)=-2a>0$. These are two degenerate minima. The system spontaneously breaks the $\psi \to -\psi$ symmetry and settles into one of these ordered states.

Crucially, as the temperature is lowered towards $T_c$, the order parameter ${\psi }_{eq}=\sqrt{-a_0(T-T_c)/b}$ grows continuously from zero. This smooth onset of order is the defining characteristic of a continuous or second-order phase transition.

The following video provides an excellent visualization and walkthrough of this exact analysis.

The Thermodynamic Signature of Transition Order

Before we explore discontinuous transitions, let's formalize this classification. In thermodynamics, the order of a phase transition is defined by which derivative of the free energy with respect to temperature is the first to be discontinuous at $T_c$.

• First-Order Transition: The first derivative, $S=-(\partial F/\partial T)$, is discontinuous. This implies a finite change in entropy, $\Delta S$, and thus a latent heat, $L=T_c\Delta S$.
• Second-Order Transition: The first derivative (entropy) is continuous, but the second derivative, the heat capacity $C_V=-T({\partial }^{2}F/\partial T^2)$, is discontinuous.

Let's check our symmetric model. The equilibrium free energy is:

where $a=a_0(T-T_c)$. You can verify that the first derivative $(\partial F_{eq}/\partial T)$ is continuous at $T=T_c$ (it's zero from both sides, relative to the background $F_0$), but the second derivative has a finite jump. This confirms our model describes a second-order transition.

First-Order Transitions: The Role of Asymmetry

How can we modify our model to produce a first-order transition? The key is to introduce terms that create a competition between the disordered state ($\psi =0$) and an ordered state with a large order parameter. The simplest way to do this is to break the $\psi \to -\psi$ symmetry by including odd powers in the expansion.

Consider a free energy of the form:

where we assume $b,c>0$ for definiteness. The cubic term, $-\frac{c}{3}{\psi }^3$, explicitly favors positive $\psi$. Such a term is forbidden in systems with up-down symmetry like the Ising model, but it is allowed in others, like the isotropic-nematic transition in liquid crystals, where the order parameter describes molecular alignment.

Let's analyze the minima of this potential. The shape of $f(\psi )$ is now more complex.

• For high temperatures ($a$ large and positive), $f(\psi )$ still has a single minimum at $\psi =0$.
• As $a$ decreases, a second, local minimum appears at a positive value of $\psi$. Initially, this new minimum has a higher free energy than the minimum at $\psi =0$.
• The phase transition occurs at a critical temperature $T_c$ where the free energies of these two minima become equal. At this point, the system can jump from the $\psi =0$ state to the new minimum with $\psi \ne 0$.
• Crucially, this transition happens while the $\psi =0$ state is still a local minimum ($a>0$). The jump to the ordered state is therefore discontinuous.

The following reading details this analysis.

The image below visually contrasts the shape of the free energy potential $G(Q)$ (equivalent to our $f(\psi )$) for the two types of transitions at their respective critical temperatures, $T=T_c$.

First-Order Transitions from Symmetric Potentials: The Tricritical Point

A fascinating prediction of Landau theory is that a first-order transition can occur even in a system that respects $\psi \to -\psi$ symmetry, meaning no odd powers are allowed. This happens if the coefficient of the quartic term, $b$, is negative. Stability then requires a positive sextic term, d/6$$\psi^6.

• If $b>0$, we recover the standard second-order transition as $a$ passes through zero.
• If $b<0$, the quartic term actually encourages a non-zero order parameter. A competition arises between the $a{\psi }^2$ term, which wants $\psi =0$ for $a>0$, and the $b{\psi }^4$ term, which wants $\psi \ne 0$. This leads to a first-order transition, similar to the case with the cubic term.

The point in the $(a,b)$ parameter space where the transition changes from second-order to first-order (i.e., at $a=0,b=0$) is called a tricritical point. This is a beautiful example of how the Landau framework can predict complex phase behavior from simple symmetry arguments and analysis of polynomial coefficients.

Conclusion

Today we have put the Landau free energy to work. By analyzing the minima of the potential as a function of its coefficients, we can predict the fundamental nature of a phase transition. This powerful method of theoretical inquiry reduces a complex many-body problem to the analysis of a simple polynomial.

Key Takeaways:

• The equilibrium state of a system corresponds to the minimum of the Landau free energy $f(\psi )$.
• Continuous (second-order) transitions are characterized by the order parameter growing smoothly from zero below $T_c$. They are typically described by a Landau expansion with only even powers of $\psi$ and a positive quartic coefficient ($b>0$).
• Discontinuous (first-order) transitions involve a sudden jump in the order parameter from zero to a finite value. They can arise from two primary mechanisms within Landau theory:
  1. The presence of odd-powered terms (e.g., a cubic term), which are allowed when the symmetry of the ordered state is not simply related by inversion to the disordered state.
  2. A negative quartic coefficient ($b<0$) in a symmetric potential, which requires a positive sextic term for stability.
• The meeting point of a line of second-order transitions and a line of first-order transitions is a tricritical point.

We have now seen how to construct the Landau model and use it to classify transitions. The next step in our methodological journey is to extract quantitative predictions. For a continuous transition, the order parameter near $T_c$ behaves as $\psi \sim (T_c-T{)}^{\beta }$, where we found the exponent $\beta =1/2$. In the next lesson, we will see how Landau theory allows for the systematic calculation of this and other critical exponents, which are universal quantities that characterize the physics near a second-order phase transition.