$\phi^4$ Theory and Continuous Phase Transitions

Welcome to your next lesson in our study of Landau theory.

In our previous session, we established the foundational principle of Landau theory: the form of the free energy expansion near a phase transition is dictated by the symmetries of the high-temperature, disordered phase. For a vast class of systems exhibiting an "up/down" or Ising-like symmetry (where the order parameter $\eta$ can be inverted, $\eta \to -\eta$), we found that the free energy must be an even function of $\eta$. Truncating at the first non-trivial order necessary for stability gives us the renowned ${\varphi }^4$ (or ${\eta }^4$) model:
$F(T,\eta )=F_0(T)+A(T){\eta }^2+B(T){\eta }^4$
Today, we will analyze this expression to achieve our learning goal: to describe a continuous phase transition and derive the temperature dependence of the order parameter. We will see how this simple, symmetry-based form elegantly captures the essence of critical phenomena.

1. The Canonical ${\varphi }^4$ Model

Let's dissect the coefficients $A(T)$ and $B(T)$ to imbue our model with physical meaning.

• The Quartic Term, $B(T)$: For the free energy to be bounded from below as $\eta \to \pm \infty$, we must have $B(T)>0$. A system with $B<0$ would be unstable, with the order parameter growing without limit. Near the critical temperature $T_c$, we can generally assume that $B(T)$ is a slowly varying function of temperature, and it is sufficient to approximate it by a positive constant, $B(T)\approx B>0$.

• The Quadratic Term, $A(T)$: This coefficient is the engine of the phase transition.

  • For $T>T_c$, in the disordered phase, we expect the free energy to have a single minimum at $\eta =0$. This requires the curvature at the origin to be positive, so $A(T)>0$.
  • For $T<T_c$, in the ordered phase, the system develops a non-zero order parameter. This means $\eta =0$ must become unstable (a local maximum), and two new minima must appear at $\eta \ne 0$. This requires $A(T)<0$.

The simplest functional form for $A(T)$ that satisfies these conditions is a linear one that passes through zero at $T=T_c$. We can write this as a Taylor expansion around $T_c$:
$A(T)\approx a(T-T_c)$
where $a$ is a positive constant.

Putting this together, our working model for the free energy density (per unit volume or per site) near a continuous phase transition is:
$f(\eta ,T)=f_0(T)+a(T-T_c){\eta }^2+B{\eta }^4$

2. Minimization and Temperature Dependence

The equilibrium state of the system corresponds to the value of the order parameter, ${\eta }_0$, that minimizes the free energy $f(\eta ,T)$. The following reading from David Tong's lecture notes on Statistical Field Theory provides a concise and clear derivation of this minimization process and its consequences.

Let's summarize the key steps and insights from the reading. To find the minimum, we solve $\frac{\partial f}{\partial \eta }=0$:
$\frac{\partial f}{\partial \eta }=2a(T-T_c)\eta +4B{\eta }^3=\eta \left(2a(T-T_c)+4B{\eta }^2\right)=0$

This equation yields two possible sets of solutions for the equilibrium order parameter ${\eta }_0$.

Case 1: High Temperature Phase ($T>T_c$)

Here, $T-T_c>0$. The only real solution to the minimization equation is ${\eta }_0=0$. The second derivative, $\frac{{\partial }^{2}f}{\partial {\eta }^2}{|}_{\eta =0}=2a(T-T_c)$, is positive, confirming this is a stable minimum. The system is in its symmetric, disordered phase.

Case 2: Low Temperature Phase ($T<T_c$)

Here, $T-T_c<0$. We now have three real solutions:

1. ${\eta }_0=0$. The second derivative is now negative ($2a(T-T_c)<0$), meaning the symmetric state has become a local maximum and is unstable.
2. ${\eta }_0^2=-\frac{2a(T-T_c)}{4B}=\frac{a(T_c-T)}{2B}$. This gives two non-zero solutions:
   ${\eta }_0(T)=\pm \sqrt{\frac{a}{2B}}(T_c-T{)}^{1/2}$
   The second derivative at these points is $4a(T_c-T)>0$, confirming they are the new stable minima.

The system must "choose" one of these two degenerate ground states, spontaneously breaking the $\eta \to -\eta$ symmetry. This change in the free energy landscape is the heart of the Landau description of a continuous phase transition.

3. The Critical Exponent $\beta$

The result from our analysis is one of the most important predictions of Landau theory. For $T<T_c$, the magnitude of the order parameter grows as:
$|{\eta }_0(T)|\propto (T_c-T{)}^{1/2}$

This behavior defines a critical exponent. The exponent $\beta$ describes how the order parameter vanishes as the critical point is approached from the ordered phase:
${\eta }_0(T)\sim (T_c-T{)}^{\beta }\text{as }T\to T_c^{-}$
From our derivation, we find that Landau's ${\varphi }^4$ theory predicts a universal value for this exponent:
$\beta =\frac{1}{2}$
This value is characteristic of all mean-field theories. While we derived it from a general symmetry argument, you will find it also emerges from specific models like the van der Waals gas or the mean-field Ising model. The following video illustrates exactly this, showing how expanding the mean-field free energy for the Ising model naturally leads to the ${\varphi }^4$ form we have been studying.

This connection is crucial: Landau theory provides a powerful, general framework whose simplest application reproduces the results of simpler, model-specific mean-field approximations. Its power lies in its independence from microscopic details, relying only on the system's symmetries.

Conclusion

Key Takeaways:

• The ${\varphi }^4$ Landau theory, $f=f_0+a(T-T_c){\eta }^2+B{\eta }^4$, is the minimal model for describing a continuous phase transition in a system with up/down ($\eta \to -\eta$) symmetry.
• The phase transition is driven by the change in sign of the coefficient of the quadratic term, $A(T)=a(T-T_c)$, which destabilizes the symmetric phase ($\eta =0$) below $T_c$.
• By minimizing the free energy, we derived the temperature dependence of the order parameter in the ordered phase: $|{\eta }_0|\propto (T_c-T{)}^{1/2}$.
• This result defines the mean-field value of the critical exponent $\beta =1/2$.

Preview of the Next Lesson:

We have successfully used the Landau free energy to derive the behavior of the order parameter and find the critical exponent $\beta$. But there are other critical exponents, such as those governing the specific heat ($\alpha$), susceptibility ($\gamma$), and the critical isotherm ($\delta$). In the next lesson, we will continue our analysis of the ${\varphi }^4$ model to demonstrate that Landau theory reproduces the full set of standard mean-field critical exponents.