Landau Expansion with an External Field

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Hello! Welcome back to our exploration of Landau theory.

In our last lessons, we constructed Landau free energy expansions based on symmetry arguments. We saw how a $ϕ^{4}$ theory with a positive quartic term describes a continuous phase transition, while a $ϕ^{6}$ theory can model a first-order transition and allows for a tricritical point. All of this analysis was for systems in the absence of an external field.

Today, we will address a crucial extension of this framework. Our goal is to incorporate an external field into the Landau expansion and derive the equation of state near criticality. The external field is a powerful tool, as it directly couples to the order parameter and explicitly breaks the underlying symmetry of the system. We will see that by adding one simple term to our free energy, we can derive a single, powerful "equation of state" that encapsulates the relationships between temperature, order parameter, and field, and from which all the mean-field critical exponents can be extracted.

1. Including the External Field

Let's return to our canonical example of a ferromagnetic transition, where the order parameter is the magnetization, $m$ . How does an external magnetic field, $h$ , affect the system's free energy?

In thermodynamics, the change in energy due to an external field coupling to its conjugate variable is linear. For a magnetic system, the interaction energy is simply $- h \cdot m$ . This term must be added to the free energy density. For a simple scalar order parameter, our $ϕ^{4}$ Landau free energy becomes:

f (m, T, h) = f_{0} (T) + \frac{1}{2} a (T) m^{2} + \frac{1}{4} b m^{4} - hm

Here, as before, $a (T) = a_{0} (T - T_{c})$ with $a_{0} > 0$ , and we assume $b > 0$ for a continuous transition.

The key new element is the linear coupling term, $- hm$ .

It is linear in $m$ because the field energy is proportional to the magnetization.
It breaks the $m \to - m$ symmetry of the free energy. If $h > 0$ , the free energy is lowered for $m > 0$ , explicitly favoring a state with positive magnetization. This aligns with our physical intuition.

The figure below, adapted from the ideas in B.D. Simons' notes, illustrates this effect. For $T < T_{c}$ (i.e., $a < 0$ ), the symmetric double-well potential at $h = 0$ is tilted by the field. One minimum becomes the stable equilibrium state, while the other becomes a metastable state. For $T > T_{c}$ (i.e., $a > 0$ ), the field shifts the minimum of the single potential well away from $m = 0$ .

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Schematic diagrams of the Landau free energy f(m) from B.D. Simons' notes. On the left (t<0, or T<T_c), the field h breaks the degeneracy of the two minima. On the right (t>0, or T>T_c), the field induces a non-zero magnetization by shifting the minimum of the potential.

2. The Equation of State

The equilibrium state of the system is the one that minimizes the free energy for a given temperature $T$ and field $h$ . To find this minimum, we set the derivative of $f$ with respect to $m$ equal to zero:

\frac{\partial f}{\partial m} = am + b m^{3} - h = 0

Rearranging this gives us the fundamental Landau equation of state:

h = am + b m^{3}

This simple cubic equation is remarkably powerful. It provides a complete mean-field description of the relationship between the external field $h$ , the order parameter $m$ , and the temperature (encoded in the coefficient $a$ ). Let's now use it to re-derive the critical exponents.

The following reading provides a concise derivation of the equation of state and the critical exponents that follow from it.

[PDF] Chapter 2 Ginzburg-Landau Phenomenology

The following chapter from 'Ginzburg-Landau Phenomenology' by B. D. Simons directly addresses today's topic. It starts from the Ginzburg-Landau Hamiltonian, applies the mean-field (saddle-point) approximation, and derives the equation of state and the associated critical exponents.

Please read section 2.2, 'Landau Mean-Field Theory'. Pay close attention to the derivation starting from the minimization of the free energy density f(m, h) to obtain the equation for the average magnetization m̄. Then, follow the calculations for the exponents β, γ, and δ.

3. Analysis of the Equation of State

As the reading demonstrates, we can analyze the equation of state $h = am + b m^{3}$ in different limits to find the critical exponents. Let's summarize the key results.

Spontaneous Magnetization ( $β$ ):
We look for a non-zero solution when the field is turned off ( $h = 0$ ).
$am + b m^{3} = 0 ⟹ m (a + b m^{2}) = 0$
Below $T_{c}$ , we have $a < 0$ . A non-trivial solution exists at $m^{2} = - a / b$ . Since $a \propto (T - T_{c})$ , we get $m \propto T_{c} - T$ , which implies the critical exponent $β = 1/2$ .
Critical Isotherm ( $δ$ ):
We examine the system right at the critical temperature, $T = T_{c}$ , which means $a = 0$ .
The equation of state simplifies to $h = b m^{3}$ .
This gives $m \propto h^{1/3}$ , which implies the critical exponent $δ = 3$ .
Zero-Field Susceptibility ( $γ$ ):
The susceptibility is defined as $χ = \frac{\partial m}{\partial h}_{h = 0}$ . It's often easier to calculate its inverse, $χ^{- 1} = \frac{\partial h}{\partial m}_{h = 0}$ .
Differentiating the equation of state gives:
$\frac{\partial h}{\partial m} = a + 3 b m^{2}$
We evaluate this at the equilibrium magnetization for $h = 0$ .
- For $T > T_{c}$ ( $a > 0$ ), the equilibrium state is $m = 0$ . Thus, $χ^{- 1} = a$ . Since $a \propto (T - T_{c})$ , we have $χ \propto (T - T_{c})^{- 1}$ , giving $γ = 1$ .
- For $T < T_{c}$ ( $a < 0$ ), the equilibrium state is $m^{2} = - a / b$ . Thus, $χ^{- 1} = a + 3 b (- a / b) = a - 3 a = - 2 a$ . Since $a \propto (T - T_{c})$ , we have $χ \propto (T_{c} - T)^{- 1}$ , again giving $γ = 1$ .

This unified derivation confirms the mean-field exponents we previously found through separate analyses. It highlights that the simple Landau free energy expansion contains the complete critical behavior in the mean-field approximation. For a slightly different pedagogical take on the same derivation, you can also consult section 7.2 of the notes by Sidney Redner (Resource ID LINK).

Test your understanding!

In the previous lesson, we saw that at a tricritical point (TCP), the coefficients $a$ and $b$ in the Landau expansion are both zero. Let's analyze the behavior right at this special point.

Start with the $ϕ^{6}$ free energy in the presence of an external field $h$ :
$f (m) = \frac{1}{2} a m^{2} + \frac{1}{4} b m^{4} + \frac{1}{6} c m^{6} - hm$

Derive the equation of state for this model.
Set $a = 0$ and $b = 0$ to describe the TCP. How does the magnetization $m$ scale with the field $h$ at the TCP?
What is the value of the critical exponent $δ$ at the tricritical point ( $δ_{TCP}$ )?

Show answer

Minimizing the free energy by setting $\frac{\partial f}{\partial m} = 0$ gives the equation of state:
$h = am + b m^{3} + c m^{5}$
At the tricritical point, $a = 0$ and $b = 0$ , so the equation of state simplifies dramatically to:
$h = c m^{5}$
This means the magnetization scales with the field as $m = (h / c)^{1/5}$ , or $m \propto h^{1/5}$ .
The critical exponent $δ$ is defined by $m \propto h^{1/ δ}$ at the critical point. By comparing this with our result, we find that at the tricritical point, $δ_{TCP} = 5$ . This is different from the standard mean-field value of $δ = 3$ , confirming that the tricritical point belongs to a different universality class.

Conclusion

Key Takeaways:

An external field $h$ that is conjugate to the order parameter $m$ is incorporated into the Landau free energy via a linear coupling term, $- hm$ .
This term explicitly breaks the symmetry of the potential and selects a preferred state.
Minimizing the free energy with respect to the order parameter yields the equation of state, which for the $ϕ^{4}$ model is $h = am + b m^{3}$ .
This single equation concisely describes the system's behavior near criticality and can be used to derive all the standard mean-field exponents ( $β = 1/2, γ = 1, δ = 3$ ).

Preview of the Next Lesson:

So far, our discussion of Landau theory has been quite abstract, focusing on general properties based on symmetry. We have established a powerful phenomenological framework, but the coefficients $a$ and $b$ have remained generic functions of temperature and other external parameters. In the next lesson, we will bridge this gap between theory and reality. We will apply Landau theory to model a specific physical system: the cubic-to-tetragonal structural phase transition in a crystal. This will involve identifying the correct multi-component order parameter and relating the abstract Landau coefficients to measurable material properties like elastic constants.