Domain Wall Structure and Energy Calculation

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Hello! Welcome to the next lesson in our exploration of Landau-Ginzburg theory.

In our previous session, we focused on the physics of small fluctuations around the uniform ground state. By making the Gaussian approximation, we were able to calculate the two-point correlation function and derive its famous Ornstein-Zernike form. This gave us a picture of a system teeming with small, random, spatially-correlated deviations from the mean.

Today, we shift our perspective from these perturbative fluctuations to a different kind of spatial variation: a domain wall. Instead of a small wobble around a single ground state, a domain wall is a large, stable structure that interpolates between two different, but equally valid, ground states. Such configurations can only exist below the critical temperature, $T < T_{c}$ , where spontaneous symmetry breaking provides multiple degenerate ground states (e.g., magnetization up and down).

Our goal for this lesson is to use the full, non-linear Landau-Ginzburg functional to calculate the physical properties of a domain wall. We will:

Derive the equation that governs the spatial profile of the order parameter across the wall.
Solve this equation to find the explicit shape and width of the wall.
Calculate the free energy cost associated with creating such a wall.

This exercise is a beautiful application of the principles of variational calculus to a concrete physical problem and reveals deep connections between the structure of these non-uniform states and the critical properties of the system.

1. The Equation of a Domain Wall

Let's begin with the Landau-Ginzburg free energy functional for a one-component order parameter $m (r)$ in the absence of an external field ( $h = 0$ ):

F [m (r)] = \int d^{d} r [\frac{1}{2} a_{0} m^{2} + \frac{1}{4} b_{0} m^{4} + \frac{1}{2} κ (\nabla m)^{2}]

We are interested in the phase where $T < T_{c}$ , which means $a_{0} = a (T - T_{c}) < 0$ . As you know, this gives a "double-well" potential with minima at $m = \pm m_{0}$ , where $m_{0} = - a_{0} / b_{0}$ .

A domain wall is a static configuration that minimizes this free energy. Since you are familiar with the principle of least action from classical field theory, you'll know that such a configuration must satisfy the Euler-Lagrange equation, $δ F / δ m (r) = 0$ .

Test your understanding!

Apply the functional derivative to the Landau-Ginzburg functional $F [m (r)]$ to derive the general time-independent Euler-Lagrange equation for the order parameter field $m (r)$ .

Show answer

Using the calculus of variations, we find:

\frac{δ F}{δ m ( r )} = a_{0} m (r) + b_{0} m (r)^{3} - κ \nabla^{2} m (r) = 0

This is a non-linear partial differential equation.

To make the problem tractable, we'll consider the simplest possible domain wall: a flat plane separating a region with $m = - m_{0}$ from a region with $m = + m_{0}$ . We can align our coordinate system so that the order parameter only varies along the $x$ -direction. Thus, $m (r) = m (x)$ , and the Laplacian $\nabla^{2}$ becomes $d^{2} / d x^{2}$ .

Our equation of motion simplifies to a non-linear ordinary differential equation (ODE):

κ \frac{d ^{2} m}{d x ^{2}} = a_{0} m + b_{0} m^{3}

We need to solve this equation subject to the boundary conditions that define the domain wall:

$m (x) \to - m_{0}$ as $x \to - \infty$
$m (x) \to + m_{0}$ as $x \to + \infty$

This setup is described clearly in the following resource.

[PDF] Statistical Field Theory - DAMTP

Please read the beginning of the section on domain walls from David Tong's lecture notes on Statistical Field Theory. This will introduce the setup and state the resulting ODE.

Read the portion of Section 1.3.2 titled 'Domain Walls'. Start from the paragraph beginning 'Suppose that we have T < Tc...' and read until just before the solution is presented in equation (1.32). Note that Tong's notation \alpha_2, \alpha_4, \gamma corresponds to our a_0, b_0, \kappa.

2. The Structure of the Domain Wall

The ODE we derived has a famous solution. The reading below presents it and gives the physical interpretation of its parameters.

[PDF] Statistical Field Theory - DAMTP

Now, continue reading in the same section of Tong's notes to see the solution for the domain wall profile.

Read the paragraph containing equation (1.32). This gives the explicit solution for m(x) and defines the domain wall width.

The solution for the profile of the order parameter across the domain wall is:

m (x) = m_{0} tanh (\frac{x}{ξ _{w}})

Here, $ξ_{w}$ is the width of the domain wall, given by:

ξ_{w} = \frac{- 2 κ}{a _{0}} = \frac{2 κ}{a ( T _{c} - T )}

domain_wall_profile

CAPTION: The profile of the order parameter m(x) for a domain wall located at x=X. The order parameter smoothly interpolates between the two degenerate ground states, -m_0 and +m_0, over a characteristic distance determined by the wall width W (denoted \xi_w in our text). Source: Figure 14 from 'Statistical Field Theory' by D. Tong.

This solution beautifully captures the physics. The $tanh$ function smoothly connects the two ground states. The transition is not instantaneous, as the gradient term $κ (\nabla m)^{2}$ penalizes sharp changes. It's also not infinitely spread out, as the potential term $(\frac{1}{2} a_{0} m^{2} + \frac{1}{4} b_{0} m^{4})$ penalizes deviations from the ground state values $\pm m_{0}$ . The profile $m (x)$ represents the optimal balance between these two competing energy costs.

Notice the crucial relationship between the wall width $ξ_{w}$ and the correlation length $ξ$ we found in the previous lesson. In the low-temperature phase ( $T < T_{c}$ ), the correlation length for fluctuations is $ξ_{T < T_{c}} = κ / (- 2 a_{0})$ . We see that:

ξ_{w} = 2 ξ_{T < T_{c}}

The width of the domain wall is directly proportional to the correlation length. As we approach the critical point from below ( $T \to T_{c}^{-}$ ), $a_{0} \to 0$ , causing both $ξ$ and $ξ_{w}$ to diverge. The domain wall becomes infinitely broad and delocalized, and the distinction between the domains themselves dissolves, which is precisely what we expect at a continuous phase transition.

3. The Energy of a Domain Wall

Creating a domain wall costs energy, because the order parameter must deviate from its minimum-energy value $m_{0}$ within the wall. This energy cost is not proportional to the volume of the system, but rather to the area of the wall. We call this energy per unit area the surface tension of the wall, denoted by $σ$ .

Let's consider a wall with area $A$ lying in the $y - z$ plane. The surface tension is:

σ = \frac{F [ m ( x )] - F [ m _{0} ]}{A} = \int_{- \infty}^{\infty} d x (f (m (x)) - f (m_{0}))

where $f (m)$ is the free energy density:

f (m (x)) = \frac{1}{2} a_{0} m (x)^{2} + \frac{1}{4} b_{0} m (x)^{4} + \frac{1}{2} κ (\frac{d m}{d x})^{2}

and $f (m_{0})$ is the density of the uniform ground state.

To calculate this integral, we substitute our solution $m (x) = m_{0} tanh (x / ξ_{w})$ . The calculation is a bit lengthy but provides a very important physical result. A key simplification arises from an argument analogous to the equipartition of energy in mechanics. For the configuration that solves the equation of motion, the integrated contribution from the gradient term is exactly equal to the integrated contribution from the potential term. We can therefore write:

σ = \int_{- \infty}^{\infty} κ (\frac{d m}{d x})^{2} d x

Let's carry this out:

First, we find the derivative: $\frac{d m}{d x} = \frac{m _{0}}{ξ _{w}} sech^{2} (\frac{x}{ξ _{w}})$ .
Then we form the integral for $σ$ : $σ = κ \int_{- \infty}^{\infty} (\frac{m _{0}}{ξ _{w}})^{2} sech^{4} (\frac{x}{ξ _{w}}) d x$
This integral can be evaluated using standard methods (e.g., the substitution $u = tanh (x / ξ_{w})$ ). The result of the definite integral $\int_{- \infty}^{\infty} sech^{4} (x / ξ_{w}) d x$ is $\frac{4}{3} ξ_{w}$ .
Putting it all together, the surface tension is: $σ = κ \frac{m _{0}^{2}}{ξ _{w}^{2}} (\frac{4}{3} ξ_{w}) = \frac{4}{3} \frac{κ m _{0}^{2}}{ξ _{w}}$

Now, let's substitute our expressions for $m_{0} = - a_{0} / b_{0}$ and $ξ_{w} = - 2 κ / a_{0}$ :

σ = \frac{4}{3} \frac{κ ( - a _{0} / b _{0} )}{- 2 κ / a _{0}} = \frac{4 κ}{3 b _{0}} \frac{( - a _{0} )}{∣ a _{0} ∣ ^{- 1/2} 2} = \frac{2 2}{3} \frac{κ}{b _{0}} (- a_{0})^{3/2}

Since $a_{0} = a (T - T_{c})$ , we have our final result for the temperature dependence of the surface tension:

σ (T) \propto (T_{c} - T)^{3/2}

This is a very significant result. It tells us that the energy cost to create domains decreases as we approach the critical point, and vanishes exactly at $T_{c}$ . This is another piece of the critical puzzle: at the critical point, it costs no energy to create interfaces, allowing fluctuations and domains of all length scales to form, which is the hallmark of criticality.

The parametric dependence of this result is also discussed in the last part of the section you read from Tong's notes.

[PDF] Statistical Field Theory - DAMTP

Finally, read the last paragraph of the section on domain walls. It provides a quick scaling argument to arrive at the parametric dependence of the wall's free energy, which matches our detailed calculation.

Read the paragraph following equation (1.32), starting with 'We can also compute the cost in free energy...'. The expression (1.33) shows the parametric dependence of the wall energy. Verify that it matches our result, recalling the correspondence between notations.

Conclusion

Key Takeaways:

Below $T_{c}$ , the Landau-Ginzburg equations admit stable, non-uniform solutions called domain walls, which interpolate between the degenerate ground states.
The spatial profile of a simple 1D domain wall is given by $m (x) = m_{0} tanh (x / ξ_{w})$ .
The width of the domain wall, $ξ_{w} = - 2 κ / a_{0}$ , is proportional to the correlation length $ξ$ and diverges as $T \to T_{c}^{-}$ .
The free energy cost per unit area of a domain wall, its surface tension $σ$ , is finite below $T_{c}$ but vanishes at the critical point as $σ \propto (T_{c} - T)^{3/2}$ .

Today's lesson demonstrates the power of the Landau-Ginzburg functional beyond describing uniform phases or small fluctuations. It allows us to calculate the structure and energetics of the topological defects—the domain walls—that separate the ordered domains.

Preview of the Next Lesson:

We have now analyzed both small fluctuations (the Gaussian approximation) and large, non-linear structures (domain walls). We've seen that mean-field theory (which corresponds to ignoring fluctuations) gives us a basic picture, but the fluctuations are clearly important, especially near $T_{c}$ . A crucial question remains: under what conditions is it actually valid to use mean-field theory and ignore these fluctuations? In our next lesson, we will answer this by deriving the Ginzburg criterion, which provides a precise condition for the validity of mean-field theory by comparing the magnitude of fluctuations to the mean value of the order parameter itself.