Mean-Field Theory of the Ising Model

Hello! Welcome to the fourth module of our course.

In the previous module, we delved into the Ising model, culminating in a look at the exact solution for the two-dimensional case. We contrasted the critical exponents derived from Onsager's solution ($\beta =1/8$, $\gamma =7/4$, etc.) with those from a "mean-field theory" ($\beta =1/2$, $\gamma =1$, etc.). We saw a dramatic disagreement, which highlighted how mean-field approximations can fail to capture the true physics of fluctuations in low dimensions. However, we used these mean-field results without deriving them.

Today, we'll put mean-field theory (MFT) under the microscope. Our goal is to formally derive its central prediction for the Ising model. This lesson addresses the learning outcome: Derive the mean-field equation for the Ising model using the molecular field approximation. We will see precisely where the approximation lies and how it transforms an intractable many-body problem into a solvable one.

1. The Interacting System and the "Molecular Field"

Let's start with the familiar Ising Hamiltonian on a lattice of $N$ sites with coordination number $z$ (the number of nearest neighbors for any given site):

Here, $s_i=\pm 1$, the first sum is over all nearest-neighbor pairs, $J$ is the exchange coupling constant ($J>0$ for ferromagnetism), and the second term represents the interaction with an external magnetic field, which we'll denote as $h={\mu }_B{H}_{ext}$ for simplicity.

The complexity comes from the interaction term $-J\sum s_i{s}_j$. The state of spin $s_i$ is coupled to the state of its neighbors, which are in turn coupled to their neighbors, and so on. To solve this, we need an approximation.

The idea, first proposed by Pierre Weiss, is to focus on a single, representative spin, say at site $k$, and ask: what is the effective magnetic field it experiences? The energy of this single spin due to its interactions is:

We can factor out $s_k$ to make this look like a Zeeman energy term:

This expression shows that the spin $s_k$ feels the external field $h$ plus an additional field from its neighbors, which Weiss termed the molecular field. The key problem is that this molecular field, $J\sum s_j$, is not a static number; it fluctuates as the neighboring spins $s_j$ flip in thermal equilibrium. This is the essence of the many-body problem.

2. The Mean-Field Approximation

Here is the crucial step. We make an approximation that replaces the complex, fluctuating environment of a spin with a simple, static, average environment.

The Approximation: We replace the fluctuating spin variables of the neighbors, $s_j$, with their thermal average value, $⟨s_j⟩$.

For a translationally invariant system (like an infinite lattice or one with periodic boundary conditions), the average value of a spin is the same on every site. We define this average value as the magnetization per spin, $m$.

The sum over the $z$ nearest neighbors of site $k$ thus becomes:

With this, our fluctuating effective field becomes a simple, non-fluctuating mean field, which is the same for every site:

This is the famous molecular field. It consists of the external field plus a contribution proportional to the average magnetization of the system itself.

The following resource provides a very clear and direct derivation of this mean field.

An alternative, slightly more formal way to understand this approximation is by considering the fluctuations directly. This approach is well-explained in the following video.

Both methods arrive at the same result: the complicated, interacting Hamiltonian is replaced by an effective, non-interacting one where each spin simply sees a constant magnetic field $h_{\text{MF}}$.

3. Deriving the Self-Consistency Equation

The mean-field approximation has converted our problem into a familiar one: a system of $N$ non-interacting spins in a uniform effective magnetic field $h_{\text{MF}}$. Since the spins are now decoupled, we can calculate the properties of a single spin and generalize.

The Hamiltonian for a single spin is $H_1=-s\cdot h_{\text{MF}}$. The partition function for this single spin is:

where $\beta =1/(k_{B}T)$.

The thermal average of this spin, which by definition is the magnetization per spin $m$, is calculated as:

This simplifies to the well-known result for a two-state paramagnet:

We are almost done. The final step is to substitute our expression for the mean field, $h_{\text{MF}}=h+Jzm$, back into this equation. This gives us:

This is the mean-field self-consistency equation. It is the central result of our lesson.

Why "self-consistent"? Because the quantity we want to find, $m$, appears on both sides of the equation. The magnetization m on the left is determined by the tanh function, whose argument depends on m itself through the mean field. A physical solution must satisfy this condition—the magnetization that generates the mean field must be the same magnetization that results from the spins aligning in that mean field.

For a final consolidation of these steps, you can refer to the following notes, which provide a compact, mathematical summary of the derivation.

Conclusion

In this lesson, we have formally derived the foundational equation of mean-field theory for the Ising model.

Key Takeaways:

• The complexity of the Ising model arises from the coupling between fluctuating spins.
• The molecular field approximation simplifies this by replacing the fluctuating field from neighboring spins with a static, average mean field.
• This approximation effectively decouples the spins, reducing the many-body problem to a solvable single-body problem of a spin in a constant effective field, $h_{\text{MF}}=h+Jzm$.
• Solving this simplified problem leads to the self-consistency equation for the magnetization: $m=\tanh (\beta (h+Jzm))$.

Preview of the Next Lesson:
We now have a single, albeit transcendental, equation for the magnetization $m$. How does this simple equation lead to a phase transition? In our next lesson, we will solve this self-consistency equation, both graphically and analytically for temperatures near the critical point. We will see how it predicts the onset of spontaneous magnetization below a critical temperature $T_c$ and use it to derive the mean-field critical exponents we've discussed previously.