Mean-Field Theory for the Classical Heisenberg Model

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Hello! Let's continue our journey into mean-field theory.

In our last lesson, we established a rigorous foundation for mean-field theory using the Bogoliubov inequality. We saw that the familiar self-consistency equation for the Ising model isn't just an ad-hoc guess but arises from a systematic variational principle that seeks to find the best non-interacting approximation for the system's free energy.

Today, we will apply this powerful framework to a different, richer model of magnetism: the classical Heisenberg model. This will allow us to explore a crucial question in the study of phase transitions: how does the underlying symmetry of the system's microscopic variables affect its macroscopic critical behavior? While the Ising model's spins are restricted to point "up" or "down", Heisenberg spins are free to point anywhere in three-dimensional space.

Our learning outcome is to apply mean-field theory to the classical Heisenberg model and compare its predictions with the Ising case. This comparison will reveal which aspects of critical phenomena are "universal" within mean-field theory and which are model-dependent.

1. The Classical Heisenberg Model

The Heisenberg model describes a system of interacting magnetic moments, which we will treat as classical 3D vectors $S_{i}$ located on the sites of a lattice. For simplicity, we assume each vector has a fixed magnitude $∣ S_{i} ∣ = S$ . The Hamiltonian for a ferromagnetic system in an external magnetic field $h$ is:

H = - J ⟨ i, j ⟩ \sum S_{i} \cdot S_{j} - h \cdot i \sum S_{i}

Here, $J > 0$ is the exchange coupling constant, and the dot product $S_{i} \cdot S_{j}$ energetically favors the parallel alignment of neighboring spins. The key difference from the Ising model is that the spin variables $S_{i}$ are continuous vectors, not discrete scalars. This gives the order parameter a continuous rotational symmetry (specifically, O(3) symmetry), unlike the discrete up/down ( $Z_{2}$ ) symmetry of the Ising model.

Following the mean-field prescription, we replace the interaction of spin $S_{i}$ with its neighbors by an interaction with an effective "molecular field" generated by the average magnetization, $m = ⟨ S ⟩$ .

j n.n. of i \sum S_{i} \cdot S_{j} \approx j n.n. of i \sum S_{i} \cdot ⟨ S_{j} ⟩ = z S_{i} \cdot m

where $z$ is the coordination number of the lattice. This decouples the spins, leading to the mean-field Hamiltonian:

H_{MF} = - i \sum S_{i} \cdot (z J m + h) = - i \sum S_{i} \cdot h_{eff}

The problem is now reduced to a system of non-interacting classical spins in a uniform effective field $h_{eff} = z J m + h$ .

2. The Self-Consistency Equation and the Langevin Function

Our next task is to solve this non-interacting problem and impose self-consistency. We need to calculate the average magnetization $m$ of a system of spins in the field $h_{eff}$ and then equate it to the $m$ that generates the field.

The calculation involves integrating over all possible orientations of a classical spin $S$ . This is a standard problem in statistical mechanics, and its solution yields the Langevin function.

To see how this is derived and how it leads to the self-consistency equation, please study the following resource.

MEAN-FIELD APPROACH TO MAGNETISM

The following text derives the mean-field theory for a system of classical vector spins. It shows how the problem is reduced to calculating the thermal average of a single spin in an effective field, leading to an implicit equation for the magnetization involving the Langevin function.

Please read section 5, 'Mean field approximation for ferromagnets', from equation (41) up to and including equation (51). The text uses 'S' for the average spin magnitude, which we are calling 'm'. Pay close attention to how the interacting Hamiltonian is approximated by the mean-field form (Eq. 42), and how this leads to a self-consistency equation (Eq. 50/51) involving the Langevin function, L(f).

As you have just read, the average magnetization $m = ∣ m ∣$ for a spin of magnitude $S$ in an effective field $h_{eff}$ is given by:

m = S \cdot L (βS h_{eff})

where $L (x) = coth (x) - \frac{1}{x}$ is the Langevin function and $β = 1/ (k_{B} T)$ .

By substituting $h_{eff} = z J m + h$ , we arrive at the self-consistency equation for the Heisenberg model:

m = S \cdot L (βS (z J m + h))

This equation is the Heisenberg model's counterpart to the $m = tanh (β (z J m + h))$ equation for the spin-1/2 Ising model.

3. Analysis and Comparison with the Ising Model

Let's analyze this equation for the case of spontaneous magnetization ( $h = 0$ ) and compare the results with those of the Ising model.

Graphical Solution and Critical Temperature

The self-consistency equation for $h = 0$ is $m = S \cdot L (βS z J m)$ . To analyze this, we can define a reduced magnetization $μ = m / S$ and a dimensionless variable $x = β S^{2} z J μ$ . The equation then becomes:

μ = L (\frac{x}{μ})

which is not very transparent. A better approach is to plot both sides of the original equation, normalized:

Left side: $y = m$
Right side: $y = S \cdot L (βS z J m)$

A non-trivial solution ( $m > 0$ ) exists if the initial slope of the Langevin function curve is greater than the slope of the line $y = m$ . The slope of $y = m$ is 1. For small arguments, the Langevin function can be expanded as $L (x) \approx x /3$ . Therefore, the initial slope of the right side is:

\frac{d}{d m} [S \cdot \frac{βS z J m}{3}]_{m = 0} = \frac{β S ^{2} z J}{3} = \frac{S ^{2} z J}{3 k _{B} T}

A phase transition occurs when this slope is exactly 1. This defines the critical temperature $T_{c}$ :

\frac{S ^{2} z J}{3 k _{B} T _{c}} = 1 ⟹ k_{B} T_{c}^{Heisenberg} = \frac{z J S ^{2}}{3}

How does this compare to the Ising model? For an Ising model with spins $\pm S$ , the self-consistency equation is $m = S tanh (β (z J m))$ . The initial slope of the right-hand side is $β S^{2} z J$ . Setting this to 1 gives:

k_{B} T_{c}^{Ising} = z J S^{2}

The critical temperature for the classical Heisenberg model is one-third of that for the Ising model with the same parameters. The continuous degrees of freedom available to the Heisenberg spins provide more ways to introduce disorder, making the ordered ferromagnetic state more fragile and lowering the temperature at which it breaks down.

Critical Exponents

Let's examine the behavior near $T_{c}$ . The procedure is identical to what we did for the Ising model, but we use the expansion of the Langevin function instead of the hyperbolic tangent.

Spontaneous Magnetization ( $β$ ): For $T \to T_{c}^{-}$ and $h = 0$ , $m$ is small. We expand the Langevin function to third order: $L (x) \approx x /3 - x^{3} /45$ .
$\frac{m}{S} = \frac{βS z J m}{3} - \frac{( βS z J m ) ^{3}}{45} + \dots$
Rearranging and using $T_{c} = z J S^{2} / (3 k_{B})$ , we find:
$m^{2} \propto (1 - \frac{T}{T _{c}}) ⟹ m \propto (T_{c} - T)^{1/2}$
This gives the critical exponent $β = 1/2$ , identical to the Ising model.
Susceptibility ( $γ$ ): For $T \to T_{c}^{+}$ and small $h$ , $m$ is small. We use the first-order expansion $L (x) \approx x /3$ .
$m \approx S \frac{βS ( z J m + h )}{3}$
The magnetic susceptibility is $χ = \partial m / \partial h ∣_{h = 0}$ . Solving for $m$ gives:
$χ = \frac{C}{T - T _{c}}$
This is the Curie-Weiss law, which gives the critical exponent $γ = 1$ , again identical to the Ising model.

Test your understanding!

Given that the critical exponents $β$ and $γ$ are the same for the mean-field Ising and Heisenberg models, what value would you predict for the critical exponent $δ$ , which describes the relation $m \sim h^{1/ δ}$ at $T = T_{c}$ ? (Hint: you don't need to do a full calculation, just think about the structure of the self-consistency equation near the critical point).

Show answer

You would predict $δ = 3$ . At $T = T_{c}$ , the linear term in the expansion of $m$ vs $m$ cancels out, leaving a relationship of the form $h \propto m^{3}$ . This cubic relationship at the critical point is a general feature of the mean-field free energy expansion ( $ϕ^{4}$ theory, which we will study later) and is independent of whether the self-consistency equation comes from a tanh or a Langevin function. Therefore, mean-field theory predicts $δ = 3$ for both models.

Summary and Synthesis

Let's collect our findings in a table:

Feature	Ising Model (spin S)	Classical Heisenberg Model (spin S)
Order Parameter Symmetry	Discrete ( $Z_{2}$ )	Continuous (O(3))
Self-consistency Eq. for $μ = m / S$	$μ = tanh (β S^{2} z J μ)$	$μ = L (β S^{2} z J μ)$
Critical Temperature $T_{c}$	$k_{B} T_{c} = z J S^{2}$	$k_{B} T_{c} = \frac{z J S ^{2}}{3}$
Exponent $β$ (Magnetization)	1/2	1/2
Exponent $γ$ (Susceptibility)	1	1
Exponent $δ$ (Critical Isotherm)	3	3

This comparison reveals a crucial insight into mean-field theory:

Universal Predictions: The critical exponents ( $α, β, γ, δ, \dots$ ) are identical for both models. Mean-field theory is insensitive to the symmetry of the order parameter (discrete vs. continuous). The exponents it predicts are "universal" for any system to which the theory is applicable.
Non-Universal Predictions: Model-specific details, such as the value of the critical temperature $T_{c}$ , do depend on the microscopic Hamiltonian. The lower $T_{c}$ for the Heisenberg model is a physically meaningful result of its continuous symmetry, which makes the ordered state more susceptible to thermal fluctuations.

The insensitivity of mean-field exponents to order parameter symmetry is a direct consequence of the theory's core approximation: it neglects spatial correlations and fluctuations. More advanced theories, like the renormalization group, will show that the exponents do depend on both the dimension of space and the symmetry of the order parameter. The breakdown of mean-field theory is most dramatic in low dimensions. For instance, the Mermin-Wagner theorem, a topic for a future lesson, proves that a continuous symmetry (like the Heisenberg model's) cannot be spontaneously broken at any finite temperature in two dimensions or less, meaning $T_{c} = 0$ . Mean-field theory incorrectly predicts a finite $T_{c}$ in all dimensions.

Conclusion

Key Takeaways:

The mean-field approximation can be readily applied to the classical Heisenberg model, leading to a self-consistency equation based on the Langevin function.
The critical temperature predicted for the Heisenberg model is lower than for the Ising model, reflecting the greater destabilizing effect of thermal fluctuations on a system with continuous symmetry.
Within the mean-field approximation, all critical exponents ( $β, γ, δ$ ) are identical for the Ising and Heisenberg models. This highlights that MFT is a "universal" theory in this sense, but it also reveals its core limitation: its inability to account for the true role of fluctuations, dimensionality, and symmetry, which differentiate real physical systems.

Preview of the Next Lesson:
We have now examined mean-field theory from two perspectives: the molecular field picture and the more formal variational principle. We've applied it to both discrete and continuous spin models. We are now prepared to discuss the limitations of this approach more formally. In the next lesson, we will explore the role of fluctuations and dimensionality in the breakdown of mean-field theory, setting the stage for more powerful theoretical tools.