Spin Waves: Ferromagnet vs. Antiferromagnet

Recommended resources

The freely available resources this lesson will direct you to.

Hello! In our last lesson, we delved into the low-energy excitations of ferromagnets and found that they are collective, quantized spin waves called magnons. The key result was that for a ferromagnet, the magnon dispersion relation is gapless and quadratic at long wavelengths: $ℏ ω_{k} \propto ∣ k ∣^{2}$ . This behavior is the signature of a Goldstone mode arising from the spontaneous breaking of continuous SU(2) rotational symmetry.

Today, we will turn our attention to antiferromagnets (AFMs), where the ground state has neighboring spins aligned in opposite directions. This seemingly simple change in the ground state configuration leads to profoundly different dynamics. Our goal is to derive the spin-wave dispersion for an antiferromagnet and, in doing so, contrast it with the ferromagnetic case. You will discover that antiferromagnetic magnons exhibit a linear dispersion, a result that stems from the more complex, quantum-fluctuating nature of the antiferromagnetic ground state.

1. The Antiferromagnetic Ground State: A Two-Sublattice Approach

Let's start with the Heisenberg Hamiltonian for nearest-neighbor interactions:
$H = J \sum_{⟨ i, j ⟩} S_{i} \cdot S_{j}$
For an antiferromagnet, the exchange constant $J > 0$ , favoring anti-parallel alignment of neighboring spins. The classical ground state on a bipartite lattice (like a 1D chain, 2D square, or 3D cubic lattice) is the Néel state, where the lattice is divided into two interpenetrating sublattices, A and B, with all spins on A pointing 'up' and all spins on B pointing 'down'.

Spin Precession in Ferromagnets and Antiferromagnets — This image illustrates the collective spin dynamics. In a ferromagnet (left), all spins precess in phase. In an easy-axis antiferromagnet (center), spins on the two sublattices precess in opposite directions (left- and right-handed), maintaining their anti-parallel nature.

Unlike the fully polarized ferromagnetic state, the Néel state is not an eigenstate of the quantum Heisenberg Hamiltonian. The term $S_{i}^{+} S_{j}^{-}$ can flip a pair of adjacent, anti-aligned spins, taking the system out of the perfect Néel configuration. This means the true quantum ground state has inherent fluctuations—what we call zero-point fluctuations—even at $T = 0$ .

To proceed with spin-wave theory, we retain the classical Néel state as our starting point and introduce separate bosonic operators for the spin deviations on each sublattice.

The Holstein-Primakoff Transformation for Two Sublattices

We apply the Holstein-Primakoff (HP) transformation, but we must treat each sublattice differently according to its ground state orientation.

For a site $p$ on sublattice A (spins 'up', $S_{p}^{z} \approx S$ ), we use the same transformation as in the ferromagnetic case, using bosonic operators $a^_{p}, a^_{p}^{†}$ :
$S^_{A p}^{z} = S - a^_{p}^{†} a^_{p}, S^_{A p}^{+} \approx 2 S a^_{p}, S^_{A p}^{-} \approx 2 S a^_{p}^{†}$
For a site $q$ on sublattice B (spins 'down', $S_{q}^{z} \approx - S$ ), a spin deviation means increasing $S_{z}$ from $- S$ . To handle this, we perform the transformation in a coordinate system rotated by $π$ around the x-axis, or more directly, define a new set of bosonic operators $b^_{q}, b^_{q}^{†}$ :
$S^_{B q}^{z} = - S + b^_{q}^{†} b^_{q}, S^_{B q}^{+} \approx 2 S b^_{q}^{†}, S^_{B q}^{-} \approx 2 S b^_{q}$

Notice the crucial difference in the expressions for $S^^{+}$ and $S^^{-}$ for sublattice B. A lowering operator $S^^{-}$ for the global axis corresponds to a raising operator in the local frame of the 'down' spin, thus it's associated with $b^_{q}$ , which annihilates a spin deviation on that site.

A rigorous derivation of this setup is presented in the following resource.

Theory of Magnetism Lecture notes

These lecture notes by Alexander Tsirlin provide a detailed derivation of spin-wave theory for antiferromagnets. We will follow its logic closely. First, familiarize yourself with how the Holstein-Primakoff transformation is adapted for the two-sublattice model.

Please read section 5.4.1, 'Linear spin-wave theory', on pages 42-43. Pay close attention to equations (5.42) and (5.43), which define the HP transformation for the two sublattices, and see how they are substituted into the Hamiltonian to arrive at the bosonic form in equation (5.46).

2. The Bogoliubov Transformation

After substituting the two HP transformations into the Heisenberg Hamiltonian and moving to momentum space via Fourier transforms, we arrive at a quadratic bosonic Hamiltonian of the form (similar to Eq. 5.46 in the notes):

H_{0} \approx z J S k \sum [(a^_{k}^{†} a^_{k} + b^_{k}^{†} b^_{k}) + γ_{k} (a^_{k} b^_{- k} + a^_{k}^{†} b^_{- k}^{†})]

where $z$ is the coordination number and $γ_{k} = \frac{1}{z} \sum_{δ} e^{i k \cdot δ}$ is the geometric factor you saw in the last lesson.

This Hamiltonian is not diagonal. The terms $a^_{k}^{†} b^_{- k}^{†}$ and $a^_{k} b^_{- k}$ are problematic; they correspond to the creation and annihilation of a pair of magnons, one on each sublattice. To find the true elementary excitations (the normal modes), we must diagonalize this Hamiltonian. This is achieved using the Bogoliubov transformation.

This powerful technique defines a new set of bosonic operators ( $α^_{k}, β^_{k}$ ) as a linear combination of the old ones:

a^_{k} b^_{- k}^{†} = u_{k} α^_{k} - v_{k} β^_{- k}^{†} = - v_{k} α^_{k} + u_{k} β^_{- k}^{†}

where $u_{k}$ and $v_{k}$ are real coefficients satisfying $u_{k}^{2} - v_{k}^{2} = 1$ to ensure the new operators obey bosonic commutation relations. This structure should be familiar to you from your project on squeezed states of light; the mathematical formalism for diagonalizing the two-mode squeezing Hamiltonian is identical. The AFM ground state is, in fact, a two-mode squeezed vacuum of $a$ and $b$ magnons.

The goal is to choose $u_{k}$ and $v_{k}$ such that the Hamiltonian becomes diagonal in the new operators:
$H = E_{g}^{'} + \sum_{k} ℏ ω_{k} (α^_{k}^{†} α^_{k} + β^_{k}^{†} β^_{k})$

The detailed algebra for this diagonalization is quite involved but follows a standard procedure. The resource below walks through it step-by-step.

Theory of Magnetism Lecture notes

Now, let's see how the Bogoliubov transformation is used to diagonalize the antiferromagnetic Hamiltonian. Focus on the strategy rather than memorizing every line of algebra.

Please read section 5.4.2, 'Bogolyubov transformation', on pages 44-45. Follow the logic of how the transformation is defined in equations (5.49)-(5.50), and how the condition that the Hamiltonian becomes diagonal leads to the expression for the magnon energy \lambda_k (our \hbar\omega_k) in equation (5.55).

3. The Antiferromagnetic Magnon Dispersion

After the Bogoliubov transformation, we find two degenerate magnon modes ( $α$ and $β$ ) for each $k$ . These correspond to the two possible chiralities (left- and right-handed precession) of the spin system, as shown in the image above. Their dispersion relation is given by:
$ℏ ω_{k} = z J S 1 - γ_{k}^{2}$

This is the central result for antiferromagnetic magnons. Let's analyze it in the long-wavelength limit ( $∣ k ∣ \to 0$ ) for a simple cubic lattice where $γ_{k} \approx 1 - \frac{1}{z} ∣ k ∣^{2} a^{2}$ .

γ_{k}^{2} \approx (1 - \frac{1}{z} ∣ k ∣^{2} a^{2})^{2} \approx 1 - \frac{2}{z} ∣ k ∣^{2} a^{2}

Substituting this into the dispersion relation:

ℏ ω_{k} \approx z J S 1 - (1 - \frac{2}{z} ∣ k ∣^{2} a^{2}) = z J S \frac{2}{z} ∣ k ∣^{2} a^{2} = J S a 2 z ∣ k ∣

The energy of an antiferromagnetic magnon is linear in its wavevector, $ℏ ω_{k} \propto ∣ k ∣$ . This is qualitatively different from the quadratic dispersion in ferromagnets. These are also Goldstone modes, but they have a "relativistic" linear dispersion, like acoustic phonons or photons, rather than a "non-relativistic" quadratic one.

4. Comparison: Ferromagnets vs. Antiferromagnets

We can now summarize the key distinctions between magnons in these two systems.

Feature	Ferromagnet (FM)	Antiferromagnet (AFM)
Ground State	Simple: all spins aligned (classical state is a true eigenstate). Corresponds to boson vacuum.	Complex: Néel state is not an eigenstate. True ground state has zero-point fluctuations.
Spin Precession	All spins precess in phase with the same chirality.	Sublattices A and B precess in opposite directions (different chiralities).
Magnon Modes	One mode per $k$ .	Two degenerate modes per $k$ , corresponding to the two chiralities.
Dispersion at low $k$	Quadratic: $\hbar\omega_{\mathbf{k}} \propto	\mathbf{k}
Ground State $S_{z}$	Saturation value: $⟨ S^{z} ⟩ = S$ at $T = 0$ .	Reduced magnetization: $⟨ S^{z} ⟩_{s u b l a tt i ce} < S$ even at $T = 0$ due to zero-point fluctuations.

The linear vs. quadratic dispersion is a direct experimental signature that distinguishes these two types of magnetic order in inelastic neutron scattering experiments.

Test your understanding!

Consider a 1D antiferromagnetic chain with lattice constant $a$ . The geometric factor is $γ_{k} = cos (k a)$ .

Write down the full magnon dispersion relation $ℏ ω_{k}$ for this chain.
Using a small- $k$ expansion, explicitly show that the dispersion is linear in $∣ k ∣$ .

Show answer

For a 1D chain, $z = 2$ . The dispersion is $ℏ ω_{k} = z J S 1 - γ_{k}^{2}$ .
$ℏ ω_{k} = 2 J S 1 - cos^{2} (k a) = 2 J S ∣ sin (k a) ∣$
For small $k$ , we use the Taylor expansion $sin (x) \approx x$ .
$ℏ ω_{k} \approx 2 J S ∣ k a ∣ = (2 J S a) ∣ k ∣$
The dispersion is indeed linear in $∣ k ∣$ , with a velocity (slope) of $v = 2 J S a /ℏ$ .

Conclusion

Today, we've explored the fascinating world of spin waves in antiferromagnets, revealing a stark contrast with their ferromagnetic counterparts. The necessity of a two-sublattice model and the inherent quantum fluctuations of the ground state lead to a more complex theoretical description, requiring the powerful Bogoliubov transformation.

Key Takeaways:

The classical Néel state is not the true quantum ground state of an antiferromagnet, which instead features zero-point spin fluctuations.
Spin-wave theory for AFMs requires a two-sublattice model and separate Holstein-Primakoff transformations for each.
The resulting bosonic Hamiltonian is non-diagonal and must be diagonalized via a Bogoliubov transformation.
The dispersion relation for antiferromagnetic magnons is linear in $k$ for small wavevectors ( $ℏ ω_{k} \propto ∣ k ∣$ ), unlike the quadratic dispersion in ferromagnets.
This linear dispersion, along with the reduction of the ordered moment at $T = 0$ , are fundamental consequences of the quantum nature of the antiferromagnetic ground state.

Preview of the Next Lesson

So far, we have considered magnetic interactions between localized spins that are immediate neighbors (direct exchange) or very close (superexchange, which can also be described by this formalism). But how do magnetic moments in a metal, separated by many lattice sites, communicate with each other? In our next lesson, we will explore the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, an indirect exchange mechanism where the conduction electrons of a metal mediate a long-range, oscillating interaction between localized magnetic moments.