218
I. Proskurin and R. L. Stamps
In the next section, we will show how these arguments can be generalized to spin
excitations in antiferromagnetic materials. Similar to the results of this section, the
symmetry analysis will play a principal role in our discussion.
9.3 Spin-Wave Chirality in Antiferromagnetic Insulators
The symmetry analysis developed in the previous section for Maxwell’s equations
can be generalized to other dynamical systems. Here, we develop such generalization
for spin-wave excitations in an antiferromagnetic insulator. A key observation that
helps us to draw the analogy between spin-wave dynamics and electrodynamics is
that the antiferromagnetic spin waves can be also characterized by two polarization
states. This stems from the fact that the magnetization dynamics in antiferromagnets
involves two coupled magnetic sublattices. We, therefore, examine the symmetry
transformation in the extended space that includes three-dimensional rotations and
transformations between the sublattices, in order to find an algebra of nongeometric
symmetries for spin waves equivalent to that of the electrodynamics.
9.3.1 Equations of Motion for Antiferromagnetic Spin Waves
We start our discussion with a simple case of a collinear antiferromagnet with two
equivalent magnetic sublattices M 1 (t, r) and M 2 (t, r). The energy for such antiferromagnet can be written in the following form
W =
d
3 r
α
2
∂ M 1
∂ x μ
·
∂ M 1
∂ x μ
+
∂ M 2
∂ x μ
·
∂ M 2
∂ x μ
+ α
∂ M 1
∂ x μ
·
∂ M 2
∂ x μ
+
δ
2
M 1 · M 2 −
β
2
(M 1 · n)
2
+ (M 2 · n)
2
, (9.40)
where α, α
, and δ are the antiferromagnetic exchange parameters and β > 0
describes the uniaxial magnetic anisotropy with n being the unit vector along the
anisotropy axis [57]. In the ground state, the anisotropy stabilizes a uniform magnetic
ordering along n where two sublattices compensate each other, M 1 = −M 2 , so that
the total magnetization vanishes.
In the semi-classical limit, magnetization dynamics are described by the Landau–
Lifshitz–Gilbert equations of motion
∂ M i
∂t
= γ M i × H
eff
i − ηM i ×
∂ M i
∂t
, (i = 1, 2),
(9.41)
I. Proskurin and R. L. Stamps
In the next section, we will show how these arguments can be generalized to spin
excitations in antiferromagnetic materials. Similar to the results of this section, the
symmetry analysis will play a principal role in our discussion.
9.3 Spin-Wave Chirality in Antiferromagnetic Insulators
The symmetry analysis developed in the previous section for Maxwell’s equations
can be generalized to other dynamical systems. Here, we develop such generalization
for spin-wave excitations in an antiferromagnetic insulator. A key observation that
helps us to draw the analogy between spin-wave dynamics and electrodynamics is
that the antiferromagnetic spin waves can be also characterized by two polarization
states. This stems from the fact that the magnetization dynamics in antiferromagnets
involves two coupled magnetic sublattices. We, therefore, examine the symmetry
transformation in the extended space that includes three-dimensional rotations and
transformations between the sublattices, in order to find an algebra of nongeometric
symmetries for spin waves equivalent to that of the electrodynamics.
9.3.1 Equations of Motion for Antiferromagnetic Spin Waves
We start our discussion with a simple case of a collinear antiferromagnet with two
equivalent magnetic sublattices M 1 (t, r) and M 2 (t, r). The energy for such antiferromagnet can be written in the following form
W =
d
3 r
α
2
∂ M 1
∂ x μ
·
∂ M 1
∂ x μ
+
∂ M 2
∂ x μ
·
∂ M 2
∂ x μ
+ α
∂ M 1
∂ x μ
·
∂ M 2
∂ x μ
+
δ
2
M 1 · M 2 −
β
2
(M 1 · n)
2
+ (M 2 · n)
2
, (9.40)
where α, α
, and δ are the antiferromagnetic exchange parameters and β > 0
describes the uniaxial magnetic anisotropy with n being the unit vector along the
anisotropy axis [57]. In the ground state, the anisotropy stabilizes a uniform magnetic
ordering along n where two sublattices compensate each other, M 1 = −M 2 , so that
the total magnetization vanishes.
In the semi-classical limit, magnetization dynamics are described by the Landau–
Lifshitz–Gilbert equations of motion
∂ M i
∂t
= γ M i × H
eff
i − ηM i ×
∂ M i
∂t
, (i = 1, 2),
(9.41)