220
I. Proskurin and R. L. Stamps
The matrix in (9.44) can be symmetrized by an appropriate choice of the
units that can be expressed in the form of the transformation ψ = N ¯
ψ, where
N = diag([ε
(m)
p ]
−1/2
, [ε
(l)
p ]
−1/2
). In the symmetric units, the equation of motion for
the antiferromagnetic spin waves is written as
i
∂ ¯
ψ(t, p)
∂t
= H 0 ( p) ¯
ψ(t, p),
(9.45)
where the matrix on the right-hand side becomes symmetric
H 0 ( p) =
0
−ω p ( ˆ
S · n)
−ω p ( ˆ
S · n)
0
= −ω p σ 1 ⊗ ( ˆ
S · n),
(9.46)
with ω p =
ε
(m)
p ε
(l)
p .
This expression has a structure similar to H( p) in (9.12) for the Maxwell’s
equations. The important difference between H 0 and H comes from their properties under spatial inversion (P) and time-reversal (T ) transformations. For example, in the case of the time-reversal transformation, φ(t, p) in (9.9) transforms as
T φ(t, p) → σ 3 φ(−t, p). The Pauli matrix σ 3 appears on the right-hand side due
to the different transformation properties of the electric and magnetic field with
respect to T . In contrast, both components of ψ(t, p) are odd under T , so that
T ψ(t, p) → −ψ(−t, p). This means that if we want to transform from the spin wave
dynamics to the electrodynamics, we should replace σ 1 in (9.46) with σ 2 = iσ 1 σ 3 to
ensure correct properties under the PT transformations.
9.3.2 Nongeometric Symmetries for Spin-Wave Dynamics
Formal analogy between the equations of motion for the antiferromagnetic spin waves
and the Maxwell’s equations enables us to generalize the concept of nongeometric
symmetries. We may ask a question about all the transformations ¯
ψ(t, p) → ¯
ψ
(t, p)
that leave the equation of motion (9.45) invariant.
In order to find all such symmetries, one can repeat the steps of Sect. 9.2.1.1. First,
we have to transform to the basis where H 0 ( p) is diagonal. For this purpose, we
make a unitary transformation ¯
ψ = U m ˜
ψ, where the transformation matrix, U m =
U 1 ⊗ ˆ
U , is given by the rotation matrix to the helicity basis in (9.14) (where p is
replaced by n) combined with the SU (2) rotation in the subspace of m p and l p
U 1 =
1
√
2
1 1
−1 1
.
(9.47)
I. Proskurin and R. L. Stamps
The matrix in (9.44) can be symmetrized by an appropriate choice of the
units that can be expressed in the form of the transformation ψ = N ¯
ψ, where
N = diag([ε
(m)
p ]
−1/2
, [ε
(l)
p ]
−1/2
). In the symmetric units, the equation of motion for
the antiferromagnetic spin waves is written as
i
∂ ¯
ψ(t, p)
∂t
= H 0 ( p) ¯
ψ(t, p),
(9.45)
where the matrix on the right-hand side becomes symmetric
H 0 ( p) =
0
−ω p ( ˆ
S · n)
−ω p ( ˆ
S · n)
0
= −ω p σ 1 ⊗ ( ˆ
S · n),
(9.46)
with ω p =
ε
(m)
p ε
(l)
p .
This expression has a structure similar to H( p) in (9.12) for the Maxwell’s
equations. The important difference between H 0 and H comes from their properties under spatial inversion (P) and time-reversal (T ) transformations. For example, in the case of the time-reversal transformation, φ(t, p) in (9.9) transforms as
T φ(t, p) → σ 3 φ(−t, p). The Pauli matrix σ 3 appears on the right-hand side due
to the different transformation properties of the electric and magnetic field with
respect to T . In contrast, both components of ψ(t, p) are odd under T , so that
T ψ(t, p) → −ψ(−t, p). This means that if we want to transform from the spin wave
dynamics to the electrodynamics, we should replace σ 1 in (9.46) with σ 2 = iσ 1 σ 3 to
ensure correct properties under the PT transformations.
9.3.2 Nongeometric Symmetries for Spin-Wave Dynamics
Formal analogy between the equations of motion for the antiferromagnetic spin waves
and the Maxwell’s equations enables us to generalize the concept of nongeometric
symmetries. We may ask a question about all the transformations ¯
ψ(t, p) → ¯
ψ
(t, p)
that leave the equation of motion (9.45) invariant.
In order to find all such symmetries, one can repeat the steps of Sect. 9.2.1.1. First,
we have to transform to the basis where H 0 ( p) is diagonal. For this purpose, we
make a unitary transformation ¯
ψ = U m ˜
ψ, where the transformation matrix, U m =
U 1 ⊗ ˆ
U , is given by the rotation matrix to the helicity basis in (9.14) (where p is
replaced by n) combined with the SU (2) rotation in the subspace of m p and l p
U 1 =
1
√
2
1 1
−1 1
.
(9.47)