222
I. Proskurin and R. L. Stamps
which leaves (9.42) and (9.43) invariant for any real parameter θ . Similar to the
electrodynamics, we have an algebraic property Q 2 Q 2 = Q 8 , which establishes a
relation between the duality, the rotation symmetry along n, and ∂ t .
9.3.3 Conserving Chirality of Spin Waves
The existence of the symmetry transformations makes possible a formulation of the
conservation laws that correspond to these symmetries. Conserving quantities can
be expressed in terms of bilinear forms similar to (9.21)
C =
1
2
d
3 pψ
†
(t, p)ρQψ(t, p),
(9.52)
where Q is a symmetry transformation, which can be expressed as a linear combination of Q i (i = 1, . . . , 8), and the measure ρ = diag(ε
(m)
p , ε
(l)
p ) is necessary for
transforming from the symmetric representation of the equations of motions in (9.45)
and (9.46) to the original units.
The conservation law for spin-wave chirality can be formulated similar to the
expression for the optical zilch in Sect. 9.2.1.3. Since the rotation symmetry is preserved only along the direction of n, we take the component of the spin wave momentum along this direction p n = ( p · n)n, and apply the conservation law in (9.52) for
the symmetry transformation p n Q 5 = ( ˆ
S · p n ). As a result, the expression for conserving spin-wave chirality is given by
C
(m)
χ
=
i
2
d
3 p
ε
(m)
p m
∗
p · ( p n × m p ) + ε
(l)
p l
∗
p · ( p n × l p )
,
(9.53)
which is a direct analogue of the Lipkin’s zilch for the electromagnetic field. In real
space, the chirality density for spin waves can be written as
ρ
(m)
χ (t, r) =
1
2
∇ n m ·
∂l
∂t
+ ∇ n l ·
∂ m
∂t
,
(9.54)
where ∇ n = ∇ · n.
Physical meaning of C
(m)
χ
becomes clear if we rewrite the expression (9.53) in
terms of circularly polarized magnon operators. In this case, total spin wave chirality
is determined by the difference between the number of left (N
(R)
p ) and right (N
(R)
p )
polarized magnons [38]
C
(m)
χ
= 2
p
p n ω p
N
(L)
p − N
(R)
p
.
(9.55)
I. Proskurin and R. L. Stamps
which leaves (9.42) and (9.43) invariant for any real parameter θ . Similar to the
electrodynamics, we have an algebraic property Q 2 Q 2 = Q 8 , which establishes a
relation between the duality, the rotation symmetry along n, and ∂ t .
9.3.3 Conserving Chirality of Spin Waves
The existence of the symmetry transformations makes possible a formulation of the
conservation laws that correspond to these symmetries. Conserving quantities can
be expressed in terms of bilinear forms similar to (9.21)
C =
1
2
d
3 pψ
†
(t, p)ρQψ(t, p),
(9.52)
where Q is a symmetry transformation, which can be expressed as a linear combination of Q i (i = 1, . . . , 8), and the measure ρ = diag(ε
(m)
p , ε
(l)
p ) is necessary for
transforming from the symmetric representation of the equations of motions in (9.45)
and (9.46) to the original units.
The conservation law for spin-wave chirality can be formulated similar to the
expression for the optical zilch in Sect. 9.2.1.3. Since the rotation symmetry is preserved only along the direction of n, we take the component of the spin wave momentum along this direction p n = ( p · n)n, and apply the conservation law in (9.52) for
the symmetry transformation p n Q 5 = ( ˆ
S · p n ). As a result, the expression for conserving spin-wave chirality is given by
C
(m)
χ
=
i
2
d
3 p
ε
(m)
p m
∗
p · ( p n × m p ) + ε
(l)
p l
∗
p · ( p n × l p )
,
(9.53)
which is a direct analogue of the Lipkin’s zilch for the electromagnetic field. In real
space, the chirality density for spin waves can be written as
ρ
(m)
χ (t, r) =
1
2
∇ n m ·
∂l
∂t
+ ∇ n l ·
∂ m
∂t
,
(9.54)
where ∇ n = ∇ · n.
Physical meaning of C
(m)
χ
becomes clear if we rewrite the expression (9.53) in
terms of circularly polarized magnon operators. In this case, total spin wave chirality
is determined by the difference between the number of left (N
(R)
p ) and right (N
(R)
p )
polarized magnons [38]
C
(m)
χ
= 2
p
p n ω p
N
(L)
p − N
(R)
p
.
(9.55)