9 Symmetry Approach to Chiral Optomagnonics in Antiferromagnetic Insulators
215
zilch conservation and duality symmetry as it was originally discussed in [16, 17],
which allows to write the conservation law above in the following equivalent form
C χ = −
i
2
d
3 pφ
†
(t, p)Q 2 ∂ t φ(t, p) =
1
2
d
3 r
B ·
∂ E
∂t
− E ·
∂ B
∂t
. (9.24)
This expression can be easily generalized to accommodate higher order terms in
space and time derivatives. By replacing Q 2 ∂ t with −(i p)
2n
Q 2 (i∂ t )
2m+1 , which is
again a symmetry operation, we can find a hierarchy of conserving zilches
C
(m,n)
χ
=
1
2
d
3 r
B · ∇
2n
∂
2m+1
t
E − E · ∇
2n
∂
2m+1
t
B
,
(9.25)
where 00-zilch corresponds to the optical chirality [23, 24, 31].
It is possible to derive the conservation law for the optical zilch using the Noether’s
formalism by applying a specific “hidden” gauge transformation to the Lagrangian of
the electromagnetic field [31], which leads to the same results as in (9.23) and (9.25).
The advantage of the approach discussed in this section, based on the symmetry
analysis of the Maxwell’s equations, is that it does not depend on any specific gauge
choice. This fact makes it easy to extend this formalism to other physical systems
with similar form of the equations of motion.
9.2.2 Optical Chirality in Gyrotropic Media
Having now a complete picture of the nongeometric symmetries in vacuum, we
discuss how this approach can be applied for the light-matter interactions. Electromagnetic field in dielectric medium is usually described by the material form of the
Maxwell equations
∇ × E = −
∂ B
∂t
,
∇ · B = 0,
(9.26)
∇ × H =
∂ D
∂t
,
∇ · D = 0,
(9.27)
supplemented by the constituent relations between the fields E, H, D, and B. The
constituent relations impose additional constraints on the form of the symmetry transformations for the electromagnetic field, which reflect the intrinsic symmetries of
the medium. This often leads to the reduction of the invariance algebra in (9.19)
to lesser number of elements [50]. In the case of common constituent relations,
D( p) = ˆ
ε( p)E( p) and B( p) = ˆ
μ( p)H( p), where ˆ
ε( p) and ˆ
μ( p) denote the electric permittivity and magnetic permeability tensors in the Fourier space, Maxwell’s
equations in (9.9) are replaced by
215
zilch conservation and duality symmetry as it was originally discussed in [16, 17],
which allows to write the conservation law above in the following equivalent form
C χ = −
i
2
d
3 pφ
†
(t, p)Q 2 ∂ t φ(t, p) =
1
2
d
3 r
B ·
∂ E
∂t
− E ·
∂ B
∂t
. (9.24)
This expression can be easily generalized to accommodate higher order terms in
space and time derivatives. By replacing Q 2 ∂ t with −(i p)
2n
Q 2 (i∂ t )
2m+1 , which is
again a symmetry operation, we can find a hierarchy of conserving zilches
C
(m,n)
χ
=
1
2
d
3 r
B · ∇
2n
∂
2m+1
t
E − E · ∇
2n
∂
2m+1
t
B
,
(9.25)
where 00-zilch corresponds to the optical chirality [23, 24, 31].
It is possible to derive the conservation law for the optical zilch using the Noether’s
formalism by applying a specific “hidden” gauge transformation to the Lagrangian of
the electromagnetic field [31], which leads to the same results as in (9.23) and (9.25).
The advantage of the approach discussed in this section, based on the symmetry
analysis of the Maxwell’s equations, is that it does not depend on any specific gauge
choice. This fact makes it easy to extend this formalism to other physical systems
with similar form of the equations of motion.
9.2.2 Optical Chirality in Gyrotropic Media
Having now a complete picture of the nongeometric symmetries in vacuum, we
discuss how this approach can be applied for the light-matter interactions. Electromagnetic field in dielectric medium is usually described by the material form of the
Maxwell equations
∇ × E = −
∂ B
∂t
,
∇ · B = 0,
(9.26)
∇ × H =
∂ D
∂t
,
∇ · D = 0,
(9.27)
supplemented by the constituent relations between the fields E, H, D, and B. The
constituent relations impose additional constraints on the form of the symmetry transformations for the electromagnetic field, which reflect the intrinsic symmetries of
the medium. This often leads to the reduction of the invariance algebra in (9.19)
to lesser number of elements [50]. In the case of common constituent relations,
D( p) = ˆ
ε( p)E( p) and B( p) = ˆ
μ( p)H( p), where ˆ
ε( p) and ˆ
μ( p) denote the electric permittivity and magnetic permeability tensors in the Fourier space, Maxwell’s
equations in (9.9) are replaced by