9 Symmetry Approach to Chiral Optomagnonics in Antiferromagnetic Insulators
237
L =
dx
ρ M ·
L ×
∂ L
∂t
−
a
2
|M|
2
−A
∂
∂ x
∂ L
∂ x
2
−
∂ M
∂ x
2
− M ·
∂ L
∂ x
+
˜
β
2
(L
z
)
2
,
(9.98)
where M =
1
2S
(S A + S B ) and L =
1
2S
(S A − S B ), which satisfy the constraints M ·
L = 0 and M
2
+ L
2
= 1. The parameters of the Lagrangian are as follows: ρ =
2S, a = 8J S
2 , = 2J S
2 a 0 , A = J S
2 a
2
0 , and β = 4K S
2 . Note that this expression
contains so-called topological term proportional to , which breaks the inversion
symmetry in the Lagrangian [79].
The expression for the spin current can be obtained applying the Noether’s theorem
to the Lagrangian transformation under the local infinitesimal rotation around z
M → M + δφ(ˆ z × M),
(9.99)
L → L + δφ(ˆ z × L),
(9.100)
where δφ(x) is the local rotation angle. The corresponding change in the Lagrangian
is given by
δL = −
dxδφ
ρ
∂
∂t
M
z
(1 − |M|
2
)
−A
∂
∂ x
(ˆ z × L) ·
∂ L
∂ x
− (ˆ z × M) ·
∂ M
∂ x
−
∂
∂ x
M · (ˆ z × L)
,
(9.101)
which gives the following expression for the spin current density
j
z
s = −Aˆ z
L ×
∂ L
∂ x
−
M ×
∂ M
∂ x
− ˆ z · (L × M).
(9.102)
The first term in this expression is consistent with the expression for the spin current
obtained from the equations of motion. The second is the contribution from the
topological terms, which has different symmetry. In particular, it changes the sign if
we interchange S A and S B .
References
1. I. Žuti´ c, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). https://doi.org/10.1103/
RevModPhys.76.323
2. S.E. Thompson, S. Parthasarathy, Mater. Today 9(6), 20 (2006). https://doi.org/10.1016/S13697021(06)71539-5