228
I. Proskurin and R. L. Stamps
relativistic language, interband effects in the dynamics of an electron wave packet
correspond to the Zitterbewegung, or the trembling motion of an ultra-relativistic
particle [74]. The Zitterbewegung effect has also been proposed for antiferromagnetic
magnons [40]. It can be easily understood by looking at the time evolution of ¯
ψ p (t)
calculated from (9.45) to (9.46)
¯
ψ p (t) =
1
2
1 + σ 1 ⊗ ( ˆ
S · n)
e
iω p t
+
1 − σ 1 ⊗ ( ˆ
S · n)
e
−iω p t
¯
ψ p (0), (9.68)
which is similar to the analogous equation for relativistic particles [74]. This expression contains the off-diagonal elements responsible for the mixing of m p and l p
components of ¯
ψ p while evolving in time.
By applying (9.68) to the time evolution of the spin current in (9.67), we find
that the spin current has two contributions, J
(n)
s (t) = J
(n)
s0 + J
(n)
s1 (t). The first contribution is conserved part of the spin current. It does not depend on time and is
proportional to the group velocity of magnons v p = ∂ω p /∂ p. In our matrix notations, it can be written as
J
(n)
s =
1
4M s
p
v p ¯
ψ
†
(0)( ˆ
S · n) ¯
ψ p (0).
(9.69)
The second term in the spin current oscillates at the double frequency, and can be
attributed to the Zitterbewegung of magnons
J
(n)
s1 (t) =
1
16M s
p
e
2iω p t K p ¯
ψ
†
p (0)
( ˆ
S · n)
1
−1 −( ˆ
S · n)
¯
ψ p (0) + H.c., (9.70)
where
K p =
1
ω p
ε
(l)
p
∂ε
(m)
p
∂ p
− ε
(m)
p
∂ε
(l)
p
∂ p
.
(9.71)
The physical meaning of these terms becomes clear if we transform to the helicity basis, ˜
ψ p = ( ˜
ψ
(R)
p , ˜
ψ
(L)
p )
T , where we have well-defined left and right polarized
magnon modes, see (9.14), (9.47) and (9.48). In this basis, the first term is determined
by the difference in numbers of magnons with opposite polarizations
J
(n)
s =
1
4M s
p
v p
˜
ψ
∗(R)
p
˜
ψ
(R)
p − ˜
ψ
∗(L)
p
˜
ψ
(L)
p
,
(9.72)
while the second term is purely off-diagonal and corresponds to the interband processes
J
(n)
s1 (t) = −
1
8M s
p
˜
ψ
†
p (0)
0
K p ˆ
S
z e
−2iω p ˆ
S
z t
K p ˆ
S
z e
2iω p ˆ
S
z t
0
˜
ψ p (0).
(9.73)
I. Proskurin and R. L. Stamps
relativistic language, interband effects in the dynamics of an electron wave packet
correspond to the Zitterbewegung, or the trembling motion of an ultra-relativistic
particle [74]. The Zitterbewegung effect has also been proposed for antiferromagnetic
magnons [40]. It can be easily understood by looking at the time evolution of ¯
ψ p (t)
calculated from (9.45) to (9.46)
¯
ψ p (t) =
1
2
1 + σ 1 ⊗ ( ˆ
S · n)
e
iω p t
+
1 − σ 1 ⊗ ( ˆ
S · n)
e
−iω p t
¯
ψ p (0), (9.68)
which is similar to the analogous equation for relativistic particles [74]. This expression contains the off-diagonal elements responsible for the mixing of m p and l p
components of ¯
ψ p while evolving in time.
By applying (9.68) to the time evolution of the spin current in (9.67), we find
that the spin current has two contributions, J
(n)
s (t) = J
(n)
s0 + J
(n)
s1 (t). The first contribution is conserved part of the spin current. It does not depend on time and is
proportional to the group velocity of magnons v p = ∂ω p /∂ p. In our matrix notations, it can be written as
J
(n)
s =
1
4M s
p
v p ¯
ψ
†
(0)( ˆ
S · n) ¯
ψ p (0).
(9.69)
The second term in the spin current oscillates at the double frequency, and can be
attributed to the Zitterbewegung of magnons
J
(n)
s1 (t) =
1
16M s
p
e
2iω p t K p ¯
ψ
†
p (0)
( ˆ
S · n)
1
−1 −( ˆ
S · n)
¯
ψ p (0) + H.c., (9.70)
where
K p =
1
ω p
ε
(l)
p
∂ε
(m)
p
∂ p
− ε
(m)
p
∂ε
(l)
p
∂ p
.
(9.71)
The physical meaning of these terms becomes clear if we transform to the helicity basis, ˜
ψ p = ( ˜
ψ
(R)
p , ˜
ψ
(L)
p )
T , where we have well-defined left and right polarized
magnon modes, see (9.14), (9.47) and (9.48). In this basis, the first term is determined
by the difference in numbers of magnons with opposite polarizations
J
(n)
s =
1
4M s
p
v p
˜
ψ
∗(R)
p
˜
ψ
(R)
p − ˜
ψ
∗(L)
p
˜
ψ
(L)
p
,
(9.72)
while the second term is purely off-diagonal and corresponds to the interband processes
J
(n)
s1 (t) = −
1
8M s
p
˜
ψ
†
p (0)
0
K p ˆ
S
z e
−2iω p ˆ
S
z t
K p ˆ
S
z e
2iω p ˆ
S
z t
0
˜
ψ p (0).
(9.73)