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H, ʳ
x
(
j
)
,
dt
x
(
j
)=
i
d
H, ʳ
y
(
j
)
,
dt
y
(
j
)=
i
d
.
H, ʳ
x
(
j
)
ʳ
z
(
k
)
,
dt
K
xy
(
j, k
)=
i
d
.
H, ʳ
z
(
j
)
ʳ
z
(
k
)
.
dt
K
zz
(
j, k
)=
i
d
(15)
As an example, let us consider the dynamics of the
ʳ
y
(
j
) basis operator.
In doing so, the following commutation property for Pauli matrices will prove
useful:
[
ʳ
a
, ʳ
b
]=2
i˃
abc
ʳ
c
,
(16)
where
˃
abc
is the Levi-Civita function and is defined as,
⊧
⊨
⊩
+1 if (
a, b, c
)is(
x, y, z
)
,
(
y, z, x
)or(
z, x, y
)
−
˃
abc
=
1if (
a, b, c
)is(
z, y, x
)
,
(
x, z, y
)or(
y, x, z
)
.
0 f
a
=
b
or
b
=
c
or
a
=
c
(17)
Using this property and Eq. (
14
), the time dependance of
ʳ
y
(
j
) is then,
H, ʳ
y
(
j
)
,
d
dt
ʳ
y
(
j
)=
i
2
ʷ
j
ʳ
z
(
j
)
ʳ
x
(
j
)
ʳ
z
(
m
)
N
1
E
j,m
k
−
.
=
(18)
m
Taking the expectation values of both sides, we get:
N
d
dt
y
(
j
)=2
ʷ
i
z
(
j
)
E
j,m
k
−
K
xz
(
j, m
)
.
(19)
m
In addition to
z
, we notice that
K
xz
also appears on the right-hand side of
Eq. (
19
), and therefore its dynamics must also be computed in order to capture
the full dynamic behaviour of
y
. Similarly, when we derive the dynamical equa-
tion for
K
xz
, various two-point correlation terms will appear in that equation,
and thus we will need to compute the dynamics for those terms as well, and so
on. At the end, we will be left with a linear system of coupled ODEs that must
all be solved in order to compute the system's dynamics.
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