Chemistry Reference
In-Depth Information
using
y
b;i
¼y
a;i
þ
1
¼ u
i
and adding the kinetic energy of X ions, we rewrite the
model (
12.1
)as
H ¼t
MM
X
i;s
:Þt
MXM
X
i;s
ðc
2
i
1
;s
c
2
i;s
þ
ðc
2
i;s
c
2
iþ
1
;s
þ
h.c
h.c
:Þ
X
u
i
ðn
2
iþ
1
n
2
i
ÞþU
M
X
i
n
i;"
n
i;#
þ K
MX
X
i
X
u
i
þðM
2
u
i
;
þ b
= Þ
i
i
(12.2)
where the
i
th binuclear unit contains two M (M
1 and 2
i
and the
mass of the X ion is denoted byM. We have obtained the stable and metastable, static
states using
t
MM
¼
1.0,
t
MXM
¼
0.8, and
K
MX
¼
6.0 as a typical parameter set. The
main results shown below are, however, not affected by the details of the chosen
parameters. Indeed, they are supported by the perturbation theory from the strong-
coupling limit as explained below. For
U
M
¼
¼
Pt) sites, 2
i
6.0, the boundary between the two
phases is located at
b
c
¼
4.4 when the 12-site lattice is exactly diagonalized [
23
] and
at
6.0 in the time-
independent
Hartree-Fock (HF) approximation. The stable
(metastable) state is CP (CDW) for
b
c
¼
b > b
c
. Because the
time-
independent
approximation does not include quantum fluctuations, the effect
of the on-site repulsion
U
M
is overestimated. Here we use
U
M
¼
b < b
c
and CDW (CP) for
4.0 instead
hereafter. The boundary is then located at
4.9.
For the photoinduced dynamics, we initially add small random numbers to the
lattice variables
u
and
b
c
¼
u
. They help nucleation of a local domain in the stable phase.
The Boltzmann distribution is employed at a fictitious temperature,
T ¼
0.01 so that
they are weakly random. During the calculations, the system is isolated and its total
energy is conserved. Unless otherwise stated, lowest energy photoexcitations are
introduced by changing the occupation [
42
] of the highest occupied and the lowest
unoccupied HF orbitals. When higher energy excitations are considered, we change
the occupation of different sets of HF orbitals.
The evolution of the HF wave functions is obtained by solving the time-
dependent Schr
¨
dinger equation. The checkerboard decomposition is used for the
exponential operator so that it is accurate to the second order with respect to the time
slice. The evolution of the lattice variables is obtained by solving the classical
equation of motion. The leapfrog method is used and accurate to the second order
with respect to the time slice. Self-consistency is imposed at each time on the spin
density by iteration and on the lattice variables through the Hellmann-Feynman
theorem. The bare optical phonon energy used here is
1
=
2
0354,
as a typical parameter. The results below are not modified by the details of the model
parameters. In the strong-coupling limit, the CDW phase is characterized by the
electron density, (
n
1
,
n
2
,
n
3
,
n
4
)
o ¼ð
2
K
MX
=MÞ
¼
0
:
¼
(1, 1, 2, 2) or (2, 2, 1, 1). The CDW order
i
parameters is defined as
r
CDW
ð
2
i
1
Þ¼ð
1
Þ
ðn
2
i
2
þ
2
n
2
i
1
þ n
2
i
3
Þ=
4 and
i
r
CDW
ð
2
iÞ¼ð
1
Þ
ðþn
2
i
1
þ
2
n
2
i
n
2
iþ
1
3
Þ=
4. In the same limit, the CP phase is
characterized by (
n
1
,
n
2
,
n
3
,
n
4
)
¼
(1, 2, 1, 2) or (2, 1, 2, 1). The CP order parameter
i
is defined as
r
A
(
i
)isso
defined that, in the strong-coupling limit of the A phase, it takes uniformly a positive
value or uniformly a negative value.
r
CP
ðiÞ¼ð
1
Þ
ðn
i
1
þ
2
n
i
n
iþ
1
Þ=
4. The order parameter
Search WWH ::
Custom Search