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the basis set makes it computationally prohibitive to model any device larger
than just a few cells. Thus, further approximations are typically required for
modelling larger arrays of QCA cells.
1.2 Two-State Approximation
The first of such approximations is the two-state approximation. As shown
in [
39
], the ground state of a single cell remains almost completely contained
within a two-dimensional subspace of the full sixteen-dimensional Hilbert space.
For simplicity, we will consider idealized cells with saturation polarizations of
±
1. A more realistic treatment would have the polarizations spanning a slightly
smaller range, however the basic argument that follows would be unchanged. We
therefore refer to our reduced basis as the polarization basis, and denote the two
states as
, as shown in Fig.
1
. The Hamiltonian in the polarization
basisforasystemof
N
interacting QCA cells, under the influence of driver cells,
is described by a 2
N
|
0
≥
and
|
1
≥
2
N
×
Ising-like Hamiltonian:
N
N−
1
N
N
1
2
ʳ
z
(
i
)
ʳ
z
(
j
)+
1
2
H
=
E
i,j
k
E
i,D
k
−
ʷ
i
ʳ
x
(
i
)
−
P
D
ʳ
z
(
i
)
,
(3)
i
=1
i
=1
j
=
i
+1
i
=1
D∈Ω
i
where
ʩ
i
represents the neighbourhood of driver cells that have appreciable
effect on cell
i
,and
P
D
is the polarization of those cells. The driver cells have
polarizations that can range from
1 to +1. The tunnelling energy,
ʷ
, takes the
place of the transverse magnetic field in the Ising spin model, and is determined
from the nature of the potential barriers between the dots in the cell. When
the potential barriers are raised,
ʷ
is small, and the cells are allowed to become
polarized (i.e., take on one of the antipodal configurations). When the potential
barriers are lowered, the opposite is true, and the electrons become delocalized.
E
i,
k
is the kink energy between cells, and is the energetic cost of two cells having
opposite polarization. In general, the interaction between cells can be described
by a quadrupole-quadrupole interaction in which the kink energy decreases as
the fifth power of the distance between cells. Thus it is typical to only consider
nearest-neighbour coupling within the Hamiltonian, as the kink energy between
next-to-nearest neighbours is reduced by a factor of
−
1
32
. The kink energy is anal-
ogous to the coupling constant,
J
, in the Ising spin model. The Pauli operators,
ʳ
a
(
i
);
a
=
x, y, z
, represent the tensor product of
N
2
2 identity operators,
with the
i
th
identity operator replaced by the usual Pauli matrix shown below.
For example,
ʳ
y
(2)
×
↕
1
↗
ʳ
y
↗
1
↗ ...↗
1
. The polarization of a cell,
i
, is defined
as
P
i
=
−≤
ʳ
z
(
i
)
.
ʳ
x
=
01
≥
,
ʳ
y
=
0
i
,
ʳ
z
=
−
.
10
01
10
−i
0
Asshownin[
39
], the ground state of a single cell within the full sixteen-
dimensional Hilbert space remains almost completely contained within the
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