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b
a
0
0
1.2
0
V=3to
0
4to
1.0
0
-
0.8
-
-4to
-
0.6
-
-(13/3)to
-20
-15
-10
-5
0
5
10
15
20
0
0
0.4
V=2to
0
0
0.2
0
-
0
-6
-4
-2
0
2
4
6
-
-
(
-U)/to
ω
-
-20
-15
-10
-5
0
5
10
15
20
Relative Position of Doublon (site)
Fig. 8.20 (a) Spectra calculated by the doublon-holon model. Inset: Schematic energy diagram
for
V ¼
3
t
0
. The lowest two discrete levels are almost degenerate. In the figure, their splitting is
exaggerated. (b) Two wave functions of the lowest two levels. The values as a function of the
doublon position relative to that of the holon are plotted
of extremely large attraction. These marked features, (1) and (2), are well under-
stood by the “semi-infiniteness” of this model. Namely, as we have already defined,
the doublon and the holon do not occupy the same site. Moreover, we neglect the
probability of the position exchange between them, because such transition occurs
via the ground state and the probability is reduced by a factor comparable to 1/
U
.
Exactly speaking, a PBC, by which the system becomes a ring, makes the exchange
possible. However, such effect is negligible when we consider a sufficiently large
system size. In any way, the substantial inhibition of the doublon-hole exchange
separate all the states into two spaces; the doublon is on the left-hand side of the
holon, and vice versa. This situation enables us to consider the problem indepen-
dently in each space. If we focus on the relative coordinate of the two particles, this
situation corresponds to a semi-infinite system, in the sense of considering only a
positive or negative region of the coordinate. It is well known that the semi-infinite
system has a critical value of the attraction for the bound state formation, and this
also applies for our problem. The energy degeneracy between the two lowest
excitons is understood naturally in this context [
33
,
34
], as illustrated in
Fig.
8.20b
. Regarding the spectral shape, the similarity between the spectra in
Fig.
8.19
and those obtained using a full hamiltonian in Fig.
8.18
tells us the
usefulness of this doublon-hole model in spite of its simpleness.
In the rest of this section, we discuss another aspect of the optical conductivity
spectrum in the Ni compounds. The point is the sharpness of the main peak. As has
been already discussed, this sharpness is nothing but an exciton effect, and, in this
sense, it is not surprising. However, even in the Mott insulators, there must be e-l
interaction to some degree, and, therefore, we must expect a certain amount of line
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