Abstract
Abstract
A conceptual quantity—the minimal effective amount of a quantum state
ϕ
(
r
j
)
in d-dimensional systems, defined by
N
∗
=
∑
j
=
1
N
min
{
N
|
ϕ
(
r
j
)
|
2
,
1
}
, is newly proposed, where system sizes
N
=
L
d
. The effective dimension d
IR
can be calculated by
N
∗
=
h
∗
(
L
)
L
d
I
R
, where
h
∗
(
L
)
does not change faster than any nonzero power. However, the nature of
h
∗
(
L
)
is unknown priori in any given model, but is at the same time very important for its numerical analysis. Hence, analytical results can provide insights on
h
∗
(
L
)
in more complex situations. In this paper, we get exact results of 1D continuous sine functions, exponential decay functions and power-law decay functions. They are used to distinguish extended and localized phases in the 1D uniform potential model, Anderson model and HMP (hopping rates modulated by a power-law function) model.
Funder
National Natural Science Foundation of China