Figure 2
(a)
in the lower Hubbard band (LHB) for
at
in the 1D Hubbard model obtained by using the dynamical density-matrix renormalization-group method [
21]. Curves indicate dominant modes obtained by using exact solutions [
21]. (b) RPA results for
obtained by using the data in (a). In (a),(b), straight solid green lines indicate
. Gaussian broadening is used with standard deviation
. (c)
[energy of the mode for
in the LHB at
] for
[Figs. 1i, 1j, 1k, 1p] (blue circles). The solid red curve indicates a fit. The dashed green curve shows results for
. (d) The same as (c) but for 1D. The solid blue curve shows
for
[
21]. In (d),(f),
,
, and
are defined in the 1D Hubbard model similarly to
,
, and
, respectively. (e)
[
extrapolated to
] (red circles) and
[Fig. 1l] (blue diamonds). The dotted green line indicates
, where the spin-wave velocity of the 2D Heisenberg model
[
27] (
). The inset shows
dependence. (f) The same as (e) but for 1D. The solid blue curve shows
[
21]. The dotted green line indicates
, where the spin-wave velocity of the 1D Heisenberg model
[
29]. (g)
(blue diamonds) and
(red circles) [Fig. 1l]. The dotted green line indicates a fit in the large-
regime, assuming
. The inset shows
dependence. (h) Spectral weight
for
in the LHB. Blue squares with a solid line show 2D results for
. Red circles with a dashed line denote 1D results for
taken from Ref.
21. The dotted green line indicates results for
[
10]. The inset shows the Mott gap
between the LHB and the upper Hubbard band at
. Blue diamonds show 2D results. The dotted red curve denotes 1D results [
33].
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