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Figure 6.2:
Static structure factor
for a gas of HR at different values of the gas parameter
(symbols) and for a TG gas (dashed line).
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Contrarily to the TG case, it is not possible to obtain analytical expressions for
and
in the HR problem. We have calculated them using configurations
generated by a Monte Carlo simulation according to the exact probability
distribution function
. The results for the static structure factor
are shown in Fig. 6.2. Compared to
in the TG regime, a clear peak is
visible for values of
of the order of twice the Fermi wave vector
and the peak is more pronounced as
increases. The change of slope for
small values of
reflects the increase of the speed of sound
with
.
The long-range behavior of
can be obtained from the hydrodynamic theory of
low-energy excitations [RC67,Sch77,Hal81]. For
, where
is the healing length of the system, one finds the
following power-law decay (1.199):
![\begin{displaymath}
g_1(z)\propto1/\vert z\vert^\alpha,
\end{displaymath}](img1450.gif) |
(6.2) |
where the exponent
is given by
. For a TG gas
, and thus
. For a HR gas one finds
and thus
. This behavior at
long range is clearly shown in Fig. 6.3 where we compare
of a gas
of HR with
, 0.2 and 0.3 to the result of a TG gas [JMMS80]. The
long-range power-law decay of
is reflected in the infrared divergence of
the momentum distribution
holding for
.
A gas of HR exhibits a weaker infrared divergence compared to a TG gas. The
correlation functions of a HR gas at
, 0.2 should accurately describe
the physical situation of a Bose gas with large and negative
. For
we expect already some deviations from the HR model, as it is evident
from the equation of state in Fig. 6.1, which should broaden the peak in
and decrease the slope of the power-law decay in
at large distances.
The analysis of correlation functions clearly shows that the super-Tonks regime
corresponds to a Luttinger liquid where short range correlations are significantly
stronger than in the TG gas.
Figure 6.3:
One-body density matrix
for a gas of HR at different values of the gas parameter
(solid lines) and for a TG gas (dashed line). Higher values of density correspond
to a faster decay of
.
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G.E. Astrakharchik
15-th of December 2004