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Consider the Hamiltonian
, Eq. 8.3, which describes a
homogeneous 1D two-component Fermi gas. The ground state energy
of
has been calculated exactly using Bethe's ansatz for
attractive [Gau67] and repulsive [Yan67] interactions, and can be
expressed in terms of the linear number density
, where
is the size
of the system,
![\begin{displaymath}
\frac{E_{hom}}{N}=\frac{\hbar^2n_{1D}^2}{2m}e(\gamma)
\end{displaymath}](img1696.gif) |
(8.6) |
The dimensionless parameter
is proportional to the coupling constant
,
, while its absolute value is inversely
proportional to the 1D gas parameter
,
.
The function
is obtained by solving a set of integral
equations8.3, which is similar to that
derived by Lieb and Liniger [LL63] for 1D bosons with repulsive contact
interactions.
To obtain the energy per particle, Eq. 8.6, we solve these integral
equations for
[Gau67] and for
[Yan67].
Figure 8.2 shows
Figure 8.2:
(solid line),
(dashed line)
and
(inset) for a homogeneous two-component 1D Fermi gas as a function of
(horizontal arrows indicate the asymptotic values of
,
and
, respectively).
|
the energy per particle,
(solid line), the chemical potential
,
(dashed line), and the velocity of sound
(inset), which is obtained from the inverse compressibility
, as a function of the interaction
strength
. In the weak coupling limit,
,
is given
by
![\begin{displaymath}
\mu_{hom}=\frac{\pi^2}{4} \; \frac{\hbar^2n_{1D}^2}{2m}+
\gamma \; \frac{\hbar^2 n_{1D}^2}{2m} +
\cdots\;,
\end{displaymath}](img1705.gif) |
(8.7) |
where the first term on the right hand side is the energy of an ideal two-component
atomic Fermi gas, and the second term is the mean-field energy, which accounts for
interactions. The chemical potential increases with increasing
, and reaches
an asymptotic value for
(indicated by a horizontal arrow
in Fig. 8.2),
![\begin{displaymath}
\mu_{hom}= \pi^2 \; \frac{\hbar^2n_{1D}^2}{2m}-
\frac{16 \pi^2 \ln(2)}{3 \gamma} \; \frac{\hbar^2n_{1D}^2}{2m} + \cdots\;.
\end{displaymath}](img1707.gif) |
(8.8) |
The first term on the right hand side coincides with the chemical potential of a
one-component ideal 1D Fermi gas with
atoms, the second term has been calculated
in [RFZZ03a]. Interestingly, for
, the strong atom-atom repulsion
between atoms with different spin plays the role of an effective Pauli
principle [RFZZ03a,RFZZ03b].
For attractive interactions and large enough
the energy per particle is
negative (see Fig. 8.2), reflecting the existence of a molecular Bose gas,
which consists of
diatomic molecules with binding energy
.
Each molecule is comprised of two atoms with different spin. In the limit
, the chemical potential becomes
![\begin{displaymath}
\mu_{hom}=-\frac{\hbar^2}{2ma_{1D}^2}+
\frac{\pi^2}{16} \; \...
...\frac{\pi^2}{12 \gamma} \;
\frac{\hbar^2n_{1D}^2}{2m} + \cdots
\end{displaymath}](img1713.gif) |
(8.9) |
The first term is simply
, one half of the binding energy of the
1D molecule, while the second term is equal to half of the chemical potential of a
bosonic Tonks-Girardeau gas with density
, consisting of
molecules
with mass
8.4.
Importantly, the compressibility remains positive for
[a horizontal arrow in the inset of Fig. 8.2 indicates the asymptotic
value of
,
], which implies that two-component 1D Fermi
gases are mechanically stable even in the strongly-attractive regime. In contrast,
the ground state of 1D Bose gases with
has negative
compressibility [McG64] and is hence mechanically unstable.
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Up: Interacting fermions in highly
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G.E. Astrakharchik
15-th of December 2004