Homogeneous system

Consider the Hamiltonian $H_{1D}^0$, Eq. 8.3, which describes a homogeneous 1D two-component Fermi gas. The ground state energy $E_{hom}$ of $H_{1D}^0$ 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 $n_{1D} = N/L$, where $L$ is the size of the system,

$$\frac{E_{hom}}{N}=\frac{\hbar^2n_{1D}^2}{2m}e(\gamma)$$ (8.6)

The dimensionless parameter $\gamma$ is proportional to the coupling constant $g_{1D}$, $\gamma=m g_{1D}/(\hbar^2n_{1D})$, while its absolute value is inversely proportional to the 1D gas parameter $n_{1D}\vert a_{1D}\vert$, $\vert\gamma\vert=2/n_{1D}\vert a_{1D}\vert$. The function $e(\gamma)$ is obtained by solving a set of integral equations^8.3, 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 $\gamma<0$ [Gau67] and for $\gamma>0$ [Yan67].

Figure 8.2 shows

Figure 8.2: $E_{hom}/N$ (solid line), $\mu _{hom}$ (dashed line) and $c$ (inset) for a homogeneous two-component 1D Fermi gas as a function of $\gamma$ (horizontal arrows indicate the asymptotic values of $E_{hom}/N$, $\mu _{hom}$ and $c$, respectively).

the energy per particle, $E_{hom}/N$ (solid line), the chemical potential $\mu _{hom}$, $\mu_{hom}=dE_{hom}/dN$ (dashed line), and the velocity of sound $c$ (inset), which is obtained from the inverse compressibility $mc^2=n_{1D}\partial\mu_{hom}/\partial n_{1D}$, as a function of the interaction strength $\gamma$. In the weak coupling limit, $\vert\gamma\vert\ll 1$, $\mu _{hom}$ is given by

$$\mu_{hom}=\frac{\pi^2}{4} \; \frac{\hbar^2n_{1D}^2}{2m}+ \gamma \; \frac{\hbar^2 n_{1D}^2}{2m} + \cdots\;,$$ (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 $\gamma$, and reaches an asymptotic value for $\gamma \rightarrow \infty$ (indicated by a horizontal arrow in Fig. 8.2),

$$\mu_{hom}= \pi^2 \; \frac{\hbar^2n_{1D}^2}{2m}- \frac{16 \pi^2 \ln(2)}{3 \gamma} \; \frac{\hbar^2n_{1D}^2}{2m} + \cdots\;.$$ (8.8)

The first term on the right hand side coincides with the chemical potential of a one-component ideal 1D Fermi gas with $N$ atoms, the second term has been calculated in [RFZZ03a]. Interestingly, for $\gamma\gg 1$, 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 $\vert\gamma\vert$ the energy per particle is negative (see Fig. 8.2), reflecting the existence of a molecular Bose gas, which consists of $N/2$ diatomic molecules with binding energy $\epsilon_{bound}$. Each molecule is comprised of two atoms with different spin. In the limit $\gamma \rightarrow -\infty$, the chemical potential becomes

$$\mu_{hom}=-\frac{\hbar^2}{2ma_{1D}^2}+ \frac{\pi^2}{16} \; \... ...\frac{\pi^2}{12 \gamma} \; \frac{\hbar^2n_{1D}^2}{2m} + \cdots$$ (8.9)

The first term is simply $\epsilon_{bound}/2$, 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 $n_{1D}/2$, consisting of $N/2$ molecules with mass $2m$^8.4. Importantly, the compressibility remains positive for $\gamma \rightarrow -\infty$ [a horizontal arrow in the inset of Fig. 8.2 indicates the asymptotic value of $c$, $c=\pi\hbar n_{1D}/(4m)$], 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 $g_{1D}<0$ has negative compressibility [McG64] and is hence mechanically unstable.