Hamiltonian of the system

We consider one-dimensional two-component fermi gas at zero temperature. Pauli principle applies to fermions of same spin (and spin orientation). We model interactions between fermions of different spins by pseudopotential of strength $g_{1D} = -2\hbar^2/ma_{1D}$ with $a_{1D}$ being the one-dimensional scattering length. We consider both negative $g_{1D}$ (attraction) and positive $g_{1D}$ (repulsion). The Hamiltonian of a homogeneous system is given by

$$\hat H_{1d}^0 = -\frac{\hbar^2}{2m}\sum\limits_{i=1}^{N} \frac{\p... ...^{N_{\uparrow}}\sum_{i=1}^{N_{\downarrow}} \delta(z_i^\uparrow-z_j^\downarrow),$$ (1)

where $N = N_\uparrow + N_\downarrow$ is total number of fermions of mass $m$. In following we consider same number of fermions of one spin state and the other spin state $N_\uparrow = N_\downarrow$. Properties of a homogeneous system is defined by the one-dimensional gas parameter $na_{1D}$ with $n$ being density of the system. In presence of external harmonic confinement the one-dimensional Hamiltonian is given by

$$\hat H_{1d} = \hat H_{1d}^0 + \sum\limits_{i=1}^N\frac{1}{2}m\omega_z^2z_i^2$$ (2)

In this system an important parameter is $Na_{1d}^2/a_z^2$ with $a_z = \sqrt{\hbar/m\omega_z}$ being oscillator length.