\sum \limits_{j=1}^{N_B}g_{AB}\delta(x_i^{A}-x_j^{B}).\label{Hamilt} \end{aligned}$$ We operate in the ultracold regime, hence $s$-wave scattering is the dominant interaction process. Consequently the interspecies interactions can be adequately modeled by contact interactions, which scale with the effective one-dimensional coupling strength $g_{AB}$ for the different fermionic species. Since $s$-wave scattering is forbidden for spin-polarized fermions [@pethick2002bose; @lewenstein2012ultracold], due to the antisymmetry of the fermionic wavefunction, fermions of the same species are considered to be non-interacting. Therefore, only interspecies interactions are relevant in the MB Hamiltonian. The effective interspecies one-dimensional coupling strength [@olshanii1998atomic] is given by ${g_{AB}} =\frac{{2{\hbar ^2}{a^s_{AB}}}}{{\mu a_ \bot ^2}}{\left( {1 - {\left|{\zeta (1/2)} \right|{a^s_{AB}}}/{{\sqrt 2 {a_ \bot }}}} \right)^{ -
1}}$, where $\zeta$ refers to the Riemann zeta function and $\mu=\frac{M}{2}$ is the corresponding reduced mass. ${a_\bot } = \sqrt{\hbar /{\mu{\omega _ \bot }}}$ is the transversal length scale with transversal confinement frequency ${{\omega _ \bot }}$ and ${a^s_{AB}}$ is the three-dimensional $s$-wave scattering length between the two distinct species. We note that $g_{AB}$ can be experimentally adjusted either by means of ${a^s_{AB}}$ with the aid of Feshbach resonances [@kohler2006production; @chin2010feshbach] or by manipulating ${{\omega _ \bot }}$ via confinement-induced resonances [@olshanii1998atomic; @kim2006suppression].

In the following our Hamiltonian is rescaled in units of 