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We introduce a generic and accessible implementation of an exact diagonalization method for studying few-fermion models. Our aim is to provide a testbed for the newcomers to the field as well as a stepping stone for trying out novel optimizations and approximations. This userguide consists of a description of the algorithm, and several examples in varying orders of sophistication. In particular, we exemplify our routine using an effective-interaction approach that fixes the low-energy physics. We benchmark this approach against the existing data, and show that it is able to deliver state-of-the-art numerical results at a significantly reduced computational cost., 22 + 4 pages

We present a numerical analysis of spin-$\frac{1}{2}$ fermions in a one-dimensional harmonic potential in the presence of a magnetic point-like impurity at the center of the trap. The model represents a few-body analogue of a magnetic impurity in the vicinity of an $s$-wave superconductor. Already for a few particles we find a ground-state level crossing between sectors with different fermion parities. We interpret this crossing as a few-body precursor of a quantum phase transition, which occurs when the impurity `breaks' a Cooper pair. This picture is further corroborated by analyzing density-density correlations in momentum space. Finally, we discuss how the system may be realized with existing cold-atoms platforms., Comment: SciPost Submission

We study the phase diagram of two different mixed-dimensional $t-J_z-J_{\perp}$-models on the square lattice, in which the hopping amplitude $t$ is only nonzero along the $x$-direction. In the first, bosonic, model, the spin exchange amplitude $J_{\perp}$ is negative and isotropic along the $x$ and $y$ directions of the lattice, and $J_z$ is isotropic and positive. The low-energy physics is characterized by spin-charge separation: the holes hop as free fermions in an easy-plane ferromagnetic background. In the second model, $J_{\perp}$ is restricted to the $x$-axis while $J_z$ remains isotropic and positive. The model is agnostic to particle statistics, and shows stripe patterns with anti-ferromagnetic N{\'e}el order at low temperature and high hole concentrations, in resemblance of the mixed-dimensional $t-J_z$ and $t-J$ models. At lower hole concentration, a very strong first order transition and hysteresis loop is seen extending to a remarkably high 14(1)% hole doping., Comment: 8 pages, 9 figures