Lattice Regularization of Non-relativistic Interacting Fermions in One Dimension

Few-body systems in one spatial dimension provide an ideal setting for testing lattice methods, since exact results are available against which lattice systematics can be assessed. We study two-component non-relativistic fermions with attractive contact interactions on a lattice, where the even ($s$-wave) interaction acts between opposite spins and the odd ($p$-wave) interaction between like spins, using exact diagonalization. The couplings are fixed entirely from two-body data. We demonstrate that the two-body couplings alone determine the few-body physics. We compute the three- and four-body systems' ground states when $s$- and $p$-wave interactions coexist and find that both of them, expressed relative to the two-body threshold, depend only on a single dimensionless parameter.