Bethe ansatz solvability and supersymmetry of the $M_2$ model of single fermions and pairs

A detailed study of a model for strongly-interacting fermions with exclusion rules and lattice $\mathcal{N}=2$ supersymmetry is presented. A submanifold in the space of parameters of the model where it is Bethe-ansatz solvable is identified. The relation between this manifold and the existence of additional, so-called dynamic, supersymmetries is discussed. The ground states are analysed with the help of cohomology techniques, and their exact finite-size Bethe roots are found. Moreover, through analytical and numerical studies it is argued that the model provides a lattice version of the $\mathcal{N}=1$ super-sine-Gordon model at a particular coupling where an additional $\mathcal{N}=(2,2)$ supersymmetry is present. The dynamic supersymmetry is shown to allow an exact determination of the gap scaling function of the model.