Supersymmetric lattice models: Field theory correspondence, integrability, defects and degeneracies

In this PhD thesis lattice models of interacting spinless fermions with an explicit supersymmetry are studied in one-dimension. The interactions between the particles in these models occur because of the restriction that at most k particles can occupy neighbouring sites. For every value of k there is a new model, called the Mk model. In this thesis the focus is on the k=1 and k=2 models but the k=3 model is mentioned as well. The Mk models have a critical point and can be tuned off-critical by a staggering perturbation. In both cases the models are integrable for the right values of their parameters. For the M2 model this parameter space is identified by use of a Bethe ansatz. Also two additional dynamical supersymmetries are found and the relation between these and the Bethe ansatz is studied. An important topic in this thesis is the relation between the lattice model and a field theory that corresponds to the continuum limit of the lattice model. At the critical point the Mk model relates to the k-th N=2 superconformal minimal model. By introducing defects into the lattice model all (para)fermion spin fields in the corresponding CFT can be identified. A spinon basis for the CFT is constructed from which properties of a corresponding lattice model can be found. Off-criticality the relation of the M2 model with the super sine-Gordon model is studied.