Lattice Models with $\mathcal{N}=2$ Supersymmetry

We introduce lattice models with explicit $\mathcal{N}=2$ supersymmetry. In these interacting models, the supersymmetry generators $Q^{\pm }$ yield the Hamiltonian $H=\{Q^{+},Q^{-}\}$ on any graph. The degrees of freedom can be described as either fermions with hard cores, or as quantum dimers; the Hamiltonian of our simplest model contains a hopping term and a repulsive potential. We analyze these models using conformal field theory, the Bethe ansatz, and cohomology. The simplest model provides a manifestly supersymmetric lattice regulator for the supersymmetric point of the massless $(1+1)$-dimensional Thirring (Luttinger) model. Generalizations include a quantum monomer-dimer model on a two-leg ladder.