Supersymmetry on the lattice: route to flat bands and their topological classification

In quantum mechanics, supersymmetry (SUSY) posits an equivalence between two elementary degrees of freedom, bosons and fermions. In this talk, I will show how this fundamental concept can be applied to connect bosonic and fermionic lattice models in the realm of condensed matter physics, e.g., to identify a variety of (bosonic) phonon and magnon lattice models which admit topologically nontrivial free fermion models as superpartners. At the single-particle level, the bosonic and the fermionic models that are generated by the SUSY are isospectral except for zero modes, such as flat bands, whose existence is undergirded by the Witten index of the SUSY theory. We develop a unifying framework to formulate these SUSY connections in terms of general lattice graph correspondences. Notably, a Hermitian form of the supercharge operator, the generator of the SUSY, can itself be interpreted as a hopping Hamiltonian on a bipartite lattice. A five-fold symmetry classification scheme of the SUSY lattice correspondences is derived, including cases with a non-zero Witten index, based on a topological classification of the underlying Hermitian supercharge operator.