Supersymmetry on the lattice: Geometry, flat bands, and topology

Supersymmetry (SUSY) posits an equivalence between two fundamental degrees of freedom: fermions and bosons. Using this, a variety of phonon and magnon models can be identified which have topologically nontrivial free fermion models as superpartners. At the single-particle level, the bosonic and the fermionic models that are generated by SUSY are isospectral except for zero modes that potentially manifest as flat bands. The existence of a flat band on a lattice is intricately related to the Witten index of the SUSY theory it is generated from and admits a five-fold classification by virtue of SUSY. Extending to real degrees of freedom, we can systematically construct topological mechanical systems by an exact SUSY. Such a construction naturally defines hitherto unexplored topological invariants for bosonic (mechanical) systems, such as bosonic Wilson loop operators that are formulated in terms of SUSY-related fermionic Berry curvature. This would be useful to identify topological bosonic models whose fermionic partner represents some symmetry-protected topological phase.