Anti-mathcal{PT} flatbands

We consider tight-binding single particle lattice Hamiltonians which are invariant under an antiunitary antisymmetry: the anti-mathcal{PT} symmetry. The Hermitian Hamiltonians are defined on d-dimensional non-Bravais lattices. For an odd number of sublattices, the anti-mathcal{PT} symmetry protects a flatband at energy E = 0. We derive the anti-mathcal{PT} constraints on the Hamiltonian and use them to generate examples of generalized kagome networks in two and three lattice dimensions. Furthermore, we show that the anti-mathcal{PT} symmetry persists in the presence of uniform DC fields and ensures the presence of flatbands in the corresponding irreducible Wannier-Stark band structure. We provide examples of the Wannier-Stark band structure of generalized kagome networks in the presence of DC fields, and their implementation using Floquet engineering.