Anti-$\mathcal{PT}$ Flatbands

We consider tight-binding single particle lattice Hamiltonians which are invariant under an anti-unitary anti-symmetry: 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.