Wannier-Stark flatbands in Bravais lattices

We systematically construct flatbands (FB) for tight-binding models on simple Bravais lattices in space dimension d geq 2 in the presence of a static uniform DC field. Commensurate DC field directions yield irreducible Wannier-Stark (WS) bands in perpendicular dimension d - 1 with d-dimensional eigenfunctions. The irreducible bands turn into dispersionless flatbands in the absence of nearest neighbor hoppings between lattice sites in any direction perpendicular to the DC field. The number of commensurate directions which yield flatbands is of measure one. We arrive at a complete halt of transport, with the DC field prohibiting transport along the field direction, and the flatbands prohibiting transport in all perpendicular directions as well. The anisotropic flatband eigenstates are localizing at least factorially (faster than exponential).