Flat bands in tight-binding lattices with anisotropic potentials

We consider tight-binding models on Bravais lattices with anisotropic onsite potentials that vary along a given direction and are constant along the transverse one. Inspired by our previous work on flat bands in anti-$\mathcal{PT}$ symmetric Hamiltonians [Mallick et al., Phys.~Rev.~A 105, L021305 (2022)], we construct an anti-$\mathcal{PT}$ symmetric Hamiltonians with an $E=0$ flat band by tuning the hoppings and the shapes of potentials. This construction is illustrated for the square lattice with bounded and unbounded potentials. Unlike flat bands in short-ranged translationally invariant Hamiltonians, we conjecture that the considered $E=0$ flat bands do not host compact localized states. Instead the flat-band eigenstates exhibit a localization transition along the potential direction upon increasing the potential strength for bounded potentials. For unbounded potentials flat-band eigenstates are always localized irrespective of the potential strength.