Controlling a quadratic band touching with an incommensurate potential

We theoretically investigate semimetals with quadratic band touching points in the presence of incomensurate potentials. Several different models with different symmetries, each realizing the quadratic band touching dispersions are considered. We combine analytical and numerical methods to examine the properties of the reconstructed quadratic touching bands such as the effective mass and whether the system remains as a semimetal. It is determined under what general conditions can this problem have a "magic-angle" where the curvature of the quadratic band touching vanishes. The experimental implications of these results are discussed.