Because a material with an incommensurate charge density wave (ICDW) is only quasiperiodic, Bloch's theorem does not apply and there is no sharply defined Fermi surface. We will show that, as a consequence, there are no quantum oscillations which are truly periodic functions of $1/B$ (where $B$ is the magnitude of an applied magnetic field). For a weak ICDW, there exist broad ranges of $1/B$ in which approximately periodic variations occur, but with frequencies that vary inexorably in an unending cascade with increasing $1/B$. For a strong ICDW, e.g., in a quasicrystal, no quantum oscillations survive at all. Rational and irrational numbers really are different.

• Received 27 October 2014

DOI:https://doi.org/10.1103/PhysRevB.91.085105