Are there quantum oscillations in an incommensurate charge density wave?

Understand

Because a material with an incommensurate charge density wave (ICDW) is only quasi-periodic, 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.