Quantum oscillation phenomena describe the periodic variation of thermodynamic and transport properties of materials as a function of magnetic field. Their observation is commonly assumed to be a definite sign for the presence of a Fermi surface (FS) in metals. Indeed, the effect forms the basis of a well-established experimental procedure for accurately measuring FS topology and geometry of metallic systems, with parameters commonly extracted by fitting to the Onsager and Lifshitz-Kosevich (LK) theories based on Fermi liquid theory.
I will discuss recent experimental and theoretical developments which challenge the canonical description of quantum oscillations. I will first review our work on quantum oscillations in insulators. Second, I will discuss the possibility of sharp quantum oscillation frequencies in metals which do not correspond to Fermi surface orbits. Finally, I will present a rigorous example of how inter-Landau level interactions in doped Mott insulators can lead to aperiodic quantum oscillations and unusual mass renormalisations.