  Quantum magnetic oscillations in crystals are typically understood in terms
of Bohr-Sommerfeld quantisation, the frequency of oscillation is given by the
area of a closed electron trajectory. However, since the 1970s, oscillations
have been observed with frequencies that do not correspond to closed electron
trajectories and this effect has remained not fully understood. Previous theory
has focused on explaining the effect using various kinetic mechanisms, however,
frequencies without a closed electron orbit have been observed in equilibrium
and so a kinetic mechanism cannot be the entire story. In this work we develop
a theory which explains these frequencies in equilibrium and can thus be used
to understand measurements of both Shubnikov-de Haas and de Haas-van Alphen
oscillations. We show, analytically, that these frequencies arise due to
multi-electron correlations. We then extend our theory to explain a recent
experiment on artificial crystals in GaAs two-dimensional electron gases, which
revealed for the first time magnetic oscillations having frequencies that are
half of those previously observed. We show that the half-frequencies arise in
equilibrium from single-particle dynamics with account of impurities. Our
analytic results are reinforced by exact numerics, which we also use clarify
prior works on the kinetic regime.
