  Quantum oscillations (QO) describe the periodic variation of physical
observables as a function of inverse magnetic field in metals. The Onsager
relation connects the basic QO frequencies with the extremal areas of closed
Fermi surface pockets, and the theory of magnetic breakdown explains the
observation of sums of QO frequencies at high magnetic fields. Here we develop
a quantitative theory of difference frequency QOs in two- and three-dimensional
metals with multiple Fermi pockets with parabolic or linearly dispersing
excitations. We show that a non-linear interband coupling, e.g. in the form of
interband impurity scattering, can give rise to otherwise forbidden QO
frequencies which can persist to much higher temperatures compared to the basis
frequencies. We discuss the experimental implications of our findings for
various material candidates, for example multi-fold fermion systems, and the
relation to magneto intersubband oscillations known for coupled two-dimensional
electron gases.
