  Quantum oscillations, conventionally thought to be a metallic property, have
recently been shown to arise in certain kinds of insulators, with properties
very different from those in metals. All departures from the canonical behavior
found so far arise only in the amplitude and the phase but not in the
frequency. Here I show that such robustness in the behavior of the frequency is
only valid for a particle-hole symmetric insulator; in a strongly particle-hole
asymmetric insulator, de Haas-van Alphen oscillations (oscillations in
magnetization and susceptibility) and Shubnikov-de Haas oscillations
(oscillations in the density of states) exhibit different frequencies, with the
frequency of the latter changing with temperature. I demonstrate these effects
with numerical calculations on a lattice model, and provide a theory to account
for the unusual behavior.
