Unusual frequency of quantum oscillations in strongly particle-hole asymmetric insulators

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 appear 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. The relevance of the theory in the context of recent experiments on ${\mathrm{SmB}}₆$ is discussed.