  We develop a theory of quantum oscillations in insulators with an emergent
fermi sea of neutral fermions minimally coupled to an emergent $U(1)$ gauge
field. As pointed out by Motrunich (Phys. Rev. B 73, 155115 (2006)), in the
presence of a physical magnetic field the emergent magnetic field develops a
non-zero value leading to Landau quantization for the neutral fermions. We
focus on the magnetic field and temperature dependence of the analogue of the
de Haas-van Alphen effect in two- and three-dimensions. At temperatures above
the effective cyclotron energy, the magnetization oscillations behave similarly
to those of an ordinary metal, albeit in a field of a strength that differs
from the physical magnetic field. At low temperatures the oscillations evolve
into a series of phase transitions. We provide analytical expressions for the
amplitude and period of the oscillations in both of these regimes and simple
extrapolations that capture well their crossover. We also describe oscillations
in the electrical resistivity of these systems that are expected to be
superimposed with the activated temperature behavior characteristic of their
insulating nature and discuss suitable experimental conditions for the
observation of these effects in mixed-valence insulators and triangular lattice
organic materials.
