  We study quantum oscillations for a system of fermionic and bosonic dimers
and compare the results to those experimentally observed in the cuprate
superconductors in their underdoped regime. Based on gauge invariance, we argue
that the charge carriers obey the Onsager quantization condition and quantum
oscillations take on a Lifshitz-Kosevich form. We obtain the effective mass and
find good qualitative agreement with experiments if we tune the model to the
point where the observed mass divergence at optimum doping is associated to a
van Hove singularity at which four free-dimer Fermi pockets touch pairwise in
the interior of the Brillouin zone. The same van Hove singularity leads to a
maximum in the d-wave superconducting pairing amplitude when anti-ferromagnetic
interactions are included. Our combined results therefore suggest that a
quantum critical point separating the underdoped and overdoped regimes is
marked by the location of the van Hove saddle point in the fermionic dimer
dispersion.
