Entropy and de Haas-van Alphen oscillations of a three-dimensional marginal Fermi liquid

We study quantum oscillations (de Haas-van Alphen effect) in a three-dimensional metal tuned to a quantum critical point. The conventional approach to this problem involves extensions of the Lifshitz-Kosevich formula, which breaks down when the correlation length exceeds the cyclotron radius, mainly due to (i) non-analytic finite-temperature dependence of the self-energy, (ii) an enhancement of the oscillatory part of the self-energy by quantum fluctuations, and (iii) non-trivial dynamical scaling laws associated with quantum criticality. By incorporating these effects, we derive the modified oscillation amplitudes of the specific heat and magnetization. Our results manifestly satisfy the third law of thermodynamics (Nernst's theorem) and remain valid in the non-Fermi liquid regime.