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Onsager rule, quantum oscillation frequencies, and the density of states in the mixed-vortex state of cuprates

The Onsager rule determines the frequencies of quantum oscillations in magnetic fields. We show that this rule remains intact to an excellent approximation in the mixed-vortex state of the underdoped cuprates even though the Landau level index $n$ may be fairly low, $n\sim 10$. The models we consider are fairly general, consisting of a variety of density wave states combined with $d$-wave superconductivity within a mean field theory. Vortices are introduced as quenched disorder and averaged over many realizations, which can be considered as snapshots of a vortex liquid state. We also show that the oscillations ride on top of a field independent density of states, $ρ(B)$, for higher fields. This feature appears to be consistent with recent specific heat measurements [C. Marcenat, et al. Nature Comm. {\bf 6}, 7927 (2015)]. At lower fields we model the system as an ordered vortex lattice, and show that its density of states follows a dependence $ρ(B)\propto \sqrt{B}$ in agreement with the semiclassical results [G. E. Volovik, JETP Lett. {\bf 58}, 469 (1993)].