The Onsager rule determines the relationship between Fermi surface area and 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 $\rho \left(B\right)$ for higher fields. This feature appears to be consistent with recent specific heat measurements [C. Marcenat et al., Nature Communications 6, 7927 (2015)]. The experimental data do not go to low fields at the lowest temperature 3 K. Thus, we cannot compare the density of state for the entire field range. Of course, the high temperature data are linear in the field at lower fields, as they should be, but our theory is only valid at very low temperatures, ideally at zero temperature. At lower fields and zero temperature we model the system as an ordered vortex lattice, and show that its density of states follows a dependence $\rho \left(B\right)\propto \sqrt{B}$ in agreement with the semiclassical results [JETP Lett 58, 469 (1993)].

• Received 4 March 2016
• Revised 2 May 2016

DOI:https://doi.org/10.1103/PhysRevB.93.184505