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

Abstract

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)].