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Quantum oscillations of the specific heat in d-wave superconductors with loop current order

We report numerical results of quantum oscillations of the specific heat in the vortex state of a $d_{x^2-y^2}$-wave superconductor in the presence of loop current order, which gives rise to Fermi pockets coexisting with nodal $d_{x^2-y^2}$-wave superconductivity. Within a lattice tight-binding model, we find that in an intermediate temperature range, the oscillations seem to approximately follow Onsager relation with an effective charge comparable to the electric charge. However, the quasiparticle spectrum does not resemble Landau levels. In order to understand the origin of the oscillations, we also perform Franz-Tesanovic transformation in the presence of the loop order and find that in addition to scalar and Berry potentials, one component of the gauge invariant superfluid velocity couples to the low lying Dirac particles as a component of a vector potential. The magnetic field associated with this vector potential vanishes on average but is highly non-uniform in the magnetic unit cell. We also compare the results with the model without the loop order but with Zeeman-like coupling which also induces Fermi pockets in the superconducting state.