Quantum Oscillations in the Zeroth Landau Level: Serpentine Landau Fan and the Chiral Anomaly

We identify an unusual mechanism for quantum oscillations in nodal semimetals, driven by a single pair of Landau levels periodically closing their gap at the Fermi energy as a magnetic field is varied. These ``zero Landau level'' quantum oscillations (ZQOs) appear in the nodal limit where the zero-field Fermi volume vanishes and have distinctive periodicity and temperature dependence. We link the Landau spectrum of a two-dimensional (2D) nodal semimetal to the Rabi model, and show by exact solution that, across the entire Landau fan, pairs of opposite-parity Landau levels are intertwined in a ``serpentine'' manner. We propose 2D surfaces of topological crystalline insulators as natural settings for ZQOs. In certain 3D nodal semimetals, ZQOs lead to oscillations of anomaly physics. We propose a transport measurement capable of observing such oscillations, which we demonstrate numerically.