Nonlinear Landau fan diagram and aperiodic magnetic oscillations in three-dimensional systems

Quantum oscillations offer a powerful probe for the geometry and topology of the Fermi surface in metals. Onsager's semiclassical quantization relation governs these periodic oscillations in 1/B, leading to a linear Landau fan diagram. However, higher-order magnetic susceptibility-induced corrections give rise to a generalized Onsager's relation, manifesting in experiments as a nonlinear Landau fan diagram and aperiodic quantum oscillations. Here, we explore the generalized Onsager's relation to three-dimensional (3D) systems to capture the B-induced corrections in the free energy and the Fermi surface. We unravel the manifestation of these corrections in the nonlinear Landau fan diagrams and aperiodic quantum oscillations by deriving the B-dependent oscillation frequency and the generalized Lifshitz-Kosevich equation, respectively. Our theory explains the necessary conditions to observe these fascinating effects and predicts the magnetic field dependence of the cyclotron mass. As a concrete example, we elucidate these effects in a 3D spin-orbit coupled system and extract zero-field magnetic response functions from analytically obtained Landau levels. Our comprehensive study deepens and advances our understanding of aperiodic quantum oscillations.