Magnetic Susceptibility of Relativistic Electrons in the Crust of Strongly Magnetized Neutron Stars

Pulsars, characterized as highly magnetized and rapidly rotating neutron stars, offer a unique laboratory for probing physics under extreme conditions. Magnetars, a subclass of pulsars powered by magnetic field energy, exhibit quantized and highly degenerate Landau levels for relativistic electrons in their crustal ultrastrong magnetic fields. The energy difference between these Landau levels and the field-free system determines the magnetic susceptibility. We first review spin degrees of freedom in relativistic electrons and magnetization mechanisms, then employ quantum statistical methods to calculate the magnetic susceptibility of relativistic electron gases in magnetar crusts. Finally, numerical simulations for the paramagnetic susceptibility oscillatory in superhigh magnetic fields in the magnetar crust was performed. Our results reveal that the magnetization under ultrastrong fields demonstrates oscillatory behavior analogous to the de Haas–van Alphen effect observed in certain low-temperature metals. The total susceptibility, $\chi$, comprises a non-oscillatory component (${\chi }_m$) and an oscillatory term (${\tilde{\chi }}_m$), where higher harmonic amplitudes of the oscillatory susceptibility grow with increasing electron density. Notably, the total paramagnetic susceptibility of electrons near the crust-core boundary does not exceed the critical magnetization threshold. However, if an ultrastrong magnetic field exists in the neutron star core, the susceptibility of the electron gas could surpass this critical value, suggesting the potential occurrence of non-equilibrium magnetization processes. This implies a first-order phase transition, akin to gas–liquid transitions, leading to coexisting stable magnetization states or metastable supercooled magnetic phases. A sudden transition from metastable to stable states may release stored magnetic energy, offering a plausible explanation for the observed excess radiation during magnetar giant flares.