Magnetized charged boson gas and two-step condensation in neutron-star matter

We study the thermodynamic properties of a noninteracting, relativistic gas of charged scalar bosons in a uniform magnetic field, including both statistical and vacuum contributions at arbitrary field strengths. Focusing on the low-temperature regime and separating the lowest Landau level (LLL) from excited states provides a clearer view of the magnetic field’s impact on thermodynamic quantities. We revisit Bose-Einstein condensation (BEC), specific heat, magnetization, and the equation of state (EoS). A central result is the diffuse character of BEC induced by the magnetic field, reflected in the specific heat, which exhibits two plateaus: the first appears when the system becomes effectively one-dimensional through magnetic confinement, while the second—associated with the true one-particle ground state—is suppressed by the field. Consequently, no critical condensation temperature arises. For magnetization, the LLL contribution shifts from diamagnetic to paramagnetic. As the field strengthens with the inclusion of the excited states, the statistical magnetization remains negative. In contrast, the vacuum contribution dominates at strong fields, driving paramagnetism. We also show that antiparticles increase specific heat, magnetization, and total pressure. These effects are illustrated for a pion gas under neutron star conditions and compared with previous results for a neutral vector boson system.