Magnetized Dense Matter in the Nambu–Jona–Lasinio Model and Its Applications to Compact Stars

We review the properties of strongly interacting matter under the influence of an external magnetic field, with emphasis on results obtained using the Nambu–Jona–Lasinio model and its variants. Strong magnetic fields, relevant for heavy-ion collisions and magnetars, introduce modifications to the phase structure of quantum chromodynamics. In the first half of the review, we discuss the Nambu–Jona–Lasinio model and its major variants, including Polyakov loop extended model, Kobayashi–Masakawa–’tHooft extended model, entangled model, and nonlocal formulations, highlighting their successes and limitations. Particular attention is given to magnetic catalysis and inverse magnetic catalysis and the tension between model predictions and lattice simulation results. We further review the emergence of phases at high density and low temperature such as the magnetic color–flavor locked phase, the magnetic two-flavor superconducting phase, the crystalline color superconductor and magnetized quarkyonic matter, along with possible mesonic π and ρ meson condensation channels. Implications for strongly magnetized neutron stars are discussed wherever relevant. This review aims to provide a focused overview of the capabilities and limitations of approaches within the Nambu–Jona–Lasinio model and to outline open challenges in understanding strongly interacting magnetized matter at intermediate densities.