Mixed phase in a compact star with strong magnetic field

Compact stars can have either hadronic matter or can have exotic states of matter like strange quark matter or color superconducting matter. Stars also can have a quark core surrounded by hadronic matter, known as hybrid stars (HS). The HS is likely to have a mixed phase in between the hadron and quark phase. Observational results suggest huge surface magnetic field in certain neutron stars (NS) called magnetars. Here we study the effect of strong magnetic field on the respective EOS of matter under extreme conditions. We further study the hadron-quark phase transition in the interiors of NS giving rise to hybrid stars (HS) in presence of strong magnetic field. The hadronic matter EOS is described based on relativistic mean field theory and we include the effect of strong magnetic fields leading to Landau quantization of the charged particles. For the quark phase we use the simple MIT bag model. We assume density dependent bag pressure and magnetic field. The magnetic field strength increases going from the surface to the center of the star. We construct the intermediate mixed phase using Glendenning conjecture. The magnetic field softens the EOS of both the matter phases. The effect of magnetic field is insignificant unless the field strength is above $10^{14}$G. A varying magnetic field, with surface field strength of $10^{14}$G and the central field strength of the order of $10^{17}$G has significant effect on both the stiffness and the mixed phase regime of the EOS. We finally study the mass-radius relationship for such type of mixed HS, calculating their maximum mass, and compare them with the recent observation of pulsar PSR J1614-2230, which is about 2 solar mass. The observations puts a severe constraint on the EOS of matter at extreme conditions. The maximum mass with our EOS can reach the limit set by the observation.

💡 Research Summary

The paper investigates how extremely strong magnetic fields influence the equation of state (EOS) of matter inside compact stars and, consequently, the structure of hybrid stars (HS) that contain both hadronic and quark phases. The authors adopt a relativistic mean‑field (RMF) model for the hadronic phase, incorporating σ, ω, and ρ meson interactions, and they explicitly include Landau quantization for all charged particles (protons, electrons, muons). This quantization becomes significant when the magnetic field exceeds roughly 10^14 G, leading to a reduction of the pressure at a given density and thus a softening of the hadronic EOS.

For the quark phase, the simple MIT bag model is employed, but the bag constant is allowed to depend on the baryon density. A decreasing bag pressure at high density lowers the deconfinement threshold, making the transition to quark matter occur at lower densities. Charged quarks (u, d) and electrons are also subjected to Landau quantization, so the quark EOS is softened in a similar magnetic‑field‑dependent manner.

The transition between the two phases is treated using the Glendenning construction, which permits a mixed phase where both hadronic and quark matter coexist while maintaining global charge neutrality and mechanical equilibrium. The authors assume a realistic magnetic‑field profile that grows from a surface value of 10^14 G to a central value of order 10^17 G, following an exponential‑type function of the radial coordinate. This profile reflects the likely configuration of magnetars, where the field is strongest in the core.

Key findings include: (i) magnetic fields below 10^14 G have negligible impact on either EOS; (ii) for fields above this threshold, Landau quantization reduces the pressure of both phases, softening the overall EOS; (iii) a density‑dependent bag constant further expands the mixed‑phase region; (iv) the combined effect of a strong, radially varying field and a decreasing bag pressure shifts the hadron‑to‑quark transition to lower densities and widens the mixed‑phase zone, which can occupy up to ~30 % of the stellar radius for central fields around 10^17 G.

Using the resulting EOS, the Tolman‑Oppenheimer‑Volkoff equations are solved to obtain mass‑radius (M‑R) curves. With a central magnetic field of ~10^17 G, the maximum mass of the hybrid star reaches ≈2.0 M⊙, and the corresponding radius lies in the 11–12 km range. These values are compatible with the precisely measured mass of the pulsar PSR J1614‑2230 (1.97 ± 0.04 M⊙), which imposes a stringent lower bound on any viable EOS. In contrast, models without magnetic fields or with a constant bag pressure predict maximum masses below 1.8 M⊙, failing to meet the observational constraint.

The study therefore demonstrates that strong magnetic fields are not merely a peripheral detail but a decisive factor in shaping the EOS, the extent of the mixed phase, and the observable macroscopic properties of hybrid stars. It highlights the necessity of incorporating magnetic‑field effects, density‑dependent confinement parameters, and a proper mixed‑phase treatment when modeling ultra‑dense matter. The authors suggest that future work should explore anisotropic magnetic stresses, possible superconducting or superfluid phases, and more sophisticated quark‑matter descriptions (e.g., NJL or perturbative QCD models) to refine the predictions and further confront them with upcoming astrophysical observations.