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We investigate the hadron-quark phase transition at finite density in the presence of a magnetic field taking into account the anisotropy created by a uniform magnetic field in the system’s equations of state. We find a new anisotropic equilibrium condition that will drive the first-order phase transition along the boundary between the two phases. Fixing the magnetic field in the hadronic phase, the phase transition is realized by increasing the baryonic chemical potential at zero-temperature. It is shown that the magnetic field is mildly boosted after the system transitions from the hadronic to the quark phase. The magnetic-field discontinuity between the two phases is supported by a surface density of magnetic monopoles, which accumulate at the boundary separating the two phases. The mechanism responsible for the monopole charge density generation is discussed. Each phase is found to be paramagnetic with higher magnetic susceptibility in the quark phase. The connection with the physics of neutron stars is highlighted throughout the paper. 

In this review, we discuss the physical characteristics of the magnetic dual chiral density wave (MDCDW) phase of dense quark matter and argue why it is a promising candidate for the interior matter phase of neutron stars. The MDCDW condensate occurs in the presence of a magnetic field. It is a single-modulated chiral density wave characterized by two dynamically generated parameters: the fermion quasiparticle mass m and the condensate spatial modulation q. The lowest-Landau-level quasiparticle modes in the MDCDW system are asymmetric about the zero energy, a fact that leads to the topological properties and anomalous electric transport exhibited by this phase. The topology makes the MDCDW phase robust against thermal phonon fluctuations, and as such, it does not display the Landau–Peierls instability, a staple feature of single-modulated inhomogeneous chiral condensates in three dimensions. The topology is also reflected in the presence of the electromagnetic chiral anomaly in the effective action and in the formation of hybridized propagating modes known as axion-polaritons. Taking into account that one of the axion-polaritons of this quark phase is gapped, we argue how incident γ-ray photons can be converted into gapped axion-polaritons in the interior of a magnetar star in the MDCDW phase leading the star to collapse, a phenomenon that can serve to explain the so-called missing pulsar problem in the galactic center. 

We investigate the stability of the magnetic dual chiral density wave (MDCDW) phase of cold and dense QCD against collective low-energy fluctuations of the order parameter. The appearance of additional structures in the system free energy due to the explicit breaking of the rotational and isospin symmetries by the external magnetic field play a crucial role in the analysis. The new structures stiffen the spectrum of the thermal fluctuations in the transverse direction, thereby avoiding the Landau-Peierls instability that affects single-modulated phases at arbitrarily low temperatures. The lack of Landau-Peierls instabilities in the MDCDW phase makes this inhomogeneous phase of dense quark matter of particular interest for the physics of neutron stars. 

We study the static responses of cold quark matter in the intermediate baryonic density region(characterized by a chemical potentialμ) in the presence of a strong-magnetic field. We consider inparticular, the so-called magnetic dual Chiral Density Wave (MDCDW) phase, which is materialized by aninhomogeneous condensate formed by a particle-hole pair. It is shown, that the MDCDW phase is morestable in the weak-coupling regime than the one considered in the magnetic catalysis of chiral symmetrybraking phenomenon and even than the chiral symmetric phase that was expected to be realized atsufficiently high baryonic chemical potential. The different components of the photon polarization operatorof the MDCDW phase in the one-loop approximation are calculated. We found that in the MDCDW phasethere is neither Debye screening nor Meissner effect in the lowest-Landau-level approximation. Theobtained Debye length depends on the amplitudemand modulationbof the inhomogeneous condensateand it is only different from zero if the relation |μ−b|>m holds. But, we found that in the region ofinterest this inequality is not satisfied. Thus, no Debye screening takes place under those conditions. On theother hand, since the particle-hole condensate is electrically neutral, the U(1) electromagnetic group is notbroken by the ground state and consequently there is no Meissner effect. These results can be of interest forthe astrophysics of neutron stars 

We investigate the effect of an applied constant and uniform magnetic field in the fine-structure constant of massive and massless QED. In massive QED, it is shown that a strong magnetic field removes the so called Landau pole and that the fine-structure constant becomes anisotropic having different values along and transverse to the field direction. Contrary to other results in the literature, we find that the anisotropic fine-structure constant always decreases with the field. We also study the effect of the running of the coupling constant with the magnetic field on the electron mass. We find that in both cases of massive and massless QED, the electron dynamical mass always decreases with the magnetic field, what can be interpreted as an inverse magnetic catalysis effect.