# Superconducting Phases in Neutron Star Cores
## 1 Introduction  
In the core of a neutron star, neutrons and protons are both expected to form condensates via Cooper pairing, leading to a neutron superfluid and a proton superconductor, respectively. Within the proton superconductor magnetic flux is quantized into microscopic filaments called fluxtubes, and the macroscopic dynamics of the star's core magnetic field depends on the microscopic dynamics of these fluxtubes. In the conventional picture [1, 2, 3], the inner core is a type-I superconductor, meaning that fluxtubes are mutually attractive and therefore coalesce into macroscopic patches of magnetic flux, whereas the outer core is a type-II superconductor, meaning that fluxtubes are mutually repulsive and therefore form a regular array. The transition between these two superconducting regimes occurs where the ratio of the London penetration length, \(\lambda\), and proton coherence length, \(\xi_{\rm p}\), the so-called Ginzburg-Landau parameter \(\kappa\equiv\lambda/\xi_{\rm p}\), takes the value \(\kappa=1/\sqrt{2}\). The inner and outer core are therefore defined to be the regions in which \(\kappa<1/\sqrt{2}\) and \(\kappa>1/\sqrt{2}\), respectively. However, the microscopic behavior of fluxtubes in the core is affected by the coupling between the proton and neutron condensates, and in particular by their mutual entrainment, which could significantly change the location of this transition, and might even result in entirely different types of superconductivity. For example, [PERSON] et al. [4, 5] showed that strong density coupling between the condensates could produce type-I superconductivity throughout the whole core, although only if the coupling is much stronger than is generally expected [6]. A more detailed study by [PERSON] and [PERSON] [7], incorporating density and density-gradient coupling, found that in some cases the transition from type-I to type-II is mediated by domains of \"type-II(\(n\))\" superconductivity, wherein each fluxtube carries \(n\) magnetic flux quanta. Subsequently, [PERSON] and [PERSON] [8] argued that these type-II(\(n\)) fluxtubes are generally unstable, and instead there is a regime of \"type-1.5\" superconductivity, in which the fluxtubes form bundles with a preferred separation. Such unconventional behavior has also been discussed in the context of terrestrial multi-band superconductors [9, 10, 11, 12, 13]. We will show later that inhomogeneous flux distributions arise in our model under a wide range of parameter conditions, as a result of coupling between the twocondensates. We adopt the term \"type-1.5\" superconductivity to maintain consistency with earlier literature on the topic. We note that there are other mechanisms that might produce bundles of fluxtubes inside neutron stars, such as a macroscopic instability of the fluxtube lattice [14] or interactions with quantized vortices in the neutron superfluid [15]. However, these scenarios do not correspond to type-1.5 superconductivity in the sense used here. The nature of superconductivity in the star's core may have observational consequences. For example, there is observational evidence for long-period precession in some neutron stars [16; 17; 18; 19], which places constraints on dissipation within the core. If the superfluid neutron vortices in the outer core are tightly pinned to a regular array of fluxtubes then precession becomes essentially impossible [20; 21]. This contradiction could be resolved if the entire core were in a type-I state [21; 22; 23], or if instabilities resulting from precession are able to unpin the vortices [24]. Pinning between vortices and fluxtubes could also significantly affect the rise time, amplitude, and relaxation rate of rotational glitches in pulsars [25; 26; 27], as well as the long-term evolution of their rotation and magnetic field [28; 29]. The goal of this paper is to extend previous studies to include all relevant couplings between the neutron and proton condensates, and to determine the precise conditions under which different types of superconductivity arise.