# Superconducting Phases in Neutron Star Cores
## 5 Discussion  
Combining analytical and numerical techniques, we have determined the ground state for a mixture of proton and neutron superfluid condensates in the neutron star core,in the presence of magnetic flux. The condensates are coupled by density and density-gradient interactions, and by mutual entrainment. Our model extends the phenomenological Ginzburg-Landau framework used in previous works to consistently satisfy Galilean invariance on small scales. In addition to the three homogeneous phases (Meissner, fluxtube lattice, and non-superconducting) that are familiar from single-component superconductors, we have shown that inhomogeneous mixtures of any two of these phases can occur in the ground state, as a result of the coupling between the condensates. The novel mixed states occur because the fluxtubes perturb the neutron condensate density, which can reduce the overall free energy of the coupled system. As described in Section 4.2, coupling between the two condensates always increases the effective coherence length of fluxtubes, and hence under a range of conditions the fluxtubes have a preferred separation distance. Under these conditions, and for relatively weak mean field values (\(\overline{B}<B_{\rm cl}\)), we have type-1.5 superconductivity, in which the fluxtubes form bundles with local hexagonal symmetry, with the rest of the domain in a flux-free Meissner state. In cases where the tendency towards fluxtube bundling is sufficiently strong, there may also be a range of mean field values (\(B_{\rm c2}<\overline{B}<H_{\rm c2^{\prime}}\)) for which there is a mixture of fluxtube and non-superconducting regions. This occurs if it is energetically favorable to confine the proton condensate within part of the domain, forming a lattice with the preferred separation, with the remaining magnetic flux concentrated into non-superconducting regions. In all cases, we find that fluxtubes preferentially adopt a singly-charged, hexagonal pattern, even in cases where a periodic lattice is not the ground state. To relate our results to neutron stars, we must provide estimates for all of the dimensionless parameters in our model. The results presented in Section 3 assumed the following dimensional values: \(n_{\rm p}=0.0251\,\
icefrac{{1}}{{m}}\), \(n_{\rm n}=0.258\,\
icefrac{{1}}{{m}}\), \(\xi_{\rm p}=31.4\,\text  
expression for the latter, it is generally quite close to \(H_{\rm c2}\), and so we have the approximate condition \(H_{\rm c}\simeq H_{\rm c2}\). In terms of our dimensionless variables, this condition becomes  
\[\kappa\simeq\left[1+\frac{h_{1}}{\epsilon}\Big{(}1+\frac{\alpha}{R^{2}}\Big{)} \right]\left(2-\frac{2\alpha^{2}}{\epsilon R^{2}}\right)^{-1/2}. \tag{53}\]  
An analogous result was obtained by [PERSON] [37]. In a neutron star we typically expect \(h_{1}/\epsilon\) to be of order unity and so, even if we neglect the density coupling parameter \(\alpha\), the type-I inner core will be significantly larger than predicted by Equation (52). We note that an alternative method to determine the point of transition to type-I superconductivity is to measure the \"surface energy\" of the interface between Meissner and non-superconducting regions [37; 62; 63]. The surface energy in a type-I superconductor is necessarily positive, because otherwise the interface is unstable to the formation of fluxtubes. However, this provides only a necessary condition for stability; the interface could still be unstable to nonlinear perturbations even if the surface energy is positive. Given that we have found the non-superconducting state to be metastable in some cases (i.e., unstable to nonlinear perturbations), we speculate that in those cases the boundary between the type-I and type-1.5 regimes does not occur exactly where the surface energy is zero. Further work will be required to confirm this however.