# Superconducting Phases in Neutron Star Cores
## 3 Superconducting Ground States
### Phase Diagrams  
Using the procedure described above, we can determine the superconducting phase transitions that occur when \(\overline{B}\) or \(|\mathbf{H}|\) is used as the control parameter. The results shown in Figure 2 demonstrate that these transitions can be qualitatively different from those seen in a single-component Ginzburg-Landau superconductor. In order to directly compare the resulting superconducting phases with those shown previously in Figure 1, we produce phase diagrams in which \(\kappa\) is used as the independent parameter and all other parameters are fixed to the same values used in Figure 2. The result is shown in Figure 5. For \(\kappa\lesssim 1.42\) we have type-I superconductivity, and for \(\kappa\gtrsim 5.36\) we have type-II superconductivity. However, within the range \(1.42\lesssim\kappa\lesssim 5.36\) we have type-I.5 superconductivity: in the experiment with \(|\mathbf{H}|\), the transition from the Meissner to the fluxtube state at \(H_{\mathrm{cl^{\prime}}}\) is first order, because the lattice first appears with a finite separation between fluxtubes; in the experiment with \(\overline{B}\), there is a critical value \(\overline{B}=B_{\mathrm{cl}}\) below which bundles of fluxtubes form alongside flux-free Meissner regions. This behavior results from the fact that fluxtubes are mutually attractive at large separation distances, and mutually repulsive at small separation distances, i.e., they have a preferred separation, as indicated by the minimum in the curve \(\mathcal{F}(a)\) seen in Figure 2.  
Over the narrow range \(1.42\lesssim\kappa\lesssim 1.58\) an additional feature is present: in the experiment with \(|\mathbf{H}|\), the transition from the fluxtube state to the non-superconducting state is first order at \(H_{\mathrm{cl^{\prime}}}\), as the proton condensate density vanishes discontinuously; in the experiment with \(\overline{B}\), there is a critical value \(\overline{B}=B_{\mathrm{cl^{\prime}}}\) above which bundles of fluxtubes form alongside non-superconducting regions. This behavior can also be understood intuitively in terms of interactions between neighboring fluxtubes: for this range of parameters, the tendency towards a preferred separation distance is so strong, and the dip in the \(\mathcal{F}(a)\) curve so pronounced, that it becomes energetically favorable to form a mixture of fluxtubes and non-superconducting regions, rather than a periodic lattice with the \"wrong\" separation distance.  
The phase diagrams shown in Figure 5 resemble those hypothesized by [PERSON] and [PERSON] [8], and demonstrate that type-I.5 superconductivity can arise over a significant range of the Ginzburg-Landau parameter \(\kappa\), when coupling between the proton and neutron condensates is present. However, to determine whether the behavior seen for this particular choice of coupling parameters is generic, we need to understand these phase transitions in more detail. In the following section we therefore seek analytical expressions for the locations of the various transitions in phase space, in order to determine which couplings between the two condensates give rise to inhomogeneous ground states.  
Figure 5: _Cont._