# Superconducting Phases in Neutron Star Cores
## 3 Superconducting Ground States
### The Numerical Model
For this particular case, there are four distinct behaviors seen across the full range of \(a\):  
* for \(0<a\lesssim 12.7\) the ground state is non-superconducting;
* for \(12.7\lesssim a\lesssim 18.5\) the ground state is a mixture of non-superconductor and a hexagonal fluxtube lattice;
* for \(18.5\lesssim a\lesssim 26\) the ground state is a hexagonal fluxtube lattice;
* for \(a\gtrsim 26\) the ground state is a mixture of a hexagonal fluxtube lattice and the Meissner state. Once the function \(\mathcal{F}_{\rm g}(a)\) is known, it is straightforward to also determine the minimum Gibbs energy as a function of \(\mathbf{H}\). In fact, the mean Gibbs energy density is  
\[\
abla^{2}\psi_{\mathrm{n}}=R^{2}(|\psi_{\mathrm{n}}|^{2}-1)\psi_ {\mathrm{n}}+\alpha(|\psi_{\mathrm{P}}|^{2}-1)\psi_{\mathrm{n}}\]4  
and the minimum of \(\overline{G}\) over all \(a\) can be interpreted graphically from the plots in \[\
abla^{2}\psi_{\mathrm{n}}=R^{2}(|\psi_{\mathrm{n}}|^{2}-1)\psi_ {\mathrm{n}}+\alpha(|\psi_{\mathrm{P}}|^{2}-1)\psi_{\mathrm{n}}\]5 Since \(\mathcal{F}_{\rm g}(a)\) is a convex and monotonically decreasing function, each point on the curve \(\mathcal{F}_{\rm g}(a)\) corresponds to a ground state with energy \(\overline{G}=\mathcal{F}_{\rm g}^{\prime}(a)\), and the corresponding value of \(|\mathbf{H}|\) can be found by extrapolating the tangent line up to the \(\mathcal{F}\)-axis, as shown in \[\
abla^{2}\psi_{\mathrm{n}}=R^{2}(|\psi_{\mathrm{n}}|^{2}-1)\psi_ {\mathrm{n}}+\alpha(|\psi_{\mathrm{P}}|^{2}-1)\psi_{\mathrm{n}}\]6 The two ranges of \(a\) for which the function \(\mathcal{F}_{\rm g}(a)\) is linear give rise to two critical values of \(|\mathbf{H}|\) (i.e., \(H_{\rm cl^{\prime}}\) and \(H_{\rm c2^{\prime}}\)) at which the ground state changes discontinuously. At these values there are first-order transitions between a hexagonal fluxtube lattice with finite mean field and either the Meissner state (at \(H_{\rm cl^{\prime}}\)) or the non-superconducting state (at \(H_{\rm c2^{\prime}}\)). \[\
abla^{2}\psi_{\mathrm{n}}=R^{2}(|\psi_{\mathrm{n}}|^{2}-1)\psi_ {\mathrm{n}}+\alpha(|\psi_{\mathrm{P}}|^{2}-1)\psi_{\mathrm{n}}\]7 The ground-state Helmholtz free energy per unit length, \(\mathcal{F}_{\rm g}(a)\), can be used to infer the minimum Gibbs free-energy density, \(\overline{G}\), as a function of the external field, \(\mathbf{H}\). The tangent to each point on the curve \(\mathcal{F}_{\rm g}(a)\) has slope \(\overline{G}\), and intersects the vertical axis at the point \(\mathcal{F}=4\pi\kappa^{2}|\mathbf{H}|\). The dotted lines indicate the transitions at \(H_{\rm cl^{\prime}}\) and \(H_{\rm c2^{\prime}}\), both of which are first-order in this example. We have used the same dimensionless parameters as for \[\
abla^{2}\psi_{\mathrm{n}}=R^{2}(|\psi_{\mathrm{n}}|^{2}-1)\psi_ {\mathrm{n}}+\alpha(|\psi_{\mathrm{P}}|^{2}-1)\psi_{\mathrm{n}}\]8  
We emphasize that the function \(\mathcal{F}_{\rm g}\) represents the minimum free energy only for a hypothetical unbounded domain, free from any geometrical constraints. In any simulation with a finite domain size, the free energy in the ground state will generally exceed this value. Nevertheless, by using a large enough computational domain, and choosing values of \(a\) within the appropriate ranges, we can obtain examples of the inhomogeneous ground states described above. Figure 4 shows two such examples, with \(a=14.5\) and \(a=52\). These values of \(a\) were chosen so that in each case approximately half of the domain contains a hexagonal lattice. As shown in Figure 2, in each case the free energy is lower than that of a pure lattice, but still significantly higher than for the true ground state in an unbounded domain.