# Superconducting Phases in Neutron Star Cores
## 3 Superconducting Ground States
### The Numerical Model
In particular, we can impose either a square or a hexagonal lattice symmetry by using the following domain shapes:  
* for a square lattice, we take \(N=1\) and \(L_{x}/L_{y}=1\);
* for a hexagonal lattice, we take \(N=2\) and \(L_{x}/L_{y}=\sqrt{3}\). In order to directly compare these two cases, we calculate the Helmholtz free energy per magnetic flux quantum per unit length:  
\[\mathcal{F}\equiv\frac{1}{N}\int_{x=0}^{L_{x}}\int_{y=0}^{L_{y}}F\,\mathrm{d}x \,\mathrm{d}y\,. \tag{22}\]  
As an example, in Figure 2 we plot \(\mathcal{F}\) as a function of the area per magnetic flux quantum,  
\[a\equiv\frac{L_{x}L_{y}}{N}=\frac{2\pi}{\overline{B}}\,, \tag{23}\]  
for one particular set of parameters, whose motivation is explained later, in Section 5. Note that in the case of a fluxtube lattice, \(a\) corresponds to the area of a single Wigner-Seitz cell. This plot was produced by finding the minimum value of \(\mathcal{F}\) for both square and hexagonal lattices for domains of various sizes. We also plot the energy in the non-superconducting state, which has \(\psi_{\mathrm{p}}=0\) and a uniform magnetic field \(\mathbf{B}=(0,0,2\pi/a)\), and is known analytically (see Section 4.1). However, as discussed in the previous section, in some cases the true ground state might be inhomogeneous, if this allows the average energy to be lower than that of any homogeneous state. Suppose, for example, that a fraction, \(\phi\), of the total magnetic flux is  
\[\
abla^{2}\psi_{\mathrm{n}}=R^{2}(|\psi_{\mathrm{n}}|^{2}-1)\psi_ {\mathrm{n}}+\alpha(|\psi_{\mathrm{P}}|^{2}-1)\psi_{\mathrm{n}}\]0 The Helmholtz free energy per flux quantum per unit length, \(\mathcal{F}\), as a function of the area per magnetic flux quantum, \(a\), with the dimensionless parameters \(\kappa=1.444\), \(R=0.371\), \(\epsilon=0.097\), \(\alpha=0\), \(h_{1}=0.102\), \(h_{2}=0.387\), and \(h_{3}=0.263\). The energy in both the square (long-dashed, cyan) and hexagonal (solid, blue) lattice states matches smoothly onto the energy of the non-superconducting state (short-dashed, purple) at \(a\simeq 12.9\) (region enlarged in the upper inset), and both converge to the same finite value as \(a\to\infty\). The dotted gray lines indicate the lower convex envelope, plotted separately in Figure 3, which is the true ground state in an unbounded domain. The two red dots indicate the values for the two simulations in \[\
abla^{2}\psi_{\mathrm{n}}=R^{2}(|\psi_{\mathrm{n}}|^{2}-1)\psi_ {\mathrm{n}}+\alpha(|\psi_{\mathrm{P}}|^{2}-1)\psi_{\mathrm{n}}\]1 We show an enlarged view of the left point in the lower inset. contained in regions with \(a=a_{1}\) and \(\mathcal{F}=\mathcal{F}_{1}\), while the rest is in regions with \(a=a_{2}\) and \(\mathcal{F}=\mathcal{F}_{2}\). In that case, the overall values of \(a\) and \(\mathcal{F}\) are given by the lever rule:  
\[\
abla^{2}\psi_{\mathrm{n}}=R^{2}(|\psi_{\mathrm{n}}|^{2}-1)\psi_ {\mathrm{n}}+\alpha(|\psi_{\mathrm{P}}|^{2}-1)\psi_{\mathrm{n}}\]2  
In this way, an energy \(\overline{\mathcal{F}}\) that is lower than \(\mathcal{F}(\overline{a})\) can be achieved in any range of \(a\) for which the function \(\mathcal{F}(a)\) is not convex. In fact, the true ground state in an unbounded domain is given by the lower convex envelope of all the homogeneous states, which is indicated by the dotted gray lines in \[\
abla^{2}\psi_{\mathrm{n}}=R^{2}(|\psi_{\mathrm{n}}|^{2}-1)\psi_ {\mathrm{n}}+\alpha(|\psi_{\mathrm{P}}|^{2}-1)\psi_{\mathrm{n}}\]3 We will use the notation \(\mathcal{F}_{\rm g}(a)\) to refer to the true ground-state energy as a function of the area \(a\).