# Superconducting Phases in Neutron Star Cores
## 3 Superconducting Ground States
### The Helmholtz and Gibbs Free Energies  
We now seek the ground state for this system in the presence of an imposed magnetic field. There are two distinct thought-experiments that can be considered. In the first experiment, we control the magnetic flux density, \({\bf B}\), by imposing a mean or net magnetic flux, and minimize the Helmholtz free energy,  
\[{\cal F}=\left\langle F\right\rangle, \tag{14}\]  
where the angled brackets represent some kind of integral over our physical domain, which could be finite or infinite. This experiment closely approximates the conditions in the core of a neutron star, which becomes superconducting as the star cools in the presence of a pre-existing magnetic flux. However, as we will discuss below, the ground state under these conditions can be inhomogeneous, i.e., macroscopic domains of distinct physical behavior can appear. For conceptual convenience, we can consider an alternative experiment in which the system is coupled to a thermodynamic external magnetic field, \({\bf H}\), by minimizing the dimensionless Gibbs free energy,  
\[{\cal G}=\left\langle F-2\kappa^{2}{\bf H}\cdot\mathbf{\
abla}\times{\bf A} \right\rangle={\cal F}-2\kappa^{2}{\bf H}\cdot\left\langle{\bf B}\right\rangle. \tag{15}\]  
In an unbounded domain, the ground state in this experiment is guaranteed to be homogeneous, and hence the phase diagram is generally simpler. For later reference, we present in Figure 1 the phase diagrams for a single-component Ginzburg-Landausuperconductor, i.e., we discuss its state as a function of the Ginzburg-Landau parameter, \(\kappa\). (For more details, we refer the reader to standard textbooks on superconductivity, e.g., [PERSON] [52]). For \(\kappa<1/\sqrt{2}\), we have a type-I superconductor; when \(\mathbf{H}\) is used as the control parameter, there is a first-order transition between the Meissner state (with \(\mathbf{B}=\mathbf{0}\)) and the non-superconducting state (with \(\mathbf{B}=\mathbf{H}\)) at the critical value \(|\mathbf{H}|=H_{\mathrm{c}}=1/(\sqrt{2}\kappa)\) in our dimensionless units. When the mean magnetic flux, \(\overline{B}\), is used as the control parameter, this discontinuity resolves into an intermediate phase for \(0<\overline{B}<H_{\mathrm{c}}\), in which Meissner regions alternate with non-superconducting ones. For \(\kappa>1/\sqrt{2}\), on the other hand, we have a type-II superconductor; for \(H_{\mathrm{c1}}<|\mathbf{H}|<H_{\mathrm{c2}}\) the magnetic flux organizes into a hexagonal lattice of discrete fluxtubes. The transitions at the lower critical field, \(H_{\mathrm{c1}}\), and the upper critical field, \(H_{\mathrm{c2}}\), are both second-order, because fluxtubes appear with infinite separation at \(|\mathbf{H}|=H_{\mathrm{c1}}\), and the superconductor density becomes vanishingly small at \(|\mathbf{H}|=H_{\mathrm{c2}}\). In our dimensionless units, \(H_{\mathrm{c2}}=1\) and \(H_{\mathrm{c1}}=\mathcal{F}_{\infty}/(4\pi\kappa^{2})\), where \(\mathcal{F}_{\infty}\) is the energy per unit length of a single fluxtube. When \(\overline{B}\) is the control parameter, there is a similar second-order transition at \(\overline{B}=H_{\mathrm{c2}}\), and a first-order transition between the intermediate and fluxtube states at \(\kappa=1/\sqrt{2}\). Figure 1: Phase diagrams for a one-component Ginzburg–Landau superconductor, for different values of the Ginzburg–Landau parameter, \(\kappa\). The top panel shows the experiment with an imposed external field, \(|\mathbf{H}|\), in our nondimensional units. The first-order and second-order transitions at the different critical fields are indicated by solid and dashed black lines, respectively, and the resulting phases labeled accordingly. Shading of the respective regions is indicative of the magnetic flux distribution. The bottom panel shows the phase transitions in the experiment with an imposed mean flux, \(\overline{B}\). For more details see the text.