# Superconducting Phases in Neutron Star Cores
## 4 Phase Transitions with \(\mathbf{H}\)
### The Lower Critical Field, \(H_{\mathrm{c1}}\) vs. \(H_{\mathrm{c1^{\prime}}}\)  
As shown in Section 3.3, the lower critical field is a first-order phase transition if the Helmholtz free energy per unit length per flux quantum \(\mathcal{F}(a)\) in the pure lattice state has a minimum at some finite value of \(a\), since then a fluxtube lattice forms with finite separation. If this minimum is \(\mathcal{F}_{\mathrm{min}}\), say, then we have \(H_{\mathrm{c1^{\prime}}}=\mathcal{F}_{\mathrm{min}}/(4\pi\kappa^{2})\). If, on the other hand, \(\mathcal{F}\) is a monotonically decreasing function of \(a\) in the lattice state, then there is a second-order transition at \(H_{\mathrm{c1}}=\mathcal{F}_{\infty}/(4\pi\kappa^{2})\), as in the case of a single-component superconductor, where  
\[\mathcal{F}_{\infty}=\lim_{a\to\infty}\mathcal{F}(a)\,. \tag{29}\]  
In this limit, interactions between the fluxtubes vanish, and \(\mathcal{F}_{\infty}\) is equivalent to the energy per unit length of a single fluxtube in an infinite domain. This energy can be computed efficiently by using polar coordinates \(r,\theta\) centered on the fluxtube core, and seeking solutions of Equations (16)-(18) in the form [7]  
\[\psi_{\mathrm{P}}=f(r)\,\mathrm{e}^{\mathrm{i}\theta}\,,\quad\psi_{\mathrm{n} }=g(r)\,,\quad\mathbf{A}=A_{\theta}(r)\,\mathbf{e}_{\theta}\,. \tag{30}\]  
This ansatz assumes that the fluxtube carries a single quantum of magnetic flux, and that there is no corresponding phase defect in the neutron condensate. It further results in a system of ordinary differential equations that can be solved numerically, yielding the value of \(\mathcal{F}_{\infty}\) to high accuracy. However, to determine whether the lower transition is second-order or first-order, we need to know whether the function \(\mathcal{F}(a)\) tends to \(\mathcal{F}_{\infty}\) from above or from below, which is equivalent to asking whether the long-range interaction between fluxtubes is repulsive or attractive. This can be derived rigorously using a method introduced by [PERSON] [54], which we describe in detail in Appendix B. In fact, the main result can be obtained heuristically by considering the perturbations produced by a fluxtube in the far-field, i.e., at a large distance from its core. It is convenient to work with the real variables  
\[f\equiv|\psi_{\mathrm{P}}|\,,\quad g\equiv|\psi_{\mathrm{n}}|\,,\quad\chi \equiv\arg\psi_{\mathrm{n}}\,,\quad\mathbf{V}\equiv\mathbf{\
abla}(\arg\psi_{ \mathrm{P}})-\mathbf{A}\,. \tag{31}\]The gauge invariance of the free energy density (12) guarantees that it can be rewritten in terms of these variables without loss of generality; for the full expression see Equation (A6). The corresponding Euler-Lagrange equations are then  
\[0=f(f^{2}-1)+\frac{\alpha}{\epsilon}f(g^{2}-1)-\
abla^{2}f+f| \mathbf{V}|^{2}\] \[+\frac{h_{1}}{\epsilon}[f|\mathbf{\
abla}g|^{2}+fg^{2}|\mathbf{V}- \mathbf{\
abla}\chi|^{2}-\mathbf{\
abla}\cdot(g^{2}\mathbf{\
abla}f)]-\frac{h_{2}}{2 \epsilon}f\
abla^{2}(g^{2})-\frac{h_{3}}{2}f\
abla^{2}(f^{2})\,, \tag{32}\] \[0=\frac{R^{2}}{\epsilon}g(g^{2}-1)+\frac{\alpha}{\epsilon}g(f^{2 }-1)-\frac{1}{\epsilon}\
abla^{2}g+\frac{1}{\epsilon}g|\mathbf{\
abla}\chi|^{2}\] \[+\frac{h_{1}}{\epsilon}[g|\mathbf{\
abla}f|^{2}+gf^{2}|\mathbf{V}- \mathbf{\
abla}\chi|^{2}-\mathbf{\
abla}\cdot(f^{2}\mathbf{\
abla}g)]-\frac{h_{2}}{2 \epsilon}g\
abla^{2}(f^{2})-\frac{h_{3}}{2\epsilon^{2}}g\
abla^{2}(g^{2})\,,\] (33) \[0=\mathbf{\
abla}\cdot[g^{2}\mathbf{\
abla}\chi+h_{1}f^{2}g^{2}(\mathbf{ \
abla}\chi-\mathbf{\
abla})]\,,\] (34) \[0=f^{2}\mathbf{V}+\frac{h_{1}}{\epsilon}f^{2}g^{2}(\mathbf{V}- \mathbf{\
abla}\chi)+\kappa^{2}\mathbf{\
abla}\times(\mathbf{\
abla}\times\mathbf{V})\,. \tag{35}\]  
The far-field structure of the fluxtube can be determined by linearizing these about the uniform state with \(f=g=1\), and \(\mathbf{V}=\mathbf{\
abla}\chi=\mathbf{0}\).