# Superconducting Phases in Neutron Star Cores
## 4 Phase Transitions with \(\mathbf{H}\)
### The Upper Critical Field, \(H_{\rm c2}\) vs. \(H_{\rm c2^{\prime}}\)  
As shown in Section 3.3, if the free energy \(\mathcal{F}(a)\) is convex (and monotonically decreasing), then the transition between the fluxtube lattice state and the non-superconducting state is second-order. This means that the order parameter \(\psi_{\rm P}\) vanishes smoothly at the transition point, \(H_{\rm c2}\), which can thus be determined analytically by considering linear perturbations to the non-superconducting state. Working in the symmetric gauge, the non-superconducting state is given by \(|\psi_{\rm P}|=0\), \(|\psi_{\rm n}|=\sqrt{1+\alpha/R^{2}}\), and \({\bf A}=\frac{1}{2}\overline{B}\mathbf{e}_{z}\times{\bf x}\), where \(\overline{B}=2\pi/a\) is the (uniform) mean magnetic flux. From the linearized version of Equation (18), we find that the perturbation \(\delta\psi_{\rm P}\) must satisfy the equation  
\[\left[1+\frac{h_{1}}{e}\Big{(}1+\frac{\alpha}{R^{2}}\Big{)}\right]\Big{(} \boldsymbol{\
abla}-\tfrac{1}{2}\overline{B}\mathbf{e}_{z}\times{\bf x}\Big{)} ^{2}\delta\psi_{\rm P}=-\Big{(}1-\frac{\alpha^{2}}{eR^{2}}\Big{)}\delta\psi_{ \rm P}\,. \tag{48}\]  
As known from single-component systems, e.g., [52], bounded solutions of this equation first appear when  
\[\overline{B}=H_{\rm c2}\equiv\frac{1-\frac{\alpha^{2}}{eR^{2}}}{1+\frac{h_{1} }{e}\Big{(}1+\frac{\alpha}{R^{2}}\Big{)}}\,. \tag{49}\]  
Therefore, the solid (blue) and long-dashed (cyan) lines in Figure 2, as highlighted in the upper inset, meet the short-dashed (purple) line at the point where \(a=2\pi/H_{\rm c2}\). However, if the function \(\mathcal{F}(a)\) is not convex at this point then the upper transition will occur not at \(|{\bf H}|=H_{\rm c2}\), but at a higher value \(|{\bf H}|=H_{\rm c2^{\prime}}\), and will be first order. To determine whether the function \(\mathcal{F}(a)\) is convex at this point, we can seek weakly nonlinear solutions in the vicinity of \(a=2\pi/H_{\rm c2}\), following the method of [PERSON] [56]. We present the details of this calculation in Appendix C. The main result is that the function \(\mathcal{F}(a)\) becomes non-convex when  
\[\left[\frac{2}{H_{\rm c2}}+h_{3}-\left(1-\frac{\alpha^{2}}{eR^{2 }}\right)^{2}\!\frac{1}{\kappa^{2}H_{\rm c2}^{3}}\right]\pi\beta eR^{2}=\] \[\sum_{i}\exp(-\tfrac{1}{2}H_{\rm c2}|{\bf x}_{i}|^{2})\,\frac{ \Big{[}(h_{2}-h_{1})\tfrac{1}{2}H_{\rm c2}|{\bf x}_{i}|^{2}+h_{1}+\alpha\tfrac {1}{H_{\rm c2}}\Big{]}^{2}}{\Big{(}\frac{1}{R^{2}+\alpha}+\tfrac{h_{3}}{eR^{2 }}\Big{)}\tfrac{1}{2}H_{\rm c2}|{\bf x}_{i}|^{2}+\tfrac{1}{H_{\rm c2}}}\,, \tag{50}\]  
where \({\bf x}_{i}\) is the location of the \(i\)-th lattice point, as in Section 4.2, and \(\beta\) is the kurtosis of the proton order parameter,  
\[\beta\equiv\frac{\overline{|\psi_{\rm P}|^{4}}}{(|\psi_{\rm P}|^{2})^{2}}=\sum _{i}\exp(-\tfrac{1}{2}H_{\rm c2}|{\bf x}_{i}|^{2})\,. \tag{51}\]  
Equation (50) is valid for both hexagonal and square lattices of singly-charged flux-tubes. (Singly-charged here means that each fluxtube carries a single quantum of magnetic flux.) Although there are cases in which other fluxtube configurations have a lower free energy (see Appendix C), the point of transition between \(H_{\rm c2}\) and \(H_{\rm c2^{\prime}}\) is always determined by the singly-charged hexagonal lattice, for which \(\beta\simeq 1.1596\)[57]. Thus from Equation (50) we can determine the value of \(\kappa\) at which \(H_{\rm c2}\), \(H_{\rm c2^{\prime}}\) and \(B_{\rm c2}\) all coincide. In the case shown in Figure 5, this yields the value \(\kappa\simeq 1.58\), in agreement with our numerical results.