Superconductivity with two critical temperatures

The weak-coupling thermodynamics in zero magnetic field is extended to the case displaying two second-order transitions. First, a free-energy functional is obtained from a generalized Bardeen-Cooper-Schrieffer (BCS) Hamiltonian by assuming a symmetry on the pairing interaction. This functional is applicable for the cases of an anisotropic Fermi surface, nonunitary states, and mixing of two irreducible representations. Second, the Landau theory of second-order transitions is extended to describe the lower transition. Here, the transition cannot occur independently without affecting the component of the upper transition. Third, a numerical calculation is performed for a model system of an elliptic Fermi surface, and theoretical curves for the specific heat with two and three discontinuities are presented. It is found from this calculation that a lower transition can occur to form a more uniform gap at T=0, provided that (1) two channels Γ=1,2 have fairly close ${\mathit{T}}_{\mathit{c}0}^{\left(\mathrm{\Gamma }\right)}$≡1.13${\mathrm{\varepsilon }}_{\mathit{c}}$ exp[-1/N(0)${\mathit{V}}^{\left(\mathrm{\Gamma }\right)}$]; (2) an isotropic gap cannot be formed within the basis functions of the upper transition. Finally, two theoretical curves of ${\mathit{A}}_{1\mathit{g}}$-${\mathit{E}}_{1\mathit{g}}$ mixing are compared with a specific-heat experiment of ${\mathrm{UPt}}_3$. They give a good fit over the entire temperature range, and a second-order transition is suggested.