In this paper, we present a mathematical analysis for the phonon-dominated multiband isotropic and anisotropic BCS gap equations at any finite temperature T. We establish the existence of a critical temperature Tc so that, when T < Tc, there exists a unique positive gap solution, representing the superconducting phase; when T > Tc, the only nonnegative gap solution is the zero solution, representing the normal phase. Furthermore, when T = Tc, we prove that the only gap solution is the zero solution and that the positive gap solution depend on the temperature T < Tc monotonically and continuously. In particular, as T → Tc, the gap solution tends to zero, which enables us to determine the critical temperature Tc. In the isotropic case where the entries of the interaction matrix K are all constants, we are able to derive an elegant Tc equation which says that Tc depends only on the largest positive eigenvalue of K but does not depend on the other details of K. In the anisotropic case, we may derive a similar Tc equation in the context of the Markowitz-Kadanoff model and we prove that the presence of anisotropic fluctuations enhances Tc as in the single-band case. A special consequence of these results is that the half-unity exponent isotope effect may rigorously be proved in the multiband BCS theory, isotropic or anisotropic.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics