Bounds on \(T_c\) in the Eliashberg Theory of Superconductivity. III: Einstein Phonons

The dispersionless limit of the standard Eliashberg theory of superconductivity is studied, in which the effective electron-electron interactions are mediated by Einstein phonons of frequency \(\Omega >0\), equipped with electron-phonon coupling strength \(\lambda \). The general results on \(T_c\) for phonons with non-trivial dispersion relation, obtained in a previous paper by the authors, (II), then become amenable to a detailed evaluation. The results are based on the traditional notion that the phase transition between normal and superconductivity coincides with the linear stability boundary \(\mathscr {S}_{\!c}\) of the normal state region against perturbations toward the superconducting region. The variational principle for \(\mathscr {S}_{\!c}\), obtained in (II), simplifies as follows: If \((\lambda ,\Omega ,T)\in \mathscr {S}_{\!c}\), then \(\lambda = 1/\mathfrak {h}(\varpi )\), where \(\varpi :=\Omega /2\pi T\), and where \(\mathfrak {h}(\varpi )>0\) is the top eigenvalue of a compact self-adjoint operator \(\mathfrak {H}(\varpi )\) on \(\ell ^2\) sequences; \(\mathfrak {H}(\varpi )\) is the dispersionless limit \(P(d\omega )\rightarrow \delta (\omega -\Omega )d\omega \) of the operator \(\mathfrak {K}(P,T)\) of (II). It is shown that when \(\varpi \le \sqrt{2}\), then the map \(\varpi \mapsto \mathfrak {h}(\varpi )\) is invertible. For sufficiently large \(\lambda \) (\(\lambda >0.77\) will do) this yields the following: (i) the existence of a critical temperature \(T_c(\lambda ,\Omega ) = \Omega f(\lambda )\); (ii) an ordered sequence of lower bounds on \(f(\lambda )\) that converges to \(f(\lambda )\). Also obtained is an upper bound on \(T_c(\lambda ,\Omega )\), which is not optimal yet agrees with the asymptotic behavior \(T_c(\lambda ,\Omega ) \sim C \Omega \sqrt{\lambda }\) for large enough \(\lambda \), given \(\Omega \), though with a constant C that is a factor \(\approx 2.034\) larger than the optimal constant \(\frac{1}{2\pi }\mathfrak {g}(2)^\frac{1}{2} =0.1827262477...\), with \(\mathfrak {g}(\gamma )>0\) the largest eigenvalue of the compact self-adjoint operator \(\mathfrak {G}(\gamma )\) for the \(\gamma \) model, determined rigorously in the first one, (I), of this series of papers on \(T_c\) by the authors.