Bounds on $T_c$ in the Eliashberg theory of Superconductivity. III: Einstein phonons

Abstract

The dispersionless limit of the standard Eliashberg theory of superconductivity is studied. The effective electron-electron interactions are mediated by Einstein phonons of frequency $\Omega >0$, equipped with electron-phonon coupling strength $\lambda$. This allows for a detailed evaluation of the general results on $T_c$ for phonons with non-trivial dispersion relation, obtained in a previous paper, (II), by the authors. The variational principle for the linear stability boundary ${\mathcal{S}}_c$ of the normal state region against perturbations toward the superconducting region, obtained in (II), simplifies as follows: If $(\lambda ,\Omega ,T)\in {\mathcal{S}}_c$, then $\lambda =1/\mathfrak{h}(\varpi )$, where $\varpi :=\Omega /2\pi T$, and where $\mathfrak{h}(\varpi )>0$ is the largest eigenvalue of a compact self-adjoint operator $\mathfrak{H}(\varpi )$ on ${\ell }^2$ sequences; $\mathfrak{H}(\varpi )$ is the dispersionless limit $P(d\omega )\to \delta (\omega -\Omega )d\omega$ 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 $\lambda >0.77$ this yields: (i) the existence of a critical temperature $T_c(\lambda ,\Omega )=\Omega f(\lambda )$; (ii) a sequence of lower bounds on $f(\lambda )$ that converges to $f(\lambda )$. Also obtained is an upper bound on $T_c(\lambda ,\Omega )$, which agrees with the asymptotic behavior $T_c(\lambda ,\Omega )\sim C\Omega \sqrt{\lambda }$ for $\lambda \sim \infty$, given $\Omega$, though with $C\approx 2.034C_{\infty }$, where $C_{\infty }:=\frac{1}{2\pi }\mathfrak{k}(2{)}^{\frac{1}{2}}=\mathrm{0.1827262477...}$ is the optimal constant, and $\mathfrak{k}(\gamma )>0$ the largest eigenvalue of a compact self-adjoint operator for the $\gamma$ model, determined in the first paper, (I), on $T_c$ by the authors.

36 pages, 2 figures, revised version, accepted for publication in J. Statist. Phys

References in corpus (5)

• Superconductivity near a quantum critical point in the extreme retardation regime
• Fundamental limits on the electron-phonon coupling and superconducting $T_c$
• Upper limit for superconducting transition temperature in Eliashberg-McMillan theory
• Instability of metals with respect to strong electron-phonon interaction
• Bounds on $T_c$ in the Eliashberg theory of Superconductivity. I: The $\gamma$-model

Cited by in corpus (1)

• Analytical solution of the Eliashberg equations for strong-coupling superconductivity in hydrides