Bounds on $T_c$ in the Eliashberg theory of Superconductivity. I: The $\gamma$-model

Abstract

Using the recent reformulation for the Eliashberg theory of superconductivity in terms of a classical interacting Bloch spin chain model, rigorous upper and lower bounds on the critical temperature $T_c$ are obtained for the $\gamma$ model -- a version of Eliashberg theory in which the effective electron-electron interaction is proportional to $(g/|{\omega }_n-{\omega }_m|{)}^{\gamma }$, where ${\omega }_n-{\omega }_m$ is the transferred Matsubara frequency, $g>0$ a reference energy, and $\gamma >0$ a parameter. The rigorous lower bounds are based on a variational principle that identifies $(T_c/g{)}^{\gamma }$ with the largest (positive) eigenvalue of an explicitly constructed compact, self-adjoint operator $\mathfrak{G}(\gamma )$. These lower bounds form an increasing sequence that converges to $T_c(g,\gamma )$. The upper bound on $T_c(g,\gamma )$ is based on fixed point theory, proving linear stability of the normal state for $T$ larger than the upper bound on $T_c(g,\gamma )$.

49 pages, 2 figures, revised preprint version; appeared with different layout in J. Statist. Phys

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Cited by in corpus (4)

• Fundamental limits on the electron-phonon coupling and superconducting $T_c$
• Bounds on $T_c$ in the Eliashberg theory of Superconductivity. III: Einstein phonons
• Analytical solution of the Eliashberg equations for strong-coupling superconductivity in hydrides
• Upper bound on $T_c$ in a strongly coupled electron-boson superconductor