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

TL;DR

This paper derives rigorous upper and lower bounds on the critical temperature in a generalized Eliashberg superconductivity model using a novel spin chain reformulation, providing insights into the stability and eigenvalue structure of the theory.

Contribution

It introduces a new reformulation of Eliashberg theory as a classical spin chain and establishes explicit bounds on the critical temperature for the $eta$-model, advancing theoretical understanding.

Findings

Lower bounds form an increasing sequence converging to $T_c$.

Upper bounds are based on fixed point theory and stability analysis.

Eigenvalue characterization of $T_c$ via a self-adjoint operator.

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 $T_c(g,\gamma)$.