Superconductivity Near a Quantum Critical Point: Bounds on the Transition Temperature in the $\gamma$-Model

Near a quantum critical point (QCP) in a metal, strong Fermion-Fermion interactions mediated by soft collective bosons give rise to two competing phenomena: non-Fermi liquid behavior and superconductivity that deviates from conventional BCS and Migdal-Eliashberg theories. We consider the problem of obtaining closed-form analytical lower and upper bounds on transition temperatures for such systems. We focus mainly on a class of models known as the gamma-model, which generalizes the Eliashberg theory of Superconductivity where the effective interaction potential scales as V(Omega) ~ 1/|Omega|^gamma. Building on a recent reformulation of Migdal-Eliashberg theory, expressed as a classical infinite spin chain with nonlocal interactions, and employing a simple linear algebra framework, we derive rigorous closed-form expressions for upper and lower bounds on the superconducting transition temperature for any gamma > 0. While our lower bounds coincide precisely with those previously in the literature, our derivation offers a more streamlined and accessible method. Moreover, our upper bounds are substantially tighter than any existing estimates and converge rapidly enough to the numerical results from various prior studies.