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

TL;DR

The paper addresses the problem of bounding the superconducting transition temperature $T_c$ in the quantum-critical gamma-model with $V(\Omega) \propto 1/|\Omega|^{\gamma}$ by mapping the strong-coupling problem to a classical spin chain and performing a linear stability analysis of the associated Hessian. It establishes a rigorous truncation method to replace the infinite Hessian with finite matrices, justified by a congruence transform to a bounded operator and Weyl's theorem. It then derives closed-form lower bounds on $T_c$ via eigenvalue interlacing (giving $\tau_{c,1}=1$ and a second bound with a explicit expression, plus higher-order bounds from $\det(H^{(N)})=0$) and an improved upper bound using Gershgorin disks and a similarity transform, yielding $\tau_{u}^{\gamma} = \sum_{n=0}^{\infty} (\tfrac{1}{2})^{2n} \zeta(\gamma+2n+1)$. The results yield bounds that converge rapidly to numerical data and tighten previous estimates, providing analytic control over $T_c$ across all $\gamma>0$.

Abstract

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.

Figures (5)

• Figure 1: Our results for the first four lower bounds $\tau_{c,1}^{\gamma},\tau_{c,2}^{\gamma},\tau_{c,3}^{\gamma},\tau_{c,4}^{\gamma}$ as obtained from Eq.s (\ref{['tau0']}),(\ref{['tau1']}),and(\ref{['eq:lower']}), and our upper bound calculated from Eq (\ref{['eq:ourbound']}) in comparison with the numerical data from PhysRevB.99.144512 and the upper bound obtained by Kiessling et al Kiessling2025. Note that we are plotting $\tau^\gamma$ instead of $\tau$ vs $\gamma$.
• Figure 2: Comparison of the first four lowest bounds $\tau_{c,1},\tau_{c,2},\tau_{c,3},\tau_{c,4}$ versus $\gamma$, plotted alongside previous numerical solutions of the generalized Eliashberg equations from PhysRevB.99.144512 (black crosses). The plot illustrates the convergence behavior of the bounds as $\gamma$ increases, with each successive bound providing a tighter estimate of $\tau_c$.
• Figure 3: The transition temperature $\tau_c$ is plotted vs $\gamma$ for different truncation values $N=100,400,1000$. for $\gamma \geq 0.5$, all truncation levels agree well with the previous numerical solutions of generalized Eliashberg equations PhysRevB.99.144512. As $\gamma \to 0$, $\tau_c \to \infty$ as expected, which shows that the optimal truncation $N \to \infty$. All lower truncation levels serve as a "lower bound".
• Figure 4: Comparison of our results (blue line) for the upper bound of the critical temperature obtained from Eq. \ref{['eq:ourbound']} as a function of $\gamma$ in the range $0.1 \leq \gamma \leq 5$ compared to the numerical data (black crosses) from PhysRevB.99.144512 in comparison to the Upper bound obtained by Kiessling et. al Kiessling2025 (red line). Our results show a significant improvement in the upper bound. Note that we are plotting $\tau_c^\gamma$ to be able to fit both bounds in the same plot.
• Figure 5: The upper bound $\tau_0^ \gamma$ vs $\gamma$ for different values of $p$ shown as legends. Dots are the numerical data. It's clear that for $p=1/6$, the upper bound is below the data. It indicates that somewhere $1/2 < p < 1/3$, the assumption that the zeroth Gershgorin disc is the lowest is no longer valid.