超导 Eliashberg 理论中 $T_c$ 的界。I：$\gamma$ 模型
Bounds on $T_c$ in the Eliashberg theory of Superconductivity. I: The $γ$-model
AI总结：
本文借助相互作用 Bloch 自旋链重构 Eliashberg 理论，用变分特征值与不动点方法为 $\gamma$ 模型的临界温度 $T_c$ 建立严格上下界。
摘要：
利用最近将超导 Eliashberg 理论表述为经典相互作用 Bloch 自旋链模型的重构，针对 $\gamma$ 模型得到了临界温度 $T_c$ 的严格上界和下界；该模型是 Eliashberg 理论的一个版本，其中有效电子—电子相互作用正比于 $(g/|\omega_n-\omega_m|)^\gamma$，这里 $\omega_n-\omega_m$ 是转移的 Matsubara 频率，$g>0$ 是参考能量，$\gamma>0$ 是参数。严格下界基于一个变分原理，该原理将 $(T_c/g)^\gamma$ 识别为显式构造的紧自伴算子 $\mathfrak{G}(\gamma)$ 的最大（正）特征值。这些下界构成一个递增序列，并收敛于 $T_c(g,\gamma)$。对 $T_c(g,\gamma)$ 的上界基于不动点理论，证明了当 $T$ 大于 $T_c(g,\gamma)$ 的上界时正常态的线性稳定性。
英文摘要：
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 $γ$ model -- a version of Eliashberg theory in which the effective electron-electron interaction is proportional to $(g/|ω_n-ω_m|)^γ$, where $ω_n-ω_m$ is the transferred Matsubara frequency, $g>0$ a reference energy, and $γ>0$ a parameter. The rigorous lower bounds are based on a variational principle that identifies $(T_c/g)^γ$ with the largest (positive) eigenvalue of an explicitly constructed compact, self-adjoint operator $\mathfrak{G}(γ)$. These lower bounds form an increasing sequence that converges to $T_c(g,γ)$. The upper bound on $T_c(g,γ)$ is based on fixed point theory, proving linear stability of the normal state for $T$ larger than the upper bound on $T_c(g,γ)$.
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