超导 Eliashberg 理论中 $T_c$ 的界限。II：色散声子
Bounds on $T_c$ in the Eliashberg theory of Superconductivity. II: Dispersive phonons
AI总结：
本文研究了色散声子介导的超导 Eliashberg 理论，通过变分原理和紧自伴算子确定了相变稳定性边界，并给出了临界温度 $T_c$ 的收敛下界及渐近上界。
摘要：
本文研究了标准的超导 Eliashberg 理论，其中有效的电子-电子相互作用由一般色散声子介导，其 Eliashberg 谱函数 $α^2 F(ω)\geq 0$ 在小 $ω>0$ 时 $\proptoω^2$，并在大 $ω$ 时趋于零。Eliashberg 函数还定义了电声子耦合强度 $λ:= 2 \int_0^\infty\frac{α^2 F(ω)}ωdω$。设 $\frac{2α^2 F(ω)}ωdω=: λP(dω)$，形式上定义一个具有紧支撑的概率测度 $P(dω)$，并如通常那样假设正常态与超导态之间的相变与正常区域在受到趋向超导区域的扰动时的线性稳定性边界 $\mathscr{S}_{\\!c}$ 重合，本文证明了 $\mathscr{S}_{\\!c}$ 是函数 $Λ(P,T)$ 的图像，该函数由一个变分原理确定：若 $(λ,P,T)\in\mathscr{S}_{\\!c}$，则 $λ= 1/\mathfrak{k}(P,T)$，其中 $\mathfrak{k}(P,T)>0$ 是本文构造的 $\ell^2$ 序列上紧自伴算子 $\mathfrak{K}(P,T)$ 的最大特征值。给定 $P$，本文给出了关于 $T$ 的充分条件，在此条件下映射 $T\mapsto λ= Λ(P,T)$ 是可逆的。对于足够大的 $λ$，这得出：(i) 临界温度 $T_c$ 作为 $λ$ 和 $P$ 的函数的存在性；(ii) 一系列收敛于 $T_c(λ,P)$ 的 $T_c(λ,P)$ 下界。本文还得到了 $T_c(λ,P)$ 的一个上界。它与给定 $P$ 时对 $λ\sim\infty$ 有效的渐近形式 $T_c(λ,P) \sim C \sqrt{\langle ω^2\rangle} \sqrtλ$ 一致，尽管常数 $C$ 比精确常数大了一个约 $2.034$ 的因子。这里，$\langleω^2\rangle := \int_0^\infty ω^2 P(dω)$。
英文摘要：
The standard Eliashberg theory of superconductivity is studied, in which the effective electron-electron interactions are mediated by generally dispersive phonons, with Eliashberg spectral function $α^2 F(ω)\geq 0$ that is $\proptoω^2$ for small $ω>0$ and vanishes for large $ω$. The Eliashberg function also defines the electron-phonon coupling strength $λ:= 2 \int_0^\infty\frac{α^2 F(ω)}ωdω$. Setting $\frac{2α^2 F(ω)}ωdω=: λP(dω)$, formally defining a probability measure $P(dω)$ with compact support, and assuming as usual that the phase transition between normal and superconductivity coincides with the linear stability boundary $\mathscr{S}_{\!c}$ of the normal region against perturbations toward the superconducting region, it is shown that $\mathscr{S}_{\!c}$ is a graph of a function $Λ(P,T)$ that is determined by a variational principle: if $(λ,P,T)\in\mathscr{S}_{\!c}$, then $λ= 1/\mathfrak{k}(P,T)$, where $\mathfrak{k}(P,T)>0$ is the largest eigenvalue of a compact self-adjoint operator $\mathfrak{K}(P,T)$ on $\ell^2$ sequences constructed in the paper. Given $P$, sufficient conditions on $T$ are stated under which the map $T\mapsto λ= Λ(P,T)$ is invertible. For sufficiently large $λ$ this yields: (i) the existence of a critical temperature $T_c$ as function of $λ$ and $P$; (ii) a sequence of lower bounds on $T_c(λ,P)$ that converges to $T_c(λ,P)$. Also obtained is an upper bound on $T_c(λ,P)$. It agrees with the asymptotic form $T_c(λ,P) \sim C \sqrt{\langle ω^2\rangle} \sqrtλ$ valid for $λ\sim\infty$, given $P$, though with a constant $C$ that is a factor $\approx 2.034$ larger than the sharp constant. Here, $\langleω^2\rangle := \int_0^\infty ω^2 P(dω)$.
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