Bounds on $T_c$ in the Eliashberg theory of Superconductivity. II: Dispersive phonons
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arXiv
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2024
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| author | Kiessling, Michael K. -H. Altshuler, Boris L. Yuzbashyan, Emil A. |
| author_facet | Kiessling, Michael K. -H. Altshuler, Boris L. Yuzbashyan, Emil A. |
| contents | 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, $\langleω^2\rangle := \int_0^\infty ω^2 P(dω)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_00532 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on $T_c$ in the Eliashberg theory of Superconductivity. II: Dispersive phonons Kiessling, Michael K. -H. Altshuler, Boris L. Yuzbashyan, Emil A. Mathematical Physics Superconductivity 82D55 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, $\langleω^2\rangle := \int_0^\infty ω^2 P(dω)$. |
| title | Bounds on $T_c$ in the Eliashberg theory of Superconductivity. II: Dispersive phonons |
| topic | Mathematical Physics Superconductivity 82D55 |
| url | https://arxiv.org/abs/2409.00532 |