Bounds on $T_c$ in the Eliashberg theory of Superconductivity. III: Einstein phonons
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| Natura: | Preprint |
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2024
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| _version_ | 1866908446907957248 |
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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 dispersionless limit of the standard Eliashberg theory of superconductivity is studied. The effective electron-electron interactions are mediated by Einstein phonons of frequency $Ω>0$, equipped with electron-phonon coupling strength $λ$. This allows for a detailed evaluation of the general results on $T_c$ for phonons with non-trivial dispersion relation, obtained in a previous paper, (II), by the authors. The variational principle for the linear stability boundary $\mathscr{S}_{\!c}$ of the normal state region against perturbations toward the superconducting region, obtained in (II), simplifies as follows: If $(λ,Ω,T)\in\mathscr{S}_{\!c}$, then $λ= 1/\mathfrak{h}(\varpi)$, where $\varpi:=Ω/2πT$, and where $\mathfrak{h}(\varpi)>0$ is the largest eigenvalue of a compact self-adjoint operator $\mathfrak{H}(\varpi)$ on $\ell^2$ sequences; $\mathfrak{H}(\varpi)$ is the dispersionless limit $P(dω)\toδ(ω-Ω)dω$ of the operator $\mathfrak{K}(P,T)$ of (II). It is shown that when $\varpi \leq \sqrt{2}$, then the map $\varpi\mapsto\mathfrak{h}(\varpi)$ is invertible. For $λ>0.77$ this yields: (i) the existence of a critical temperature $T_c(λ,Ω) = Ωf(λ)$; (ii) a sequence of lower bounds on $f(λ)$ that converges to $f(λ)$. Also obtained is an upper bound on $T_c(λ,Ω)$, which agrees with the asymptotic behavior $T_c(λ,Ω) \sim C Ω\sqrtλ$ for $λ\sim\infty$, given $Ω$, though with $C\approx 2.034 C_\infty$, where $C_\infty := \frac{1}{2π}\mathfrak{k}(2)^\frac12 =0.1827262477...$ is the optimal constant, and $\mathfrak{k}(γ)>0$ the largest eigenvalue of a compact self-adjoint operator for the $γ$ model, determined in the first paper, (I), on $T_c$ by the authors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_02121 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on $T_c$ in the Eliashberg theory of Superconductivity. III: Einstein phonons Kiessling, Michael K. -H. Altshuler, Boris L. Yuzbashyan, Emil A. Superconductivity Mathematical Physics 82D55 The dispersionless limit of the standard Eliashberg theory of superconductivity is studied. The effective electron-electron interactions are mediated by Einstein phonons of frequency $Ω>0$, equipped with electron-phonon coupling strength $λ$. This allows for a detailed evaluation of the general results on $T_c$ for phonons with non-trivial dispersion relation, obtained in a previous paper, (II), by the authors. The variational principle for the linear stability boundary $\mathscr{S}_{\!c}$ of the normal state region against perturbations toward the superconducting region, obtained in (II), simplifies as follows: If $(λ,Ω,T)\in\mathscr{S}_{\!c}$, then $λ= 1/\mathfrak{h}(\varpi)$, where $\varpi:=Ω/2πT$, and where $\mathfrak{h}(\varpi)>0$ is the largest eigenvalue of a compact self-adjoint operator $\mathfrak{H}(\varpi)$ on $\ell^2$ sequences; $\mathfrak{H}(\varpi)$ is the dispersionless limit $P(dω)\toδ(ω-Ω)dω$ of the operator $\mathfrak{K}(P,T)$ of (II). It is shown that when $\varpi \leq \sqrt{2}$, then the map $\varpi\mapsto\mathfrak{h}(\varpi)$ is invertible. For $λ>0.77$ this yields: (i) the existence of a critical temperature $T_c(λ,Ω) = Ωf(λ)$; (ii) a sequence of lower bounds on $f(λ)$ that converges to $f(λ)$. Also obtained is an upper bound on $T_c(λ,Ω)$, which agrees with the asymptotic behavior $T_c(λ,Ω) \sim C Ω\sqrtλ$ for $λ\sim\infty$, given $Ω$, though with $C\approx 2.034 C_\infty$, where $C_\infty := \frac{1}{2π}\mathfrak{k}(2)^\frac12 =0.1827262477...$ is the optimal constant, and $\mathfrak{k}(γ)>0$ the largest eigenvalue of a compact self-adjoint operator for the $γ$ model, determined in the first paper, (I), on $T_c$ by the authors. |
| title | Bounds on $T_c$ in the Eliashberg theory of Superconductivity. III: Einstein phonons |
| topic | Superconductivity Mathematical Physics 82D55 |
| url | https://arxiv.org/abs/2409.02121 |