Bounds on $T_c$ in the Eliashberg theory of Superconductivity. I: The $γ$-model
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866918048942784512 |
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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 | 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 $T_c(g,γ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00533 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on $T_c$ in the Eliashberg theory of Superconductivity. I: The $γ$-model Kiessling, Michael K. -H. Altshuler, Boris L. Yuzbashyan, Emil A. Mathematical Physics Superconductivity 82D55 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 $T_c(g,γ)$. |
| title | Bounds on $T_c$ in the Eliashberg theory of Superconductivity. I: The $γ$-model |
| topic | Mathematical Physics Superconductivity 82D55 |
| url | https://arxiv.org/abs/2409.00533 |