Superconductivity Near a Quantum Critical Point: Bounds on the Transition Temperature in the $γ$-Model
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912785214996480 |
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| author | Elezaby, Ahmed Abanov, Artem |
| author_facet | Elezaby, Ahmed Abanov, Artem |
| contents | Near a quantum critical point (QCP) in a metal, strong Fermion-Fermion interactions mediated by soft collective bosons give rise to two competing phenomena: non-Fermi liquid behavior and superconductivity that deviates from conventional BCS and Migdal-Eliashberg theories. We consider the problem of obtaining closed-form analytical lower and upper bounds on transition temperatures for such systems. We focus mainly on a class of models known as the gamma-model, which generalizes the Eliashberg theory of Superconductivity where the effective interaction potential scales as V(Omega) ~ 1/|Omega|^gamma. Building on a recent reformulation of Migdal-Eliashberg theory, expressed as a classical infinite spin chain with nonlocal interactions, and employing a simple linear algebra framework, we derive rigorous closed-form expressions for upper and lower bounds on the superconducting transition temperature for any gamma > 0. While our lower bounds coincide precisely with those previously in the literature, our derivation offers a more streamlined and accessible method. Moreover, our upper bounds are substantially tighter than any existing estimates and converge rapidly enough to the numerical results from various prior studies. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_20009 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Superconductivity Near a Quantum Critical Point: Bounds on the Transition Temperature in the $γ$-Model Elezaby, Ahmed Abanov, Artem Superconductivity Strongly Correlated Electrons Near a quantum critical point (QCP) in a metal, strong Fermion-Fermion interactions mediated by soft collective bosons give rise to two competing phenomena: non-Fermi liquid behavior and superconductivity that deviates from conventional BCS and Migdal-Eliashberg theories. We consider the problem of obtaining closed-form analytical lower and upper bounds on transition temperatures for such systems. We focus mainly on a class of models known as the gamma-model, which generalizes the Eliashberg theory of Superconductivity where the effective interaction potential scales as V(Omega) ~ 1/|Omega|^gamma. Building on a recent reformulation of Migdal-Eliashberg theory, expressed as a classical infinite spin chain with nonlocal interactions, and employing a simple linear algebra framework, we derive rigorous closed-form expressions for upper and lower bounds on the superconducting transition temperature for any gamma > 0. While our lower bounds coincide precisely with those previously in the literature, our derivation offers a more streamlined and accessible method. Moreover, our upper bounds are substantially tighter than any existing estimates and converge rapidly enough to the numerical results from various prior studies. |
| title | Superconductivity Near a Quantum Critical Point: Bounds on the Transition Temperature in the $γ$-Model |
| topic | Superconductivity Strongly Correlated Electrons |
| url | https://arxiv.org/abs/2512.20009 |