Numerical Solution of the Bardeen-Cooper-Schrieffer Equation for Unconventional Superconductors
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917278763712512 |
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| author | Buchheit, Andreas A. Keßler, Torsten Rjasanow, Sergej |
| author_facet | Buchheit, Andreas A. Keßler, Torsten Rjasanow, Sergej |
| contents | In this work, we consider the analytical properties and the efficient numerical solution of the Bardeen-Cooper-Schrieffer equation for unconventional superconductivity incorporating long-range power-law electron-electron interactions within a tight-binding model on a $d$-dimensional lattice. It is a nonlinear convolution equation for the complex matrix-valued superconducting gap under symmetry constraints imposed by the fermionic anticommutation rules. The long-range interaction enters in momentum space in the form of the now efficiently computable Epstein zeta function, which exhibits a power-law singularity at zero momentum. This needs to be accounted when evaluating the convolution. After a brief overview of some of the equation's analytical properties, we discuss its efficient numerical solution using a Galerkin method with B-splines. We present numerical results for a nodal superconductor on a two-dimensional square lattice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15911 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Numerical Solution of the Bardeen-Cooper-Schrieffer Equation for Unconventional Superconductors Buchheit, Andreas A. Keßler, Torsten Rjasanow, Sergej Mathematical Physics Superconductivity Numerical Analysis 45E10, 46E35, 35S05, 65N30 In this work, we consider the analytical properties and the efficient numerical solution of the Bardeen-Cooper-Schrieffer equation for unconventional superconductivity incorporating long-range power-law electron-electron interactions within a tight-binding model on a $d$-dimensional lattice. It is a nonlinear convolution equation for the complex matrix-valued superconducting gap under symmetry constraints imposed by the fermionic anticommutation rules. The long-range interaction enters in momentum space in the form of the now efficiently computable Epstein zeta function, which exhibits a power-law singularity at zero momentum. This needs to be accounted when evaluating the convolution. After a brief overview of some of the equation's analytical properties, we discuss its efficient numerical solution using a Galerkin method with B-splines. We present numerical results for a nodal superconductor on a two-dimensional square lattice. |
| title | Numerical Solution of the Bardeen-Cooper-Schrieffer Equation for Unconventional Superconductors |
| topic | Mathematical Physics Superconductivity Numerical Analysis 45E10, 46E35, 35S05, 65N30 |
| url | https://arxiv.org/abs/2602.15911 |