Finsler geometry in anisotropic superconductivity: a Ginzburg-Landau approach
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914224236658688 |
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| author | Fakhri, Y. Alipour |
| author_facet | Fakhri, Y. Alipour |
| contents | We present a rigorous generalization of the classical Ginzburg--Landau model to smooth, compact Finsler manifolds without boundary. This framework provides a natural analytic setting for describing anisotropic superconductivity within Finsler geometry. The model is constructed via the Finsler--Laplacian, defined through the Legendre transform associated with the fundamental function F, and by employing canonical Finsler measures such as the Busemann--Hausdorff and Holmes--Thompson volume forms. We introduce an anisotropic Ginzburg--Landau functional for complex scalar fields coupled to gauge potentials and establish the existence of minimizers in the appropriate Finsler--Sobolev spaces by the direct method in the calculus of variations. Furthermore, we analyze the asymptotic regime as the Ginzburg--Landau parameter epsilon to 0 and prove a precise Gamma--convergence result: the rescaled energies converge to the Finslerian length functional associated with the limiting vortex filaments. In particular, the limiting vortex energy is shown to equal $π$ times the Finslerian length of the corresponding current, thereby extending the classical Bethuel--Brezis--He'lein result to anisotropic settings. These findings demonstrate that Finsler geometry unifies metric anisotropy and variational principles in gauge-field models, broadening the geometric scope of the Ginzburg--Landau theory beyond the Riemannian framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19720 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finsler geometry in anisotropic superconductivity: a Ginzburg-Landau approach Fakhri, Y. Alipour Mathematical Physics 53B40, 35J60, 35J20, 49J45, 58E50, 35Q56 We present a rigorous generalization of the classical Ginzburg--Landau model to smooth, compact Finsler manifolds without boundary. This framework provides a natural analytic setting for describing anisotropic superconductivity within Finsler geometry. The model is constructed via the Finsler--Laplacian, defined through the Legendre transform associated with the fundamental function F, and by employing canonical Finsler measures such as the Busemann--Hausdorff and Holmes--Thompson volume forms. We introduce an anisotropic Ginzburg--Landau functional for complex scalar fields coupled to gauge potentials and establish the existence of minimizers in the appropriate Finsler--Sobolev spaces by the direct method in the calculus of variations. Furthermore, we analyze the asymptotic regime as the Ginzburg--Landau parameter epsilon to 0 and prove a precise Gamma--convergence result: the rescaled energies converge to the Finslerian length functional associated with the limiting vortex filaments. In particular, the limiting vortex energy is shown to equal $π$ times the Finslerian length of the corresponding current, thereby extending the classical Bethuel--Brezis--He'lein result to anisotropic settings. These findings demonstrate that Finsler geometry unifies metric anisotropy and variational principles in gauge-field models, broadening the geometric scope of the Ginzburg--Landau theory beyond the Riemannian framework. |
| title | Finsler geometry in anisotropic superconductivity: a Ginzburg-Landau approach |
| topic | Mathematical Physics 53B40, 35J60, 35J20, 49J45, 58E50, 35Q56 |
| url | https://arxiv.org/abs/2510.19720 |