Time-reversal invariant vortex in topological superconductors and gravitational $\mathbb{Z}_2$ topology
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arXiv
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2025
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| author | Yamamoto, Kazuki Kan, Naoto Fukaya, Hidenori |
| author_facet | Yamamoto, Kazuki Kan, Naoto Fukaya, Hidenori |
| contents | We study a time-reversal invariant vortex, namely a spin vortex, in helical superconductors by focusing on its emergent gravitational structure. The topology of the time-reversal invariant vortex is classified by a $\mathbb{Z}_2$ invariant: helical Majorana zero modes appear at the vortex core when the winding number is odd, while no such zero modes exist when it is even. We provide a formal mapping to the theory of gravity to describe this $\mathbb{Z}_2$ topological structure. Identifying a superconducting order parameter as a vielbein in the theory of gravity, we explicitly convert the Bogoliubov-de-Genne Hamiltonian into the Dirac Hamiltonian coupled to a nontrivial gravitational field. Then we find that a gravitational curvature is induced at the vortex core, with its total flux quantized in integer multiples of $π$, reflecting the $\mathbb{Z}_2$ topology. Although the curvature vanishes everywhere except at the vortex core, the energy spectrum remains sensitive to the total curvature flux, owing to the gravitational Aharonov-Bohm effect. We further demonstrate that our gravitational framework can be applied to the topological phase transition driven by the vortex-linking precess in three-dimensional helical superconductors such as the He-B phase. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_06586 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Time-reversal invariant vortex in topological superconductors and gravitational $\mathbb{Z}_2$ topology Yamamoto, Kazuki Kan, Naoto Fukaya, Hidenori Mesoscale and Nanoscale Physics Quantum Gases Superconductivity High Energy Physics - Lattice High Energy Physics - Theory We study a time-reversal invariant vortex, namely a spin vortex, in helical superconductors by focusing on its emergent gravitational structure. The topology of the time-reversal invariant vortex is classified by a $\mathbb{Z}_2$ invariant: helical Majorana zero modes appear at the vortex core when the winding number is odd, while no such zero modes exist when it is even. We provide a formal mapping to the theory of gravity to describe this $\mathbb{Z}_2$ topological structure. Identifying a superconducting order parameter as a vielbein in the theory of gravity, we explicitly convert the Bogoliubov-de-Genne Hamiltonian into the Dirac Hamiltonian coupled to a nontrivial gravitational field. Then we find that a gravitational curvature is induced at the vortex core, with its total flux quantized in integer multiples of $π$, reflecting the $\mathbb{Z}_2$ topology. Although the curvature vanishes everywhere except at the vortex core, the energy spectrum remains sensitive to the total curvature flux, owing to the gravitational Aharonov-Bohm effect. We further demonstrate that our gravitational framework can be applied to the topological phase transition driven by the vortex-linking precess in three-dimensional helical superconductors such as the He-B phase. |
| title | Time-reversal invariant vortex in topological superconductors and gravitational $\mathbb{Z}_2$ topology |
| topic | Mesoscale and Nanoscale Physics Quantum Gases Superconductivity High Energy Physics - Lattice High Energy Physics - Theory |
| url | https://arxiv.org/abs/2507.06586 |