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: a zero-energy Majorana Kramers pair appears 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 Gennes 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 $\pi$, 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.

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• Dirac equation
• Gravitation
• Majorana bound states
• Quantum fields in curved spacetime
• Topological superconductors
• Bogoliubov-de Gennes equations
• Superconductivity
• Superconductors