Time-reversal invariant vortex in topological superconductors and gravitational Z2 topology

Abstract

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 Z2 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 Z2 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 π, reflecting the Z2 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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