Time-reversal invariant vortex in topological superconductors and gravitational $\mathbb{Z}_2$ topology

TL;DR

This paper explores a gravitational analogy in topological superconductors, revealing a $bZ_2$ invariant that classifies vortex states and induces quantized curvature at vortex cores, impacting Majorana zero modes and phase transitions.

Contribution

It introduces a novel gravitational framework to classify and analyze time-reversal invariant vortices in topological superconductors, linking topology with gravitational curvature and Majorana modes.

Findings

Majorana zero modes appear at odd-winding vortices

Gravitational curvature is quantized at vortex cores

The framework applies to topological phase transitions in 3D superconductors

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 $\mathbb{Z}_2$ 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 $\pi$, 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.