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

TL;DR

This paper investigates time-reversal invariant vortices in class $DIII$ topological superconductors by mapping the Bogoliubov–de Gennes (BdG) Hamiltonian to a Dirac Hamiltonian in curved spacetime, identifying the superconducting order parameter with a vielbein. The analysis reveals an emergent gravitational curvature concentrated at the vortex core with flux $n\pi$ and a gravitational Aharonov–Bohm phase that is quantized as $(-1)^n$, establishing a $\mathbb{Z}_2$ topology: Majorana Kramers zero modes appear for odd winding $n$ while they do not for even $n$, with edge modes coupling to vortex modes. In three dimensions, vortex rings can be linked or unlinked, and linking induces a $0$ to $\pi$ transition in the gravitational AB phase, corresponding to the presence of helical Majorana zero modes along the line defect. The work provides a gravity-inspired framework for understanding defect-bound states in topological superconductors and suggests broader implications for emergent gravity in condensed-matter systems.

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 $π$, 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.

Figures (3)

• Figure 1: Schematic illustration of (a) a time-reversal invariant vortex in a two-dimensional helical superconductor and (b) the corresponding induced gravitational flux.
• Figure 2: Schematic figure of the energy spectrum of the edge states [Eq. \ref{['eq:energy spectrum of edge state']}] and vortex bound states [Eq. \ref{['eq:energy spectrum of vortex bound state']}] when the winding number of the time reversal invariant vortex is (a) odd integer and (b) even integer. The red (blue) circle represents the energy eigenvalue corresponding to the spin-up (spin-down) state for edge states and spin-down (spin-up) state for vortex bound states. Here $j$ represents the eignevalue of the effective angular momentum $J$.
• Figure 3: A time-reversal invariant vortex in a three-dimensional helical superconductor. (a) The one-dimensional vortex structure $\vb{r}_0(z')$ is depicted by the green curve. For every fixed $z'$, one can define a local coordinate system whose $x'y'$-plane is perpendicular to the local orientation $\hat{\vb{n}}$ of the vortex. (b,c) In a closed three-dimensional system, the topology of a vortex pair configuration is determined by whether the two vorteex rings are linked or unlinked. When a linking occurs, the system undergoes a topological phase transition, manifested as a discontinuous jump of the gravitational AB phase along one vortex from $0$ to $\pi$.