# Spin-controlled topological phase transition in non-Euclidean space

## TL;DR

Researchers demonstrate a new way to control topological phase transitions using Möbius rings in non-Euclidean space with circularly polarized light.

## Contribution

The paper introduces spin-controlled topological phase transitions in non-Euclidean systems using Möbius rings.

## Key findings

- Möbius rings with an 8π period enable spin-controlled transitions between topological edge and bulk states.
- Circularly polarized light can excite and control topological edge states in non-Euclidean configurations.
- The approach works for both Hermitian and non-Hermitian systems in 2D.

## Abstract

Modulation of topological phase transition has been pursued by researchers in both condensed matter and optics research fields, and has been realized in Euclidean systems, such as topological photonic crystals, topological metamaterials, and coupled resonator arrays. However, the spin-controlled topological phase transition in non-Euclidean space has not yet been explored. Here, we propose a non-Euclidean configuration based on Möbius rings, and we demonstrate the spin-controlled transition between the topological edge state and the bulk state. The Möbius ring, which is designed to have an 8π period, has a square cross section at the twist beginning and the length/width evolves adiabatically along the loop, accompanied by conversion from transverse electric to transverse magnetic modes resulting from the spin-locked effect. The 8π period Möbius rings are used to construct Su–Schrieffer–Heeger configuration, and the configuration can support the topological edge states excited by circularly polarized light, and meanwhile a transition from the topological edge state to the bulk state can be realized by controlling circular polarization. In addition, the spin-controlled topological phase transition in non-Euclidean space is feasible for both Hermitian and non-Hermitian cases in 2D systems. This work provides a new degree of polarization to control topological photonic states based on the spin of Möbius rings and opens a way to tune the topological phase in non-Euclidean space.

The online version contains supplementary material available at 10.1007/s12200-024-00110-w.