# Effects of geometry on spin-orbit Kramers states in semiconducting nanorings

## TL;DR

This paper explores how the shape and curvature of semiconducting nanorings influence spin-orbit Kramers states, revealing that geometric shape and inhomogeneous curvature can induce topological quantum phase transitions detectable via tunneling measurements.

## Contribution

It demonstrates that nanoring shape and curvature control can induce non-trivial spin-orbit state mixing and topological phase transitions, extending understanding of geometric effects in quantum spin systems.

## Key findings

- Shape symmetry constrains quantum evolution.
- Deformed rings can induce dynamical quantum phase transitions.
- Topological transitions can be detected via tunneling amplitude variations.

## Abstract

The holonomic manipulation of spin-orbital degenerate states, encoded in the Kramers doublet of narrow semiconducting channels with spin-orbit interaction, is shown to be intimately intertwined with the geometrical shape of the nanostructures. The presence of doubly degenerate states is not sufficient to guarantee a non-trivial mixing by only changing the Rashba spin-orbit coupling. We demonstrate that in nanoscale quantum rings the combination of arbitrary inhomogeneous curvature and adiabatic variation of the spin-orbit amplitude, e.g. through electric-field gating, can be generally employed to get non-trivial combinations of the degenerate states. Shape symmetries of the nanostructure act to constrain the adiabatic quantum evolution. While for circular rings the geometric phase is not generated along a non-cyclic path in the parameters space, remarkably, for generic mirror-symmetric shape deformed rings the spin-orbit driving can lead to a series of dynamical quantum phase transitions. We explicitly show this occurrence and propose a route to detect such topological transitions by measuring a variation of the electron tunneling amplitude into the semiconducting channel.