Parallel spin transport and holonomy in non-Euclidean curved circuits on a spherical two-dimensional electron gas

TL;DR

This work shows that curvature in spherical 2DEGs with Rashba SOC introduces spin holonomy that breaks the symmetric AC interference observed in flat 1D Rashba circuits. By deriving geodesic- and polygon-based 1D Hamiltonians and identifying a parallel-transport condition, the authors connect SU(2) spin holonomy to geometric phases that modify conductance in curved geometries. The conductance, treated with disorder averaging and a semiclassical framework, reveals an offset-shift mechanism that can restore symmetry in the AC pattern when parallel transport is accounted for, and highlights the role of the solid-angle holonomy in the spin texture evolution. The results point to feasible experimental platforms, such as curved 2DEG caps, and open avenues for exploring spin dynamics in curved spaces and curved-space simulators using condensed-matter systems.

Abstract

The quantum conductance of one-dimensional (1D) circuits built on flat (Euclidean) two-dimensional electron gases (2DEGs) is known to display a symmetric response to the inversion of Rashba spin-orbit coupling fields in Aharonov-Casher (AC) interference patterns. Here, we show that this symmetry breaks down in curved (non-Euclidean) 1D circuits defined on spherical 2DEGs. We demonstrate that this is a consequence of parallel transport and holonomy of the electronic spin on the surface of the sphere, and that a symmetric response can be recovered when considering the parallel transport condition as an offset shifting the AC pattern. We discuss 1D triangular circuits defined along geodesic arcs on the sphere as a case study, and generalize it to regular polygons and parallel curves of given latitude.

Figures (10)

• Figure 1: 1D curve $\mathcal{C}$ embedded in 3D space and parametrized by $\mathbf{r}(\ell)$, with $\ell$ the arclength. It displays the local Frenet-Serret triad $\{\hat{T}(\ell),\hat{N}(\ell),\hat{B}(\ell)\}$.
• Figure 2: Geodesic curve $\mathcal{G}$ (great circle) on the surface of a sphere displaying the effective Rashba field $\mathbf{B}_{\text{R}}$ (produced by a radial electric field $\mathbf{E}\propto \alpha_\text{R}$) and the local Frenet-Serret triad $\{\hat{T}(\ell),\hat{N}(\ell),\hat{B}(\ell)\}$, parametrized by the arclength $\ell$: $\hat{N}(\ell)$ points to the sphere's center while $\{\hat{T}(\ell),\hat{B}(\ell)\}$ generate a tangent plane. Importantly, notice that $\hat{B}(\ell)$ and $\mathbf{B}_{\text{R}}$ are constant and antiparallel/parallel for positive/negative Rashba SOC strengths $\alpha_{\text{R}}$.
• Figure 3: Regular elliptic triangle on a spherical surface of radius $R$ with geodesic sides of length $L$. It is characterized by the polar angle $\eta$, the arc angle $\theta_3=L/R$, and the interior angle $\gamma$. The local tangent bases read $\{\hat{T}_n,\hat{B}_n\}$ with $n=0,1,2$. Notice that $\hat{B}_n$ is constant along the corresponding geodesic segment.
• Figure 4: Top view of a triangular circuit on a Rashba sphere. It displays the effective Rashba field texture $\{\mathbf{B}_{\text{R}n}\}$ (antiparallel to the unit vectors $\{\hat{B}_n\}$). The field texture is tangent to the sphere along the geodesic segments and locally perpendicular to the propagation directions $\{\hat{T}_n\}$. In the Euclidean limit, it reduces to a coplanar texture.
• Figure 5: Response of the global spin phase $\phi$ to the curvature (in terms of $\theta_3=L/R$) and Rashba SOC strength ($k_{\text{R}}L$) in elliptic triangular circuits. The dashed line corresponds to an octant triangle (interior angles $\gamma=\pi/2$). The solid line indicates the parallel spin transport condition. The global spin phase vanishes along the dotted lines ($\phi=0$).