Non-trivial topology generation by a precessional Rashba field confined in the central-force potential

The two-dimensional quantum confinement system with a central-force potential causes the degeneracy of the eigenstates due to the orbital angular momentum l. Here, we apply the Rashba field (Ξ) precessionally and study the topological tuning via Berry’s parameters of connection A, curvature B and phase γ.



The C₂ operation connects eigenstates degenerated among angular momenta and spins. By employing the zenith θ_Ξ, and azimuthal angles φ_Ξ of the Rashba field, we introduce the spherical parameter space (Ξ sphere) having a latitude and longitude. Both at the north and south poles of Ξ sphere, we have A=0, whereas a finite A arises along the latitude, leading to the maximum at the equator. At the equator (θ_Ξ=π/2), the direction A reverses oppositely, and then leads to zero at the south pole (θ_Ξ=π). The increase/decrease and inversion in A cause the Berry curvature B=▽×A. As such, B results in the distribution like a "sea urchin," leading to the maximum at the equator.



We finally calculate the Berry phase γ by performing a line integration of A along the equi-energy line (latitude). The applied Rashba field of θ_Ξ=0 and π conserves the native (trivial) topology of the system. Although the in-plane application (θ_Ξ=π/2) generates the finite value in γ, the native topology does not change because of 2π jump in γ. Contrary, the Rashba field except for those above applications generates the finite (non-trivial) γ, being nonintegral multiples of π with varying in accordance with the zenith angle.