3D quaternionic condensations, Hopf invariants, and skyrmion lattices with synthetic spin-orbit coupling

TL;DR

This paper explores the topological properties of two-component bosonic condensates with 3D Weyl spin-orbit coupling, revealing quaternionic phase structures, skyrmion lattices, and diverse spin textures influenced by interaction strength and spin-orbit coupling regimes.

Contribution

It introduces a quaternionic framework for topological analysis of condensates with Weyl spin-orbit coupling, uncovering new skyrmion lattice structures and spin textures.

Findings

Quaternionic phase manifold is S^3, spin orientations form S^2 via Hopf mapping.

3D skyrmion configurations emerge in condensate distributions.

Strong spin-orbit coupling leads to Landau-level quantization and skyrmion lattice formation.

Abstract

We study the topological configurations of the two-component condensates of bosons with the $3$D $\vec{\sigma}\cdot \vec{p}$ Weyl-type spin-orbit coupling subject to a harmonic trapping potential. The topology of the condensate wavefunctions manifests in the quaternionic representation. In comparison to the $U(1)$ complex phase, the quaternionic phase manifold is $S^3$ and the spin orientations form the $S^2$ Bloch sphere through the 1st Hopf mapping. The spatial distributions of the quaternionic phases exhibit the 3D skyrmion configurations, and the spin distributions possess non-trivial Hopf invariants. Spin textures evolve from the concentric distributions at the weak spin-orbit coupling regime to the rotation symmetry breaking patterns at the intermediate spin-orbit coupling regime. In the strong spin-orbit coupling regime, the single-particle spectra exhibit the Landau-level type quantization. In this regime, the three-dimensional skyrmion lattice structures are formed when interactions are below the energy scale of Landau level mixings. Sufficiently strong interactions can change condensates into spin-polarized plane-wave states, or, superpositions of two plane-waves exhibiting helical spin spirals.