三维四元数凝聚、Hopf不变量和合成自旋轨道耦合的Skyrmion晶格
3D quaternionic condensations, Hopf invariants, and skyrmion lattices with synthetic spin-orbit coupling
摘要：
本文研究了在简谐囚禁势作用下，具有$3$ D $\vecσ\cdot \vec{p}$ Weyl型自旋轨道耦合的两组分玻色子凝聚体的拓扑构型。凝聚波函数的拓扑结构在四元数表示中得到了体现。与$U(1)$复相相比，四元数相流形是$S^3$，并且自旋取向通过第一霍普夫映射形成$S^2$ Bloch球。四元数相位的空间分布呈现三维Skyrmion组态，自旋分布具有非平凡的Hopf不变量。自旋织构从弱自旋轨道耦合区的同心分布演化到中等自旋轨道耦合区的旋转对称性破缺。在强自旋轨道耦合区，单粒子谱表现出朗道能级型量子化。在这种情况下，当相互作用低于朗道能级混合的能量标度时，形成三维Skyrmion晶格结构。足够强的强相互作用可以将凝聚体转变为自旋极化的平面波态，或者，呈现螺旋状自旋螺旋的两个平面波的叠加。
英文摘要：
We study the topological configurations of the two-component condensates of bosons with the $3$D $\vecσ\cdot \vec{p}$ 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.
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