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Physical Review A

Three-dimensional quaternionic condensations, Hopf invariants, and skyrmion lattices with synthetic spin-orbit coupling

Yi Li, Xiang-Fa Zhou, Congjun Wu

Published Mar 14, 2016• 26 citations
Publisher

Abstract

We study the topological configurations of the two-component condensates of bosons with the three-dimensional $(3\mathrm{D}) \stackrel{P\vec}{\ensuremath{\sigma}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{p}$ Weyl-type spin-orbit coupling subject to a harmonic trapping potential. The topology of the condensate wave functions manifests in the quaternionic representation. In comparison to the $\text{U}(1)$ complex phase, the quaternionic phase manifold is ${S}^{3}$ and the spin orientations form the ${S}^{2}$ Bloch sphere through the first Hopf mapping. The spatial distributions of the quaternionic phases exhibit the 3D skyrmion configurations, and the spin distributions possess nontrivial 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.

DOI
10.1103/physreva.93.033628

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We study the topological configurations of the two-component condensates of bosons with the three-dimensional (3\mathrm{D}) \stackrel{P\vec}{\ensuremath{\sigma}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{p} Weyl-type spin-orbit coupling subject to a harmonic trapping potential. The topology of the condensate wave functions manifests in the quaternionic representation. In comparison to the U(1)\text{U}(1)U(1) complex phase, the quaternionic phase manifold is S3{S}^{3}S3 and the spin orientations form the S2{S}^{2}S2 Bloch sphere through the first Hopf mapping. The spatial distributions of the quaternionic phases exhibit the 3D skyrmion configurations, and the spin distributions possess nontrivial 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.
We study the topological configurations of the two-component condensates of bosons with the 333D σ⃗⋅p⃗\vec{\sigma}\cdot \vec{p}σ⋅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)U(1)U(1) complex phase, the quaternionic phase manifold is S3S^3S3 and the spin orientations form the S2S^2S2 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.
We study unconventional condensations of two-component bosons in a harmonic trap subject to the 3D σ⃗⋅p⃗\vec{\sigma}\cdot \vec{p}σ⋅p​-type spin-orbit (SO) coupling. The topology of condensate wavefunctions manifests in the quaternionic representation. The spatial distributions of the S3S^3S3 quaternionic phase exhibit 3D skyrmion configurations, while those of the S2S^2S2 spin orientation possess non-zero Hopf invariants. As increasing SO coupling strength, spin textures evolve from concentric distributions to lattice structures at weak interactions. Strong interactions change condensates into spin-polarized plane-wave states, or, superpositions of two plane-waves exhibiting helical spin spirals.
We study unconventional condensations of two-component bosons in a harmonic trap subject to the 3D ~σ · ~p-type spin-orbit (SO) coupling. The topology of condensate wavefunctions manifests in the quaternionic representation. The spatial distributions of the S quaternionic phase exhibit 3D skyrmion configurations, while those of the S spin orientation possess non-zero Hopf invariants. As increasing SO coupling strength, spin textures evolve from concentric distributions to lattice structures at weak interactions. Strong interactions change condensates into spin-polarized planewave states, or, superpositions of two plane-waves exhibiting helical spin spirals.
The spin–orbit coupling with bosons gives rise to novel properties that are absent in usual bosonic systems. Under very general conditions, the conventional ground state wavefunctions of bosons are constrained by the ‘no-node’ theorem to be positive definite. In contrast, the linear dependence of the spin–orbit coupling leads to complex-valued condensate wavefunctions beyond this theorem. In this paper, we review the study of this class of unconventional Bose–Einstein condensations focusing on their topological properties. Both the 2D Rashba and 3D -type Weyl spin–orbit couplings give rise to Landau-level-like quantization of single-particle levels in the harmonic trap. Interacting condensates develop the half-quantum vortex structure spontaneously breaking the time-reversal symmetry and exhibit topological spin textures of the skyrmion type. In particular, the 3D Weyl coupling generates topological defects in the quaternionic phase space as an SU(2) generalization of the usual U(1) vortices. Rotating spin–orbit-coupled condensates exhibit rich vortex structures due to the interplay between vorticity and spin texture. In the Mott-insulating states in optical lattices, quantum magnetism is characterized by the Dzyaloshinskii–Moriya-type exchange interactions.
We study ultracold bosonic atoms with synthetic three-dimensional spin-orbit (SO) coupling in a cubic optical lattice. In the superfluidity phase, the lowest energy band exhibits one, two, or four pairs of degenerate single-particle ground states depending on the SO-coupling strengths, which can give rise to condensate states with spin stripes for weak atomic interactions. In the deep Mott-insulator regime, the effective spin Hamiltonian of the system combines three-dimensional Heisenberg exchange interactions, anisotropy interactions, and Dzyaloshinskii-Moriya interactions. Based on Monte Carlo simulations, we numerically demonstrate that the resulting Hamiltonian with an additional Zeeman field has a rich phase diagram with spiral, stripe, vortex crystal, and especially Skyrmion crystal spin textures in each xyxyxy-plane layer. The obtained Skyrmion crystals can be tunable with square and hexagonal symmetries in a columnar manner along the zzz axis, and moreover are stable against the interlayer spin-spin interactions in a large parameter region.
We present a variational study of pseudo-spin-1/21/21/2 Bose gases in a harmonic trap with weak three-dimensional (3D) spin-orbit coupling of $\mathbf{\ensuremath{\sigma}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ type. This spin-orbit coupling mixes states with different parities, which inspires us to approximate the single-particle state with the eigenstates of the total angular momentum, i.e., superposition of harmonic sss-wave and ppp-wave states. As the time-reversal symmetry is protected by two-body interaction, we set the variational order parameter as the combination of two mutually time-reversal symmetric eigenstates of the total angular momentum. The variational results essentially reproduce the 3D skyrmionlike ground state recently identified by Kawakami et al. [T. Kawakami, T. Mizushima, M. Nitta, and K. Machida, Phys. Rev. Lett. 109, 015301 (2012)]. We show that these skyrmionlike ground states emerging in this model are primarily caused by the ppp-wave spatial mode involved in the variational order parameter that drives two spin components spatially separated. We find the ground state of this system falls into two phases with different density distribution symmetries depending on the relative magnitude of intraspecies and interspecies interaction: phase I has parity symmetric and axisymmetric density distributions, while phase II is featured with special joint symmetries of discrete rotational and time-reversal symmetry. With the increasing interaction strength the transition occurs between two phases with distinct density distributions, while the topological 3D skyrmionlike spin texture is symmetry protected.