Symmetry-protected skyrmions in three-dimensional spin-orbit-coupled Bose gases

We present a variational study of pseudo-spin-$1/2$ Bose gases in a harmonic trap with weak three-dimensional (3D) spin-orbit coupling of $\mathbf{\sigma }·\mathbf{p}$ 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 $s$-wave and $p$-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 $p$-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.

Figure 1

Phase diagram of weakly SO-coupled two-component bosons with coupling strength $\lambda =0.2\sqrt{\hslash \omega /m}$, which shows two skyrmionlike phases I and II. Phase I is a skyrmion ground state of order parameter $exp[-i\mathrm{\Omega }(\mathbf{r})·\mathbf{S}]\mathbf{\zeta }$ with ${\mathbf{\zeta }}_z={(1,0)}^T$ and phase II is a skyrmion state with ${\mathbf{\zeta }}_x=\frac{1}{\sqrt{2}}{(1,1)}^T$. The interaction parameter $g$ is in units of $\hslash \omega l_T^3/N$. Density distribution and spin texture of these two phases are shown in Figs. 2 and 3, respectively.

Figure 2

Density distribution and spin texture of phase I for $\alpha =-1.6$. All coordinates are in units of $l_T=\sqrt{\hslash /m\omega }$. Top: Three rows are densities in $xy, yz$, and $xz$ planes, respectively; three columns are for up and down components and the total density as explicitly labeled above each column. Density distributions in $xz$ and $yz$ planes are the same due to the $z$-axis rotational symmetry. Bottom: 3D skyrmion spin texture $\mathbf{s}\left(\mathbf{r}\right)=\mathbf{S}\left(\mathbf{r}\right)/n\left(\mathbf{r}\right)$ of phase I. The streamline plot of $\mathbf{s}$ in a selected region is shown.

Figure 3

Density distribution and spin texture of phase II for $\alpha =-1.6$. All coordinates are in units of $l_T=\sqrt{\hslash /m\omega }$. Top: Three rows are densities in $xy, yz$, and $xz$ planes, respectively; three columns are for up and down components and the total density as explicitly labeled above each column. Though the total density is isotropic again, the density distribution for the two components exhibits more complex symmetry as described in the text. Bottom: 3D skyrmion spin texture $\mathbf{s}\left(\mathbf{r}\right)=\mathbf{S}\left(\mathbf{r}\right)/n\left(\mathbf{r}\right)$ of phase II, which is roughly a $\pi /2$ rotation about the $y$ axis of that in phase I. The topological structure of the spin texture is protected by the time-reversal symmetry.