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

Abstract

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 $\mathbf{\ensuremath{\sigma}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ 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 $s$-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.