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arXiv:1512.06394 · New Journal of Physics

Magnetic and nematic phases in a Weyl type spin–orbit-coupled spin-1 Bose gas

Guanjun Chen, Li Chen, Yunbo Zhang

Published Jun 8, 2016• 6 citations
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Abstract

We present a variational study of the spin-1 Bose gases in a harmonic trap with three-dimensional spin–orbit (SO) coupling of Weyl type. For weak SO coupling, we treat the single-particle ground states as the form of perturbational harmonic oscillator states in the lowest total angular momentum manifold with j = 1, m j = 1, 0, −1. When the two-body interaction is considered, we set the trail order parameter as the superposition of three degenerate single-particle ground-states and the weight coefficients are determined by minimizing the energy functional. Two ground state phases, namely the magnetic and the nematic phases, are identified depending on the spin-independent and the spin-dependent interactions. Unlike the non-SO-coupled spin-1 Bose–Einstein condensate for which the phase boundary between the magnetic and the nematic phase lies exactly at zero spin-dependent interaction, the boundary is modified by the SO-coupling. We find the magnetic phase is featured with phase-separated density distributions, 3D skyrmion-like spin textures and competing magnetic and biaxial nematic orders, while the nematic phase is featured with miscible density distributions, zero magnetization and spatially modulated uniaxial nematic order. The emergence of higher spin order creates new opportunities for exploring spin-tensor-related physics in SO coupled superfluid.

DOI
10.1088/1367-2630/18/6/063010

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We present a variational study of the spin-1 Bose gases in a harmonic trap with three-dimensional spin–orbit (SO) coupling of Weyl type. For weak SO coupling, we treat the single-particle ground states as the form of perturbational harmonic oscillator states in the lowest total angular momentum manifold with j = 1, m j = 1, 0, −1. When the two-body interaction is considered, we set the trail order parameter as the superposition of three degenerate single-particle ground-states and the weight coefficients are determined by minimizing the energy functional. Two ground state phases, namely the magnetic and the nematic phases, are identified depending on the spin-independent and the spin-dependent interactions. Unlike the non-SO-coupled spin-1 Bose–Einstein condensate for which the phase boundary between the magnetic and the nematic phase lies exactly at zero spin-dependent interaction, the boundary is modified by the SO-coupling. We find the magnetic phase is featured with phase-separated density distributions, 3D skyrmion-like spin textures and competing magnetic and biaxial nematic orders, while the nematic phase is featured with miscible density distributions, zero magnetization and spatially modulated uniaxial nematic order. The emergence of higher spin order creates new opportunities for exploring spin-tensor-related physics in SO coupled superfluid.
Starting from the Hamiltonian of the second quantization form, the weakly interacting Bose–Einstein condensate with spin–orbit coupling of Weyl type is investigated. It is found that the SU (2) nonsymmetric term, i.e., the spin-dependent interaction, can lift the degeneracy of the ground states with respect to the z component of the total angular momentum J z , casting the ground condensate state into a configuration of zero J z . This ground state density profile can also be affirmed by minimizing the full Gross–Pitaevskii energy functional. The spin texture of the zero J z state indicates that it is a knot structure, whose fundamental group is π 3 ( M ) ≅ π 3 ( S 2 ) = Z .
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.
We consider a SU(3) spin-orbit-coupled Bose gas with nonlocal soft-core interactions confined in a harmonic trapping potential. The presence of SU(3) spin-orbit coupling induces three minima of the single-particle spectrum. In the absence of spin-exchange interaction, our results show that regardless of the sign of the soft-core interaction, the ground state occupies one single minimum in the momentum space and corresponds to a threefold-degenerate magnetized phase. For the ferromagnetic condensate, the ground state shows an amplitude-modulated plane-wave state, where its modulated amplitude and frequency are related to the strength of the soft-core interaction and SU(3) spin-orbit coupling, respectively. In particular, due to the subtle balance of the soft-core interaction and SU(3) spin-orbit coupling, a stable higher-order spin vortex lattice can be formed in the lattice phase of an antiferromagnetic condensate, where the density distributions of each component and the total system show spatial configurations with threefold and sixfold symmetries, respectively. We also give an experimental protocol to observe such states within current experimental capacity.
Abstract We investigate the ground state phases of the two-component Bose–Einstein condensates with three-dimensional Weyl-type spin–orbit coupling (SOC) loaded in a cubic optical lattice. The single-particle lowest energy band exhibits one, two or four pairs of degenerate ground states depending on the configuration of the SOC. We find two types of the supersolid (SS) phases with the isotropic SOC, and the SS phases are signaled by the momentum peaks distribution. We also discuss the spin textures of the SS phases, the spin vectors consist alternatively of two sets of continuous textures at the even and the odd lattice sites.
We consider a SU(3) spin–orbit coupled Bose–Einstein condensate confined in a harmonic plus quartic trap. The ground-state wave functions of such a system are obtained by minimizing the Gross–Pitaevskii energy functional, and the effects of the spin-dependent interaction and spin–orbit coupling are investigated in detail. For the case of ferromagnetic spin interaction, the SU(3) spin–orbit coupling induces a threefold-degenerate plane wave ground state with nontrivial spin texture. For the case of antiferromagnetic spin interaction, the system shows phase separation for weak SU(3) spin–orbit coupling, where three discrete minima with unequal weights in momentum space are selected, while hexagonal honeycomb lattice structure for strong SU(3) SOC, where three discrete minima with equal weights are selected.
We theoretically establish the mean-field phase diagram of a homogeneous spin-1, spin-orbit-coupled Bose gas as a function of the spin-dependent interaction parameter, the Raman coupling strength, and the quadratic Zeeman shift. We find that the interplay between spin-orbit coupling and spin-dependent interactions leads to the occurrence of ferromagnetic or ferronematic phases which also break translational symmetry. For weak Raman coupling, increasing attractive spin-dependent interactions (as in 87Rb^{87}\mathrm{Rb}87Rb or 7Li^{7}\mathrm{Li}7Li) induces a transition from a uniform to a stripe XYXYXY ferromagnet (with no nematic order). For repulsive spin-dependent interactions (as in 23Na^{23}\mathrm{Na}23Na), however, we find a transition from an XYXYXY spin spiral phase ($\ensuremath{\langle}{S}_{z}\ensuremath{\rangle}=0$ and uniform total density) with uniaxial nematic order to a biaxial ferronematic, where the total density, spin vector, and nematic director oscillate in real space. We investigate the stability of these phases against the quadratic Zeeman effect, which generally tends to favor uniform phases with either ferromagnetic or nematic order but not both. We discuss the relevance of our results to ongoing experiments on spin-orbit-coupled, spinor Bose gases.