According to the "no-node" theorem, many-body ground state wavefunctions of conventional Bose-Einstein condensations (BEC) are positive-definite, thus time-reversal symmetry cannot be spontaneously broken. We find that multi-component bosons with spin-orbit coupling provide an unconventional type of BECs beyond this paradigm. We focus on the subtle case of isotropic Rashba spin-orbit coupling and the spin-independent interaction. In the limit of the weak confining potential, the condensate wavefunctions are frustrated at the Hartree-Fock level due to the degeneracy of the Rashba ring. Quantum zero-point energy selects the spin-spiral type condensate through the "orderfrom-disorder" mechanism. In a strong harmonic confining trap, the condensate spontaneously generates a half-quantum vortex combined with the skyrmion type of spin texture. In both cases, time-reversal symmetry is spontaneously broken. These phenomena can be realized in both cold atom systems with artificial spin-orbit couplings generated from atom-laser interactions and exciton condensates in semi-conductor systems.

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 spin-orbit coupling leads to complex-valued condensate wavefunctions beyond this theorem. In this article, we review the study of this class of unconventional Bose-Einstein condensations focusing on their topological properties. Both the 2D Rashba and 3D σ · p-type Weyl spin-orbit couplings give rise to Landau-level-like quantization of single-particle levels in the harmonic trap. The interacting condensates develop the half-quantum vortex structure spontaneously breaking 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 investigate magnetic properties in the superfluid and Mott-insulating states of two-component bosons with spin-orbit (SO) coupling in 2D square optical lattices. The spin-independent hopping integral t and SO coupled one λare fitted from band structure calculations in the continuum, which exhibit oscillations as increasing SO coupling strength. The magnetic superexchange model is derived in the Mott-insulating state with one-particle per-site, characterized by the DzyaloshinskyMoriya (DM) interaction. In the limit of |λ| ≪ |t|, we find a spin spiral Mott state whose pitch value is the same as that in the incommensurate superfluid state, while in the opposite limit |t| ≪ |λ|, the ground state can be found by a dual transformation to the |λ| ≪ |t| limit. Quantum many-body states with spontaneous incommensurate modulated structure have attracted considerable interests in the past decades, and occur in many settings of condensed matter and ultracold atom physics, such as frustrated magnetism, unconventional superconductor and superfluid and so on. Celebrated examples include the incommensurate magnetism with long-range and short range magnetic order 1,2 , the Fulde-FerrellLarkin-Ovchinnikov (FFLO) pairing states 3,4 . Recently, the Bose-Einstein condensations (BEC) with spin-orbit (SO) coupling introduce a new member to this family. The SO coupled BECs are genuinely new phenomena due to the fact that the kinetic energy is not just a Laplacian but also linearly depends on momentum, which gives rise the complex-valued condensate wavefunctions beyond Feynman's no-node theorem5 . An interesting property of SO coupled condensates of bosons is that they can spontaneously break time-reversal symmetry which is absent in conventional BECs of both superfluid 4 He and many experiments of ultra-cold alkali bosons 6 . For example, it is predicted that such condensates can spontaneously develop half-quantum vortex coexisting with 2D skyrmion-type spin textures in the harmonic trap 7 . Experimentally, spin textures of the SO coupled bosons have been observed in exciton condensations, which is a solid state boson systems with relativistic SO coupling 8 . Theoretically, extensive studies have been performed for SO coupled bosons which exhibit various spin orderings and textures from competitions among SO coupling, interaction, and confining trap energy 7,11-18 . In the optical lattice, the SO coupled bosons are even more interesting. Early investigations have showed that the characteristic incommensurate wavevectors are incommensurate with the lattice 19 . In this article, we study the SO coupled Bose-Hubbard model, focusing on the magnetic properties. The tight binding model is constructed and the spin-independent hopping integral t and SO coupled hopping integral λ are calculated as functions of the SO coupling strength in the continuum. Magnetic superexchange models are derived characterized by the Dzyaloshinsky-Moriya (DM) interaction 23,24 . In the Mott-insulating phase, single particle condensation is suppressed but the spi...

We consider the quasi-2D two-component Bose-Einstein condensates with Rashba spin-orbit (SO) coupling in a rotating trap. The rotation angular velocity couples to the mechanical angular momentum which contains a non-canonical part arising from SO coupling. The effects of an external Zeeman term favoring spin polarization along the radial direction is also considered, which has the same form as the non-canonical part of the mechanical angular momentum. The rotating condensate exhibits a variety of rich structures as varying the strengths of the trapping potential and interaction. With a strong trapping potential, the condensate exhibits a half-quantum vortex-lattice configuration. Such a configuration is driven to the normal one by introducing the external radial Zeeman field. In the case of a weak trap potential, the condensate exhibits a multi-domain pattern of plane-wave states under the external radial Zeeman field.