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

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 spindependent 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.

Orientational order parameters for arbitrary quantum systems

“…( 12) with the singleparticle kernel (17) gives the many-particle order parameter as a normalized sum over the single-particle order parameters. These definitions agree, for s = 1, up to a normalization with the standard definitions of the polarization [64] and nematic tensor [36,107,110]. They are usually obtained by a state multipole expansion of the density operator [64], a procedure that is less general than the formalism discussed here, since it is restricted to angular momenta.…”

“…[118,119] for explicit forms). Nematic order in quantum molecular systems can therefore not be described with the usual methods, such as the construction of symmetric traceless tensors used for Fermi liquids [62] and spins [110]. The typical approach for the definition of order parameters for such molecular systems is through angular averages [64], although Cartesian order parameters are more closely linked to experiments [19,20].…”

Orientational Order Parameters for Arbitrary Quantum Systems

“…[20]. The tensor operators (33) can be expressed in terms of the spin operator ⃗ S. [64,[108][109][110][111][112] The transformation rules (for sharp angular momentum s) read [64] T (s)…”

“…These definitions agree, for s = 1, up to a normalization with the standard definitions of the polarization [64] and nematic tensor. [36,109,112] They are usually obtained by a state multipole expansion of the density operator, [64] a procedure that is less general than the formalism discussed here, since it is restricted to angular momenta. Thus, we can recover the definitions used in the literature, but also generalize them toward larger spins.…”

“…Nematic order in quantum molecular systems can therefore not be described with the usual methods, such as the construction of symmetric traceless tensors used for Fermi liquids [62] and spins. [112] The typical approach for the definition of order parameters for such molecular systems is through an angular expansions of the orientational distribution function of the molecular ensemble. For general nonlinear molecules, this function depends on the three Euler angles.…”