## (a) Dependence of δ on η with κ = 2.0

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**Figure 3.** (a) Dependence of δ on η with κ = 2.0. (b) Dependence of δ on κ with η = 0.0096. The black dotted line corresponds to the case when only the SO part of the mean-field energy is minimized. The solid red line corresponds to the case of counting in both the SO and the external potential parts of the mean-field energy. The units of δ and κ are a_{\perp }=\sqrt{\hbar /M\omega } and \sqrt{\hbar M\omega }, respectively.

**Abstract**

We analytically and numerically investigate the ground state of spin–orbit coupled spin-1 Bose–Einstein condensates in an external parabolic potential. When the spin–orbit coupling is introduced, spatial displacement exists between the atom densities of components with different magnetic quantum numbers. The analytical calculations show this displacement reaches a maximum when the spin–orbit coupling strength is comparable with that of the trapping potential. As the spin–orbit coupling strength gets larger and larger, the spatial displacement decreases at a rate inversely proportional to the spin–orbit coupling strength. Correspondingly, periphery half-skyrmion textures arise; this displacement can be reflected by the non-uniform magnetic moment in the *z* direction. With the manipulation of the external trap, the local magnitude of the non-uniform magnetic moment can be increased evidently. This kind of increase of the local magnetic moment is also observed in the square vortex lattice phase of the condensate.