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Electric field effects in one-dimensional spin-1/2 $K_1J_1Γ_1Γ_1^\prime K_2J_2$ model with ferromagnetic Kitaev coupling

We perform a systematic study on the effects of electric fields in the Luttinger liquid phase of the one-dimensional spin-$1/2$ $K_1J_1Γ_1Γ_1^\prime K_2J_2$ model in the region of ferromagnetic nearest-neighboring Kitaev coupling. We find that while electric fields along $(1,1,1)$-direction maintain the Luttinger liquid behavior, fields along other directions drive the system to a dimerized state. An estimation is made on how effective a $(1,1,1)$-field is for tuning the Luttinger parameter in real materials. Our work is useful for understanding the effects of electric fields in one-dimensional generalized Kitaev spin models, and provides a starting point for exploring the electric-field-related physics in two dimensions based on a quasi-one-dimensional approach.