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We formulate a Majorana mean-field theory for the extended $JK\Gamma$ Kitaevmodel in a magnetic Zeeman field of arbitrary direction, and apply it forstudying spatially inhomogeneous states harboring vortices. This mean-fieldtheory is exact in the pure Kitaev limit and captures the essential physicsthroughout the Kitaev spin liquid phase. We determine the charge profile aroundvortices and the corresponding quadrupole tensor. The quadrupole-quadrupoleinteraction between distant vortices is shown to be either repulsive orattractive, depending on parameters. We predict that electrically biasedscanning probe tips enable the creation of vortices at preselected positions.Our results open new perspectives for the electric manipulation of Ising anyonsin Kitaev spin liquids.

Electric polarization near vortices in the extended Kitaev model