  We theoretically show that the Kitaev interaction generates a novel class of
spin texture in the excitation spectrum of the antiferromagnetic insulator
found in the Kitaev-Heisenberg-$\Gamma$ model. In conducting electronic
systems, there is a series of vortex type of spin texture along the Fermi
surface induced by Rashba and Dresselhaus spin-orbit coupling. Such spin
textures are rarely found in magnetic insulators, since there had been no
systematic ways to control the kinetics of its quasi-particle called magnon
using a magnetic field or spacially asymmetric exchange couplings. Here, we
propose a general framework to explore such spin textures in arbitrary
insulating antiferromagnets. We introduce an analytical method to transform any
complicated Hamiltonian to the simple representation based on pseudo-spin
degrees of freedom. The direction of the pseudo-spin on a Bloch sphere
describes the degree of contributions from the two magnetic sublattices to the
spin moment carried by the magnon. The momentum dependent fictitious "Zeeman
field" determines the direction of the pseudo-spin and thus becomes the control
parameter of the spin texture, which is explicitly described by the original
model parameters. The framework enabled us to clarify the uncovered aspect of
the Kitaev interaction, and further provides a tool to easily design or explore
materials with intriguing magnetic properties. Since these spin textures can be
a source of a pure spin current, the Kitaev materials $A_{2}$PrO$_{3}$ ($A$
=Li, Na) shall become a potential platform of power-saving spintronics devices.
