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Vector chirality for effective total momentum $J_{\textrm{eff}}$ in a nonfrustrated Mott insulator: Effects of strong spin-orbit coupling and broken inversion symmetry

I propose the emergence of the spin-orbital-coupled vector chirality in a non-frustrated Mott insulator with the strong spin-orbit coupling due to $ab$-plane's inversion-symmetry (IS) breaking. I derive the superexchange interactions for a $t_{2g}$-orbital Hubbard model on a square lattice with the strong spin-orbit coupling and the IS-breaking-induced hopping integrals, and explain the microscopic origins of the Dzyaloshinsky-Moriya (DM) -type and the Kitaev-type interactions. Then, by adopting the mean-field approximation to a minimal model including only the Heisenberg-type and the DM-type nearest-neighbor interactions, I show that the IS breaking causes the spin-orbital-coupled chirality as a result of stabilizing the screw state. I also highlight the limit of the hard-pseudospin approximation in discussing the stability of the screw states in the presence of both the DM-type and the Kitaev-type interactions, and discuss its meaning. I finally discuss the effects of tetragonal crystal field and $J_{\textrm{eff}}=\frac{3}{2}$ states, and the application to the iridates near the $[001]$ surface of Sr$_{2}$IrO$_{4}$ and the interface between Sr$_{2}$IrO$_{4}$ and Sr$_{3}$Ir$_{2}$O$_{7}$.