Interplay of the Jahn--Teller Effect and Spin-Orbit Coupling: The Case of Trigonal Vibronic Modes

We study an interplay between the orbital degeneracy and spin-orbit coupling giving rise to spin-orbital entangled states. As a specific example, we analyze the interaction of electrons occupying triply degenerate single-ion $t_{2g}$ levels with trigonal vibronic modes (the $t\otimes T$ problem). A more general problem of the electron-lattice interaction involving both tetragonal and trigonal vibrations is also considered. It is shown that the result of such interaction crucially depends on the occupation of $t_{2g}$ levels leading to either the suppression or enhancement of the Jahn-Teller effect by the spin-orbit coupling.