Spin-orbit entanglement and j=1/2 state in CuAl₂O₄

Spin-orbit (SO) Mott insulators are regarded as a new paradigm of magnetic materials, whose properties are largely influenced by the SO coupling and featured by highly anisotropic bond-dependent exchange interactions, as manifested in 4d and 5d systems. We show that a very similar situation can be realized in cuprates, when the Cu²⁺ ions reside in a tetrahedral environment. A special attention will be paid to CuAl₂O₄, which was experimentally found to retain cubic structure and does not show any long-range magnetic order down to T=0.5 K. These are the strong Coulomb correlations and the spin-orbit coupling, which conspire to suppress the Jahn-Teller distortions in CuAl₂O₄. The spin-orbit-entangled j_eff=1/2 state is then naturally realizes in the situation of t_2g⁵ configuration and degenerate t_2g subshell. This in turn explains unusual magnetic properties of CuAl₂O₄. Using first-principles calculations, we construct a realistic spin model and show that the magnetic properties of this compound are largely controlled by anisotropic compass-type exchange interactions that dramatically modify the magnetic ground state by lifting the spiral spin-liquid degeneracy and stabilizing a commensurate single-q spiral.