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Spin-orbit (SO) Mott insulators are regarded as a new paradigm of magneticmaterials, whose properties are largely influenced by SO coupling and featuredby highly anisotropic bond-dependent exchange interactions between thespin-orbital entangled Kramers doublets, as typically manifested in $5d$iridates. Here, we propose that a very similar situation can be realized incuprates when the Cu$^{2+}$ ions reside in a tetrahedral environment, like inspinel compounds. Using first-principles electronic structure calculations, weconstruct a realistic model for the diamond lattice of the Cu$^{2+}$ ions inCuAl$_2$O$_4$ and show that the magnetic properties of this compound arelargely controlled by anisotropic compass-type exchange interactions thatdramatically modify the magnetic ground state by lifting the spiral spin-liquiddegeneracy and stabilizing a commensurate single-$\boldsymbol{q}$ spiral.

Realization of anisotropic compass model on the diamond lattice of Cu$^{2+}$ in CuAl$_2$O$_4$