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

Spin-orbit (SO) Mott insulators are regarded as a new paradigm of magnetic materials, whose properties are largely influenced by SO coupling and featured by highly anisotropic bond-dependent exchange interactions between the spin-orbital entangled Kramers doublets, as typically manifested in $5d$ iridates. Here, we propose that a very similar situation can be realized in cuprates when the Cu${}^{2+}$ ions reside in a tetrahedral environment, like in spinel compounds. Using first-principles electronic structure calculations, we construct a realistic model for the diamond lattice of the Cu${}^{2+}$ ions in CuAl${}_2$O${}_4$ 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-$\mathit{q}$ spiral.