Theoretical evidence of spin-orbital-entangled $J_{mathbf{eff}}$=1/2 state in the 3$d$ transition metal oxide CuAl$_2$O$_4$

Transition metal oxides exhibit various competing phases and exotic phenomena depending on how their reaction to the rich degeneracy of the $d$-orbital. Large spin-orbit coupling (SOC) reduces this degeneracy in a unique way by providing a spin-orbital-entangled ground state for 4$d$ and 5$d$ transition metal compounds. In particular, the spin-orbital-entangled Kramers doublet, known as the $J_{mathbf{eff}}$=1/2 pseudospin, appears in layered iridates and $alpha$-RuCl$_3$, manifesting a relativistic Mott insulating phase. Such entanglement, however, seems barely attainable in 3$d$ transition metal oxides, where the SOC is small and the orbital angular momentum is easily quenched. From experimental and theoretical evidence, here we report on the CuAl$_2$O$_4$ spinel as the first example of a $J_{mathbf{eff}}$=1/2 Mott insulator in 3$d$ transition metal compounds. Based on the experimental study, including synthesis of the cubic CuAl$_2$O$_4$ single crystal, density functional theory and dynamical mean field theory calculations reveal that the $J_{mathbf{eff}}$=1/2 state survives the competition with an orbital-momentum-quenched $S$=1/2 state. The electron-addition spectra probing unoccupied states are well described by the $j_{mathbf{eff}}$=1/2 hole state, whereas electron-removal spectra have a rich multiplet structure. The fully relativistic entity found in CuAl$_2$O$_4$ provides new insight into the untapped regime where the spin-orbital-entangled Kramers pair coexists with strong electron correlation.