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Strong Rashba spin-orbit coupling (SOC) of the two-dimensional electron gas(2DEG) at the oxide interface $\mathrm{LaAlO_{3}/SrTiO_{3}}$ underlies avariety of exotic physics, but its nature is still under debate. We derive aneffective Hamiltonian for the 2DEG at the oxide interface$\mathrm{LaAlO_{3}/SrTiO_{3}}$ and find a different anisotropic Rashba SOC forthe $d_{xz}$ and $d_{yz}$ orbitals. This anisotropic Rashba SOC leads toanisotropic static spin susceptibilities and also distinctive behavior of thespin Hall conductivity. These unique spin responses may be used to determinethe nature of the Rashba SOC experimentally and shed light on the orbitalorigin of the 2DEG.

Spin responses and effective Hamiltonian for the two dimensional electron gas at oxide interface {LaAlO}$_3$/{SrTiO}$_3$