Theoretical study of novel superconductivity in Ir oxides with large spin-orbit coupling

Recently, the 5$d$ transition metal oxide Sr$_2$IrO$_4$ has attracted much attention. In this material, three $t_{2g}$ orbitals of Ir atoms are hybridized with each other by the spin-orbit coupling of 5$d$ electrons. As a result of the quantum entanglement of spin and orbital degrees of freedom, an anomalous $J_{\mathrm{eff}}$=$|L-S|=1/2$ state is realized, which causes interesting properties. To clarify the properties of this system, we have studied the ground state of the three-orbital Hubbard model with a spin-orbit coupling term using variational Monte Carlo method. Here, we study the electronic states when carriers are doped in this three-orbital system and discuss the possibility of superconductivity. The obtained ground state phase diagram reveals the antiferromagnetic state, stable around the electron density $n=5$, is destabilized by carrier doping and the ground state turns to be superconducting under a certain condition. Similar to the high-$T_{\mathrm{c}}$ cuprates, a large asymmetry between electron doping ($n>5$) and hole doping ($n<5$) is also observed. Due to the large spin-orbit coupling, the spin is no longer a good quantum number. Instead, the pseudospins form a Cooper pair and a $d_{x^2-y^2}$-wave ``pseudospin-singlet'' superconductivity is realized.