Odd-parity triplet superconductivity in multi-orbital materials with strong spin-orbit coupling: applications to doped Sr2IrO4

We explore possible superconducting states in $t_{2g}$ multi-orbital correlated electron systems with strong spin-orbit coupling (SOC). In order to study such systems in a controlled manner, we employ large-scale dynamical mean-field theory (DMFT) simulations with the hybridization expansion continuous-time Quantum Monte Carlo (CTQMC) impurity solver. To determine the pairing symmetry, we go beyond the local DMFT formalism using parquet equations to introduce the momentum dependence in the two-particle vertex and correlation functions. In the strong SOC limit, a singlet, $d$-wave pairing state in the electron-doped side of the phase diagram is observed at weak Hund's coupling, which is triggered by antiferromagnetic fluctuations. When the Hund's coupling is comparable to SOC, a two-fold degenerate, triplet $p$-wave pairing state with relatively high $T_c$ emerges in the hole-doped side of the phase diagram, which is associated with enhanced charge fluctuations. Experimental implications to doped Sr$_{2}$IrO$_{4}$ are discussed.