Magnetic superexchange couplings in Sr2IrO4

We investigate the magnetic couplings in ${\mathrm{Sr}}₂{\mathrm{IrO}}₄$ in the Mott-insulating picture, combining density-functional theory, dynamical mean-field theory, and many-body perturbation theory. We first determine the form of the ${j}_{\mathrm{eff}}=1/2$ pseudospin via the local-density-approximation $+$ dynamical mean-field theory approach. Next we study the magnetic interactions in the strong-to-intermediate coupling regime. To this end, we calculate the superexchange pseudospin tensors ${\mathrm{\ensuremath{\Gamma}}}₁, {\mathrm{\ensuremath{\Gamma}}}₂$, and ${\mathrm{\ensuremath{\Gamma}}}₃$ up to fourth order and analyze their dependence on the screened Coulomb interaction integrals $U$ and $J$. We show that, due to term cancellations, the experimental nearest-neighbor coupling ${\mathrm{\ensuremath{\Gamma}}}₁$ is reasonably well reproduced for a whole range of realistic $(U,J)$ values. We show that increasing the Hund's rule coupling $J$ (within the window of realistic values) can lead to large fourth-order contributions, which could explain the ferromagnetic next-nearest-neighbor coupling ${\mathrm{\ensuremath{\Gamma}}}₂$ extracted from the spin-wave dispersion. This regime is characterized by a sizable ring exchange $K$. For $(U,J)$ values that yield a Mott insulator with a half-filled ${j}_{\mathrm{eff}}=1/2$ state, however, fourth-order terms remain minor even if the gap is small. For no realistic parameters, we find a sizable next-next-nearest-neighbor coupling ${\mathrm{\ensuremath{\Gamma}}}₃\ensuremath{\sim}|{\mathrm{\ensuremath{\Gamma}}}₂|$. Possible implications are discussed.