Spin Hamiltonian of hyper-kagome ${\text{Na}}_4{\text{Ir}}_3{\text{O}}_8$

We derive the spin Hamiltonian for the quantum spin liquid ${\text{Na}}_4{\text{Ir}}_3{\text{O}}_8$, and then estimate the direct and superexchange contributions between near neighbor iridium ions using a tight-binding parametrization of the electronic structure. We find a magnitude of the exchange interaction comparable to experiment for a reasonable value of the on-site Coulomb repulsion. For one of the two tight-binding parametrizations we have studied, the direct exchange term, which is isotropic, dominates the total exchange. This provides support for those theories proposed to describe this quantum spin liquid that assume an isotropic Heisenberg model.

Figure 1

(Color online) Superexchange pathway between two Ir ions as marked by the arrows. The Ir ions are the large (red) spheres and the oxygen ions the small ones. The lower left (green) pathway is via an O2 (blue) ion, the upper right (purple) pathway via an O1 (brown) ion.

Figure 2

Energy bands ($t_{2g}$ spin-orbit doublet) near the Fermi energy $\left(E_F\right)$ from the 21-parameter tight-binding fit to the electronic structure of ${\text{Na}}_4{\text{Ir}}_3{\text{O}}_8$. The horizontal line marks $E_F$.

Figure 3

(Color online) Diagonal elements of the exchange interaction as a function of the on-site repulsion $\left(U_d\right)$ on the iridium sites, assuming zero repulsion $\left(U_p\right)$ on the oxygen sites; (a) is from the 18-parameter tight-binding fit and (b) from the 21-parameter fit. The horizontal dashed line is the experimental value for $J$ (Ref. 1).

Figure 4

(Color online) Diagonal elements of the exchange interaction as a function of the on-site repulsion $\left(U_p\right)$ on the oxygen sites, with a repulsion $\left(U_d\right)$ of 1.5 eV on the iridium sites; (a) is from the 18-parameter tight-binding fit and (b) from the 21-parameter fit. The horizontal dashed line is the experimental value for $J$ (Ref. 1).