Anisotropic exchange interactions of spin-orbit-integrated states in ${\text{Sr}}_2{\text{IrO}}_4$

We present a microscopic model for the anisotropic exchange interactions in ${\text{Sr}}_2{\text{IrO}}_4$. A direct construction of Wannier functions from first-principles calculations proves the $j_{\text{eff}}=1/2$ character of the spin-orbit integrated states at the Fermi level. An effective $j_{\text{eff}}$-spin Hamiltonian explains the observed weak ferromagnetism and anisotropy of antiferromagnetically ordered magnetic state, which arise naturally from the $j_{\text{eff}}=1/2$ state with a rotation of ${\text{IrO}}_6$ octahedra. It is suggested that ${\text{Sr}}_2{\text{IrO}}_4$ is a unique class of materials with effective exchange interactions in the spin-orbital Hilbert space.

Figure 1

(Color online) Calculated Wannier functions of the $|j_{\text{eff}}=1/2,+1/2⟩$ state: (a) the real part of up-spin $|\uparrow ⟩$ component, (b) the real part of the down-spin $|\downarrow ⟩$ component, and (c) the imaginary part of the down-spin $|\downarrow ⟩$ component. Blue (dark gray) and yellow (light gray) colors in the Wannier function represent negative and positive values, respectively. The inset shows the $\text{LDA}+\text{SO}+U$ band structure near $E_{\text{F}}=0\text{eV}$, emphasizing the upper and lower Hubbard bands of the $|j_{\text{eff}}=1/2,m⟩$ band above and below $E_{\text{F}}$, respectively.

Figure 2

(Color online) (a) Tight-binding band structure with $\left({\lambda }_{\text{SO}},U\right)=\left(0.4\text{eV},2.0\text{eV}\right)$, which is well compared with the $\text{LDA}+\text{SO}+U$ band structure of Figs. 1c, 1b the decomposition of the upper Hubbard band wave function [marked by a thin (red) solid line in (a)] into the $\left\{|j_{\text{eff}},m_j⟩\right\}$ basis, where the dotted line represents a $|\frac{1}{2},\frac{1}{2}⟩$ component with $U=0$, and (c) a schematic energy diagram of the $t_{2g}$ manifold in the atomic limit. Due to the large crystal field splitting by ${\Delta }_c$, the $t_{2g}$ levels can be mapped into the effective $l_{\text{eff}}=1$ states where the tetragonal field splitting by ${\Delta }_t$ is relatively insignificant.

Figure 3

(Color online) (a) Magnetic configuration and (b) DM vectors of the calculated $\text{LDA}+\text{SO}+U$ ground state of ${\text{Sr}}_2{\text{IrO}}_4$. Blue (gray) arrows in (a) represent a noncollinear ordering of the local Ir moments, consisting of both spin and orbital components, in a canted AFM configuration. The DM vectors, ${\mathbf{D}}_{AB}$ and ${\mathbf{D}}_{BC}$, in (b) are aligned along the $c$ axis with alternating signs and consistent with the DM rule.