Microscopic Mechanism for a Higher-Spin Kitaev Model

The spin $S=\frac{1}{2}$ Kitaev honeycomb model has attracted significant attention since emerging candidate materials have provided a playground to test non-Abelian anyons. The Kitaev model with higher spins has also been theoretically studied, as it may offer another path to a quantum spin liquid. However, a microscopic route to achieve higher spin Kitaev models in solid state materials has not been rigorously derived. Here we present a theory of the spin $S=1$ Kitaev interaction in two-dimensional edge-shared octahedral systems. Essential ingredients are strong spin-orbit coupling in anions and strong Hund’s coupling in transition metal cations. The $S=1$ Kitaev and ferromagnetic Heisenberg interactions are generated from superexchange paths. Taking into account the antiferromagnetic Heisenberg term from direct-exchange paths, the Kitaev interaction dominates the physics of the $S=1$ system. Using an exact diagonalization technique, we show a finite regime of $S=1$ spin liquid in the presence of the Heisenberg interaction. Candidate materials are proposed, and generalization to higher spins is discussed.

Figure 1

Indirect hopping integrals between $M$ and $A$ sites are denoted by the colored curved lines. The red, green, and blue colors represent $t_1$, $t_2$, and $t_3$, respectively, and the sign of the hopping integrals is ignored for simplicity. The $M$ sites with $e_g$ orbitals are located in the center of each octahedral cage formed by $A$ sites occupied by three $p$ orbitals. The Kitaev bond-dependent interactions $X$, $Y$, and $Z$ bonds are, respectively, represented by the red, green, and blue shaded regions. For clarity, every $A$ site is drawn by two separated $A$ sites to represent different hopping contributions from different $p$ and $e_g$ orbitals. The global coordinates of the $x$, $y$, and $z$ axes are shown in the center of the honeycomb plane.

Figure 2

(a) The phase diagram of the $S=1$ KH model. By tuning the ratio of $J/K$, two transitions signaled by the singular behavior of first (blue) and second (red) derivative of the ground state energy density $u_{\mathrm{GS}}$ are found on both the 12- and 18-site ED clusters shown in (b) and (c), respectively. Energy density units are $\sqrt{J^2+K^2}/N$. There are three phases identified by spin-spin correlators as discussed in the main text. The Kitaev spin liquid (SL) appears near $J/K\sim 0$, and AFM and ZZ orderings are, respectively, found in the antiferromagnetic and ferromagnetic Heisenberg interaction regions.