Co₂/₃Mg₁/₃I₂: Kitaev Spin-Liquid on Triangular Lattice

• Co₂/₃Mg₁/₃I₂ is a van der Waals layered material configured via partial Mg substitution in CoI₂ to realize a triangular-lattice Kitaev spin-liquid candidate.
• The compound exhibits strong bond-dependent ferromagnetic Kitaev exchange and suppressed geometric frustration, as confirmed by DFT, ED, and DMRG studies.
• Its nearly pure J_eff = 1/2 state and tunable exchange parameters suggest that moderate external fields or pressure could drive quantum phase transitions.

Co$_{2/3}$Mg$_{1/3}$I$_2$ is a van der Waals layered transition-metal iodide that has emerged as a leading candidate for realizing the Kitaev quantum spin-liquid (QSL) physics on a triangular lattice. This compound, generated by partial non-magnetic Mg$^{2+}$ substitution in CoI$_2$, is distinguished by its near-ideal combination of strong bond-dependent (Kitaev) exchange, suppressed geometric frustration, and minimal distortion of the magnetic ion environment. These features position Co$_{2/3}$Mg$_{1/3}$I$_2$ at the forefront of materials research targeting the realization of Kitaev spin liquids in 3$d$-ion, two-dimensional platforms (Ma et al., 27 Dec 2025).

1. Crystal Structure and Chemical Composition

Co$_{2/3}$Mg$_{1/3}$I$_2$ crystallizes in the trigonal space group P$\overline{3}$m1 as a van der Waals (vdW) layered material. The structure consists of layers of edge-sharing CoI$_6$ octahedra, where the magnetic Co$^{2+}$ (3$d^7$) and non-magnetic Mg$^{2+}$ ions are distributed on the vertices of a triangular network. The partial substitution involves one third of Co sites replaced by Mg, resulting in the formal composition.

The Mg$^{2+}$ sites (ionic radius $\approx$ 0.72 Å, close to that of Co$^{2+}$ at 0.745 Å) act primarily as non-magnetic "blockers" for third-nearest-neighbor (3NN) Co–Co exchanges. Structural metrics remain effectively unchanged by substitution: the in-plane lattice parameters retain $a = b \approx 3.87$ Å, and the interlayer spacing is $c \approx 7.06$ Å. The CoI$_6$ octahedra thus maintain their nearly regular geometry, and the local ligand field acting on Co centers is only minimally perturbed (Ma et al., 27 Dec 2025).

2. Spin–Orbital Ground State and Single-Ion Physics

Each Co$^{2+}$ ion in Co$_{2/3}$Mg$_{1/3}$I$_2$ resides in an octahedral iodine environment, resulting in a substantial $t_{2g}$–$e_g$ crystal field splitting $\Delta \approx 1.0$ eV. This splitting stabilizes a high-spin $S=3/2$ configuration, as the low-spin alternative would require a crossover threshold $2J_H \approx 1.8$ eV.

The $t_{2g}$ shell is not fully filled, yielding an effective orbital angular momentum $L_{\text{eff}} = 1$. Strong spin–orbit coupling (SOC) with $\lambda \approx 26.6$ meV splits the $L_{\text{eff}}=1$, $S=3/2$ manifold into $J_{\text{eff}} = 1/2, 3/2, 5/2$ multiplets. In Co$_{2/3}$Mg$_{1/3}$I$_2$, the trigonal crystal field distortion is weak ($\delta_{\text{tri}} \approx 4.5$ meV), ensuring that the ground doublet remains a nearly pure $J_{\text{eff}}=1/2$ state, as confirmed by exact diagonalization (approximately 98% purity by weight). The $g$-factor exhibits moderate anisotropy ($g_\parallel \approx 4.6$ in-plane, $g_\perp \approx 3.4$ out-of-plane), but the $J_{\text{eff}}=1/2$ character is robust under these perturbations (Ma et al., 27 Dec 2025).

3. Minimal Effective Hamiltonian and Exchange Mechanisms

Projection onto the $J_{\text{eff}}=1/2$ lowest-energy Kramers doublets yields a minimal effective spin Hamiltonian of the “Kitaev–Heisenberg” type on a triangular lattice:

$$H = \sum_{\langle ij \rangle_1} \Big[ J_1\,\mathbf{S}_i \cdot \mathbf{S}_j + K_1\,S_i^\gamma S_j^\gamma \Big] + J_3 \sum_{\langle\langle\langle ij\rangle\rangle\rangle_3} \mathbf{S}_i \cdot \mathbf{S}_j$$

where:

• $\langle ij \rangle_1$ runs over 1st-nearest-neighbor (1NN) Co–Co bonds,
• $\gamma \in \{x, y, z\}$ marks the three symmetry-inequivalent Kitaev bond types,
• $J_1$ is the isotropic 1NN Heisenberg exchange (AFM),
• $K_1$ is the bond-dependent 1NN Kitaev interaction (FM),
• $J_3$ is the 3NN Heisenberg exchange, responsible for geometric frustration.

Mg substitution selectively disrupts the 3NN Co paths, resulting in a pronounced reduction in $J_3$ without appreciable impact on $J_1$ or $K_1$, thereby isolating the bond-dependent frustration intrinsic to the Kitaev limit (Ma et al., 27 Dec 2025).

4. Numerical Exchange Parameters and Frustration Hierarchy

State-of-the-art density-functional theory (GGA) calculations, combined with maximally-localized Wannier orbital construction and exact diagonalization of multi-orbital Hubbard models, provide the following exchange couplings for Co$_{2/3}$Mg$_{1/3}$I$_2$ (in meV):

Compound	$J_1$	$K_1$	$J_3$	$|K_1/J_1|$	$|K_1/J_3|$
CoI$_2$	+0.63	–4.17	+2.16	6.6	1.9
Co$_{2/3}$Mg$_{1/3}$I$_2$	+0.89	–3.15	+0.53	3.5	5.9

In Co$_{2/3}$Mg$_{1/3}$I$_2$, the ratio $|K_1/J_3| \approx 5.9$ highlights a regime where bond-dependent interactions far outweigh both the nearest-neighbor Heisenberg and geometric frustration terms, an arrangement not achieved in pristine CoI$_2$. The $|K_1/J_1|$ value remains large, furnishing a strong Kitaev (FM) exchange domination on the triangular network (Ma et al., 27 Dec 2025).

5. Computational Methodology

The theoretical investigation employs a combination of first-principles electronic structure calculations, exact diagonalization (ED), and density-matrix renormalization group (DMRG) simulations:

• DFT (GGA) and maximally localized Wannier functions are used to extract the crystal field environments and hopping matrices up to 3NN.
• ED of multi-orbital Hubbard clusters, with explicit Kanamori $U$, Hund’s coupling $J_H$, and SOC, is performed to validate the $J_{\text{eff}}=1/2$ ground state and quantify two-site exchange interactions (via Löwdin projection).
• DMRG simulations on $6 \times 6$ triangular clusters are used to explore the phase diagram of the reduced $J_1$–$K_1$–$J_3$ model, clarifying conditions leading to helical versus spin-liquid magnetic regimes.

These techniques establish a robust, multi-scale characterization of the local, effective, and collective quantum properties of Co$_{2/3}$Mg$_{1/3}$I$_2$ (Ma et al., 27 Dec 2025).

6. Relevance to Kitaev Quantum Spin-Liquid Physics

Co$_{2/3}$Mg$_{1/3}$I$_2$ is notable for combining (i) a robust ferromagnetic Kitaev interaction ($K_1$), (ii) a weak 1NN Heisenberg term ($J_1$), and (iii) a strongly suppressed long-range (3NN) Heisenberg exchange ($J_3$). This interaction hierarchy ($|K_1| \gg |J_1|, |J_3|$) positions the compound in close proximity to the pure Kitaev limit for a triangular-lattice system.

Magnetic structure-factor calculations for the Mg-substituted system display only diffuse correlations, with the absence of sharp peaks, consistent with a proximate QSL phase. The application of moderate external magnetic fields ($\sim$2–4 T) or pressure could induce a transition into a bona fide Kitaev QSL. Thus, Co$_{2/3}$Mg$_{1/3}$I$_2$ emerges as a vdW-layered, $3d$-ion Kitaev candidate with minimal geometric frustration and maximal manifestation of bond-dependent frustration, making it an optimal platform for studies of Kitaev spin-liquid phenomena (Ma et al., 27 Dec 2025).