[Paper Review] Magnetic Orders Proximal to the Kitaev Limit in Frustrated Triangular Systems: Application to Ba$_3$IrTi$_2$O$_9$

Andrei Catuneanu, Jeffrey G. Rau+2 more|arXiv (Cornell University)|Jul 22, 2015

Atomic and Subatomic Physics Research4 citations

TL;DR

This paper derives a generalized $j_{\text{eff}}=1/2$ spin model on a triangular lattice including Heisenberg ($J$), Kitaev ($K$), and symmetric off-diagonal ($\Gamma$) exchanges, applying it to Ba$_3$IrTi$_2$O$_9$ via ab-initio calculations. The classical phase diagram predicts a stripy magnetic order ground state, which is confirmed by tight-binding parameter estimates and Monte Carlo simulations, resolving limitations of prior models that omitted $\Gamma$ terms.

ABSTRACT

Frustrated transition metal compounds in which spin-orbit coupling (SOC) and electron correlation work together have attracted much attention recently. In the case of 5$d$ transition metals, where SOC is large, $j_ ext{eff}=1/2$ bands near the Fermi level are thought to encompass the essential physics of the material, potentially leading to a concrete realization of exotic magnetic phases such as the Kitaev spin liquid. Here we derive a spin model on a triangular lattice based on $j_ ext{eff} = 1/2$ pseudospins that interact via antiferromagnetic Heisenberg ($J$) and Kitaev ($K$) exchanges, and crucially, an anisotropic $(Γ)$ exchange. Our classical analysis of the spin model reveals that, in addition to small regions of 120$^\circ$, $\mathbb{Z}_2$ / dual-$\mathbb{Z}_2$ vortex crystal and nematic phases, the stripy and ferromagnetic phases dominate the $J$-$K$-$Γ$ phase diagram. We apply our model to the 5$d$ transition metal compound, Ba$_3$IrTi$_2$O$_9$, in which the Ir$^{4+}$ ions form layered two-dimensional triangular lattices. We compute the band structure and nearest-neighbor hopping parameters using ab-initio calculations. By combining our ab-initio and classical analyses, we predict that Ba$_3$IrTi$_2$O$_9$ has a stripy ordered magnetic ground state.

Motivation & Objective

• To develop a comprehensive spin model for frustrated triangular iridates that includes $\Gamma$ exchange, which is missing in prior Heisenberg-Kitaev models.
• To address the limitations of existing models in describing the magnetic ground state of Ba$_3$IrTi$_2$O$_9$, particularly the absence of $\Gamma$-driven physics.
• To apply ab-initio calculations to extract tight-binding parameters and map the effective spin Hamiltonian to the $J$-$K$-$\Gamma$ phase diagram.
• To predict the magnetic ground state of Ba$_3$IrTi$_2$O$_9$ based on the derived model and ab-initio inputs, resolving discrepancies in prior theoretical predictions.

Proposed method

• Derives a nearest-neighbor spin Hamiltonian on a triangular lattice using $j_{\text{eff}}=1/2$ pseudo-spins, incorporating $J$, $K$, and $\Gamma$ exchanges from first principles.
• Uses the Luttinger-Tisza method and classical Monte Carlo simulations to map the classical magnetic phase diagram across $J$-$K$-$\Gamma$ parameter space.
• Performs ab-initio density functional theory (DFT) calculations to determine the band structure and nearest-neighbor hopping parameters ($t_1$, $t_2$, $t_3$, $t_2'$) in Ba$_3$IrTi$_2$O$_9$.
• Estimates effective exchange couplings ($J$, $K$, $\Gamma$, $D$) from hopping parameters and on-site Coulomb interactions ($U$, $J_H$) using second-order perturbation theory.
• Analyzes the effects of octahedral distortions on spin exchanges, showing splitting of $J_x$, $J_y$, $J_z$ and emergence of Dzyaloshinskii-Moriya interactions.
• Compares the predicted phase with experimental observations and discusses testable signatures such as spin configurations and order parameters.

Experimental results

Research questions

• RQ1 What is the complete magnetic phase diagram of the $j_{\text{eff}}=1/2$ spin model on a triangular lattice including $J$, $K$, and $\Gamma$ terms?
• RQ2 How do octahedral distortions in Ba$_3$IrTi$_2$O$_9$ modify the effective spin exchanges and influence the ground state?
• RQ3 Does the inclusion of $\Gamma$ exchange stabilize a different magnetic order than predicted by the $J$-$K$ model alone?
• RQ4 What is the predicted magnetic ground state of Ba$_3$IrTi$_2$O$_9$ based on ab-initio-derived parameters and the full spin model?
• RQ5 Can the model explain the absence of $\mathbb{Z}_2$ vortex crystal order in Ba$_3$IrTi$_2$O$_9$ despite proximity to the Kitaev limit?

Key findings

• The inclusion of $\Gamma$ exchange significantly alters the phase diagram, stabilizing stripy and ferromagnetic phases as dominant ground states, in contrast to the $J$-$K$ model.
• Ab-initio calculations yield tight-binding parameters: $t_1 = 7.4$ meV, $t_2 = -13$ meV, $t_2' = -32$ meV, $t_3 = -119$ meV, with $U = 2.0$ eV and $J_H/U = 0.2$.
• Effective exchange couplings are estimated as $J^x \simeq 2$ meV, $J^y \simeq 2$ meV, $J^z \simeq 6$ meV, $\Gamma \simeq -2$ meV, and $D \simeq 2$ meV in the distorted case.
• In the ideal octahedral limit ($t_2 = t_2' = -23.5$ meV), the model yields $J \simeq 2$ meV, $K \simeq 4$ meV, and $\Gamma \simeq -2$ meV, confirming the relevance of $\Gamma$ terms.
• The classical phase diagram shows that the $J$-$K$-$\Gamma$ model predicts a stripy magnetic order as the ground state for Ba$_3$IrTi$_2$O$_9$'s parameter regime.
• The model explains the absence of $\mathbb{Z}_2$ vortex crystal order in Ba$_3$IrTi$_2$O$_9$ due to the dominance of $\Gamma$-driven stripy order over Kitaev-induced topological phases.