Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1710.10193 · Physical review. B./Physical review. B

Pseudospin exchange interactions in d7 cobalt compounds: Possible realization of the Kitaev model

Huimei Liu, Giniyat Khaliullin

Published Jan 9, 2018• 279 citations
arXivHTMLPDF

Abstract

The current efforts to find the materials hosting Kitaev model physics have been focused on Mott insulators of ${d}^{5}$ pseudospin-1/2 ions ${\mathrm{Ir}}^{4+}$ and ${\mathrm{Ru}}^{3+}$ with ${t}_{2g}^{5}(S=1/2,L=1)$ electronic configuration. Here we propose that the Kitaev model can be realized in materials based on ${d}^{7}$ ions with ${t}_{2g}^{5}{e}_{g}^{2}(S=3/2,L=1)$ configuration such as ${\mathrm{Co}}^{2+}$, which also host the pseudospin-1/2 magnetism. Considering possible exchange processes, we have derived the ${d}^{7}$ pseudospin-1/2 interactions in ${90}^{\ensuremath{\circ}}$ bonding geometry. The obtained Hamiltonian comprises the bond-directional Kitaev $K$ and isotropic Heisenberg $J$ interactions as in the case of ${d}^{5}$ ions. However, we find that the presence of additional spin-active ${e}_{g}$ electrons radically changes the balance between Kitaev and Heisenberg couplings. Most remarkably, we show that the exchange processes involving ${e}_{g}$ spins are highly sensitive to whether the system is in Mott ($U<\mathrm{\ensuremath{\Delta}}$) or charge-transfer ($U>\mathrm{\ensuremath{\Delta}}$) insulating regime. In the latter case, to which many cobalt compounds do actually belong, the antiferromagnetic Heisenberg coupling $J$ is strongly suppressed and spin-liquid phase can be stabilized. The results suggest cobalt-based materials as promising candidates for the realization of the Kitaev model.

DOI
10.1103/physrevb.97.014407

Related papers

Spin interaction and magnetism in cobaltate Kitaev candidate materials: An ab initio and model Hamiltonian approach

Shishir Kumar Pandey, Ji Feng

In the quest for materials hosting Kitaev spin liquids, much of the efforts have been focused on the fourth- and fifth-row transition metal compounds, which are spin-orbit coupling--assisted Mott insulators. Here we study the structural and magnetic properties of $3d$ transition metal oxides, ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$ and ${\mathrm{Na}}_{3}{\mathrm{Co}}_{2}{\mathrm{SbO}}_{6}$. The partial occupancy of sodium in former compound is addressed using a cluster expansion, and a honeycomb lattice of sodiums is found to be energetically favored. Starting from the ab initio band structures, a many-body second-order perturbation theory leads to a pseudospin-$\frac{1}{2}$ Hamiltonian with estimated magnetic interactions. We show that the experimentally observed zigzag magnetic state is stabilized only when the first-neighbor Kitaev coupling dominates over the Heisenberg term, both of which are highly suppressed due to presence of ${e}_{g}$ orbitals. A third-neighbor Heisenberg interaction is found dominant in both these compounds. We also present a phase diagram for ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$ by varying the electron-electron and spin-orbit interactions. The computed spin excitation spectra are found to capture essential features of recent experimental magnon spectrum, lending support to our results.

arXiv:2205.03836•Physical review. B./Physical review. B•Nov 14, 2022•18 citations

Kitaev-Heisenberg Hamiltonian for high-spin d7 Mott insulators

Ryoya Sano, Yasuyuki Kato, Yukitoshi Motome

In the search for quantum spin liquids, candidate materials for the Kitaev model and its extensions have been intensively explored during the past decade, as the models realize the exact quantum spin liquids in the ground state. Thus far, insulating magnets in the low-spin ${d}^{5}$ electron configuration under the strong spin-orbit coupling have been studied for realizing the Kitaev-type bond-dependent anisotropic interactions between the spin-orbital entangled Kramers doublets. To extend the candidates, here we investigate the systems in a high-spin ${d}^{7}$ electron configuration, whose ground state is described by the spin-orbital entangled Kramers doublet. By the second-order perturbation in terms of the ${t}_{2g}\ensuremath{-}{t}_{2g}$ and ${t}_{2g}\ensuremath{-}{e}_{g}$ hoppings, we show that the effective spin model possesses the anisotropic Kitaev interactions as well as the isotropic Heisenberg ones. While the Kitaev interaction is always ferromagnetic, the Heisenberg interaction can become either ferromagnetic or antiferromagnetic depending on the Coulomb interactions and the crystalline electric fields. We also derive the effective model for the low-spin ${d}^{5}$ electron configuration within the same perturbation scheme, in which the Kitaev interaction becomes both ferromagnetic and antiferromagnetic, while the Heisenberg one is always ferromagnetic. Referring to the previous study for the Kitaev-Heisenberg model, we find that the quantum spin liquid phase exists in the reasonable parameter region in both ${d}^{7}$ and ${d}^{5}$ cases, while the former has a richer structure of the phase diagram. We discuss the advantages of the ${d}^{7}$ case in comparison with the ${d}^{5}$ case. Our results indicate that the high-spin ${d}^{7}$ state provides another platform for the Kitaev-type quantum spin liquid.

arXiv:1710.11357•Physical review. B./Physical review. B•Jan 9, 2018•215 citations

Can we realize Kitaev quantum spin liquids in the $J_\text{eff}=1/2$ honeycomb cobaltates?

Shreya Das, Sreekar Voleti, Tanusri Saha‐Dasgupta +1 more

The quest for Kitaev quantum spin liquids has led to great interest in honeycomb quantum magnets with strong spin-orbit coupling. It has been recently proposed that even Mott insulators with $3d$ transition metal ions, having nominally weak spin-orbit coupling, can realize such exotic physics. Motivated by this, we study the rhombohedral honeycomb cobaltates CoTiO$_3$, BaCo$_2$(PO$_4$)$_2$, and BaCo$_2$(AsO$_4$)$_2$, using $\textit{ab initio}$ density functional theory, which takes into account realistic crystal field distortions and chemical information, in conjunction with exact diagonalization numerics. We show that these Co$^{2+}$ magnets host $j_\text{eff}=1/2$ local moments with highly anisotropic $g$-factors, and we extract their full spin Hamiltonians including longer-range and anisotropic exchange couplings. For CoTiO$_3$, we find a nearest-neighbor easy-plane ferromagnetic $XXZ$ model with additional bond-dependent anisotropies and interlayer exchange, which supports three-dimensional (3D) Dirac nodal line magnons. In contrast, for BaCo$_2$(PO$_4$)$_2$ and BaCo$_2$(AsO$_4$)$_2$, we find a strongly suppressed interlayer coupling, and significant frustration from additional third-neighbor antiferromagnetic exchange mediated by P/As. Such bond-anisotropic $J_1$-$J_3$ spin models can support collinear zig-zag or coplanar spiral ground states; we discuss their dynamical spin correlations which reveal a gapped Goldstone mode, and argue that the effective parameters of these pseudospin-$1/2$ models may be strongly renormalized by coupling to a low energy spin-exciton. Our results call for re-examining proposals for realizing Kitaev spin liquids in the honeycomb cobaltates.

arXiv:2106.11982•Jun 22, 2021•0 citations

Antiferromagnetic Kitaev interactions in polar spin-orbit Mott insulators

Yusuke Sugita, Yasuyuki Kato, Yukitoshi Motome

A bond-directional anisotropic exchange interaction, called the Kitaev interaction, is a promising route to realize quantum spin liquids. The Kitaev interactions were found in Mott insulators with the strong spin-orbit coupling, in the presence of quantum interference between indirect electron transfers. Here we theoretically propose a different scenario by introducing a polar structural asymmetry that unbalances the quantum interference. We show that the imbalance activates additional exchange processes and gives rise to a dominant antiferromagnetic Kitaev interaction, in stark contrast to the conventional ferromagnetic ones. We demonstrate by ab initio calculations that polar Ru trihalides with multiple anions, $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuH}}_{3/2}{X}_{3/2}$ ($X=\text{Cl}$ and Br), exhibit dominant antiferromagnetic Kitaev interactions by this mechanism. Our proposal opens the way for materializing the Kitaev spin liquids in unexplored parameter regions.

arXiv:1905.12139•Physical review. B./Physical review. B•Mar 31, 2020•20 citations

Impact of gapped spin-orbit excitons on low-energy pseudospin exchange interactions

Sreekar Voleti, F. David Wandler, Arun Paramekanti

The quest for exotic quantum magnetic ground states, including the Kitaev spin liquid and quantum spin ices, has led to the discovery of several quantum materials where low-energy pseudospin-$1/2$ doublets arise from the splitting of spin-orbit entangled multiplets with higher degeneracy. Such systems include $d$-orbital and $f$-orbital Mott insulators. When the gap between the low-energy pseudospin-$1/2$ levels and the excited levels of the multiplet or ``excitons'' is not large, the effective low-energy exchange interactions between the low-energy pseudospin-$1/2$ moments can acquire significant corrections from coupling to the excitons. We extract these corrections using an extended perturbation theory that takes into account the most relevant higher-order perturbations. Such corrections can impact the exchange matrix for the low-energy pseudospin-$1/2$ levels by renormalizing the strength and the sign of Heisenberg exchange or Ising anisotropies, and potentially even inducing bond-anisotropic couplings such as Kitaev-$\mathrm{\ensuremath{\Gamma}}$ exchange interactions. We discuss recent experiments on various cobaltate and osmate materials, which hint at the ubiquity and importance of this physics. Our study suggests a distinct direction in the search for exotic spin liquids---quantum pseudospin-1/2 systems with low-energy spin-orbit excitons.

arXiv:2303.04169•Physical review. B./Physical review. B•Mar 18, 2024•3 citations

Non-Kitaev versus Kitaev honeycomb cobaltates

Xiaoyu Liu, Hae‐Young Kee

Recently, honeycomb cobaltates with $3{d}^{7}$ were proposed to display Kitaev physics despite weak spin-orbit coupling. However, other theoretical and experimental works found leading XXZ Heisenberg and negligible Kitaev interactions in ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$ (BCAO), which calls for further study to clarify the origin of the discrepancies. Here we derive the analytical expressions of the spin model using strong-coupling perturbation theory. With tight-binding parameters obtained with ab initio calculations for idealized honeycomb BCAO, we find that the largest intraorbital ${t}_{2g}\ensuremath{-}{t}_{2g}$ exchange path, which was assumed to be small in the earlier theory proposal, leads to a ferromagnetic (FM) Heisenberg interaction. This becomes the dominant interaction, as other ${t}_{2g}\ensuremath{-}{e}_{g}$ and ${e}_{g}\ensuremath{-}{e}_{g}$ contributions almost cancel each other. Exactly the same assumed-to-be-small channel also generates an antiferromagnetic Kitaev interaction, which then cancels the FM Kitaev interaction from ${t}_{2g}\ensuremath{-}{e}_{g}$ paths, resulting in a small Kitaev interaction. Under the trigonal distortion, the preeminent isotropic Heisenberg becomes an anisotropic XXZ model, as expected, which is the case in BCAO. However, when ${t}_{2g}\ensuremath{-}{e}_{g}$ and intraorbital ${t}_{2g}\ensuremath{-}{t}_{2g}$ hoppings are similar in size such as in ${\mathrm{Na}}_{3}{\mathrm{Co}}_{2}{\mathrm{SbO}}_{6}$, the Kitaev interaction may become comparable to the Heisenberg interaction. A way to achieve the Kitaev cobaltates is also discussed.

arXiv:2211.03737•Physical review. B./Physical review. B•Feb 16, 2023•45 citations
The current efforts to find the materials hosting Kitaev model physics have been focused on Mott insulators of d5{d}^{5}d5 pseudospin-1/2 ions Ir4+{\mathrm{Ir}}^{4+}Ir4+ and Ru3+{\mathrm{Ru}}^{3+}Ru3+ with t2g5(S=1/2,L=1){t}_{2g}^{5}(S=1/2,L=1)t2g5​(S=1/2,L=1) electronic configuration. Here we propose that the Kitaev model can be realized in materials based on d7{d}^{7}d7 ions with t2g5eg2(S=3/2,L=1){t}_{2g}^{5}{e}_{g}^{2}(S=3/2,L=1)t2g5​eg2​(S=3/2,L=1) configuration such as Co2+{\mathrm{Co}}^{2+}Co2+, which also host the pseudospin-1/2 magnetism. Considering possible exchange processes, we have derived the d7{d}^{7}d7 pseudospin-1/2 interactions in 90\ensuremath∘{90}^{\ensuremath{\circ}}90\ensuremath∘ bonding geometry. The obtained Hamiltonian comprises the bond-directional Kitaev KKK and isotropic Heisenberg JJJ interactions as in the case of d5{d}^{5}d5 ions. However, we find that the presence of additional spin-active eg{e}_{g}eg​ electrons radically changes the balance between Kitaev and Heisenberg couplings. Most remarkably, we show that the exchange processes involving eg{e}_{g}eg​ spins are highly sensitive to whether the system is in Mott (U<\ensuremathΔU<\mathrm{\ensuremath{\Delta}}U<\ensuremathΔ) or charge-transfer (U>\ensuremathΔU>\mathrm{\ensuremath{\Delta}}U>\ensuremathΔ) insulating regime. In the latter case, to which many cobalt compounds do actually belong, the antiferromagnetic Heisenberg coupling JJJ is strongly suppressed and spin-liquid phase can be stabilized. The results suggest cobalt-based materials as promising candidates for the realization of the Kitaev model.
In the quest for materials hosting Kitaev spin liquids, much of the efforts have been focused on the fourth- and fifth-row transition metal compounds, which are spin-orbit coupling--assisted Mott insulators. Here we study the structural and magnetic properties of 3d3d3d transition metal oxides, Na2Co2TeO6{\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}Na2​Co2​TeO6​ and Na3Co2SbO6{\mathrm{Na}}_{3}{\mathrm{Co}}_{2}{\mathrm{SbO}}_{6}Na3​Co2​SbO6​. The partial occupancy of sodium in former compound is addressed using a cluster expansion, and a honeycomb lattice of sodiums is found to be energetically favored. Starting from the ab initio band structures, a many-body second-order perturbation theory leads to a pseudospin-12\frac{1}{2}21​ Hamiltonian with estimated magnetic interactions. We show that the experimentally observed zigzag magnetic state is stabilized only when the first-neighbor Kitaev coupling dominates over the Heisenberg term, both of which are highly suppressed due to presence of eg{e}_{g}eg​ orbitals. A third-neighbor Heisenberg interaction is found dominant in both these compounds. We also present a phase diagram for Na2Co2TeO6{\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}Na2​Co2​TeO6​ by varying the electron-electron and spin-orbit interactions. The computed spin excitation spectra are found to capture essential features of recent experimental magnon spectrum, lending support to our results.
In the search for quantum spin liquids, candidate materials for the Kitaev model and its extensions have been intensively explored during the past decade, as the models realize the exact quantum spin liquids in the ground state. Thus far, insulating magnets in the low-spin d5{d}^{5}d5 electron configuration under the strong spin-orbit coupling have been studied for realizing the Kitaev-type bond-dependent anisotropic interactions between the spin-orbital entangled Kramers doublets. To extend the candidates, here we investigate the systems in a high-spin d7{d}^{7}d7 electron configuration, whose ground state is described by the spin-orbital entangled Kramers doublet. By the second-order perturbation in terms of the ${t}_{2g}\ensuremath{-}{t}_{2g}$ and ${t}_{2g}\ensuremath{-}{e}_{g}$ hoppings, we show that the effective spin model possesses the anisotropic Kitaev interactions as well as the isotropic Heisenberg ones. While the Kitaev interaction is always ferromagnetic, the Heisenberg interaction can become either ferromagnetic or antiferromagnetic depending on the Coulomb interactions and the crystalline electric fields. We also derive the effective model for the low-spin d5{d}^{5}d5 electron configuration within the same perturbation scheme, in which the Kitaev interaction becomes both ferromagnetic and antiferromagnetic, while the Heisenberg one is always ferromagnetic. Referring to the previous study for the Kitaev-Heisenberg model, we find that the quantum spin liquid phase exists in the reasonable parameter region in both d7{d}^{7}d7 and d5{d}^{5}d5 cases, while the former has a richer structure of the phase diagram. We discuss the advantages of the d7{d}^{7}d7 case in comparison with the d5{d}^{5}d5 case. Our results indicate that the high-spin d7{d}^{7}d7 state provides another platform for the Kitaev-type quantum spin liquid.
Can we realize Kitaev quantum spin liquids in the Jeff=1/2J_\text{eff}=1/2Jeff​=1/2 honeycomb cobaltates?
The quest for Kitaev quantum spin liquids has led to great interest in honeycomb quantum magnets with strong spin-orbit coupling. It has been recently proposed that even Mott insulators with 3d3d3d transition metal ions, having nominally weak spin-orbit coupling, can realize such exotic physics. Motivated by this, we study the rhombohedral honeycomb cobaltates CoTiO3_33​, BaCo2_22​(PO4_44​)2_22​, and BaCo2_22​(AsO4_44​)2_22​, using ab initio\textit{ab initio}ab initio density functional theory, which takes into account realistic crystal field distortions and chemical information, in conjunction with exact diagonalization numerics. We show that these Co2+^{2+}2+ magnets host jeff=1/2j_\text{eff}=1/2jeff​=1/2 local moments with highly anisotropic ggg-factors, and we extract their full spin Hamiltonians including longer-range and anisotropic exchange couplings. For CoTiO3_33​, we find a nearest-neighbor easy-plane ferromagnetic XXZXXZXXZ model with additional bond-dependent anisotropies and interlayer exchange, which supports three-dimensional (3D) Dirac nodal line magnons. In contrast, for BaCo2_22​(PO4_44​)2_22​ and BaCo2_22​(AsO4_44​)2_22​, we find a strongly suppressed interlayer coupling, and significant frustration from additional third-neighbor antiferromagnetic exchange mediated by P/As. Such bond-anisotropic J1J_1J1​-J3J_3J3​ spin models can support collinear zig-zag or coplanar spiral ground states; we discuss their dynamical spin correlations which reveal a gapped Goldstone mode, and argue that the effective parameters of these pseudospin-1/21/21/2 models may be strongly renormalized by coupling to a low energy spin-exciton. Our results call for re-examining proposals for realizing Kitaev spin liquids in the honeycomb cobaltates.
A bond-directional anisotropic exchange interaction, called the Kitaev interaction, is a promising route to realize quantum spin liquids. The Kitaev interactions were found in Mott insulators with the strong spin-orbit coupling, in the presence of quantum interference between indirect electron transfers. Here we theoretically propose a different scenario by introducing a polar structural asymmetry that unbalances the quantum interference. We show that the imbalance activates additional exchange processes and gives rise to a dominant antiferromagnetic Kitaev interaction, in stark contrast to the conventional ferromagnetic ones. We demonstrate by ab initio calculations that polar Ru trihalides with multiple anions, $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuH}}_{3/2}{X}_{3/2}$ (X=ClX=\text{Cl}X=Cl and Br), exhibit dominant antiferromagnetic Kitaev interactions by this mechanism. Our proposal opens the way for materializing the Kitaev spin liquids in unexplored parameter regions.
The quest for exotic quantum magnetic ground states, including the Kitaev spin liquid and quantum spin ices, has led to the discovery of several quantum materials where low-energy pseudospin-1/21/21/2 doublets arise from the splitting of spin-orbit entangled multiplets with higher degeneracy. Such systems include ddd-orbital and fff-orbital Mott insulators. When the gap between the low-energy pseudospin-1/21/21/2 levels and the excited levels of the multiplet or ``excitons'' is not large, the effective low-energy exchange interactions between the low-energy pseudospin-1/21/21/2 moments can acquire significant corrections from coupling to the excitons. We extract these corrections using an extended perturbation theory that takes into account the most relevant higher-order perturbations. Such corrections can impact the exchange matrix for the low-energy pseudospin-1/21/21/2 levels by renormalizing the strength and the sign of Heisenberg exchange or Ising anisotropies, and potentially even inducing bond-anisotropic couplings such as Kitaev-$\mathrm{\ensuremath{\Gamma}}$ exchange interactions. We discuss recent experiments on various cobaltate and osmate materials, which hint at the ubiquity and importance of this physics. Our study suggests a distinct direction in the search for exotic spin liquids---quantum pseudospin-1/2 systems with low-energy spin-orbit excitons.
Recently, honeycomb cobaltates with 3d73{d}^{7}3d7 were proposed to display Kitaev physics despite weak spin-orbit coupling. However, other theoretical and experimental works found leading XXZ Heisenberg and negligible Kitaev interactions in BaCo2(AsO4)2{\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}BaCo2​(AsO4​)2​ (BCAO), which calls for further study to clarify the origin of the discrepancies. Here we derive the analytical expressions of the spin model using strong-coupling perturbation theory. With tight-binding parameters obtained with ab initio calculations for idealized honeycomb BCAO, we find that the largest intraorbital ${t}_{2g}\ensuremath{-}{t}_{2g}$ exchange path, which was assumed to be small in the earlier theory proposal, leads to a ferromagnetic (FM) Heisenberg interaction. This becomes the dominant interaction, as other ${t}_{2g}\ensuremath{-}{e}_{g}$ and ${e}_{g}\ensuremath{-}{e}_{g}$ contributions almost cancel each other. Exactly the same assumed-to-be-small channel also generates an antiferromagnetic Kitaev interaction, which then cancels the FM Kitaev interaction from ${t}_{2g}\ensuremath{-}{e}_{g}$ paths, resulting in a small Kitaev interaction. Under the trigonal distortion, the preeminent isotropic Heisenberg becomes an anisotropic XXZ model, as expected, which is the case in BCAO. However, when ${t}_{2g}\ensuremath{-}{e}_{g}$ and intraorbital ${t}_{2g}\ensuremath{-}{t}_{2g}$ hoppings are similar in size such as in Na3Co2SbO6{\mathrm{Na}}_{3}{\mathrm{Co}}_{2}{\mathrm{SbO}}_{6}Na3​Co2​SbO6​, the Kitaev interaction may become comparable to the Heisenberg interaction. A way to achieve the Kitaev cobaltates is also discussed.