arXiv:2211.03737 · Physical review. B./Physical review. B

Non-Kitaev versus Kitaev honeycomb cobaltates

Xiaoyu Liu, Hae‐Young Kee

Published Feb 16, 2023• 45 citations

Abstract

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.

DOI: 10.1103/physrevb.107.054420

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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

arXiv:2106.11982Jun 22, 20210 citations

Ab initio guided minimal model for the “Kitaev” material BaCo2 ( AsO4)2 : Importance of direct hopping, third-neighbor exchange, and quantum fluctuations

P. A. Maksimov, Alexey V. Ushakov, Z. V. Pchelkina +3 more

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Proximate Dirac spin liquid in the honeycomb lattice J1−J3 XXZ model: Numerical study and application to cobaltates

Anjishnu Bose, Manodip Routh, Sreekar Voleti +4 more

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

Shishir Kumar Pandey, Ji Feng

arXiv:2205.03836Physical review. B./Physical review. BNov 14, 202218 citations

Kitaev-Heisenberg Hamiltonian for high-spin d7 Mott insulators

Ryoya Sano, Yasuyuki Kato, Yukitoshi Motome

arXiv:1710.11357Physical review. B./Physical review. BJan 9, 2018215 citations

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

Huimei Liu, Giniyat Khaliullin

arXiv:1710.10193Physical review. B./Physical review. BJan 9, 2018279 citations

arXiv:2211.03737 · Physical review. B./Physical review. B

Non-Kitaev versus Kitaev honeycomb cobaltates

Xiaoyu Liu, Hae‐Young Kee

Published Feb 16, 2023• 45 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevb.107.054420

Related papers

Can we realize Kitaev quantum spin liquids in the $J_\text{eff}=1/2$ honeycomb cobaltates?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.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:2106.11982Jun 22, 20210 citations

Ab initio guided minimal model for the “Kitaev” material BaCo2 ( AsO4)2 : Importance of direct hopping, third-neighbor exchange, and quantum fluctuations

P. A. Maksimov, Alexey V. Ushakov, Z. V. Pchelkina +3 more

By considering two ab initio-based complementary approaches, we analyze the electronic structure and extract effective spin models of ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$ , a honeycomb material which has been proposed as a candidate for Kitaev physics. Both methods show that the dominant direct hopping makes the bond-dependent Kitaev term negligible, diverting the material away from the sought-after spin-liquid regime. As a result, we present a simple three-parameter exchange model to describe the interactions of the lowest doublet of the honeycomb cobaltate ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$ . Remarkably, it is the third-neighbor interactions, both isotropic and anisotropic, that are responsible for the standout double-zigzag ground state of ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$ , stabilized by quantum fluctuations. A significantly large third-nearest-neighbor hopping, observed ab initio, supports the importance of the third-neighbor interactions in the stabilization of the unique ground state of ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$ .By considering two ab initio-based complementary approaches, we analyze the electronic structure and extract effective spin models of ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$, a honeycomb material which has been proposed as a candidate for Kitaev physics. Both methods show that the dominant direct hopping makes the bond-dependent Kitaev term negligible, diverting the material away from the sought-after spin-liquid regime. As a result, we present a simple three-parameter exchange model to describe the interactions of the lowest doublet of the honeycomb cobaltate ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$. Remarkably, it is the third-neighbor interactions, both isotropic and anisotropic, that are responsible for the standout double-zigzag ground state of ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$, stabilized by quantum fluctuations. A significantly large third-nearest-neighbor hopping, observed ab initio, supports the importance of the third-neighbor interactions in the stabilization of the unique ground state of ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$.

arXiv:2204.09695Physical review. B./Physical review. BOct 31, 202254 citations

Proximate Dirac spin liquid in the honeycomb lattice J1−J3 XXZ model: Numerical study and application to cobaltates

Anjishnu Bose, Manodip Routh, Sreekar Voleti +4 more

Recent theoretical and experimental work suggests that the honeycomb cobaltates, initially proposed as candidate Kitaev quantum magnets, are in fact described by a pseudospin- $1/2$ easy-plane spin Hamiltonian with nearest-neighbor ferromagnetic (FM) exchange ${J}_{1}$ being frustrated by antiferromagnetic third-neighbor exchange ${J}_{3}$ and weaker compass anisotropies. Using exact diagonalization and density-matrix renormalization group (DMRG) calculations, we show that this model exhibits FM order at small ${J}_{3}/{J}_{1}$ and zigzag (ZZ) order at large ${J}_{3}/{J}_{1}$ , separated by an intermediate phase, which we label as $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}$. This $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}$ phase is shown to exhibit spin-liquid-like correlations in DMRG, although we cannot preclude weak broken symmetries, e.g., weak Ising type N\'eel order, given the limits on our explored system sizes. Using a modified parton mean field theory and variational Monte Carlo on Gutzwiller projected wave functions, we show that the optimal FM and ZZ orders as well as the intermediate $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}$ state are proximate to a ``parent'' Dirac spin liquid (SL). This Dirac SL is shown to capture the broad continuum in the temperature and magnetic field dependent terahertz spectroscopy of ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$ , and the reported low-temperature metallic thermal conductivity in ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$ and ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$ upon incorporating disorder-induced broadening.Recent theoretical and experimental work suggests that the honeycomb cobaltates, initially proposed as candidate Kitaev quantum magnets, are in fact described by a pseudospin-$1/2$ easy-plane spin Hamiltonian with nearest-neighbor ferromagnetic (FM) exchange ${J}_{1}$ being frustrated by antiferromagnetic third-neighbor exchange ${J}_{3}$ and weaker compass anisotropies. Using exact diagonalization and density-matrix renormalization group (DMRG) calculations, we show that this model exhibits FM order at small ${J}_{3}/{J}_{1}$ and zigzag (ZZ) order at large ${J}_{3}/{J}_{1}$, separated by an intermediate phase, which we label as $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}$. This $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}$ phase is shown to exhibit spin-liquid-like correlations in DMRG, although we cannot preclude weak broken symmetries, e.g., weak Ising type N\'eel order, given the limits on our explored system sizes. Using a modified parton mean field theory and variational Monte Carlo on Gutzwiller projected wave functions, we show that the optimal FM and ZZ orders as well as the intermediate $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}$ state are proximate to a ``parent'' Dirac spin liquid (SL). This Dirac SL is shown to capture the broad continuum in the temperature and magnetic field dependent terahertz spectroscopy of ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$, and the reported low-temperature metallic thermal conductivity in ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$ and ${\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}$ upon incorporating disorder-induced broadening.

arXiv:2212.13271Physical review. B./Physical review. BNov 16, 202317 citations

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.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:2205.03836Physical review. B./Physical review. BNov 14, 202218 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.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:1710.11357Physical review. B./Physical review. BJan 9, 2018215 citations

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

Huimei Liu, Giniyat Khaliullin

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.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.

arXiv:1710.10193Physical review. B./Physical review. BJan 9, 2018279 citations

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}

{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}

{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.

Can we realize Kitaev quantum spin liquids in the

J_\text{eff}=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

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.

By considering two ab initio-based complementary approaches, we analyze the electronic structure and extract effective spin models of

{\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}

, a honeycomb material which has been proposed as a candidate for Kitaev physics. Both methods show that the dominant direct hopping makes the bond-dependent Kitaev term negligible, diverting the material away from the sought-after spin-liquid regime. As a result, we present a simple three-parameter exchange model to describe the interactions of the lowest doublet of the honeycomb cobaltate

{\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}

. Remarkably, it is the third-neighbor interactions, both isotropic and anisotropic, that are responsible for the standout double-zigzag ground state of

{\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}

, stabilized by quantum fluctuations. A significantly large third-nearest-neighbor hopping, observed ab initio, supports the importance of the third-neighbor interactions in the stabilization of the unique ground state of

{\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}

Recent theoretical and experimental work suggests that the honeycomb cobaltates, initially proposed as candidate Kitaev quantum magnets, are in fact described by a pseudospin-

1/2

easy-plane spin Hamiltonian with nearest-neighbor ferromagnetic (FM) exchange

{J}_{1}

being frustrated by antiferromagnetic third-neighbor exchange

{J}_{3}

and weaker compass anisotropies. Using exact diagonalization and density-matrix renormalization group (DMRG) calculations, we show that this model exhibits FM order at small

{J}_{3}/{J}_{1}

and zigzag (ZZ) order at large

{J}_{3}/{J}_{1}

, separated by an intermediate phase, which we label as $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}$. This $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}$ phase is shown to exhibit spin-liquid-like correlations in DMRG, although we cannot preclude weak broken symmetries, e.g., weak Ising type N\'eel order, given the limits on our explored system sizes. Using a modified parton mean field theory and variational Monte Carlo on Gutzwiller projected wave functions, we show that the optimal FM and ZZ orders as well as the intermediate $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SL}}$ state are proximate to a ``parent'' Dirac spin liquid (SL). This Dirac SL is shown to capture the broad continuum in the temperature and magnetic field dependent terahertz spectroscopy of

{\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}

, and the reported low-temperature metallic thermal conductivity in

{\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}

and

{\mathrm{BaCo}}_{2}{({\mathrm{AsO}}_{4})}_{2}

upon incorporating disorder-induced broadening.

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.

{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.

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.