Non-Kitaev vs. Kitaev Honeycomb Cobaltates

arXiv:2211.03737 · cond-mat.str-el · Submitted 2022-11-07 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Non-Kitaev vs. Kitaev Honeycomb Cobaltates".

Mira: The gist: The study investigates whether honeycomb cobaltates exhibit Kitaev physics by deriving analytical expressions for spin models using strong-coupling perturbation theory and ab-initio calculations,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So to wrap up this paper "Non-Kitaev vs. Kitaev Honeycomb Cobaltates," the main point is that the interaction between different hopping channels, specifically involving t3, determines whether a honeycomb cobaltate ends up being classified as non-Kitaev or actually displaying Kitaev physics.

Mira: They showed that assuming small intraorbital t3 hopping was wrong; instead, this channel generates both FM Heisenberg and AFM Kitaev interactions that ultimately combine to make the FM Heisenberg interaction dominant in the idealized BCAO system.

Lev: And they showed that in materials like NCSO, where hopping parameters change—like t2g minus eg hopping becoming comparable to t3—that shift allows for a boost in the FM Kitaev interaction while reducing the Heisenberg component.

Kai: So what this means for the world is that figuring out these specific exchange processes isn't just theoretical fine-tuning; it’s essential because it tells us exactly how to design or engineer a material to get Kitaev physics when you only have weak spin-orbit coupling.

Mira: They are essentially showing that the competition between these specific paths is what dictates the final magnetic character, and that we need more experimental work on materials where these hopping integrals are comparable.

Conclusion: Kai: So, this paper "Non-Kitaev vs. Kitaev Honeycomb Cobaltates" is essentially looking at why we can't always tell if these cobaltates have Kitaev physics when the spin-orbit coupling is weak.

Mira: Yeah, it’s about the competition between different exchange paths and how those dictate the final magnetic character, whether it ends up being Heisenberg or Kitaev.

Lev: From where I sit with error correction, it means that if you're building something to exploit Kitaev physics in a real chip, you can't just assume the interactions are clean; they get complicated fast.

Kai: It’s about understanding how those intraorbital hopping channels, like t3 we saw earlier, actually generate both FM Heisenberg and AFM Kitaev terms at the same time.

Mira: Right, so even when we think a certain path is negligible because of the weak spin-orbit coupling, it ends up being the one driving the main interaction in some cases.

Lev: That means our simulation models have to be super detailed because those paths that look small on paper can actually dominate what happens in a real lattice structure.

Kai: And they point out that under trigonal distortion, which is pretty common experimentally, that dominant Heisenberg term stays as an XXZ model, which fits what we see in BaCo two (AsO four) two.

Mira: But the authors also show how the competition shifts dramatically in other materials like Na three Co two SbO six where those hopping parameters are almost equal.

Lev: So for hardware design, that means you can't just rely on one simple interaction type; you have to account for all these competing terms and their relative strengths.

Kai: Basically, the paper is saying that the origin of the discrepancy we’ve seen in experiments comes down to these specific exchange processes being non-zero even when they seem suppressed.

Mira: It really changes how we interpret those experimental results; it suggests that Kitaev physics isn't just about having spin-orbit coupling, but about which specific hopping terms get emphasized.

Lev: And for the next step, if you want to actually engineer a Kitaev material, you’d need to focus on reducing that t3 hopping while boosting the p orbital mediated hoppings.

Xiaoyu Liu, Hae-Young Kee

Department of Physics, University of Toronto · Canadian Institute for Advanced Research

cond-mat.str-el

Submitted: 2022-11-07

Updated: 2022-11-12

Comments: 10 pages, 4 figures

Journal ref: Phys. Rev. B 107, 054420 (2023)

DOI: 10.1103/PhysRevB.107.054420

License: http://creativecommons.org/licenses/by-nc-sa/4.0/

Importance score: 82/100

The gist: The gist: The study investigates whether honeycomb cobaltates exhibit Kitaev physics by deriving analytical expressions for spin models using strong-coupling perturbation theory and ab-initio

Key concepts

Strong-Coupling Perturbation Theory: This is a mathematical method used to analyze complex quantum systems by treating the strong interactions between electrons separately from weaker ones. It helps derive simplified spin models from the full electronic Hamiltonian, allowing researchers to calculate how different hopping paths and exchange processes contribute to the overall magnetic behavior.
$t_3$ Hopping Channel: This refers to a specific intraorbital hopping path within the honeycomb lattice that was initially assumed to be small. The analysis reveals this channel is actually significant because it generates a strong ferromagnetic (FM) Heisenberg interaction, which becomes the main driving force in the system's magnetic properties.
Trigonal Distortion: This describes a structural distortion of the ideal honeycomb lattice symmetry. This change modifies the spin Hamiltonian, transforming an isotropic model into an anisotropic XXZ model. This distortion is crucial for understanding why BCAO behaves as expected and how it affects the relative strengths of Heisenberg versus Kitaev interactions.

Summary

The gist: The study investigates whether honeycomb cobaltates exhibit Kitaev physics by deriving analytical expressions for spin models using strong-coupling perturbation theory and ab-initio calculations, revealing that intraorbital hopping channels play a crucial role in determining the dominant Heisenberg and Kitaev interactions.

Model Derivation and Exchange Processes

The research begins by establishing the onsite Hamiltonian for honeycomb cobaltates, which includes terms like Htot = HCoulomb + Hcubic + HSOC + Htrig (Page 2). The analysis then proceeds to consider hopping paths to determine exchange processes, such as bond M1 − M2 having C2v local symmetry and the allowed hoppings Tdd (Page 3). The spin model is built using strong-coupling perturbation theory by projecting ground states doublets, resulting in a generic nearest neighbor (n.n.) spin model Hspin = X hiji∈αβ(γ) JSi · Sj + KSγ i Sγ j + Γ(Sα i Sβ j + Sβ i Sα j) (Page 4).

Dominant Interactions and Discrepancies

The study focuses on determining the relative strength among various exchange processes to identify the major interaction (Page 4). Key findings include:

The largest intraorbital hopping channel, denoted by t3, is significant and leads to a ferromagnetic (FM) Heisenberg interaction that becomes the dominant interaction because other contributions almost cancel each other (Page 2).
This same assumed-to-be-small channel also generates an antiferromagnetic (AFM) Kitaev interaction which then cancels a FM Kitaev interaction from t2g − eg paths, resulting in a small Kitaev interaction (Page 2).
Under the trigonal distortion, the preeminent isotropic Heisenberg becomes an anisotropic XXZ model as expected for BCAO (Page 2).
In Na3Co2SbO6 (NCSO), the t2g − eg hopping becomes comparable to t3 as t3 decreases, leading to a delicate competition between enhanced Kitaev and reduced Heisenberg interactions (Page 2).

Application to BCAO

Tight binding parameters for BaCo2(AsO4)2 (BCAO) are obtained using DFT calculations, confirming that t3 is the largest hopping integral (Page 5). The analysis of exchange parameters for an idealized honeycomb BCAO shows that the second-order intersite U process gives rise to a FM Heisenberg interaction, and the only significant AFM Heisenberg interaction comes from the two-hole processes (B2) with the two holes at the same orbital (Page 5). The cancellation of Heisenberg interactions is incomplete, implying that Heisenberg interaction remains dominant (Page 5).

Effect of Trigonal Distortion

The trigonal distortion modifies the ideal Jeff = 1/2 picture, leading to additional anisotropic spin interactions (Page 6). The spin Hamiltonian in the global XY Z coordinate is written as Hz spin = (JXY + D E F E JXY − D G F G JZ) (Page 6). Numerical results under trigonal distortion show that the dominant FM XXZ Heisenberg terms (JXY and JZ) are found, while contributions from fourth-order processes are non-negligible compared to the inter-site U second order contribution (Page 6). The inclusion of the trigonal distortion further weakens the Kitaev term in each process, and overall, the FM Kitaev is negligible as shown in Table III (Page 6).

Conclusion and Future Directions

The analysis suggests that while previous theories assumed negligible intraorbital t3 hopping, this channel generates FM Heisenberg and AFM Kitaev interactions which combine to make the FM Heisenberg interaction dominant (Page 7). The study concludes that a target Kitaev cobaltate displaying the KSL can be engineered with effectively reduced intraorbital t3 and enhanced p orbital mediated hoppings (Page 7). Future theoretical and experimental works are needed to discover Kitaev honeycomb cobaltates (Page 7).

How it works

The derivation of spin interactions relies on strong-coupling perturbation theory applied to projected ground states doublets (Page 4). The key mechanism involves the t3 exchange channel, which generates FM Heisenberg and AFM Kitaev interactions, ultimately leading to the dominance of the FM Heisenberg interaction (Page 2).

Key Exchange Contributions

The paper categorizes exchange processes into three groups based on hopping direction:

Inter-site U process (second-order): This gives rise to a FM Heisenberg interaction (Page 5).
Two-hole process: The dominant AFM Heisenberg interaction comes from the two-hole processes where the two holes are at the same orbital (Page 5).
Cyclic exchange process: This is always FM and is generally comparable in magnitude to other paths, meaning perfect cancellation of Heisenberg cannot be expected (Page 4).

Role of Hopping Integrals

The relative importance of hopping integrals is critical; for ideal BCAO, t3 is the largest hopping integral, which supports the dominance of the FM Heisenberg interaction (Page 5). In contrast, for Na3Co2SbO6 (NCSO), intersite t2g − eg (denoted by t6) hopping increases while t3 decreases, shifting the balance toward a Kitaev regime (Page 7).

Impact of Symmetry Breaking

The trigonal distortion breaks the ideal Jeff = 1/2 picture, modifying the spin Hamiltonian to include terms like D, E, and F in the XY Z coordinate (Page 6). This distortion is experimentally suggested by strong anisotropic g-factors in BCAO (Page 6). The resulting model under trigonal distortion is given by Hz spin = (JXY + D E F E JXY − D G F G JZ) (Page 6).

Summary of Findings

The investigation clarifies the origin of discrepancies between previous works by showing that the t3 exchange channel, previously assumed negligible, is significant and drives the dominant interaction to be FM Heisenberg (Page 7). The competition between different exchange paths determines whether cobaltates are classified as non-Kitaev or Kitaev materials (Page 7). The inclusion of trigonal distortion further weakens the Kitaev term in each process, reinforcing the dominance of the XXZ type Heisenberg interaction (Page 6).

References

[1] A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006), January Special Issue (Page 7).

[2] G. Jackeli and G. Khaliullin, Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models, Phys. Rev. Lett. 102, 017205 (2009) (Page 7).

[3] Y. Singh and P. Gegenwart, Antiferromagnetic Mott insulating state in single crystals of the honeycomb lattice material Na2IrO3, Phys. Rev. B 82, 064412 (Page 7).

[4] Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku, S. Trebst, and P. Gegenwart, Relevance of the Heisenberg-Kitaev Model for the Honeycomb Lattice Iridates A2IrO3 Phys Rev Lett 108 127203 (Page 7).

[5] K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V. Shankar, Y. F. Hu, K.-S Burch, H.-Y Kee, and Y.-J Kim α − RuCl3 A spin-orbit assisted Mott insulator on a honeycomb lattice Phys Rev B 90 041112 (Page 5).

[6] L. J. Sandilands, Y. Tian, K. W. Plumb, Y.-J Kim, and K.-S Burch Scattering continuum and possible fractionalized excitations in α−rucl3 Phys Rev Lett 114 147201 (Page 5).

[7] H.-S Kim, V. S. V., A. Catuneanu, and H.-Y Kee Kitaev magnetism in honeycomb RuCl3 with intermediate spin-orbit coupling Phys Rev B 91 241110 (Page 7).

[8] A. Banerjee, C. Bridges, J.-Q Yan, A. Aczel, L. Li, M. Stone, G. Granroth, M. Lumsden, Y.-Yiu, J Knolle et al Proximate Kitaev quantum spin liquid be Nature materials 10384604 (Page 7).

[9] X Liu, T Berlijn W.-G Yin W Ku A Tsvelik Y.-J Kim H Gretarsson Y Singh P Gegenwart and J P Hill Long-range magnetic ordering in Na2IrO3 Phys Rev B 83 220403 (Page 7).

[10] F Ye, S Chi H Cao B C Chakoumakos J A Fernandez-Baca R Custelcean T F Qi O B Korneta and G Cao Direct evidence of a zigzag spin-chain structure in the honeycomb lattice A neutron and X-ray diffraction investigation of single crystal Na2IrO3 Phys Rev B 85 180403 (Page 7).

[11] J A Sears M Songvilay K W Plumb J P Clancy Y Qiu Y Zhao D Parshall and Y.

Improvements for AI systems

A system could perform quantitative comparison between theoretical predictions for Kitaev physics in different cobaltates by calculating spin Hamiltonian parameters using strong-coupling perturbation theory and ab-initio calculations. This would allow the system to determine which exchange path is dominant, as shown by the finding that the largest intraorbital t3 hopping integral was reported in several honeycomb cobaltates in Ref. [33, 35].
The improved AI could model magnetic phase transitions by simulating the effect of trigonal distortion on spin interactions. Specifically, it could calculate how the distortion modifies the Hamiltonian into an anisotropic XXZ interaction which dominates over other interactions, as described in Section VI.
The system can predict material suitability for Kitaev spin liquids by analyzing competition between exchange channels. It can identify cases where the almost-cancellation of the Kitaev interaction discussed above no longer occurs in systems like Na3Co2SbO6, leading to a conclusion that the Kitaev interaction is comparable to or even larger than the reduced Heisenberg interaction.

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 BaCo 2 (AsO 4) 2 (BCAO), which calls for a 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 by ab-initio calculations for idealized honeycomb BCAO, we find that the largest intraorbital t 2g-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-e g and e g-e g contributions almost cancel each other. Exactly the same assumed-to-be-small channel also generates an antiferromagnetic Kitaev interaction, which then cancels a FM Kitaev interaction from t 2g-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 for BCAO. However, in Na 3 Co 2 SbO 6 the intraorbital t 2g-t 2g hopping is smaller and comparable to the t 2g-e g hopping, leading to a delicate competition between enhanced Kitaev and reduced Heisenberg interactions.