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REVIEW 3 major objections 5 minor 53 references

Magnetically ordered yet topologically robust phases emerging in concurrent Kitaev spin liquids

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper shows that in the Yao-Lee model with additional Kitaev and Heisenberg couplings, strong Kitaev interactions can freeze the spin degrees of freedom into ferromagnetic or antiferromagnetic order while the orbital degrees of…

desk verdict Useful new phase diagram for YL+Kitaev+Heisenberg, but the 'topologically robust' coexistence claim is assumed rather than demonstrated because the mean-field never leaves the zero-flux sector. read the letter →

arxiv 2507.21226 v1 pith:EIWRUKVA submitted 2025-07-28 cond-mat.str-el cond-mat.other

Muhammad Akram , Aayush Vijayvargia , Hae-Young Kee , Onur Erten This is my paper · ORCID
classification cond-mat.str-elcond-mat.other
keywords Yao-LeemodelKitaevspinliquidspin-orbitalmagneticfragmentationMajoranamean-fieldtheorytopologicalorderhoneycomblatticequantum
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that topological spin-liquid order need not be destroyed by magnetic order: in a Yao-Lee model on the honeycomb lattice supplemented by Kitaev and Heisenberg couplings, the spin sector can develop conventional ferromagnetic or antiferromagnetic order while the orbital sector keeps its liquid character. The authors find that this coexistence occurs for dominant Kitaev interactions, whereas both sectors remain liquids when the Yao-Lee interaction dominates. Heisenberg exchange can either promote or suppress the magnetic order, and the paper maps out the resulting phase diagram. If correct, the results give a concrete route to materials in which local magnetic order and topological order occupy the same lattice, a state called magnetic fragmentation.

What carries the argument

The argument runs through a Majorana-fermion representation of the four-dimensional local Hilbert space, in which the spin and orbital Pauli operators are written as products of Majorana fermions and bond operators $u_{ij}$. The plaquette operator $W_p$, built from bond operators, commutes with the full Hamiltonian, so eigenstates can still be labeled by flux sectors even though the model is no longer exactly solvable. When the Kitaev coupling is large, first-order perturbation theory projects the Yao-Lee and Heisenberg terms onto the degenerate Kitaev-liquid ground states and produces an effective Heisenberg exchange $J_{\rm eff} = J_H - \operatorname{sgn}(J_K) \, 0.525 \, J_{YL}$ in the spin sector, which decides between ferromagnetism and antiferromagnetism. Away from that limit, a self-consistent Majorana mean-field theory decouples the six- and four-Majorana terms in magnetic and non-magnetic channels, with the flux sector fixed to zero ($u_{ij}=1$), and yields the phase diagrams in $J_K/J_{YL}$ and $J_H/J_{YL}$.

What would settle it

An unbiased finite-cluster calculation, such as exact diagonalization or a tensor-network simulation on a 24-site honeycomb cluster at $J_K/J_{YL} = 2$ and $J_H = 0$, could measure the plaquette expectation $\langle W_p \rangle$ and the spin structure factor: if the ground state has nonzero magnetization but $\langle W_p \rangle$ is not close to $+1$, or if the vison gap closes, the claimed coexistence of magnetic and topological order fails. Adding the omitted $\tau_i \cdot \tau_j$ term to the Hamiltonian and repeating the calculation would test whether the predicted FM-2 phase for the microscopic parameters survives.

Watch

Extended reading notes

Core claim

The central discovery is that two interactions that each separately support a quantum spin liquid can, when combined, produce a state in which one sector orders and the other does not. For $J_K/J_{YL} > 1.38$ the spin sector is ferromagnetic and for $J_K/J_{YL} < -0.85$ it is antiferromagnetic, while the orbital sector remains a topological liquid in both cases; for $-0.85 < J_K/J_{YL} < 1.38$ the Yao-Lee spin-orbital liquid survives. The ferromagnetic transition is first order and creates a Fermi surface of Majorana fermions that disappears at a Lifshitz transition around $J_K/J_{YL} \simeq 1.91$, while the antiferromagnetic transition is second order and gaps two of the three Majorana flavors, leaving a single Dirac cone. Heisenberg exchange enlarges or shrinks the ordered regions, and for the microscopic realization highlighted in the paper the ground state is predicted to be the Yao-Lee liquid for one sign of the coupling and a ferromagnetic phase with negative orbital correlations (FM-2) for the other.

Load-bearing premise

The whole coexistence picture rests on fixing the zero-flux sector and assuming the orbital sector stays a gapped topological liquid inside the ordered phases; the vison gap is never computed there, and the flux-nonconserving $\tau_i \cdot \tau_j$ term omitted from the microscopic model could destroy that assumption.

Editorial extensions

If this is right

  • For $J_K/J_{YL} > 1.38$ the ground state is a ferromagnet in the spin sector with a Majorana Fermi surface that persists up to a Lifshitz transition at $J_K/J_{YL} \simeq 1.91$.
  • For $J_K/J_{YL} < -0.85$ the ground state is an antiferromagnet with a single gapless Majorana flavor carrying a Dirac dispersion.
  • Heisenberg exchange with one sign favors order and with the other expands the Yao-Lee liquid region; in the large-Kitaev limit the FM-AFM boundary sits at $J_H/J_{YL} = \pm 0.525$.
  • For the microscopic realization discussed in the paper, the predicted ground state is the Yao-Lee spin-orbital liquid for positive superexchange and the FM-2 phase for negative superexchange.
  • All magnetically ordered phases found in the mean-field theory retain topological order in the orbital degrees of freedom, giving a lattice realization of magnetic fragmentation driven by two spin-liquid-supporting interactions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the microscopic model also contains a $\tau_i \cdot \tau_j$ term that breaks flux conservation, the quantitative predictions, especially FM-2 for negative superexchange, deserve a stability check; a perturbative inclusion of this term is the natural next step.
  • The first-order FM transition and the Majorana Fermi surface are mean-field results; unbiased numerics could reveal that the true transition is continuous or that the Fermi surface is replaced by a different low-energy structure, without changing the coexistence claim.
  • If the vison gap in the ordered phases turns out to be small, thermal fluctuations could destroy the topological order well below the magnetic ordering temperature, meaning the coexistence region in temperature would be narrower than the zero-temperature phase diagram suggests.
  • A similar coexistence might be engineered in other spin-orbital Kitaev models, since the mechanism only requires two commuting liquid-supporting interactions acting on separate degrees of freedom, so the YL/Kitaev combination may be one member of a larger family.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This manuscript studies the Yao-Lee (YL) spin-orbital model on the honeycomb lattice with additional Kitaev and Heisenberg interactions. The YL plaquette operator remains conserved in the presence of both perturbations, but the model is no longer exactly solvable. The authors combine first-order perturbation theory in the large-Kitaev limit with a self-consistent Majorana mean-field theory. Their central claim is that, for dominant Kitaev coupling, the spin sector develops FM or AFM order depending on the sign of the coupling, while the orbital sector remains a topological spin liquid; for dominant YL coupling both sectors remain liquids. They also show that the Heisenberg interaction can stabilize or suppress the magnetic order, and they identify two FM and two AFM phases distinguished by orbital correlations. The paper ends with a quantitative prediction for the microscopic model of Ref. 41: for J>0 the ground state is the YL spin-orbital liquid, and for J<0 it is the FM-2 phase.

Significance. If fully established, the coexistence of magnetic order in one sector and topological order in another is a noteworthy result: it goes beyond the usual fragility of Kitaev spin liquids and provides a concrete setting for magnetic fragmentation in spin-orbital systems. The paper has real strengths: the large-JK perturbation theory uses the exact Kitaev bond expectation value from Baskaran et al., giving a controlled benchmark; the mean-field solution has no fitted parameters and reproduces the perturbative FM-AFM boundary at large |JK|; and the authors are explicit about the conserved plaquette operator and the limitations of their treatment. The main weakness is that the topological-liquid claim in the ordered phases is not actually demonstrated by the calculation, which is performed entirely in the zero-flux sector.

major comments (3)
  1. [Majorana mean field theory, text after Eq. (9), and paragraphs after Figs. 2-3] The central claim that the FM and AFM phases retain topological order in the tau sector is an assumption, not an output of the calculation. The self-consistent mean-field is performed with u_ij=1 on every bond, i.e., strictly in the zero-flux sector, which removes the vison dynamics needed to establish a gapped Z2 spin liquid. The statement that 'the FM and AFM phases, whose vison gaps are the same as the Kitaev model' appears without derivation, and it cannot be inferred from a calculation that never leaves the zero-flux sector. Moreover, the Majorana spectrum changes qualitatively across the phase diagram (the AFM gaps two flavors, the FM hybridizes them and creates a Fermi surface), so the flux gap should in general change as well. I ask the authors to compute the vison gap (the energy cost of flipping a plaquette) in each ordered phase, or to compare self-consistent energies between different flux sectors; without such a computation, the phrase 'topologically robust' overstates what the manuscript establishes.
  2. [Microscopic model, Eq. (10), and final quantitative prediction] The quantitative prediction for the model of Ref. 41 omits the (tau_i dot tau_j) and (sigma_i dot sigma_j)(tau_i dot tau_j) terms from Eq. (10), which are stated to be beyond scope because they break flux conservation. These terms are not obviously small compared with J, and they can change both the magnetic and the topological character of the ground state. The stability argument in the 'Before concluding' paragraph applies to small perturbations around an established gapped spin liquid; for the predicted FM-2 phase at J<0, no flux gap has been computed, so the argument does not protect the prediction. Please quantify the effect of the omitted terms, or soften the material-specific claim accordingly.
  3. [Decoupling of the six-Majorana term, Eqs. (7)-(9)] The six-Majorana Kitaev term is decoupled only in the magnetic channel m_z and the diagonal bond channels chi_alpha. This is a restricted variational ansatz: other channels, such as staggered-flux or bond-nematic order parameters, are not tested, and the SU(2) symmetry of the spin sector is broken by hand along z. The phase boundaries (for example the values -0.85, 1.38, and 1.91 quoted in the text) and even the existence of the FM-1/FM-2 and AFM-1/AFM-2 subphases depend on this choice. A small-cluster exact-diagonalization or DMRG benchmark at representative parameters, or an explicit check with a more general decoupling, would materially strengthen the phase diagram. Without such a check, the mean-field results should be presented as variational rather than definitive.
minor comments (5)
  1. [Eq. (6)] The third term in the Majorana form of the Heisenberg interaction appears to read c^x_i c^y_i c^x_j c^z_j; for sigma^z_i sigma^z_j it should be c^x_i c^y_i c^x_j c^y_j. Please correct this typo or clarify the convention.
  2. [Equation after Eq. (8)] The expression J_eff = J_H - sgn(J_K) 0.525 J_YL would be clearer with parentheses and with an explicit statement of the sign convention for the bond expectation value <tau_i^alpha tau_j^alpha> from Ref. 42, since the overall sign matters for FM versus AFM order.
  3. [Fig. 2 and Fig. 3 captions] The phase labels FM-1, FM-2, AFM-1, and AFM-2 are introduced only in the text; the captions should state explicitly that these are distinguished by the sign of the nearest-neighbor orbital correlation <tau_i^alpha tau_j^alpha> and by the presence or absence of a Fermi surface.
  4. [Definition of W_p in text after Eq. (3)] The plaquette operator is written with a repeated index (tau^z_m appears twice); the sixth bond should be labeled by a distinct lattice site, for example tau^z_n, to avoid confusion.
  5. [Second paragraph after Fig. 2] The statement that beyond the Lifshitz transition 'the FM phase becomes fully polarized, with m_i^z = 1' is asserted without showing the self-consistent solution; since m_i^z is bounded by ±1, saturation is plausible but should be demonstrated explicitly or shown in a plot.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the phase diagram is solved self-consistently from the stated Hamiltonian, with external benchmarks (Baskaran exact bond expectation, independent J_K=0 results). The zero-flux-sector assumption is a correctness gap, not a circular reduction.

full rationale

The derivation chain is self-contained. The phase diagram is obtained from a self-consistent Majorana mean-field decoupling of the stated Hamiltonian; the order parameters and bond expectation values are solved from the model, not fitted to reproduce a target phase. The large-|J_K| phase boundary is checked against perturbation theory that uses the exact Kitaev-model bond expectation value from Baskaran et al. (Ref. 42), an external benchmark, and the J_K=0 limit is compared to independent work (Ref. 27). Self-citations (Refs. 29-31, 39-40, 49) provide context and prior analogous models but are not load-bearing for the new coexistence result. The main caveat, which the text itself discloses, is that the calculation fixes the flux sector to zero flux (u_ij=1 for all bonds) and never computes the vison gap in the magnetically ordered phases, so the statement that the tau sector retains topological order rests on an assumption about flux-sector energetics rather than being derived. This is a missing-support or correctness concern, not a circular reduction: no equation is equivalent to its input by construction, and no fitted parameter is renamed as a prediction. The self-consistent mean-field parameters are outputs of the calculation, and the perturbative check uses an externally derived exact result. Therefore no circularity step can be exhibited, and the paper's central derivation is not circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the exact solvability of the YL and Kitaev limits, the known bond expectation value, and the assumption that the zero-flux sector remains the ground state and that the chosen mean-field channels are sufficient. No entirely new theoretical entities are introduced.

assumptions (6)
  • standard math The Yao-Lee model is exactly solvable and its ground state is a spin-orbital liquid.
    Used throughout; the paper relies on the known exact solution of the YL model (Ref. 22).
  • domain assumption The Kitaev model in the orbital sector has a QSL ground state and a 2^N degenerate spin paramagnet.
    Invoked in the large-JK perturbation theory to set up the effective Hamiltonian (Eq. 8).
  • standard math The bond expectation value <tau^alpha_i tau^alpha_j> = 0.525 sgn(JK) in the Kitaev ground state.
    Taken from Ref. 42 (Baskaran et al.) and used in Eq. 8 to derive J_eff.
  • domain assumption The ground state flux sector is the zero-flux sector (u_ij=1) for all parameters studied.
    The mean-field calculation fixes u_ij=1; the paper does not compare flux sectors.
  • ad hoc to paper The mean-field decoupling of the six-Majorana Kitaev term into the chosen channels captures the relevant physics.
    The decoupling is a standard approximation but not controlled; the choice of channels biases the phase diagram.
  • domain assumption The (tau_i dot tau_j) term from the microscopic model can be neglected because the flux gap protects the spin liquid.
    The paper states this in the concluding section; it is plausible but not demonstrated for finite couplings.

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Pith. "Pith review of Magnetically ordered yet topologically robust phases emerging in concurrent Kitaev spin liquids." pith.science (2026). https://pith.science/paper/EIWRUKVA

@misc{pith2026250721226,
  author       = {Pith},
  title        = {Pith review of: Magnetically ordered yet topologically robust phases emerging in concurrent Kitaev spin liquids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EIWRUKVA}},
  note         = {Machine review of arXiv:2507.21226}
}
read the original abstract

Spin-orbital generalizations of Kitaev model, such as Yao-Lee model, have attracted recent attention due to their enhanced stability of spin liquid phases against perturbations. Motivated by microscopic calculations for the realization of Yao-Lee model showing additional interactions, we study the phase diagram of the Yao-Lee model with added Kitaev and Heisenberg terms. While the plaquette operator is conserved even in the presence of added perturbations, the model becomes no longer exactly solvable. Using perturbation and Majorana mean-field theory, we find magnetic order can arise in the spin sector while the orbital sector remains a liquid for dominant Kitaev interactions, whereas both sectors form liquid phases when Yao-Lee interactions dominate. Additional Heisenberg exchange can enhance or suppress the magnetic order, revealing a rich coexistence of magnetic and topological phases.

Figures

Figures reproduced from arXiv: 2507.21226 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic of the model: A section of the honeycomb [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Majorana mean field phase diagram as a function [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Majorana fermion mean-field phase diagram of the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

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Works this paper leans on

53 extracted references · 44 canonical work pages

  1. [1]

    Broholm, R

    C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Quantum spin liquids, Science 367, eaay0668 (2020)

    2020

  2. [2]

    Our self-consistent mean-field analysis reveals that the YL spin-orbital liquid remains stable within the range −0.85 < JK/JY L < 1.38. Beyond this win- dow, the system undergoes magnetic transitions in the à sector, leading to either ferromagnetic (FM) or anti- ferromagnetic (AFM) order, depending on the sign of JK/JY L. In agreement with perturbative res...

    work page 2022

  3. [3]

    Savary and L

    L. Savary and L. Balents, Quantum spin liquids: a re- view, Reports on Progress in Physics 80, 016502 (2016)

    2016

  4. [4]

    Balents, Spin liquids in frustrated magnets, Nature 464, 199 (2010)

    L. Balents, Spin liquids in frustrated magnets, Nature 464, 199 (2010)

    2010

  5. [5]

    Jackeli and G

    G. Jackeli and G. Khaliullin, Mott insulators in the strong spin-orbit coupling limit: from Heisenberg to a quantum compass and Kitaev models, Physical Review Letters 102, 017205 (2009)

    work page 2009

  6. [6]

    Kitaev, Anyons in an exactly solved model and be- yond, Annals of Physics 321, 2 (2006)

    A. Kitaev, Anyons in an exactly solved model and be- yond, Annals of Physics 321, 2 (2006)

    2006

  7. [7]

    S. M. Winter, Y. Li, H. O. Jeschke, and R. Valent ´ ı, Chal- lenges in design of Kitaev materials: Magnetic interac- tions from competing energy scales, Physical Review B 93, 214431 (2016)

    work page 2016

  8. [8]

    J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Generic spin model for the honeycomb iridates beyond the kitaev limit, Phys. Rev. Lett. 112, 077204 (2014)

    work page 2014

Show all 53 references
  1. [9]

    Hermanns, I

    M. Hermanns, I. Kimchi, and J. Knolle, Physics of the Kitaev model: Fractionalization, dynamic correlations, and material connections, Annual Review of Condensed Matter Physics 9, 17 (2018)

    2018

  2. [10]

    J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Spin-orbit physics giving rise to novel phases in correlated systems: Iridates and related materials, Annual Review of Con- densed Matter Physics 7, 195 (2016)

    2016

  3. [11]

    Takayama, J

    T. Takayama, J. Chaloupka, A. Smerald, G. Khaliullin, and H. Takagi, Spin–orbit-entangled electronic phases in 4d and 5 d transition-metal compounds, J. Phys. Soc. Jpn. 90, 062001 (2021)

    2021

  4. [12]

    Motome and J

    Y. Motome and J. Nasu, Hunting Majorana fermions in Kitaev magnets, Journal of the Physical Society of Japan 89, 012002 (2020)

    2020

  5. [13]

    Rousochatzakis, N

    I. Rousochatzakis, N. B. Perkins, Q. Luo, and H.-Y. Kee, Beyond Kitaev physics in strong spin-orbit coupled mag- nets, Reports on Progress in Physics 87, 026502 (2024)

    2024

  6. [14]

    Trebst and C

    S. Trebst and C. Hickey, Kitaev materials, Physics Re- ports 950, 1 (2022)

    2022

  7. [15]

    Takagi, T

    H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Concept and realization of kitaev quantum spin liquids, Nature Reviews Physics 1, 264 (2019)

    2019

  8. [16]

    Plumb, J

    K. Plumb, J. Clancy, L. Sandilands, V. V. Shankar, Y. Hu, K. Burch, H.-Y. Kee, and Y.-J. Kim, α -RuCl3: A spin-orbit assisted Mott insulator on a honeycomb lat- tice, Physical Review B 90, 041112 (2014)

    2014

  9. [17]

    Singh, S

    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, Physical Review Letters 108, 127203 (2012)

    2012

  10. [18]

    Matsuda, T

    Y. Matsuda, T. Shibauchi, and H.-Y. Kee, Kitaev quan- tum spin liquids (2025), arXiv:2501.05608 [cond-mat.str- el]

    2025

  11. [19]

    Hwan Chun, J.-W

    S. Hwan Chun, J.-W. Kim, J. Kim, H. Zheng, C. C. Stoumpos, C. D. Malliakas, J. F. Mitchell, K. Mehlawat, Y. Singh, Y. Choi, T. Gog, A. Al-Zein, M. M. Sala, M. Krisch, J. Chaloupka, G. Jackeli, G. Khaliullin, and B. J. Kim, Direct evidence for dominant bond-directional interact...

    2015

  12. [20]

    Gegenwart and S

    P. Gegenwart and S. Trebst, Kitaev matter, Nature Physics 11, 444 (2015)

    2015

  13. [21]

    C. Wu, D. Arovas, and H.-H. Hung, Γ-matrix generaliza- tion of the kitaev model, Phys. Rev. B 79, 134427 (2009)

    2009

  14. [22]

    Kitagawa, T

    K. Kitagawa, T. Takayama, Y. Matsumoto, A. Kato, R. Takano, Y. Kishimoto, S. Bette, R. Dinnebier, G. Jackeli, and H. Takagi, A spin–orbital-entangled quan- tum liquid on a honeycomb lattice, Nature 554, 341 (2018)

    2018

  15. [23]

    V. S. de Carvalho, H. Freire, E. Miranda, and R. G. Pereira, Edge magnetization and spin transport in an 6 su(2)-symmetric kitaev spin liquid, Phys. Rev. B 98, 155105 (2018)

    2018

  16. [24]

    Yao and D.-H

    H. Yao and D.-H. Lee, Fermionic magnons, non-abelian spinons, and the spin quantum hall effect from an ex- actly solvable spin-1 / 2 kitaev model with su(2) symme- try, Phys. Rev. Lett. 107, 087205 (2011)

    2011

  17. [25]

    W. M. H. Natori and J. Knolle, Dynamics of a two- dimensional quantum spin-orbital liquid: Spectroscopic signatures of fermionic magnons, Phys. Rev. Lett. 125, 067201 (2020)

    2020

  18. [26]

    Chulliparambil, U

    S. Chulliparambil, U. F. P. Seifert, M. Vojta, L. Janssen, and H.-H. Tu, Microscopic models for kitaev’s sixteenfold way of anyon theories, Phys. Rev. B 102, 201111 (2020)

    2020

  19. [27]

    U. F. P. Seifert, X.-Y. Dong, S. Chulliparambil, M. Vo- jta, H.-H. Tu, and L. Janssen, Fractionalized fermionic quantum criticality in spin-orbital mott insulators, Phys. Rev. Lett. 125, 257202 (2020)

    2020

  20. [28]

    Chulliparambil, L

    S. Chulliparambil, L. Janssen, M. Vojta, H.-H. Tu, and U. F. P. Seifert, Flux crystals, majorana metals, and flat bands in exactly solvable spin-orbital liquids, Phys. Rev. B 103, 075144 (2021)

    2021

  21. [29]

    E. Nica, M. Akram, A. Vijayvargia, R. Moessner, and O. Erten, Kitaev spin-orbital bilayers and their moir´ e superlattices, npj Quantum Mater. 8, 9 (2023)

    2023

  22. [30]

    A. M. Tsvelik and P. Coleman, Order fractionalization in a kitaev-kondo model, Phys. Rev. B 106, 125144 (2022)

    2022

  23. [31]

    Vijayvargia, E

    A. Vijayvargia, E. M. Nica, R. Moessner, Y.-M. Lu, and O. Erten, Magnetic fragmentation and fractionalized goldstone modes in a bilayer quantum spin liquid, Phys. Rev. Res. 5, L022062 (2023)

    2023

  24. [32]

    Akram, E

    M. Akram, E. M. Nica, Y.-M. Lu, and O. Erten, Vi- son crystals, chiral, and crystalline phases in the Yao-Lee model, Phys. Rev. B 108, 224427 (2023)

    2023

  25. [33]

    Poliakov, W.-H

    V. Poliakov, W.-H. Kao, and N. B. Perkins, Topological transitions in the Yao-Lee spin-orbital model and effects of site disorder, Phys. Rev. B 110, 054418 (2024)

    2024

  26. [34]

    M. A. Keskiner, O. Erten, and M. O. Oktel, Kitaev-type spin liquid on a quasicrystal, Phys. Rev. B 108, 104208 (2023)

    2023

  27. [35]

    Dutta, A

    I. Dutta, A. Mukherjee, O. Erten, and K. Saha, Exactly solvable spin liquids in kitaev bilayers and moir´ e super- lattices (2025), arXiv:2506.04181 [cond-mat.str-el]

    2025

  28. [36]

    Majumder, O

    R. Majumder, O. Erten, and A. Mukherjee, Thermody- namics of vison crystals in an anisotropic quantum spin liquid, Phys. Rev. B 110, 125155 (2024)

    2024

  29. [37]

    K. I. Kugel and D. I. Khomskii, The jahn-teller effect and magnetism: transition metal compounds, Soviet Physics Uspekhi 25, 231 (1982)

    1982

  30. [38]

    Neehus, A

    A. Neehus, A. Rosch, J. Knolle, and U. F. P. Seifert, Projectively implemented altermagnetism in an exactly solvable quantum spin liquid (2025), arXiv:2504.12298 [cond-mat.str-el]

    2025

  31. [39]

    Keskiner, M

    M. Keskiner, M. Oktel, N. B. Perkins, and O. Erten, Magnetic order through kondo coupling to quantum spin liquids, Materials Today Quantum 6, 100038 (2025)

    2025

  32. [40]

    Coleman, A

    P. Coleman, A. Panigrahi, and A. Tsvelik, Solvable 3d kondo lattice exhibiting pair density wave, odd-frequency pairing, and order fractionalization, Phys. Rev. Lett. 129, 177601 (2022)

    2022

  33. [41]

    #!$ # ï

    As such, the local Hilbert space is four- dimensional, spanned by the S = 3/2 operators or Γ ma- trices. Note that the Kitaev ( HK) and Heisenberg term (HH ) only act on the orbital and spin DOF respectively while the YL term ( HY L) acts on both. The plaque- tte operator, Wp ...

  34. [42]

    Akram and O

    M. Akram and O. Erten, Emergent phases in the yao-lee model via coupling to topological spin textures (2025), arXiv:2504.08735 [cond-mat.str-el]

    2025 arXiv

  35. [43]

    Churchill, E

    D. Churchill, E. Z. Zhang, and H.-Y. Kee, Microscopic roadmap to a kitaev-yao-lee spin-orbital liquid, npj Quantum Materials 10, 26 (2025)

    2025

  36. [44]

    Baskaran, S

    G. Baskaran, S. Mandal, and R. Shankar, Exact results for spin dynamics and fractionalization in the kitaev model, Phys. Rev. Lett. 98, 247201 (2007)

    2007

  37. [45]

    M. E. Brooks-Bartlett, S. T. Banks, L. D. C. Jaubert, A. Harman-Clarke, and P. C. W. Holdsworth, Magnetic- moment fragmentation and monopole crystallization, Phys. Rev. X 4, 011007 (2014)

    2014

  38. [46]

    Petit, E

    S. Petit, E. Lhotel, B. Canals, M. Ciomaga Hatnean, J. Ollivier, H. Mutka, E. Ressouche, A. R. Wildes, M. R. Lees, and G. Balakrishnan, Observation of magnetic frag- mentation in spin ice, Nature Physics 12, 746 (2016)

    2016

  39. [47]

    Lefran¸ cois, V

    E. Lefran¸ cois, V. Cathelin, E. Lhotel, J. Robert, P. Le- jay, C. V. Colin, B. Canals, F. Damay, J. Ollivier, B. F ˚ ak, L. C. Chapon, R. Ballou, and V. Simonet, Fragmentation in spin ice from magnetic charge injection, Nature Com- munications 8, 209 (2017)

    2017

  40. [48]

    Zorko, M

    A. Zorko, M. Pregelj, M. Klanjˇ sek, M. Gomilˇ sek, Z. Jagliˇ ci´ c, J. S. Lord, J. A. T. Verezhak, T. Shang, W. Sun, and J.-X. Mi, Coexistence of magnetic order and persistent spin dynamics in a quantum kagome anti- ferromagnet with no intersite mixing, Phys. Rev. B 99, 214441 (2019)

    2019

  41. [49]

    Mauws, A

    C. Mauws, A. M. Hallas, G. Sala, A. A. Aczel, P. M. Sarte, J. Gaudet, D. Ziat, J. A. Quilliam, J. A. Lussier, M. Bieringer, H. D. Zhou, A. Wildes, M. B. Stone, D. Abernathy, G. M. Luke, B. D. Gaulin, and C. R. Wiebe, Dipolar-octupolar ising antiferromagnetism in sm2ti2o7: A mo...

    2018

  42. [50]

    Y.-Q. Wang, C. Liu, and Y.-M. Lu, Theory of topo- logical defects and textures in two-dimensional quan- tum orders with spontaneous symmetry breaking, arXiv e-prints , arXiv:2211.13207 (2022) , arXiv:2211.13207 [cond-mat.str-el]

    2022 arXiv

  43. [51]

    Vijayvargia, E

    A. Vijayvargia, E. Day-Roberts, A. S. Botana, and O. Erten, Altermagnets with topological order in kitaev bilayers (2025), arXiv:2503.09705 [cond-mat.str-el]

    2025 arXiv

  44. [52]

    Fornoville and L

    M. Fornoville and L. Janssen, Fractionalized fermionic multicriticality in anisotropic kitaev spin-orbital liquids (2025), arXiv:2505.01493 [cond-mat.str-el]

    2025

  45. [53]

    Corboz, M

    P. Corboz, M. Lajk´ o, A. M. L¨ auchli, K. Penc, and F. Mila, Spin-orbital quantum liquid on the honeycomb lattice, Phys. Rev. X 2, 041013 (2012) . Supplementary material for ‘Magnetically ordered yet topologicall y robust phases emerging in concurrent Kitaev spin liquids’ Muh...

    2012

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.