Pith. sign in

REVIEW 3 major objections 5 minor 74 references

Multiple Dirac Spin-Orbital Liquids in SU(4) Heisenberg Antiferromagnets on the Honeycomb Lattice

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read At multiple hopping-parameter sub-manifolds, the J=3/2 Hubbard model of d1 honeycomb trihalides reaches the same SU(4) Heisenberg antiferromagnet, but yields distinct Dirac spin-orbital liquids whose different symmetry implementations show

desk verdict A solid SU(4) extension paper: new hopping manifolds and symmetry fingerprints are real, but the DSOL ground state is imported, so the title overpromises a bit. read the letter →

arxiv 2508.18372 v1 pith:VT7REIAW submitted 2025-08-25 cond-mat.str-el cond-mat.mes-hallcond-mat.mtrl-scicond-mat.other

Manoj Gupta , Arijit Haldar , Subhro Bhattacharjee , Tanusri Saha-Dasgupta This is my paper · ORCID
classification cond-mat.str-elcond-mat.mes-hallcond-mat.mtrl-scicond-mat.other
keywords spin-orbitalliquidSU(4)HeisenbergantiferromagnethoneycomblatticeDiracquantumspinJ=3/2Hubbardmodeldynamicalstructurefactortransitionmetaltrihalidespartonmeanfield
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 argues that SU(4) symmetry in the J=3/2 Hubbard model of d1 honeycomb trihalides is not a one-off accident of indirect hopping, as previously thought. Solving the loop-product condition shows the symmetry appears on eight sub-manifolds of the four-hopping-parameter space, and three representative cases — direct, indirect, and the material-relevant r = −1/2 limit — all reduce to the same nearest-neighbour SU(4) Heisenberg antiferromagnet. The twist is that the site-dependent rotations needed to expose the symmetry differ (4-, 8-, and 12-site patterns), so the microscopic symmetries act differently even though the local spin Hamiltonian is identical. Each case is therefore a different symmetry-enriched U(1) Dirac spin-orbital liquid, with parton Dirac points in different places and distinct dynamical dipole structure factors. Because these S(q,ω) patterns are measurable, the paper gives a concrete route to identify which liquid a material actually hosts.

What carries the argument

The load-bearing object is the directed loop product of the four-by-four hopping matrices around a hexagon, ∏ T_ij = W0 Σ0 + Σ_{α≠0} W_α Σ_α; SU(4) symmetry emerges exactly when all fifteen non-identity polynomials W_α vanish. The argument then uses site-dependent unitary rotations ψ_i = g_i φ_i, chosen in 4-, 8-, or 12-site repeating patterns, to bring each SU(4) sub-manifold to the same local nearest-neighbour SU(4) Heisenberg antiferromagnet. The parton mean-field ansatz — four fermionic partons per site with a π-flux sign configuration around each hexagon — turns the model into four copies of graphene in a π flux, and the different g_i patterns determine where the resulting Dirac cones s

What would settle it

The claim would be settled by an unbiased numerical ground-state study of the NN SU(4) Heisenberg model on the honeycomb lattice: if it finds a gapped topological state with no gapless Dirac cones, the DSOL premise fails. Experimentally, inelastic neutron scattering on a d1 honeycomb trihalide in the r ≈ −1/2 regime that does not show the predicted extra K-point spectral weight and bifurcated third peak at Γ′, X and M1 would falsify the case-III fingerprint.

Watch

Extended reading notes

Core claim

The central claim is that the SU(4)-symmetric Heisenberg antiferromagnet on the honeycomb lattice can be reached from several inequivalent microscopic hopping limits of the J=3/2 Hubbard model, and the resulting Dirac spin-orbital liquids are physically distinct. For the direct limit (tπ = tσ, tm = 0), the identity rotation works on a 4-site unit cell; for the indirect limit (tm only), an 8-site pattern of rotations is required; for the realistic r = −1/2 direct-hopping limit, a 12-site pattern is required. In the local basis all three give the same NN SU(4) Heisenberg antiferromagnet, but the global-basis symmetries are implemented non-trivially and differently in each case. Within a parton

Load-bearing premise

The paper assumes the nearest-neighbour SU(4) Heisenberg antiferromagnet on the honeycomb lattice really has a gapless U(1) Dirac spin liquid as its ground state — a result imported from earlier numerics, not re-derived here — and if the gapped topological phase proposed elsewhere wins, the three DSOL fingerprints would not describe the material.

Editorial extensions

If this is right

  • If the central claim is right, SU(4) Dirac spin-orbital liquids are not confined to the indirect-hopping limit; direct-hopping compounds with tπ/tσ ≈ −1/2, the ratio found in ab initio studies, are also candidate DSOL materials.
  • Each case is a distinct symmetry-enriched phase, so a material's DSOL can be identified by the momentum pattern of its dipole response rather than by its local spin Hamiltonian.
  • Inelastic neutron scattering can in principle distinguish the three cases: case-I has zero dipolar response at Γ, case-II develops a six-fold star pattern at ω ≈ 2Jχ, and case-III shows extra K-point weight and bifurcated high-energy peaks.
  • Momentum-integrated probes (S(ω)) cannot tell the three liquids apart, while Raman or infrared probes at q = 0 can.
  • Tuning the hopping hierarchy — by changing metal or halide, or by strain — could move a material between different DSOLs, turning the parameter space into a playground for accessible quantum liquids.

Reading between the lines

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

  • The same logic should apply to the other five SU(4) sub-manifolds identified by the loop-product equations; each likely defines another symmetry-enriched DSOL with its own rotation pattern and scattering fingerprint, which the paper does not work out.
  • The symmetry-enrichment mechanism — identical local Hamiltonian, different implementation of microscopic symmetries — is general and could occur in other SU(N) magnets and on other bipartite lattices, not only honeycomb d1 trihalides.
  • A direct test of the case-III fingerprint could be made by computing S(q,ω) for the Gutzwiller-projected wave function rather than the unprojected mean-field state; the paper argues it qualitatively survives but does not show the projected spectra.
  • Because the ground-state competition with the gapped phase is unresolved, the fingerprints are conditional: if future numerics put the gapped phase lower, the same rotations would still define interesting symmetry-enriched states, just not gapless Dirac liquids.
Share X Bluesky LinkedIn Reddit HN

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. The paper studies the strong-coupling J=3/2 Hubbard model for d1 transition-metal trihalides on the honeycomb lattice, with four nearest-neighbor hopping pathways (tσ, tπ, tm, tm′). Using the hexagon loop-product criterion of Eq. (3), the authors identify several sub-manifolds of the hopping-parameter space on which the effective spin Hamiltonian becomes an SU(4) Heisenberg antiferromagnet. They select three representative cases—direct (tπ=tσ), indirect (tm only), and the material-relevant r=tπ/tσ=-1/2 limit—which require site-dependent unitary rotations with 4-, 8-, and 12-site patterns, respectively. Assuming the nearest-neighbor SU(4) honeycomb antiferromagnet has a U(1) Dirac spin-liquid ground state, the paper computes parton mean-field band structures and dynamical dipole structure factors S(q,ω) for the three cases and finds distinct momentum-resolved fingerprints. The central claim is that these are three distinct symmetry-enriched Dirac spin-orbital liquids with observable spectroscopic signatures.

Significance. If the central claim holds, the result is significant: it substantially generalizes the SU(4) DSOL proposal of Yamada et al. beyond the restrictive indirect-hopping limit, connects to the ab-initio hopping hierarchy with case III closest to realistic materials, and shows that the microscopic implementation of symmetries—not just the SU(4) algebra—can be diagnosed through S(q,ω). The algebraic identification of the SU(4) sub-manifolds is derived from constraint solving rather than fitting, and the S(q,ω) predictions are concrete and falsifiable. A clear strength is the detailed Supplementary Material, which provides the hopping matrices, rotation matrices u(i), and the parton mean-field structure-factor derivation in enough detail to reproduce the calculations. The main caveat is that the physical realization of all three DSOLs is inherited from an externally assumed ground state of the SU(4) Heisenberg model, and the paper does not independently establish that ground state.

major comments (3)
  1. [Eq. (3) and Fig. 2] The enumeration of the eight SU(4) sub-manifolds is asserted but not demonstrated. Eq. (3) defines fifteen polynomials Wα in terms of the four hopping parameters, and the Supplementary Materials state that setting Wα=0 yields eight independent solutions, but neither the polynomials nor the full solution set are shown. The main text lists only three cases (Eq. (4)), and the Fig. 2 caption gives partial parameterizations. Since the existence of 'multiple DSOLs' and the representativeness of the three selected cases rest on the completeness and correctness of this enumeration, please provide the explicit Wα polynomials and all eight solution branches (including any with tm′≠0) in the SM, or a reproducible computer-algebra script.
  2. ['The U(1)-Dirac spin-orbital liquid', third paragraph] The central physical conclusion is conditional on the nearest-neighbor SU(4) honeycomb Heisenberg antiferromagnet having a gapless U(1) Dirac spin-liquid ground state. This premise is imported from Refs. [30,31], while Ref. [44] is acknowledged to propose an alternative gapped topological phase. The paper neither resolves this competition nor quantifies the regime of stability. If the gapped phase is the true ground state, the computed band structures and S(q,ω) fingerprints for all three cases do not describe the model. I request a more quantitative assessment—for example, a variational comparison of the projected DSOL with the gapped ansatz, or a statement of the parameter regime where the DSOL is stable and whether case III lies in that regime. At minimum, the title and abstract should present the three DSOLs as candidate phases conditional on this premise.
  3. [SM, Eq. (S6.14)] The structure-factor formula appears to contain an inconsistency in the Γ-matrix product. In Eq. (S6.11) and in the intermediate expression after the Wick contraction (with k1=k2=k, a=d, b=c), the product is Γ(k+q)*_{Ma} Γ(k)_{Mb} Γ(k)*_{Nb} Γ(k+q)_{Na}. Eq. (S6.14), however, contains Γ(k)*_{Ma} Γ(k)_{Na} Γ(k+q)_{Mb} Γ(k+q)*_{Nb}. These are not obviously equal. Since this formula is the basis for the central S(q,ω) predictions in Fig. 4, please correct the typo or clarify the rearrangement.
minor comments (5)
  1. [Eq. (3)] The product symbol '⟨ij⟩∈7' is unclear; it should denote the six bonds of a hexagon, perhaps using a hexagon symbol or 'H'.
  2. [Abstract and main text] The spelling 'tri-halides' and 'trihalides' is used inconsistently; please unify.
  3. [Full text] There is a typo: 'antiferromagnmetn' should be 'antiferromagnetic'.
  4. [Reference [21]] Reference [21] is incomplete: the first author is missing a name ('B. and S. N. Flengas').
  5. [Fig. 4] The color contrast adjustment for the low-energy panel in case III is mentioned but not apparent from the figure; consider adding an inset or separate panel with a linear scale to make the grey-scale features visible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SU(4) sub-manifolds are obtained by solving algebraic constraints on the hopping matrices, and the S(q,ω) fingerprints are derived consequences of the parton mean-field ansatz, not re-fitted inputs.

full rationale

The paper's central new step is the identification of multiple hopping-parameter sub-manifolds with SU(4) symmetry. This is done by solving the fifteen polynomial equations Wα=0 coming from the hexagon loop product condition (Eq. 3), which is a constraint-solving calculation, not a fit. The three cases in Eq. 4 and Fig. 2 are specified directly by hopping parameters, and the corresponding local rotations g_i are constructed to bring each hopping model to the same local SU(4) Heisenberg form. The dynamical structure factor is then computed from these definitions via the explicit form factors in SM S6, using standard mean-field Wick contractions; no observable is used to tune the g_i or the mean-field parameters. The statement that S(ω) is the same for all three cases while the momentum-resolved S(q,ω) differs due to the static rotations g_i confirms that the spectral differences are derived consequences, not circular inputs. The DSOL ground-state premise is imported from external VMC/DMRG studies (Refs. [30,31]), and the paper explicitly acknowledges the alternative gapped phase of Ref. [44]; this is a correctness risk or conditional statement, not a circular reduction, because the present work does not use measured data or its own conclusions to force that premise. The self-citations to Refs. [32,33] supply the ab-initio hopping hierarchy and the J=3/2 formalism; these are prior calculations with stated assumptions, independent of the present paper's new result, and are not used to define the target conclusion. Overall, no load-bearing step reduces by construction to its own inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new particles or forces are introduced; the emergent U(1) gauge field and partons are standard tools. The central claim rests on the material model of Ref [32] (the authors' own ab initio work), the imported DSOL ground state, and the standard large-N QED3 stability argument. Free parameters are limited to the chosen case-III ratio and the internally self-consistent mean-field χ.

free parameters (2)
  • r = tπ/tσ = -1/2 (case III ratio) = -1/2
    Chosen by hand to match the ab initio hopping hierarchy of Ref [32]; it defines the materials-realistic SU(4) point. The claim that case III is realistic depends on this input.
  • Mean-field gap parameter χ = self-consistent
    Determined internally by the mean-field gap equation at 1/4 filling (Eq. S5.16), not fitted to external data; it sets the energy scale of the S(q,ω) plots.
assumptions (4)
  • domain assumption The low-energy physics of d^1 alpha-MX3 is captured by the four-orbital J=3/2 Hubbard model with the hopping structure of Ref [32] (Eq. 1).
    The entire search for SU(4) points is performed inside this model; if the ab initio hopping hierarchy were inaccurate, the claimed sub-manifolds would not map to real materials. The hierarchy originates in the authors' own previous calculation.
  • domain assumption In the strong-SOC, strong-U limit the t2g manifold projects onto the J=3/2 quartet and the exchange reduces to the NN SU(4) Heisenberg term of Eq. 6.
    Standard strong-coupling expansion invoked before Eq. 6; higher-order ring exchanges and J=1/2 admixture are neglected.
  • domain assumption The ground state of the NN SU(4) Heisenberg antiferromagnet on the honeycomb lattice is a U(1) Dirac spin liquid.
    Imported from VMC and DMRG studies (Refs [30,31]); the paper explicitly leaves open the competing gapped phase of Ref [44], making this premise load-bearing for the DSOL interpretation.
  • standard math Compact U(1) gauge-theory instanton events are irrelevant at four fermion flavors.
    Assumed to justify dropping instanton amplitudes after Eq. 11, citing Refs [48,49]; needed for the gapless DSOL to remain stable.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Multiple Dirac Spin-Orbital Liquids in SU(4) Heisenberg Antiferromagnets on the Honeycomb Lattice." pith.science (2026). https://pith.science/paper/VT7REIAW

@misc{pith2026250818372,
  author       = {Pith},
  title        = {Pith review of: Multiple Dirac Spin-Orbital Liquids in SU(4) Heisenberg Antiferromagnets on the Honeycomb Lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VT7REIAW}},
  note         = {Machine review of arXiv:2508.18372}
}
abstract

We study the strong coupling model of $d^1$ transition metal tri-halides in the large spin-orbit coupled limit. By considering ab-initio-calculation-inspired hierarchy of hopping pathways of these compounds, SU(4) symmetry is found to emerge at multiple points in the parameter space of the hopping parameters. The resultant Dirac spin-orbital liquids, within the parton mean field description, are distinct. The calculated dynamical structure factor fingerprints this distinctive nature, giving rise to observable effects. This opens up a playground for SU(4) Dirac Spin-Orbital liquid in $d^1$ Honeycomb lattice systems.

Figures

Figures reproduced from arXiv: 2508.18372 by the authors.

Figure 2
Figure 2. FIG. 2. The sub-space of hopping pathways with enhanced [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Site-dependent unitary rotation matrices, [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Dynamical dipole structure factor [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

74 extracted references · 56 canonical work pages

  1. [44]

    M. G. Yamada, M. Oshikawa, and G. Jackeli, Phys. Rev. B 104, 224436 (2021)

    work page 2021

  2. [1]

    Witczak-Krempa, G

    W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Ba- 7 lents, Annual Review of Condensed Matter Physics5, 57 (2014)

    work page 2014

  3. [2]

    J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Annual Review of Condensed Matter Physics7, 195 (2016)

    2016

  4. [3]

    Rousochatzakis, N

    I. Rousochatzakis, N. B. Perkins, Q. Luo, and H.-Y. Kee, Reports on Progress in Physics87, 026502 (2024)

    2024

  5. [4]

    tπ = −7+ √ 3 5 tσ, tm = 0; [5] tπ = −7− √ 3 5 tσ, tm = 0. (b) In hyper-plane of tσ = 1: [6] tm ∈ R, tπ = − 1 2, t′ m = −tm ± √ 8t2m+3 2 √ 2 ; [7] tm ∈ R, tπ = − 1 2, t′ m = − tm 2 ; [8] tπ ∈ R, tm = ± 1 3 q 2 3 (5t2π + 14tπ − 1), t′ m = − tm 2 . The planes tm = 0 (shaded light blue), tπ = − tσ 2 (shaded gray) andt′ m = − tm 2 (shaded gray), tπ = − 1 2(sha...

    work page 2020

  6. [5]

    Khomskii, ECS Journal of Solid State Science and Technology11, 054004 (2022)

    D. Khomskii, ECS Journal of Solid State Science and Technology11, 054004 (2022)

    work page 2022

  7. [6]

    Kitaev, Annals of Physics321, 2 (2006), january Spe- cial Issue

    A. Kitaev, Annals of Physics321, 2 (2006), january Spe- cial Issue

    work page 2006

  8. [7]

    Jackeli and G

    G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009)

    2009

Show all 74 references
  1. [8]

    K. A. Ross, L. Savary, B. D. Gaulin, and L. Balents, Phys. Rev. X1, 021002 (2011)

    2011

  2. [9]

    Shannon, O

    N. Shannon, O. Sikora, F. Pollmann, K. Penc, and P. Fulde, Phys. Rev. Lett.108, 067204 (2012)

    2012

  3. [10]

    J. G. Rau and M. J. Gingras, Annual Review of Con- densed Matter Physics10, 357 (2019)

    2019

  4. [11]

    J. S. Gardner, M. J. P. Gingras, and J. E. Greedan, Rev. Mod. Phys.82, 53 (2010)

    2010

  5. [12]

    M. J. Harris, S. T. Bramwell, D. F. McMorrow, T. Zeiske, and K. W. Godfrey, Phys. Rev. Lett.79, 2554 (1997)

    1997

  6. [13]

    Balents, Nature464, 199 (2010)

    L. Balents, Nature464, 199 (2010)

    2010

  7. [14]

    Savary and L

    L. Savary and L. Balents, Reports on Progress in Physics 80, 016502 (2017)

    2017

  8. [15]

    P. A. Lee, Science321, 1306 (2008)

    2008

  9. [16]

    Broholm, R

    C. Broholm, R. J. Cava, S. Kivelson, D. Nocera, M. Nor- man, and T. Senthil, Science367, eaay0668 (2020)

    2020

  10. [17]

    Khaliullin, Progress of Theoretical Physics Supple- ment 160, 155 (2005)

    G. Khaliullin, Progress of Theoretical Physics Supple- ment 160, 155 (2005)

    2005

  11. [18]

    Banerjee, C

    A. Banerjee, C. Bridges, J.-Q. Yan, A. Aczel, L. Li, M. Stone, G. Granroth, M. Lumsden, Y. Yiu, J. Knolle, et al., Nature materials15, 733 (2016)

    2016

  12. [19]

    Takagi, T

    H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Nature Reviews Physics1, 264 (2019)

    2019

  13. [20]

    Hermanns, I

    M. Hermanns, I. Kimchi, and J. Knolle, Annual Review of Condensed Matter Physics9, 17 (2018)

    2018

  14. [21]

    Trebst and C

    S. Trebst and C. Hickey, Physics Reports950, 1 (2022)

    2022

  15. [22]

    B. and S. N. Flengas, Canadian Journal of Chemistry 42, 1495 (1964)

    1964

  16. [23]

    Swaroop and S

    B. Swaroop and S. N. Flengas, Canadian Journal of Physics 42, 1886 (1964)

    1964

  17. [24]

    Brauer, Handbuch der Präparativen Anorgan- ischen Chemie, Bd

    G. Brauer, Handbuch der Präparativen Anorgan- ischen Chemie, Bd. II (Ferdinand Enke Verlag, Stuttgart, 1978) available at https://archive.org/ details/handbuchderprpar02brau

    1978

  18. [25]

    M. G. Yamada, M. Oshikawa, and G. Jackeli, Phys. Rev. Lett. 121, 097201 (2018)

    2018

  19. [26]

    Affleck and J

    I. Affleck and J. B. Marston, Phys. Rev. B 37, 3774 (1988)

    1988

  20. [27]

    J. B. Marston and I. Affleck, Phys. Rev. B39, 11538 (1989)

    1989

  21. [28]

    Read and S

    N. Read and S. Sachdev, Nuclear Physics B316, 609 (1989)

    1989

  22. [29]

    Read and S

    N. Read and S. Sachdev, Phys. Rev. B42, 4568 (1990)

    1990

  23. [30]

    Hermele and V

    M. Hermele and V. Gurarie, Phys. Rev. B84, 174441 (2011)

    2011

  24. [31]

    Corboz, M

    P. Corboz, M. Lajkó, A. M. Läuchli, K. Penc, and F. Mila, Phys. Rev. X2, 041013 (2012)

    2012

  25. [32]

    H.-K. Jin, W. M. H. Natori, and J. Knolle, Phys. Rev. B 107, L180401 (2023)

    2023

  26. [33]

    Gupta, B

    M. Gupta, B. Mondal, S. Bhattacharjee, and T. Saha Dasgupta, Phys. Rev. Res.5, 043219 (2023)

    2023

  27. [34]

    Mondal, V

    B. Mondal, V. B. Shenoy, and S. Bhattacharjee, Phys. Rev. B108, 245106 (2023)

    2023

  28. [35]

    Supplementary Material includes discussion on Low en- ergy electronic physics of α-MX3, Fermionic represen- tation of SU(4) spins, Rotation of SU(4) operators from globaltolocalbasis, SU(4)spinHamiltonianinglobalba- sis, mean-field decoupling and dynamical structure factor cal...

  29. [36]

    (2) refers to expansion of the hopping matrix in J-3/2 basis asΣ’s, while tσ, tπ, tm and t′ m denote the hopping pathways in t2g basis

    Though we used the same notations,t(α) ij in Eq. (2) refers to expansion of the hopping matrix in J-3/2 basis asΣ’s, while tσ, tπ, tm and t′ m denote the hopping pathways in t2g basis

  30. [37]

    Wang and A

    F. Wang and A. Vishwanath, Phys. Rev. B80, 064413 (2009)

    2009

  31. [38]

    Yamashita, N

    Y. Yamashita, N. Shibata, and K. Ueda, Phys. Rev. B 58, 9114 (1998)

    1998

  32. [39]

    Frischmuth, F

    B. Frischmuth, F. Mila, and M. Troyer, Phys. Rev. Lett. 82, 835 (1999)

    1999

  33. [40]

    Corboz, A

    P. Corboz, A. M. Läuchli, K. Penc, M. Troyer, and F. Mila, Phys. Rev. Lett.107, 215301 (2011)

    2011

  34. [41]

    Vernay, K

    F. Vernay, K. Penc, P. Fazekas, and F. Mila, Phys. Rev. B 70, 014428 (2004)

    2004

  35. [42]

    K. Penc, M. Mambrini, P. Fazekas, and F. Mila, Phys. Rev. B68, 012408 (2003)

    2003

  36. [43]

    W. M. H. Natori, E. C. Andrade, and R. G. Pereira, Phys. Rev. B98, 195113 (2018)

    2018

  37. [45]

    M. G. Yamada and S. Fujimoto, Phys. Rev. B 105, L201115 (2022)

    2022

  38. [46]

    Wen, Phys

    X.-G. Wen, Phys. Rev. B65, 165113 (2002)

    2002

  39. [47]

    Vörös and K

    D. Vörös and K. Penc, Phys. Rev. B108, 214407 (2023)

    2023

  40. [48]

    Jakab, E

    D. Jakab, E. Szirmai, M. Lewenstein, and G. Szirmai, Phys. Rev. B93, 064434 (2016)

    2016

  41. [49]

    Murthy and S

    G. Murthy and S. Sachdev, Nuclear Physics B344, 557 (1990)

    1990

  42. [50]

    A. W. Sandvik, Phys. Rev. Lett.98, 227202 (2007)

    2007

  43. [51]

    Note that these dipole correlations are different from the SU(4) color correlations plotted in Ref- PhysRevX.2.041013

  44. [52]

    G. Chen, R. Pereira, and L. Balents, Phys. Rev. B82, 174440 (2010)

    2010

  45. [53]

    Ishikawa, T

    H. Ishikawa, T. Takayama, R. K. Kremer, J. Nuss, R. Dinnebier, K. Kitagawa, K. Ishii, and H. Takagi, Phys. Rev. B100, 045142 (2019)

    2019

  46. [54]

    Romhányi, L

    J. Romhányi, L. Balents, and G. Jackeli, Phys. Rev. Lett. 118, 217202 (2017)

    2017

  47. [55]

    Bhattacharjee, S.-S

    S. Bhattacharjee, S.-S. Lee, and Y. B. Kim, New Journal of Physics14, 073015 (2012)

    2012

  48. [56]

    K. I. Kugel and D. Khomski˘ ı, Soviet Physics Uspekhi25, 231 (1982)

    1982

  49. [57]

    Y. Q. Li, M. Ma, D. N. Shi, and F. C. Zhang, Phys. Rev. Lett. 81, 3527 (1998)

    1998

  50. [58]

    K. I. Kugel, D. I. Khomskii, A. O. Sboychakov, and S. V. Streltsov, Phys. Rev. B91, 155125 (2015)

    2015

  51. [59]

    M.A.McGuire,Crystals 7(2017),10.3390/cryst7050121

    2017 doi

  52. [60]

    Ogawa, Journal of the Physical Society of Japan15, 1901 (1960)

    S. Ogawa, Journal of the Physical Society of Japan15, 1901 (1960)

    1901

  53. [61]

    Troyanov, E

    S. Troyanov, E. Snigireva, and V. Rybakov, Russian journal of inorganic chemistry36, 634 (1991)

    1991

  54. [62]

    S. I. Troyanov, E. M. Snigireva, A. P. Pirarevskii, A. I. Yanovskii, and Y. T. Struchkov, Russian Journal of In- organic Chemistry39, 360 (1994)

    1994

  55. [63]

    J. J. Yang, Y. J. Choi, Y. S. Oh, A. Hogan, Y. Horibe, K. Kim, B. I. Min, and S.-W. Cheong, Phys. Rev. Lett. 8 108, 116402 (2012)

    2012

  56. [64]

    Yoshida, K

    M. Yoshida, K. Kudo, M. Nohara, and Y. Iwasa, Nano Letters 18, 3113 (2018)

    2018

  57. [65]

    Chaubey, B

    A. Chaubey, B. Mondal, V. B. Shenoy, and S. Bhat- tacharjee, arXiv preprint arXiv:2505.04945 (2025)

    2025 arXiv

  58. [66]

    Lajkó and K

    M. Lajkó and K. Penc, Phys. Rev. B87, 224428 (2013)

    2013

  59. [67]

    U. F. Seifert, J. Willsher, M. Drescher, F. Pollmann, and J. Knolle, nature communications15, 7110 (2024)

    2024

  60. [68]

    U. F. P. Seifert, X.-Y. Dong, S. Chulliparambil, M. Vojta, H.-H. Tu, and L. Janssen, Phys. Rev. Lett.125, 257202 (2020)

    2020

  61. [69]

    van Den Bossche, F.-C

    M. van Den Bossche, F.-C. Zhang, and F. Mila, The Eu- ropean Physical Journal B-Condensed Matter and Com- plex Systems17, 367 (2000)

    2000

  62. [70]

    van den Bossche, P

    M. van den Bossche, P. Azaria, P. Lecheminant, and F. Mila, Physical Review Letters86, 4124 (2001)

    2001

  63. [71]

    W. M. H. Natori, H.-K. Jin, and J. Knolle, Phys. Rev. B 108, 075111 (2023)

    2023

  64. [72]

    W. M. H. Natori, R. Nutakki, R. G. Pereira, and E. C. Andrade, Phys. Rev. B100, 205131 (2019)

    2019

  65. [73]

    Murakami, N

    S. Murakami, N. Nagosa, and S.-C. Zhang, Phys. Rev. B 69, 235206 (2004)

    2004

  66. [74]

    Haber, SciPost Physics Lecture Notes (2021), 10.21468/scipostphyslectnotes.21

    H. Haber, SciPost Physics Lecture Notes (2021), 10.21468/scipostphyslectnotes.21. 1 Supplementary Materials Multiple Dirac Spin-Orbital Liquids in SU(4) Heisenberg Antiferromagnets on the Honeycomb Lattice Manoj Gupta1, Arijit Haldar1, Subhro Bhattacharjee2, and Tanusri Saha-D...

    2021 doi

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.