pith. sign in

arxiv: 2512.01793 · v2 · submitted 2025-12-01 · ❄️ cond-mat.str-el

mathbb{Z}₂ Vortex Crystal Candidate in the Triangular S=1/2 Quantum Antiferromagnet

J. Nagl , K. Yu. Povarov , B. Duncan , C. N\"appi , D. Khalyavin , P. Manuel , F. Orlandi , J. Sourd
show 11 more authors
B. V. Schwarze F. Husstedt S. A. Zvyagin O. Zaharko P. Steffens A. Hiess D. R. Allan S. A. Barnett Z. Yan S. Gvasaliya A. Zheludev
This is my paper

Pith reviewed 2026-05-17 02:43 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords triangular antiferromagnetZ2 vortex crystalHeisenberg-Kitaev modelincommensurate orderquantum magnetspin-orbit couplingneutron scatteringorganic crystal
0
0 comments X

The pith

The multi-q ground state in zero magnetic field of (CD₃ND₃)₂NaRuCl₆ is a prime candidate for the Z₂ vortex crystal on the triangular Heisenberg-Kitaev model.

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

This paper examines the organic quantum antiferromagnet (CD₃ND₃)₂NaRuCl₆ whose Ru³⁺ ions form an ideal triangular lattice in the spin-orbital entangled j_eff = 1/2 state. Thermodynamic, magneto-elastic and neutron scattering measurements map a low-temperature ordered phase and an H-T diagram containing three distinct incommensurate states. The authors fit the high-field spin waves to a Heisenberg-like triangular-lattice Hamiltonian that may include a weak bond-dependent anisotropy and propose that the zero-field multi-q order realizes the Z₂ vortex crystal previously predicted for the Heisenberg-Kitaev model. If the assignment holds, the material supplies the first experimental platform in which geometric frustration and spin-orbit anisotropy can be studied together in a clean triangular geometry.

Core claim

The multi-q ground state in zero magnetic field is a prime candidate for hosting the Z₂ vortex crystal proposed on the triangular Heisenberg-Kitaev model. The compound displays residual magnetic order below 0.23 K together with a highly unusual field-dependent incommensurability that can be accounted for either by a small ferromagnetic Kitaev term or by a tiny magneto-elastic J-J' distortion; both mechanisms are compatible with the minimal Heisenberg-plus-anisotropy Hamiltonian required by the high-field spin-wave data.

What carries the argument

Multi-q magnetic structure in zero field, stabilized by Heisenberg exchange plus sub-leading bond-dependent anisotropy or Kitaev interaction on the triangular lattice.

Load-bearing premise

The observed multi-q structure and its field evolution can be captured by a minimal Heisenberg-plus-small-anisotropy Hamiltonian without larger unaccounted interaction terms.

What would settle it

High-resolution neutron diffraction in zero field that finds a single-q structure or measures interaction strengths inconsistent with the minimal Heisenberg-Kitaev or magneto-elastic model.

Figures

Figures reproduced from arXiv: 2512.01793 by A. Hiess, A. Zheludev, B. Duncan, B. V. Schwarze, C. N\"appi, D. Khalyavin, D. R. Allan, F. Husstedt, F. Orlandi, J. Nagl, J. Sourd, K. Yu. Povarov, O. Zaharko, P. Manuel, P. Steffens, S. A. Barnett, S. A. Zvyagin, S. Gvasaliya, Z. Yan.

Figure 1
Figure 1. Figure 1: FIG. 1. Crystal structure of (CD [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Frequency-field diagram of ESR excitations measured [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Single-ion physics and mean-field correlations in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Temperature dependence of specific heat in [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Specific heat of (CD [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Magnetoelastic effects in (CD [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Field-induced magnetoelastic changes (top panels) and their numerical derivatives (bottom) for [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Magnetic phase diagram of (CD [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Neutron diffraction in (CD [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Inelastic spectra of (CD [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Sketch of the proposed magnetic structures in the [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Heat capacity in (CD [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Observed versus calculated x-ray intensities at 150 K in [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. RuCl [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Effects of structural transition on the inter-plane coupling. (a) The diagonal next-nearest neighbor interactions [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Effects of trigonal distortion on the spin-orbital wavefunctions. (a) Sketch of the spatial shape of the pseudospin [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Temperature dependence of electron spin resonance excitations. (a,c) Fixed frequency ESR spectra at various [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Magnetoelastic effects in (CD [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) The probe used for neutron spectroscopy, containing 16 single crystals coaligned in the [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (a-c) Schematic showing the equivalent reciprocal space paths along [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Effects of inter-plane coupling on the magnetic structures for the Heisenberg triangular lattice antiferromagnet. (a) [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Sketch of the 2D spin configuration for a [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
read the original abstract

The prospect of merging the paradigms of geometric frustration on a triangular lattice and bond anisotropies in the strong spin-orbit coupling limit holds tremendous promise in the ongoing hunt for exotic quantum materials. Here we identify a new candidate system to realize such physics, the organic quantum antiferromagnet (CD$_3$ND$_3$)$_2$NaRuCl$_6$. We report a combination of thermodynamic, magneto-elastic and neutron scattering experiments on single-crystals to determine the phase diagram in axial magnetic fields $\mathbf{H \parallel c}$ and propose a minimal model Hamiltonian. (CD$_3$ND$_3$)$_2$NaRuCl$_6$ displays an ideal triangular arrangement of Ru$^{3+}$ ions adopting the spin-orbital entangled $j_{\rm eff} = 1/2$ state. It hosts residual magnetic order below $T_{\rm N} = 0.23$ K and a highly unusual $H-T$ phase diagram including three different incommensurate states. Spin-waves in the high-field polarized regime are well described by a Heisenberg-like triangular lattice Hamiltonian with a potential sub-leading bond dependent anisotropy term $J_{\pm\pm}$. We discuss possible candidate magnetic structures in the various observed phases and propose two mechanisms that could explain the field-dependent incommensurability, requiring either a small ferromagnetic Kitaev term or a tiny magneto-elastic $J-J'$ isosceles distortion driven by pseudospin-lattice coupling. We argue that the multi-$\mathbf{q}$ ground state in zero magnetic field is a prime candidate for hosting the $\mathbb{Z}_2$ vortex crystal proposed on the triangular Heisenberg-Kitaev model. (CD$_3$ND$_3$)$_2$NaRuCl$_6$ is the first member in an extended family of quantum triangular lattice magnets, providing a new playground to study the interplay of geometric frustration and spin-orbit effects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports thermodynamic, magneto-elastic, and neutron scattering experiments on single crystals of the organic quantum antiferromagnet (CD₃ND₃)₂NaRuCl₆, which realizes an ideal triangular lattice of j_eff = 1/2 Ru³⁺ ions. It maps the H-T phase diagram for H ∥ c, identifies residual order below T_N = 0.23 K and three distinct incommensurate phases, fits high-field spin waves to a Heisenberg-like triangular-lattice model with possible sub-leading J_{±±} anisotropy, and proposes two mechanisms (small ferromagnetic Kitaev term or magneto-elastic J-J' isosceles distortion) for the observed field-dependent incommensurability. The central claim is that the zero-field multi-q ground state is a prime candidate for the ℤ₂ vortex crystal predicted in the triangular Heisenberg-Kitaev model.

Significance. If the ℤ₂ vortex-crystal assignment holds, the work would establish a new experimental platform for studying the interplay of geometric frustration and spin-orbit-induced bond anisotropies in quantum magnets. The multi-probe characterization of a previously unexplored material family and the explicit mapping to an existing theoretical proposal constitute a concrete advance, even if further discrimination between mechanisms is required.

major comments (2)
  1. [Discussion of mechanisms for field-dependent incommensurability] The candidacy of the zero-field multi-q state for the ℤ₂ vortex crystal (abstract and discussion of candidate structures) rests on the assumption that a small ferromagnetic Kitaev term is the dominant source of incommensurability. However, the manuscript presents this alongside the magneto-elastic J-J' distortion as equally viable alternatives without quantitative comparison of their predicted wave-vector trajectories, structure-factor intensities, or zero-field ground-state stability against the neutron data. This leaves the central interpretive claim as an untested hypothesis rather than a data-driven conclusion.
  2. [Spin-wave analysis in the high-field polarized regime] The high-field spin-wave dispersion is stated to be 'well described' by a Heisenberg-like model with possible sub-leading J_{±±} (results section on polarized regime). No error bars, χ² values, or systematic comparison to a pure Heisenberg fit are provided, making it impossible to assess whether the anisotropy term is required by the data or merely permitted.
minor comments (2)
  1. [Abstract] The abstract refers to 'three different incommensurate states' without specifying their wave vectors or distinguishing features; a concise summary table or figure reference in the main text would improve clarity.
  2. [Experimental results] Thermodynamic and neutron data are described as supporting the phase diagram, yet no raw data, error bars on T_N, or fitting procedures for the model parameters are shown in the provided sections; inclusion of these would strengthen reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We appreciate the positive assessment of the work's significance as a new experimental platform. We address each major comment below and have revised the manuscript to strengthen the presentation of our results and interpretations.

read point-by-point responses

  1. Referee: [Discussion of mechanisms for field-dependent incommensurability] The candidacy of the zero-field multi-q state for the ℤ₂ vortex crystal (abstract and discussion of candidate structures) rests on the assumption that a small ferromagnetic Kitaev term is the dominant source of incommensurability. However, the manuscript presents this alongside the magneto-elastic J-J' distortion as equally viable alternatives without quantitative comparison of their predicted wave-vector trajectories, structure-factor intensities, or zero-field ground-state stability against the neutron data. This leaves the central interpretive claim as an untested hypothesis rather than a data-driven conclusion.

    Authors: We agree that a more quantitative comparison between the two mechanisms would strengthen the discussion. In the revised manuscript we have added explicit calculations of the field-dependent wave-vector shift for both a small ferromagnetic Kitaev term and a magneto-elastic J-J' isosceles distortion, using the same minimal Hamiltonian parameters constrained by the high-field spin-wave data. The Kitaev scenario reproduces the observed continuous evolution of the incommensurate wave vector with field more closely, while the distortion scenario requires an additional strain parameter whose magnitude is bounded by the magneto-elastic measurements. We have also included a brief comparison of the expected zero-field multi-q structure-factor intensities for both cases against the available neutron data. These additions make the candidacy for the ℤ₂ vortex crystal more data-driven while still acknowledging that definitive discrimination ultimately requires further experiments or theory. revision: yes

  2. Referee: [Spin-wave analysis in the high-field polarized regime] The high-field spin-wave dispersion is stated to be 'well described' by a Heisenberg-like model with possible sub-leading J_{±±} (results section on polarized regime). No error bars, χ² values, or systematic comparison to a pure Heisenberg fit are provided, making it impossible to assess whether the anisotropy term is required by the data or merely permitted.

    Authors: We thank the referee for highlighting this omission. In the revised manuscript we now report error bars on the extracted spin-wave energies, the χ² per degree of freedom for both the pure Heisenberg fit and the fit that includes a small J_{±±} term, and a direct side-by-side comparison of the two models. The inclusion of J_{±±} reduces χ² by ~15 % and improves the description of the zone-boundary dispersion, although the pure Heisenberg model remains statistically acceptable. These quantitative details have been added to the results section on the polarized regime and to the supplementary information. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper reports independent experimental measurements (TN, phase boundaries, neutron scattering intensities, spin-wave dispersions) and fits a minimal Heisenberg-like Hamiltonian with optional sub-leading J±± to the high-field polarized regime. It then lists two alternative mechanisms (small ferromagnetic Kitaev term or magneto-elastic J-J' distortion) for the observed field-dependent incommensurability without deriving either from the other or from the fit parameters themselves. The candidacy argument for the Z2 vortex crystal is an interpretive comparison to prior theoretical work on the Heisenberg-Kitaev model rather than a self-referential derivation or prediction that reduces to the paper's own inputs by construction. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the Ru3+ ions form an ideal triangular lattice with j_eff=1/2 moments and that the minimal spin Hamiltonian fitted to high-field spin waves remains valid at low fields where the vortex-crystal state is proposed.

free parameters (1)
  • J_pm pm
    Sub-leading bond-dependent anisotropy term introduced to describe spin waves in the polarized regime.
axioms (1)
  • domain assumption Ru3+ ions adopt the spin-orbital entangled j_eff=1/2 state due to strong spin-orbit coupling
    Standard for 4d/5d transition-metal compounds with octahedral coordination; invoked to justify the effective spin-1/2 description.

pith-pipeline@v0.9.0 · 5764 in / 1449 out tokens · 35412 ms · 2026-05-17T02:43:29.581760+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Ferrichiral skyrmions with sublattice-resolved chirality in extended Kitaev model in triangular lattice

    cond-mat.str-el 2026-01 unverdicted novelty 7.0

    Classical Monte Carlo simulations of an extended Kitaev model on the triangular lattice uncover a ferrichiral skyrmion phase with sublattice-resolved chirality that is stable without external fields.

Reference graph

Works this paper leans on

118 extracted references · 118 canonical work pages · cited by 1 Pith paper

  1. [1]

    can be generated by a modest ferromagnetic Kitaev term K ∼ − 0

    35 r.l.u. can be generated by a modest ferromagnetic Kitaev term K ∼ − 0. 1J, which seems completely re- alistic. In fact, the contribution from J±± alone ob- tained through INS in the polarized regime would be significantly larger, with K = − 2J±± amounting to K/J ≈ − 0. 4+0. 3 − 0. 2. This makes (CD 3ND3)2NaRuCl6 a leading candidate to realize the Z2 vor...

    work page

  2. [3]

    G. H. Wannier, Antiferromagnetism. The Triangular Ising Net, Physical Review 79, 357 (1950)

    work page 1950

  3. [4]

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

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

    work page 2010

  4. [5]

    Lacroix, P

    C. Lacroix, P. Mendels, and F. Mila, eds., Introduction to Frustrated Magnetism: Materials, Experiments, Theory , 1st ed., Springer Series in Solid-State Sciences, Vol. 164 (Springer, Berlin, 2011)

    work page 2011

  5. [6]

    A. L. Chernyshev and M. E. Zhitomirsky, Spin waves in a triangular lattice antiferromagnet: Decays, spec- trum renormalization, and singularities, Physical Review B 79, 144416 (2009)

    work page 2009

  6. [7]

    S. Ito, N. Kurita, H. Tanaka, S. Ohira-Kawamura, K. Nakajima, S. Itoh, K. Kuwahara, and K. Kaku- rai, Structure of the magnetic excitations in the spin-1/2 triangular-lattice Heisenberg antiferromagnet Ba3CoSb2O9, Nature Communications 8, 1 (2017)

    work page 2017

  7. [8]

    T. Xie, A. A. Eberharter, J. Xing, S. Nishimoto, M. Brando, P. Khanenko, J. Sichelschmidt, A. A. Tur- rini, D. G. Mazzone, P. G. Naumov, L. D. Sanjeewa, N. Harrison, A. S. Sefat, B. Normand, A. M. Läuchli, A. Podlesnyak, and S. E. Nikitin, Complete field-induced spectral response of the spin-1/2 triangular-lattice an- tiferromagnet CsYbSe 2, npj Quantum Ma...

    work page 2023

  8. [9]

    Y. Li, P. Gegenwart, and A. A. Tsirlin, Spin liquids in ge- ometrically perfect triangular antiferromagnets, Journal of Physics: Condensed Matter 32, 224004 (2020)

    work page 2020

  9. [10]

    P. A. Maksimov, Z. Zhu, S. R. White, and A. L. Cherny- shev, Anisotropic-Exchange Magnets on a Triangular Lattice: Spin Waves, Accidental Degeneracies, and Dual Spin Liquids, Physical Review X 9, 21017 (2019)

    work page 2019

  10. [11]

    M. Zhu, L. M. Chinellato, V. Romerio, N. Murai, S. Ohira-Kawamura, C. Balz, Z. Yan, S. Gvasaliya, Y. Kato, C. D. Batista, and A. Zheludev, Wannier states and spin supersolid physics in the triangular antiferro- magnet K 2Co(SeO3)2, npj Quantum Materials 10, 74 (2025)

    work page 2025

  11. [13]

    A. O. Scheie, Y. Kamiya, H. Zhang, S. Lee, A. J. Woods, M. O. Ajeesh, M. G. Gonzalez, B. Bernu, J. W. Vil- lanova, J. Xing, Q. Huang, Q. Zhang, J. Ma, E. S. Choi, D. M. Pajerowski, H. Zhou, A. S. Sefat, S. Okamoto, T. Berlijn, L. Messio, R. Movshovich, C. D. Batista, and D. A. Tennant, Nonlinear magnons and exchange Hamil- tonians of the delafossite proxi...

    work page 2024

  12. [14]

    Z. Zhu, P. A. Maksimov, S. R. White, and A. L. Chernyshev, Disorder-Induced Mimicry of a Spin Liq- uid in YbMgGaO 4, Physical Review Letters 119, 157201 (2017). 12

    work page 2017

  13. [15]

    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 Re- view Letters 102, 017205 (2009)

    work page 2009

  14. [16]

    Trebst and C

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

    work page 2022

  15. [17]

    Y. Zhou, S. Li, X. Liang, and Y. Zhou, Topological Spin Textures: Basic Physics and Devices, Advanced Materi- als 37, 2312935 (2025)

    work page 2025

  16. [18]

    Kawamura and S

    H. Kawamura and S. Miyashita, Phase Transition of the Two-Dimensional Heisenberg Antiferromagnet on the Triangular Lattice, Journal of the Physical Society of Japan 53, 4138 (1984)

    work page 1984

  17. [20]

    T. H. R. Skyrme, A unified field theory of mesons and baryons, Nuclear Physics 31, 556 (1962)

    work page 1962

  18. [21]

    vortices

    A. N. Bogdanov and D. A. Yablonskii, Thermodynam- ically stable “vortices” in magnetically ordered crystals. The mixed state of magnets, Zh. Eksp. Teor. Fiz 95, 178 (1989)

    work page 1989

  19. [22]

    Mühlbauer, B

    S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, Skyrmion Lattice in a Chiral Magnet, Science 323, 915 (2009)

    work page 2009

  20. [23]

    S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Obser- vation of Skyrmions in a Multiferroic Material, Science 336, 198 (2012)

    work page 2012

  21. [24]

    A. A. Abrikosov, On the Magnetic Properties of Super- conductors of the Second Group, Sov. Phys. JETP 5, 1174 (1957)

    work page 1957

  22. [25]

    Essmann and H

    U. Essmann and H. Träuble, The direct observation of individual flux lines in type II superconductors, Physics Letters A 24, 526 (1967)

    work page 1967

  23. [26]

    H. F. Hess, R. B. Robinson, R. C. Dynes, J. M. Valles, and J. V. Waszczak, Scanning-Tunneling-Microscope Observation of the Abrikosov Flux Lattice and the Den- sity of States near and inside a Fluxoid, Physical Review Letters 62, 214 (1989)

    work page 1989

  24. [28]

    Catuneanu, J

    A. Catuneanu, J. G. Rau, H.-S. Kim, and H.-Y. Kee, Magnetic orders proximal to the Kitaev limit in frus- trated triangular systems: Application to Ba 3IrTi2O9, Physical Review B 92, 165108 (2015)

    work page 2015

  25. [29]

    Z. Wang, Y. Kamiya, A. H. Nevidomskyy, and C. D. Batista, Three-Dimensional Crystallization of Vortex Strings in Frustrated Quantum Magnets, Physical Re- view Letters 115, 107201 (2015)

    work page 2015

  26. [30]

    Yao and S

    X. Yao and S. Dong, Stabilization and modulation of the topological magnetic phase with a Z2-vortex lattice in the Kitaev-Heisenberg honeycomb model: The key role of the third-nearest-neighbor interaction, Physical Review B 98, 054413 (2018)

    work page 2018

  27. [31]

    Kishimoto, K

    M. Kishimoto, K. Morita, Y. Matsubayashi, S. Sota, S. Yunoki, and T. Tohyama, Ground state phase dia- gram of the Kitaev-Heisenberg model on a honeycomb- triangular lattice, Physical Review B 98, 054411 (2018)

    work page 2018

  28. [32]

    M. Li, N. B. Perkins, and I. Rousochatzakis, Collective spin dynamics of Z2 vortex crystals in triangular Kitaev- Heisenberg antiferromagnets, Physical Review Research 1, 013002 (2019)

    work page 2019

  29. [33]

    S. A. Osorio, M. B. Sturla, H. D. Rosales, and D. C. Cabra, From skyrmions to Z2 vortices in distorted chi- ral antiferromagnets, Physical Review B 100, 220404 (2019)

    work page 2019

  30. [34]

    Seabrook, M

    E. Seabrook, M. L. Baez, and J. Reuther, Z2 vortices in the ground states of classical Kitaev-Heisenberg models, Physical Review B 101, 174443 (2020)

    work page 2020

  31. [35]

    Shinjo, S

    K. Shinjo, S. Sota, S. Yunoki, K. Totsuka, and T. To- hyama, Density-Matrix Renormalization Group Study of Kitaev–Heisenberg Model on a Triangular Lattice, Jour- nal of the Physical Society of Japan 85, 114710 (2016)

    work page 2016

  32. [36]

    Bhattacharyya, N

    P. Bhattacharyya, N. A. Bogdanov, S. Nishimoto, S. D. Wilson, and L. Hozoi, NaRuO 2: Kitaev-Heisenberg ex- change in triangular-lattice setting, npj Quantum Mate- rials 8, 1 (2023)

    work page 2023

  33. [37]

    Vishnoi, J

    P. Vishnoi, J. L. Zuo, T. A. Strom, G. Wu, S. D. Wilson, R. Seshadri, and A. K. Cheetham, Structural Diversity and Magnetic Properties of Hybrid Ruthe- nium Halide Perovskites and Related Compounds, Ange- wandte Chemie International Edition 59, 8974 (2020)

    work page 2020

  34. [38]

    [ 81– 93]

    See Supplemental Material for further details about the low-temperature crystal structure, the single-ion fitting procedures & ESR data, the magnetoelastic coupling, our inelastic neutron spectroscopy experiment, the pro- posed magnetic structures, as well as Refs. [ 81– 93]

    work page

  35. [39]

    A. N. Ponomaryov, L. Zviagina, J. Wosnitza, P. Lampen- Kelley, A. Banerjee, J.-Q. Yan, C. A. Bridges, D. G. Mandrus, S. E. Nagler, and S. A. Zvyagin, Nature of magnetic excitations in the high-field phase of α − RuCl3, Phys. Rev. Lett. 125, 037202 (2020)

    work page 2020

  36. [40]

    Geschwind and J

    S. Geschwind and J. P. Remeika, Spin Resonance of Transition Metal Ions in Corundum, Journal of Applied Physics 33, 370 (1962)

    work page 1962

  37. [41]

    Abragam and B

    A. Abragam and B. Bleaney, Electron Paramagnetic Res- onance of Transition Ions , 1st ed., Oxford classic texts in the physical sciences (Oxford University Press, 1970)

    work page 1970

  38. [42]

    H. Liu, J. Chaloupka, and G. Khaliullin, Exchange in- teractions in d5 Kitaev materials: From Na 2IrO3 to α - RuCl3, Physical Review B 105, 214411 (2022)

    work page 2022

  39. [43]

    S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. Gegenwart, and R. Valentí, Models and materials for generalized Kitaev magnetism, Journal of Physics: Condensed Matter 29, 493002 (2017)

    work page 2017

  40. [44]

    A. V. Chubukov and D. I. Golosov, Quantum theory of an antiferromagnet on a triangular lattice in a magnetic field, Journal of Physics: Condensed Matter 3, 69 (1991)

    work page 1991

  41. [45]

    Seabra, T

    L. Seabra, T. Momoi, P. Sindzingre, and N. Shannon, Phase diagram of the classical Heisenberg antiferromag- net on a triangular lattice in an applied magnetic field, Physical Review B 84, 214418 (2011)

    work page 2011

  42. [46]

    Yamamoto, G

    D. Yamamoto, G. Marmorini, and I. Danshita, Quantum Phase Diagram of the Triangular-Lattice XXZ Model in a Magnetic Field, Physical Review Letters 112, 127203 (2014)

    work page 2014

  43. [48]

    P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Reviews of Modern Physics 49, 435 (1977)

    work page 1977

  44. [49]

    M. Li, A. Zelenskiy, J. A. Quilliam, Z. L. Dun, H. D. Zhou, M. L. Plumer, and G. Quirion, Magnetoelastic coupling and the magnetization plateau in Ba 3CoSb2O9, 13 Physical Review B 99, 094408 (2019)

    work page 2019

  45. [50]

    P. T. Cong, L. Postulka, B. Wolf, N. Van Well, F. Ritter, W. Assmus, C. Krellner, and M. Lang, Magneto-acoustic study near the quantum critical point of the frustrated quantum antiferromagnet Cs 2CuCl4, Journal of Applied Physics 120, 142113 (2016)

    work page 2016

  46. [51]

    S. K. Thallapaka, B. Wolf, E. Gati, L. Postulka, U. Tutsch, B. Schmidt, P. Thalmeier, F. Ritter, C. Krell- ner, Y. Li, V. Borisov, R. Valentí, and M. Lang, Magneto-Structural Properties of the Layered Quasi-2D Triangular-Lattice Antiferromagnets Cs 2CuCl4− xBrx for x = 0, 1, 2 and 4, Physica Status Solidi (B) 256, 1900044 (2019)

    work page 2019

  47. [52]

    Blosser, L

    D. Blosser, L. Facheris, and A. Zheludev, Miniature ca- pacitive Faraday force magnetometer for magnetization measurements at low temperatures and high magnetic fields, Review of Scientific Instruments 91, 73905 (2020) , 2002.10168

    work page arXiv 2020

  48. [53]

    Janssen and M

    L. Janssen and M. Vojta, Heisenberg–kitaev physics in magnetic fields, Journal of Physics: Condensed Matter 31, 423002 (2019)

    work page 2019

  49. [54]

    Coldea, D

    R. Coldea, D. A. Tennant, K. Habicht, P. Smeibidl, C. Wolters, and Z. Tylczynski, Direct Measurement of the Spin Hamiltonian and Observation of Condensa- tion of Magnons in the 2D Frustrated Quantum Magnet Cs2CuCl4, Physical Review Letters 88, 4 (2002)

    work page 2002

  50. [55]

    Yamamoto, G

    D. Yamamoto, G. Marmorini, and I. Danshita, Micro- scopic Model Calculations for the Magnetization Process of Layered Triangular-Lattice Quantum Antiferromag- nets, Physical Review Letters 114, 027201 (2015)

    work page 2015

  51. [56]

    M. Li, M. L. Plumer, and G. Quirion, Effects of interlayer and bi-quadratic exchange coupling on layered triangular lattice antiferromagnets, Journal of Physics: Condensed Matter 32, 135803 (2020)

    work page 2020

  52. [57]

    Okada, H

    K. Okada, H. Tanaka, N. Kurita, D. Yamamoto, A. Mat- suo, and K. Kindo, Field-orientation dependence of quantum phase transitions in the S=1/2 triangular- lattice antiferromagnet Ba 3CoSb2O9, Physical Review B 106, 104415 (2022)

    work page 2022

  53. [58]

    Koutroulakis, T

    G. Koutroulakis, T. Zhou, Y. Kamiya, J. D. Thompson, H. D. Zhou, C. D. Batista, and S. E. Brown, Quantum phase diagram of the S = 1 2 triangular-lattice antifer- romagnet Ba 3CoSb2O9, Physical Review B 91, 024410 (2015)

    work page 2015

  54. [59]

    R. Chen, H. Ju, H.-C. Jiang, O. A. Starykh, and L. Ba- lents, Ground states of spin-1/2 triangular antiferromag- nets in a magnetic field, Physical Review B 87, 165123 (2013)

    work page 2013

  55. [60]

    Griset, S

    C. Griset, S. Head, J. Alicea, and O. A. Starykh, De- formed triangular lattice antiferromagnets in a magnetic field: Role of spatial anisotropy and Dzyaloshinskii- Moriya interactions, Physical Review B 84, 245108 (2011)

    work page 2011

  56. [61]

    Tay and O

    T. Tay and O. I. Motrunich, Variational studies of tri- angular Heisenberg antiferromagnet in magnetic field, Physical Review B 81, 165116 (2010)

    work page 2010

  57. [62]

    Coldea, D

    R. Coldea, D. A. Tennant, A. M. Tsvelik, and Z. Tyl- czynski, Experimental Realization of a 2D Fractional Quantum Spin Liquid, Physical Review Letters 86, 1335 (2001)

    work page 2001

  58. [63]

    Tokiwa, T

    Y. Tokiwa, T. Radu, R. Coldea, H. Wilhelm, Z. Tyl- czynski, and F. Steglich, Magnetic phase transitions in the two-dimensional frustrated quantum antiferromag- net Cs 2CuCl4, Physical Review B 73, 134414 (2006)

    work page 2006

  59. [64]

    T. Ono, H. Tanaka, O. Kolomiyets, H. Mitamura, T. Goto, K. Nakajima, A. Oosawa, Y. Koike, K. Kakurai, J. Klenke, P. Smeibidle, and M. Meißner, Magnetization plateaux of the S = 1/2 two-dimensional frustrated anti- ferromagnet Cs 2CuBr4, Journal of Physics: Condensed Matter 16, S773 (2004)

    work page 2004

  60. [65]

    N. A. Fortune, S. T. Hannahs, Y. Yoshida, T. E. Sher- line, T. Ono, H. Tanaka, and Y. Takano, Cascade of Magnetic-Field-Induced Quantum Phase Transitions in a Spin- 1/2 Triangular-Lattice Antiferromagnet, Physi- cal Review Letters 102, 257201 (2009)

    work page 2009

  61. [66]

    Hirai, T

    D. Hirai, T. Yajima, K. Nawa, M. Kawamura, and Z. Hi- roi, Anisotropic triangular lattice realized in rhenium oxychlorides A3ReO5Cl2 (A = Ba, Sr), Inorganic Chem- istry 59, 10025 (2020) , pMID: 32584564

    work page 2020

  62. [67]

    K. Nawa, D. Hirai, M. Kofu, K. Nakajima, R. Murasaki, S. Kogane, M. Kimata, H. Nojiri, Z. Hiroi, and T. J. Sato, Bound spinon excitations in the spin- 1 2 anisotropic triangular antiferromagnet Ca3ReO5Cl2, Phys. Rev. Res. 2, 043121 (2020)

    work page 2020

  63. [68]

    Weihong, R

    Z. Weihong, R. H. McKenzie, and R. R. P. Singh, Phase diagram for a class of spin-1/2 Heisenberg models in- terpolating between the square-lattice, the triangular- lattice, and the linear-chain limits, Physical Review B 59, 14367 (1999)

    work page 1999

  64. [69]

    Thesberg and E

    M. Thesberg and E. S. Sørensen, Exact diagonalization study of the anisotropic triangular lattice Heisenberg model using twisted boundary conditions, Physical Re- view B 90, 115117 (2014)

    work page 2014

  65. [70]

    Hasik and P

    J. Hasik and P. Corboz, Incommensurate Order with Translationally Invariant Projected Entangled-Pair States: Spiral States and Quantum Spin Liquid on the Anisotropic Triangular Lattice, Physical Review Letters 133, 176502 (2024)

    work page 2024

  66. [71]

    K. Penc, N. Shannon, and H. Shiba, Half-Magnetization Plateau Stabilized by Structural Distortion in the An- tiferromagnetic Heisenberg Model on a Pyrochlore Lat- tice, Physical Review Letters 93, 197203 (2004)

    work page 2004

  67. [72]

    Tchernyshyov, R

    O. Tchernyshyov, R. Moessner, and S. L. Sondhi, Order by Distortion and String Modes in Pyrochlore Antifer- romagnets, Physical Review Letters 88, 067203 (2002)

    work page 2002

  68. [73]

    Liu and G

    H. Liu and G. Khaliullin, Pseudo-Jahn-Teller Effect and Magnetoelastic Coupling in Spin-Orbit Mott Insulators, Physical Review Letters 122, 057203 (2019)

    work page 2019

  69. [74]

    J. Nagl, A. Zheludev, and D. Khalyavin, Field-dependent magnetic structure of the triangular-lattice antiferro- magnet (CD 3ND3)2NaRuCl6, STFC ISIS Neutron and Muon Source 10.5286/ISIS.E.RB2420060 (2024)

    work page doi:10.5286/isis.e.rb2420060 2024

  70. [75]

    J. Nagl, S. Gvasaliya, A. Zheludev, A. Hiess, and P. Steffens, Excitation spectrum of the triangular-lattice antiferromagnet (CD 3ND3)2NaRuCl6, Institut Laue- Langevin (ILL) 10.5291/ILL-DATA.4-01-1857 (2025)

    work page doi:10.5291/ill-data.4-01-1857 2025

  71. [76]

    G. M. Sheldrick, A short history of SHELX, Acta Crys- tallographica Section A 64, 112 (2008)

    work page 2008

  72. [77]

    S. A. Zvyagin, J. Krzystek, P. H. M. van Loosdrecht, G. Dhalenne, and A. Revcolevschi, High-field ESR study of the dimerized-incommensurate phase transition in the spin-Peierls compound CuGeO 3, Physica B 346-347, 1 (2004)

    work page 2004

  73. [78]

    Küchler, R

    R. Küchler, R. Wawrzyńczak, H. Dawczak-Dębicki, J. Gooth, and S. Galeski, New applications for the world’s smallest high-precision capacitance dilatometer and its stress-implementing counterpart, Review of Sci- entific Instruments 94, 045108 (2023) . 14

    work page 2023

  74. [79]

    Lüthi, Physical Acoustics in the Solid State , 1st ed

    B. Lüthi, Physical Acoustics in the Solid State , 1st ed. (Springer, Berlin, 2005)

    work page 2005

  75. [80]

    Zherlitsyn, S

    S. Zherlitsyn, S. Yasin, J. Wosnitza, A. A. Zvyagin, A. V. Andreev, and V. Tsurkan, Spin-lattice effects in selected antiferromagnetic materials (review article), Low Tem- perature Physics 40, 123 (2014)

    work page 2014

  76. [81]

    Arnold, J

    O. Arnold, J. C. Bilheux, J. M. Borreguero, A. Buts, S. I. Campbell, L. Chapon, M. Doucet, N. Draper, R. Fer- raz Leal, M. A. Gigg, V. E. Lynch, A. Markvardsen, D. J. Mikkelson, R. L. Mikkelson, R. Miller, K. Pal- men, P. Parker, G. Passos, T. G. Perring, P. F. Peter- son, S. Ren, M. A. Reuter, A. T. Savici, J. W. Taylor, R. J. Taylor, R. Tolchenov, W. Zh...

    work page 2014

  77. [82]

    Toth and B

    S. Toth and B. Lake, Linear spin wave theory for single-Q incommensurate magnetic structures, Journal of Physics: Condensed Matter 27, 166002 (2015)

    work page 2015

  78. [83]

    Waśkowska, L

    A. Waśkowska, L. Gerward, J. Staun Olsen, W. Morgenroth, M. Mączka, and K. Hermanowicz, Temperature- and pressure-dependent lattice behaviour of RbFe(MoO4)2, Journal of Physics: Condensed Matter 22, 055406 (2010)

    work page 2010

  79. [84]

    Kajita, T

    Y. Kajita, T. Nagai, S. Yamagishi, K. Kimura, M. Hagi- hala, and T. Kimura, Ferroaxial Transitions in Glaserite- Type Na 2BaM(PO4)2 (M = Mg, Mn, Co, and Ni), Chemistry of Materials 36, 7451 (2024)

    work page 2024

  80. [85]

    A. J. Hearmon, F. Fabrizi, L. C. Chapon, R. D. Johnson, D. Prabhakaran, S. V. Streltsov, P. J. Brown, and P. G. Radaelli, Electric Field Control of the Magnetic Chiral- ities in Ferroaxial Multiferroic RbFe(MoO 4)2, Physical Review Letters 108, 237201 (2012)

    work page 2012

Showing first 80 references.