Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Non-linear spin wave theory in the strong easy-axis limit of the triangular XXZ model

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read In the strong easy-axis limit, the triangular XXZ spin model reduces exactly to soft-core bosons on the honeycomb lattice.

desk verdict Solid one-loop spin-wave analysis with a genuinely new honeycomb boson reduction and pseudo-Goldstone gap formula, but the order-by-disorder selection of the Y state is asserted, not derived, and the renormalization scheme stays one-shot. read the letter →

arxiv 2511.08179 v2 pith:K3VZKPJW submitted 2025-11-11 cond-mat.str-el

Achille Mauri , Siebe Roose , Frédéric Mila This is my paper
classification cond-mat.str-el
keywords triangularXXZmodeleasy-axisanisotropyspin-wavetheoryhoneycombbosonorder-by-disorderpseudo-Goldstonemodeone-loopself-energyK2Co(SeO3)2
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

Taking the limits S→∞ and J_xy→0 at fixed V=J_zz/(SJ_xy), the triangular XXZ antiferromagnet is shown to map exactly to a honeycomb-lattice model of soft-core bosons with nearest-neighbour repulsion V. At zero field this boson model has a one-parameter family of classically degenerate ground states; quantum fluctuations lift the degeneracy (order-by-disorder) and open a gap to the pseudo-Goldstone mode, epsilon_g≈3.05√(V). Computing the one-loop self-energy requires a self-consistent renormalization of a magnetic-field-like term and of the energy scale to cure infrared divergences. In this one-loop theory the spectrum develops no roton minimum at the M point for V up to about 1.4. Applied to K2Co(SeO3)2, where V≈25–29, the spin-wave framework is not adequate, motivating strong-coupling analyses.

What carries the argument

The central object is the soft-core boson model on the honeycomb lattice, obtained by the double limit with V fixed and C-sublattice spins frozen at S^z=-S. Its classical energy has an exact staggered-rescaling symmetry at zero field, φ_A→λφ_A, φ_B→φ_B/λ, which is broken at the quantum level and generates the pseudo-Goldstone gap. The one-loop calculation uses the Beliaev-Dyson self-energy (normal plus anomalous parts) in a cartesian basis where the sublattice and spin spaces diagonalize separately; the pseudo-Goldstone gap ϵ_g=√(18V(3I_{-1}-1))≈3.05√V follows from projecting the self-energy onto the would-be zero modes.

What would settle it

Compute the single-magnon spectrum of the original triangular XXZ model for S large (say S=5/2 or 10) and α small by a method that does not assume frozen C spins (e.g., exact diagonalization of a cluster or tensor network), and check whether the pseudo-Goldstone gap follows 3.05√V with V=1/(Sα). A significant deviation, or a visible roton minimum at V≈1, would falsify the claim that the one-loop honeycomb boson model captures the limit.

Watch

Extended reading notes

Core claim

The paper establishes a controlled reduction: in the simultaneous limit 1/S→0 and J_xy→0 with V fixed, the original spin Hamiltonian becomes H=Σ⟨i,j⟩(a_i†a_j+a_j†a_i+V n_i n_j)+bΣ n_i on the honeycomb lattice. Because the average boson density stays O(1), the Holstein-Primakoff expansion can be truncated at quadratic order. The classical energy of this model at b=0 is invariant under a staggered rescaling φ_i→ξ_i φ_i with ξ_A=λ, ξ_B=1/λ, which explains the accidental degeneracy; the rescaling is not a quantum symmetry, so fluctuations select the Y state and give the pseudo-Goldstone mode a finite gap (Eq. 33). The one-loop self-energy is computed analytically; infrared divergences from the g

Load-bearing premise

The derivation assumes the C-sublattice spins are strictly frozen in the S^z=-S state and that keeping only quadratic terms in the boson expansion is valid; if the C-spin fluctuations are not negligible, the honeycomb model and its one-loop results are uncontrolled.

Editorial extensions

If this is right

  • The honeycomb boson model is an exact effective description in the fixed-V limit, providing a simpler starting point than the original triangular spin problem.
  • The pseudo-Goldstone gap scales as S^{1/2} in physical units, confirming earlier analyses of order-by-disorder.
  • The self-consistent two-parameter renormalization removes the IR divergences and predicts a correction that behaves like -V ln V at small V.
  • Within one-loop order, no roton minimum appears at the M point for V up to ≈1.4, so a spin-wave mechanism alone is insufficient to explain KCSO.
  • The rapid growth of the gap with V (≈19% of bandwidth already at V≈0.05) suggests the gap in the strong-coupling regime must behave non-monotonically or arise from beyond-spin-wave physics.

Reading between the lines

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

  • A natural next test is to solve the honeycomb soft-core boson model numerically for moderate V (e.g., 0.5–5) and compare the pseudo-Goldstone gap and M-point dispersion against the one-loop predictions; deviation would signal where the scheme breaks down.
  • The paper leaves implicit that the same fixed-V mapping may extend to finite fields and to the 1/3 plateau boundary, where the honeycomb model with b≈3 could be studied with the same self-energy machinery.
  • If C-sublattice spin fluctuations are included, the pseudo-Goldstone gap may be lowered substantially; this could be tested by comparing against quantum Monte Carlo of the quantum dimer model, which includes those fluctuations.
  • The absence of a roton at one loop suggests that a roton-like minimum in KCSO likely requires either strong-coupling/hard-core-boson effects or fractionalized excitations, not simply higher-order spin-wave corrections.
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 / 4 minor

Summary. The paper analyzes the strongly easy-axis limit of the triangular-lattice XXZ model by taking S→∞ and J_xy→0 at fixed V=J_zz/(S J_xy). It derives an effective soft-core boson model on the honeycomb lattice with nearest-neighbour repulsion, H=Σ(a_i^† a_j+h.c.)+V Σ n_i n_j+b Σ n_i, and argues this reduction is controlled in the double limit. At zero field the classical energy has an accidental degeneracy; the authors state that quantum fluctuations select the Y configuration and then compute the one-loop self-energy around it. They obtain a pseudo-Goldstone gap ε_g=sqrt(18V(3I_{-1}-1))≃3.05√V and a renormalized spectrum using two self-consistently chosen parameters (b̃,z). The paper concludes that no roton minimum is generated for V up to about 1.4 and that the nonlinear spin-wave framework is inadequate for describing the zero-field spectrum of KCSO.

Significance. If the central reduction and the gap formula (33) are correct, the paper provides a clean, controlled effective model that interpolates between the semiclassical and strong-coupling regimes, with an explicit one-loop calculation of the pseudo-Goldstone gap and the spectrum. The derivation of the self-energy in Sec. V and the appendices is detailed and transparent, and the gap (33) is a parameter-free prediction in terms of V. The paper also correctly and explicitly acknowledges the limitations of applying the S→∞ result to S=1/2 compounds such as KCSO. These are genuine strengths. However, the selection of the Y state is asserted rather than derived, and the finite-k spectrum depends on a two-parameter renormalization ansatz whose robustness is not demonstrated.

major comments (3)
  1. [Sec. IV, after Eq. (8)] The classical energy (8) at b=0 depends only on ρ=φ_B^* φ_A, so there is a one-parameter family of minima, e.g. φ_A=λ/√V, φ_B=-1/(λ√V) with λ∈R^+. The paper states 'We checked that the quantum corrections select as ground state the Y configuration' without showing the calculation. This selection is load-bearing: the self-energy in Sec. V is evaluated at the Y point, and the pseudo-Goldstone gap (33) is the curvature at λ=1. Please provide the one-loop zero-point energy as a function of λ (or of |φ_A/φ_B|) and show that it is minimized at λ=1. If the selected point is not λ=1, Eqs. (21)-(33) must be rederived at the actual minimum. Citing Ref. [2] is not sufficient because the effective boson model has a different degeneracy manifold.
  2. [Sec. VII, Eqs. (37)-(40)] The renormalized field b̃ and the scale factor z are fixed by the two self-consistency conditions (39) and (40). Condition (39) enforces ω_{+1}(0)=z√(12 b̃), so b̃ is determined by requiring the gap to match the one computed in Sec. VI. Consequently the finite-k spectrum is a prediction of a specific ansatz, not a closed calculation. Please quantify the scheme dependence: for example, repeat the calculation with z=1 and b̃ fixed by (39) only, or with a different second matching point, and show that ω_λ(k) and the presence/absence of an M-point minimum are unchanged within the stated accuracy. Without this, the conclusion 'no roton minimum up to V≈1.4' is conditional on the two-parameter ansatz.
  3. [Sec. VII A and Sec. VII B] The paper concludes from Fig. 5 that no roton minimum appears for V up to about 1.37, but it also states that for V≈1 'the large corrections ... may indicate that the first order approximation becomes invalid at V≃1'. Since the one-loop expansion is controlled only for V≪1, the extrapolation to V≈1.4 is not a supported quantitative prediction. Please either restrict the no-roton claim to the controlled small-V regime or supply a quantitative estimate of the error (for example, the size of the next-order corrections) before using this result to argue against spin-wave explanations of the KCSO spectrum.
minor comments (4)
  1. [Sec. II] Typo: 'satifsfactory' should be 'satisfactory' in the paragraph discussing the linear-spin-wave approximation.
  2. [Fig. 8 caption] 'a part from' should be 'apart from'.
  3. [Sec. IV] The phrase 'We checked' also appears in the discussion of the ground-state selection. A one-sentence description of the check, or a pointer to an appendix, would be useful and would avoid leaving an unsupported claim in the main text.
  4. [Appendix C / Eq. (27)] The notation for the frequency arguments in the off-diagonal matrix elements in Eq. (43) is introduced only later in the text; a brief definition immediately after Eq. (43) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained, with renormalization parameters fixed internally and no load-bearing self-citation.

full rationale

The central derivation chain is internally generated rather than fitted. The reduction of the triangular XXZ model to the honeycomb soft-core boson model (Sec. III, Eq. (6)) is a controlled limit S→∞, J_xy→0 at fixed V=J_zz/(SJ_xy), with the C-sublattice freezing stated as a justified simplification in that limit; this is an assumption about the limit, not a definition that presupposes the result. The classical degeneracy and Y-state selection are asserted in Sec. IV ('We checked that the quantum corrections select as ground state the Y configuration...'), with the selection attributed to the external Ref. [2]; the proof of selection is omitted, but omission of a proof is a correctness/rigor concern, not circularity. The pseudo-Goldstone gap Eq. (33), ε_g = sqrt(18V(3I_{-1}-1)) ≈ 3.05√V, is an explicit one-loop result from the zero-frequency self-energy, with the IR-divergent I_{-3} term canceling after projection; it is not matched to external data. In Sec. VII the renormalization parameters b̃ and z are fixed by internal self-consistency conditions (Eqs. (39)-(40)), not by fitting to the target spectrum or to experiment, and the reported ε_g/ΔE ≈ 19% is a computed output of that scheme. No step reduces a prediction to its input by construction, and the load-bearing citations ([2], [3], [45]) are external works rather than self-citations. The unproven Y-selection and the one-loop truncation limit the paper's rigor, but they do not constitute circularity under the stated criteria.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the double-scaling limit, the freezing of C spins, and the assumed Y-state selection. No new physical entities are introduced; the two renormalization constants and the numerical broadening are the main ad hoc parameters.

free parameters (3)
  • renormalized magnetic field b-tilde = 0.0001, 0.03, 0.2, 0.5
    Introduced ad hoc as an IR regulator in the renormalized perturbation theory; fixed by the self-consistency condition at the Gamma point (Eq. 39). Maps to coupling V via Eq. (E15).
  • energy-scale renormalization z = 0.9995, 0.9399, 0.7981, 0.6954
    Added to capture bandwidth renormalization; fixed by matching at the Y point (Eq. 40).
  • numerical broadening delta = 5e-3 (2e-3 at b-tilde=0.0001)
    Chosen by hand to replace the infinitesimal i0+ in the self-energy integral; affects the treatment of van Hove singularities.
assumptions (4)
  • domain assumption The C-sublattice spins are frozen in S^z = -S, reducing the triangular lattice to a honeycomb lattice.
    Invoked in Sec. III to eliminate the C sites; required for the entire effective model. Justified in the S->infinity, alpha->0 limit but uncontrolled at S=1/2.
  • domain assumption The Holstein-Primakoff boson expansion can be truncated at quadratic order (S^+ ~ sqrt(2S) a).
    Justified by the O(1) average boson density (Eq. 4) in the fixed-V limit; underlies the quartic-only effective Hamiltonian.
  • domain assumption Quantum order-by-disorder selects the Y configuration.
    Asserted in Sec. IV ('We checked...') without derivation; this choice of classical ground state is the starting point for the spin-wave expansion and the gap calculation.
  • ad hoc to paper The one-loop self-energy with two renormalization parameters (b-tilde, z) is sufficient to compute the zero-field spectrum.
    This is the approximate self-consistent scheme of Sec. VII; the authors acknowledge it is not fully self-consistent and may fail at V ~ 1.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Non-linear spin wave theory in the strong easy-axis limit of the triangular XXZ model." pith.science (2026). https://pith.science/paper/K3VZKPJW

@misc{pith2026251108179,
  author       = {Pith},
  title        = {Pith review of: Non-linear spin wave theory in the strong easy-axis limit of the triangular XXZ model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K3VZKPJW}},
  note         = {Machine review of arXiv:2511.08179}
}
abstract

Motivated by recent experimental studies, we investigate the spectrum of the nearest-neighbour triangular XXZ model within the $1/S$ expansion, in the limit in which the exchange couplings present a strong easy-axis anisotropy $J_{xy}/J_{zz} \ll 1$. We show that in the limit in which $1/S \to 0$ and $J_{xy} \to 0$ at fixed $V = J_{zz}/(S J_{xy})$, the triangular spin model can be reduced to an effective boson model with quartic interactions on the honeycomb lattice. This effective model interpolates between a spin-wave ($V \to 0$) and a strong-coupling limit ($V \to \infty$) and encodes in a simple framework the regimes discussed by Kleine~\emph{et al.}~[Z. Phys. B Condens. Matter~{\bf 86}, 405 (1992);~{\bf 87}, 103 (1992)]. For zero field, the classical ground state of the model presents an accidental degeneracy, which can be traced to a simple symmetry of the classical energy. The model thus offers a transparent realization of a theory with quantum order-by-disorder and a pseudo-Goldstone mode. We analyze the spectrum at zero magnetic field by calculating the self-energy at one-loop order. In the calculation, we introduce a self-consistent renormalization of the energy scale and of the pseudo-Goldstone energy gap; the latter renormalization is essential to remove infrared divergences in the on-shell corrections to the energy dispersion. Finally, we discuss qualitatively the structure of the one-loop corrections in comparison with the spectrum observed experimentally in K$_{2}$Co(SeO$_{3}$)$_{2}$.

Figures

Figures reproduced from arXiv: 2511.08179 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Structure of the “Y” configuration. (b) Brillouin [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Dispersions of the two lowest LSWT branches, cal [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The ground state for [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Self-energy diagrams contributing at one-loop order. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Red solid lines: dispersion relation at zero phys [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Same plots reported in Fig [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Red solid lines: spectra calculated with Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Tadpole diagrams determining the correction to the [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Emergent Spin Supersolids in Frustrated Quantum Materials

    cond-mat.str-el 2026-01 unverdicted novelty 2.0 of 10

    Spin supersolids featuring coexisting longitudinal spin order breaking lattice symmetry and transverse order breaking spin U(1) symmetry have been established in frustrated quantum magnets through consistent experimen...

Reference graph

Works this paper leans on

60 extracted references · 3 canonical work pages · cited by 1 Pith paper

  1. [2]

    Hartree-F ock diagramΣ b The second self-energy graph, Σ b, can be calculated conveniently in real space and involves the same static averages which were calculated in Sec. A. Explicitly we find Σαβ bij(t−t ′) = 3V 2 δijδαβM γδ ⟨δmγ kδmδ k⟩0 +V tij⟨δmα k δmβ l ⟩0 δ(t−t ′) (C4) wherekandlare any pair of nearest-neighbour sites on the honeycomb lattice andt...

  2. [1]

    Calculating the second deriva- tive explicitly from Eq

    T adpole termΣ a The contribution of the tadpole graph 4a can be cal- culated in the one-loop approximation by calculating the variation of∂ 2H/(∂m α i ∂mβ j )| ¯mα i due to the shift of the or- der parameter ¯mα i −m α i0. Calculating the second deriva- tive explicitly from Eq. (7) and expanding about the av- erage configuration ¯mα i = √ 2φri we obtain:...

  3. [3]

    (C6) where Γ is the three-particle vertex in real space

    F requency-dependent self-energy termΣ c The last term Σ c reads can be expressed in real space as: Σαβ cij(t−t ′) =− i 2 X l,m,l′,m′ Γαλµ ilm Γνρβ l′m′jGλν 0ll′(t−t ′) ×G µρ 0mm′(t−t ′) . (C6) where Γ is the three-particle vertex in real space. The vertex Γ is defined via the cubic part inδmof the Hamil- tonian, expanded at the classical minimumm=m 0. It...

  4. [4]

    −3 + ϵλ(k, b)ω 3 + b 3 λ|fk| + 1− b 3 |fk|2 3 + b 3 λ|fk| # δλλ′ −zΣ λλ′(k, eλλ′(k), b). (E11) The equation detB= 0 can be recast equivalently as det

    Thus the divergence disappears from the correction to the energy spectrum. Consider now the contribution of pseudo-Goldstone modes forb= 0. In this case,λ 1 = +1,λ 2 =−λ, and N λν λ1 (0) = 6δ λ1δν1. Then we get a term which diverges on-shell: Σ(div,pseudo−Goldstone) λ (k, ω)≃ 27V 2ϵλ(k) Z |k1|≪1 d2k1 Ω N αβ −λ(k)−ϵ −λ(k)σαβ y ϵ1(k) (ω−ϵ 1(k1)−ϵ λ(k−k 1) +...

  5. [5]

    Fazekas and P

    P. Fazekas and P. W. Anderson, On the ground state properties of the anisotropic triangular antiferromagnet, Philos. Mag.30, 423 (1974)

    1974

  6. [6]

    Kleine, E

    B. Kleine, E. M¨ uller-Hartmann, K. Frahm, and P. Fazekas, Spin-wave analysis of easy-axis quantum an- tiferromagnets on the triangular lattice, Z. Phys. B Con- dens. Matter87, 103 (1992)

    1992

  7. [7]

    Kleine, P

    B. Kleine, P. Fazekas, and E. M¨ uller-Hartmann, Pertur- bation theory for the triangular Heisenberg antiferromag- net with Ising-like anisotropy, Z. Phys. B86, 405 (1992)

    1992

  8. [8]

    Murthy, D

    G. Murthy, D. Arovas, and A. Auerbach, Superfluids and supersolids on frustrated two-dimensional lattices, Phys. Rev. B55, 3104 (1997)

    1997

Show all 60 references
  1. [9]

    Wessel and M

    S. Wessel and M. Troyer, Supersolid hard-core bosons on the triangular lattice, Phys. Rev. Lett.95, 127205 (2005)

    2005

  2. [10]

    R. G. Melko, A. Paramekanti, A. A. Burkov, A. Vish- wanath, D. N. Sheng, and L. Balents, Supersolid order from disorder: hard-core bosons on the triangular lattice, Phys. Rev. Lett.95, 127207 (2005)

    2005

  3. [11]

    Heidarian and K

    D. Heidarian and K. Damle, Persistent supersolid phase of hard-core bosons on the triangular lattice, Phys. Rev. Lett.95, 127206 (2005)

    2005

  4. [12]

    A. A. Burkov and L. Balents, Superfluid-insulator tran- sitions on the triangular lattice, Phys. Rev. B72, 134502 (2005)

    2005

  5. [13]

    A. Sen, P. Dutt, K. Damle, and R. Moessner, Variational wave-function study of the triangular lattice supersolid, Phys. Rev. Lett.100, 147204 (2008)

    2008

  6. [14]

    F. Wang, F. Pollmann, and A. Vishwanath, Extended supersolid phase of frustrated hard-core bosons on a tri- angular lattice, Phys. Rev. Lett.102, 017203 (2009)

    2009

  7. [15]

    Heidarian and A

    D. Heidarian and A. Paramekanti, Supersolidity in the triangular lattice spin-1/2 XXZ model: a variational per- spective, Phys. Rev. Lett.104, 015301 (2010)

    2010

  8. [16]

    Zhang, R

    X.-F. Zhang, R. Dillenschneider, Y. Yu, and S. Eggert, Supersolid phase transitions for hard-core bosons on a triangular lattice, Phys. Rev. B84, 174515 (2011)

    2011

  9. [17]

    Yamamoto, G

    D. Yamamoto, G. Marmorini, and I. Danshita, Quantum phase diagram of the triangular-lattice XXZ model in a magnetic field, Phys. Rev. Lett.112, 127203 (2014)

    2014

  10. [18]

    Ulaga, T

    M. Ulaga, T. T. Kokalj, J., and P. Prelovˇ sek, Easy-axis Heisenberg model on the triangular lattice: from super- solid to gapped solid, Phys. Rev. B111, 174442 (2025)

    2025

  11. [19]

    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 antifer- romagnet K 2Co(SeO3)2, npj Quantum Mater.10, 74 (2025)

    2025

  12. [20]

    C. A. Gallegos, S. Jiang, S. R. White, and A. L. Cherny- shev, Phase diagram of the easy-axis triangular-lattice J1 −J 2 model, Phys. Rev. Lett.134, 196702 (2025)

    2025

  13. [21]

    Miyashita and H

    S. Miyashita and H. Kawamura, Phase transitions of anisotropic Heisenberg antiferromagnets on the triangu- lar lattice, J. Phys. Soc. Jpn.54, 3385 (1985)

    1985

  14. [22]

    A. Bose, K. Hart, R. Sutcliffe, and A. Paramekanti, Modified large-napproach to gapless spin liquids, magnetic orders, and dynamics: application to triangular lattice antiferromagnets, arXiv:2503.09695 10.48550/arXiv.2503.09695 (2025)

    2025 doi arXiv

  15. [23]

    Gao, Y.-C

    Y. Gao, Y.-C. Fan, H. Li, F. Yang, X.-T. Zeng, X.-L. Sheng, R. Zhong, Y. Qi, Y. Wan, and W. Li, Spin su- persolidity in nearly ideal easy-axis triangular quantum antiferromagnet Na 2BaCo(PO4)2, npj Quantum Mater. 7, 89 (2022)

    2022

  16. [24]

    Y. Xu, J. Hasik, B. Ponsioen, and A. H. Nevidomskyy, Simulating spin dynamics of supersolid states in a quan- tum Ising magnet, Phys. Rev. B111, L060402 (2025)

    2025

  17. [25]

    Ulaga, J

    M. Ulaga, J. Kokalj, A. Wietek, A. Zorko, and P. Prelovˇ sek, Finite-temperature properties of the easy- axis Heisenberg model on frustrated lattices, Phys. Rev. B109, 035110 (2024)

    2024

  18. [26]

    Moessner and S

    R. Moessner and S. L. Sondhi, Ising models of quantum frustration, Phys. Rev. B63, 224401 (2001)

    2001

  19. [27]

    Kamiya, L

    Y. Kamiya, L. Ge, T. Hong, Y. Qiu, D. L. Quintero- Castro, Z. Lu, H. B. Cao, M. Matsuda, E. S. Choi, C. D. Batista, M. Mourigal, H. D. Zhou, and J. Ma, The nature of spin excitations in the one-third magnetization plateau phase of Ba 3CoSb2O9, Nature Commun.9, 2666 (2018)

    2018

  20. [28]

    T. Arh, B. Sana, M. Pregelj, P. Khuntia, Z. Jagliˇ ci´ c, M. D. Le, P. K. Biswas, P. Manuel, L. Mangin-Thro, A. Ozarowski, and A. Zorko, The Ising triangular-lattice antiferromagnet neodymium heptatantalate as a quan- tum spin liquid candidate, Nat. Mater.21, 416 (2022)

    2022

  21. [29]

    T. Chen, A. Ghasemi, J. Zhang, L. Shi, Z. Tagay, L. Chen, E.-S. Choi, M. Jaime, M. Lee, Y. Hao, H. Cao, B. Winn, R. Zhong, X. Xu, N. P. Armitage, R. Cava, and C. Broholm, Phase diagram and spectroscopic evidence of supersolids in quantum Ising magnet K 2Co(SeO3)2, arXiv:2402.1...

    2024 doi

  22. [30]

    M. Zhu, V. Romerio, N. Steiger, S. D. Nabi, N. Mu- rai, S. Ohira-Kawamura, K. Y. Povarov, Y. Skourski, R. Sibille, L. Keller, Z. Yan, S. Gvasaliya, and A. Zhe- ludev, Continuum excitations in a spin supersolid on a triangular lattice, Phys. Rev. Lett.133, 186704 (2024)

    2024

  23. [31]

    H. Jia, B. Ma, Z. Wang, and G. Chen, Quantum spin su- persolid as a precursory Dirac spin liquid in a triangular lattice antiferromagnet, Phys. Rev. Research6, 033031 (2024)

    2024

  24. [32]

    R. Chi, J. Hu, H.-J. Liao, and T. Xiang, Dynamical spec- tra of spin supersolid states in triangular antiferromag- nets, Phys. Rev. B110, L180404 (2024)

    2024

  25. [33]

    Sheng, L

    J. Sheng, L. Wang, W. Jiang, H. Ge, N. Zhao, T. Li, M. Kofu, D. Yu, W. Zhu, J.-W. Mei, Z. Wang, and L. Wu, Continuum of spin excitations in an ordered magnet, In- novation6, 100769 (2025)

    2025

  26. [34]

    Woodland, R

    L. Woodland, R. Okuma, J. R. Stewart, C. Balz, and R. Coldea, From continuum excitations to sharp magnons via transverse magnetic field in the spin-1/2 Ising- like triangular lattice antiferromagnet Na 2BaCo(PO4)2, 25 arXiv:2505.06398 10.48550/arXiv.2505.06398 (2025)

    2025 doi arXiv

  27. [35]

    Zhong, S

    R. Zhong, S. Guo, and R. J. Cava, Frustrated magnetism in the layered triangular lattice materials K 2Co(SeO3)2 and Rb2Co(SeO3)2, Phys. Rev. Mater.4, 084406 (2020)

    2020

  28. [36]

    Zhang, T

    H. Zhang, T. Huang, A. O. Scheie, M. Zhu, T. Xie, N. Murai, S. Ohira-Kawamura, A. Zheludev, A. M. L¨ auchli, and C. D. Batista, Nonperturbative semi- classical spin dynamics for ordered quantum magnets, arXiv:2508.21142 10.48550/arXiv.2508.21142 (2025)

    2025 doi

  29. [37]

    J. Ma, Y. Kamiya, T. Hong, H. B. Cao, G. Ehlers, W. Tian, C. D. Batista, Z. L. Dun, D. H. Zhou, and M. Matsuda, Static and dynamical properties of the spin-1/2 equilateral triangular-lattice antiferromag- net Ba3CoSb2O9, Phys. Rev. Lett.116, 087201 (2016)

    2016

  30. [38]

    Macdougal, S

    D. Macdougal, S. Williams, D. Prabhakaran, R. I. Be- wley, D. J. Voneshen, and R. Coldea, Avoided quasi- particle decay and enhanced excitation continuum in the spin-1/2 near-Heisenberg triangular antiferromagnet Ba3CoSb2O9, Phys. Rev. B102, 064421 (2020)

    2020

  31. [39]

    R. Chi, Y. Liu, Y. Wan, H.-J. Liao, and T. Xiang, Spin excitation spectra of anisotropic spin-1/2 triangular lat- tice Heisenberg antiferromagnets, Phys. Rev. Lett.129, 227201 (2022)

    2022

  32. [40]

    Y. Gao, C. Zhang, J. Xiang, D. Yu, X. Lu, P. Sun, W. Jin, G. Su, and W. Li, Spin supersolid phase and double magnon-roton excitations in a cobalt-based trian- gular lattice, Phys. Rev. B110, 214408 (2024)

    2024

  33. [41]

    A. O. Scheie, E. A. Ghioldi, J. Xing, J. A. M. Paddi- son, N. E. Sherman, M. Dupont, L. D. Sanjeewa, S. Lee, A. J. Woods, D. Abernathy, D. M. Pajerowski, T. J. Williams, S.-S. Zhang, L. O. Manuel, A. E. Trumper, C. D. Pemmaraju, A. S. Sefat, D. S. Parker, T. P. Dev- ereaux, R....

    2024

  34. [42]

    Zheng, J

    W. Zheng, J. O. Fjaerestad, R. R. P. Singh, R. H. McKen- zie, and R. Coldea, Anomalous excitation spectra of frus- trated quantum antiferromagnets, Phys. Rev. Lett.96, 057201 (2006)

    2006

  35. [43]

    Zheng, J

    W. Zheng, J. O. Fjaerstad, R. R. P. Singh, R. H. McKenzie, and R. Coldea, Excitation spectra of the spin- 1/2 triangular-lattice Heisenberg antiferromagnet, Phys. Rev. B74, 224420 (2006)

    2006

  36. [44]

    O. A. Starykh, A. V. Chubukov, and A. G. Abanov, Flat spin-wave dispersion in a triangular antiferromag- net, Phys. Rev. B74, 180403(R) (2006)

    2006

  37. [45]

    A. L. Chernyshev and M. E. Zhitomirsky, Magnon decay in noncollinear quantum antiferromagnets, Phys. Rev. Lett.97, 207202 (2006)

    2006

  38. [46]

    A. L. Chernyshev and M. E. Zhitomirsky, Spin waves in a triangular lattice antiferromagnet: decays, spec- trum renormalization, and singularities, Phys. Rev. B79, 144416 (2009)

    2009

  39. [47]

    Powalski, G

    M. Powalski, G. S. Uhrig, and K. P. Schmidt, Roton min- imum as a fingerprint of magnon-Higgs scattering in or- dered quantum antiferromagnets, Phys. Rev. Lett.115, 207202 (2015)

    2015

  40. [48]

    Powalski, K

    M. Powalski, K. P. Schmidt, and G. S. Uhrig, Mutually attracting spin waves in the square-lattice quantum an- tiferromagnet, SciPost Phys.4, 001 (2018)

    2018

  41. [49]

    J. G. Rau, P. A. McClarty, and R. Moessner, Pseudo- Goldstone gaps and order-by-quantum disorder in frus- trated magnets, Phys. Rev. Lett.121, 237201 (2018)

    2018

  42. [50]

    P. A. Maksimov and A. L. Chernyshev, Field-induced dynamical properties of the XXZ model on a honeycomb lattice, Phys. Rev. B93, 014418 (2016)

    2016

  43. [51]

    Miyashita, Magnetic properties of Ising-like Heisen- berg antiferromagnets on the triangular lattice, J

    S. Miyashita, Magnetic properties of Ising-like Heisen- berg antiferromagnets on the triangular lattice, J. Phys. Soc. Jpn55, 3605 (1986)

    1986

  44. [52]

    Sheng and C

    Q. Sheng and C. L. Henley, Ordering due to disorder in a triangular Heisenberg antiferromagnet with exchange anisotropy, J. Phys.: Condens. Matter4, 2937 (1992)

    1992

  45. [53]

    G. H. Wannier, Antiferromagnetism. the triangular Ising net, Phys. Rev.79, 357 (1950)

    1950

  46. [54]

    Zinn-Justin,Quantum field theory and critical phe- nomena, 4th ed., International series of monographs on physics, Vol

    J. Zinn-Justin,Quantum field theory and critical phe- nomena, 4th ed., International series of monographs on physics, Vol. 113 (Oxford University Press, 2002)

    2002

  47. [55]

    S. T. Beliaev, Application of the methods of quantum field theory to a system of bosons, Sov. Phys. JETP34, 289 (1958)

    1958

  48. [56]

    Castellani, C

    C. Castellani, C. Di Castro, F. Pistolesi, and G. C. Stri- nati, Infrared behavior of interacting bosons at zero tem- perature, Phys. Rev. Lett.78, 1612 (1997)

    1997

  49. [57]

    Auerbach,Interacting electrons and quantum mag- netism(Springer-Verlag, 1994)

    A. Auerbach,Interacting electrons and quantum mag- netism(Springer-Verlag, 1994)

    1994

  50. [58]

    A. V. Chubukov, S. Sachdev, and T. Senthil, Large-S expansion for quantum antiferromagnets on a triangular lattice, J. Phys. Condens. Matter6, 8891 (1994)

    1994

  51. [59]

    P. A. Maksimov, M. E. Zhitomirsky, and A. L. Cherny- shev, Field-induced decays in XXZ triangular-lattice an- tiferromagnets, Phys. Rev. B94, 140407(R) (2016)

    2016

  52. [60]

    Lin and T

    X. Lin and T. Shi, Pseudo-Goldstone modes at finite tem- perature, arXiv:2505.07229 10.48550/arXiv.2505.07229 (2025)

    2025 doi arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.