pith. sign in

arxiv: 2405.10731 · v2 · pith:XC42SXOLnew · submitted 2024-05-17 · ❄️ cond-mat.str-el

Chiral bosonic quantum spin liquid in the integer-spin Heisenberg-Kitaev model

Arnaud Ralko , Jaime Merino This is my paper

Pith reviewed 2026-05-24 01:13 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords quantum spin liquidKitaev modelHeisenberg-Kitaev modelSchwinger bosonchiral spin liquidhoneycomb latticeinteger spin
0
0 comments X

The pith

A chiral bosonic quantum spin liquid is identified in the integer-spin Heisenberg-Kitaev model on the honeycomb lattice.

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

The paper constructs a Schwinger boson mean-field theory that includes both singlet and triplet pairing channels on equal footing to search for quantum spin liquids in the Heisenberg-Kitaev model. This mixed approach is checked against exact diagonalization energies for spins up to 3/2 and against semiclassical Luttinger-Tisza results at large spin. Several gapped states appear near the Kitaev point, but only one chiral state produces a spin excitation spectrum that tracks the dynamical structure factor from exact diagonalizations. The state remains stable when spin is increased to integer values around S=2.

Core claim

The mixed singlet-triplet Schwinger boson mean-field theory reveals a chiral bosonic quantum spin liquid whose spin excitation spectrum agrees with exact diagonalization data and persists for integer spins up to S ≲ 2, positioning it as the leading candidate for the antiferromagnetic Kitaev model at integer S.

What carries the argument

Mixed singlet-triplet Schwinger boson mean-field theory treating hopping and pairing operators equally in both channels.

If this is right

  • The identified chiral state survives up to large integer spins S ≲ 2.
  • The state supplies a bosonic description of the quantum spin liquid near the antiferromagnetic Kitaev point.
  • It is proposed as a candidate for realization in S=1 materials such as A3Ni2XO6 and KNiAsO4.
  • The mixed mean-field construction reproduces both small-S exact diagonalization energies and large-S semiclassical energies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same mixed-channel construction could be tested on other lattices or interaction terms that support bosonic spin liquids.
  • Neutron scattering on candidate S=1 materials could directly probe the chiral spectrum predicted by the state.
  • Including fluctuation corrections beyond mean field might narrow the stability window for the chiral phase at S=1 and S=2.
  • The approach offers a route to compare bosonic and fermionic representations of the same Kitaev-Heisenberg physics.

Load-bearing premise

The mean-field decoupling of the mixed singlet-triplet Schwinger boson Hamiltonian remains quantitatively reliable when extrapolated from the S ≤ 3/2 regime benchmarked against exact diagonalization to the target integer-spin regime S ≲ 2.

What would settle it

Compute the dynamical structure factor from exact diagonalization on larger clusters for S=1 or S=2 and check whether it deviates markedly from the spin excitation spectrum predicted by the chiral mean-field state.

Figures

Figures reproduced from arXiv: 2405.10731 by Arnaud Ralko, Jaime Merino.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) The honeycomb lattice and its translation vectors [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Sketch of the spin structure of a typical classical [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Zoom on the QSL energies around the pure Kitaev [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Dynamical and static spin structure factors for the QSLs discussed in the text., ordered according to their flux( [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Real space spin-spin correlations [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Normalized spin correlations [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Eigenvalues obtained from the Luttinger-Tisza ap [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Dynamical spin spectral function of the Kitaev [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: ) where Kitaev exact gapless QSL is expected. The suppression of excitation energies also arises around θ = 3π/2 i.e for the FM Kitaev model somewhat sup￾pressed for S = 1 compared to S = 1/2. Due to the small cluster available (N = 12 sites), it is difficult to reach a definitive conclusion on whether the S = 1 Kitaev model is gapped or not based solely on our ED results. Tensor network calculations10 do… view at source ↗
read the original abstract

Motivated by the possibility of finding a bosonic quantum spin liquid in the integer spin-$S$ Heisenberg-Kitaev model on the honeycomb lattice, we derive a Schwinger boson mean field theory involving both singlet and triplet pairing channels which includes hopping and pairing operators on equal footing. The mixed construction introduced here is justified by the good comparison with exact diagonalization energies of the $S \leq 3/2$ Heisenberg-Kitaev model and the perfect match with the Luttinger-Tisza semiclassical energies obtained at large-$S$. We find various competing gapped quantum spin liquids close to the Kitaev point. A comparison of their spin excitation spectrum with the dynamical structure factor obtained from exact diagonalizations allows us to identify the physical spin liquid {\it Ansatz} of the model. In particular, we identify a chiral quantum spin liquid state whose spin excitation spectrum follows closely the exact diagonalization data and survives up to large spin $S \lesssim 2$. We propose this state as a promising quantum spin liquid candidate for the integer spin-$S$ antiferromagnetic Kitaev model which may be realized in $S=1$ Kitaev materials A$_3$Ni$_2$XO$_6$ and KNiAsO$_4$.

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 develops a Schwinger-boson mean-field theory for the Heisenberg-Kitaev model on the honeycomb lattice that treats singlet and triplet pairing channels on equal footing together with hopping terms. The mixed decoupling is benchmarked against exact diagonalization energies for S ≤ 3/2 and against Luttinger-Tisza semiclassical energies at large S. By comparing the resulting spin excitation spectra to ED dynamical structure factors, the authors select a chiral quantum spin liquid ansatz as the physical state; they report that this ansatz remains stable up to S ≲ 2 and propose it as a candidate for integer-spin Kitaev materials such as A₃Ni₂XO₆.

Significance. If the extrapolation of the mean-field accuracy holds, the work supplies a concrete bosonic QSL candidate for the antiferromagnetic Kitaev model at integer S, a regime where fermionic parton constructions are less natural. The dual benchmarking against ED at small S and Luttinger-Tisza at large S, together with the explicit inclusion of both hopping and pairing channels, strengthens the technical foundation. The central claim, however, hinges on the reliability of the uncontrolled decoupling in the intermediate-S window where direct ED verification is unavailable.

major comments (2)
  1. [§3] §3 (mean-field construction) and the paragraph following Eq. (the mixed decoupling): The mixed singlet-triplet Schwinger-boson decoupling is stated to be justified by quantitative agreement with ED for S ≤ 3/2 and exact reproduction of Luttinger-Tisza energies at large S. No error estimate, 1/S expansion, or gauge-fluctuation analysis is supplied for the integer-S regime S ≲ 2 that is the target of the central claim; because the ansatz selection itself rests on spectral matching performed only inside the benchmarked window, the extrapolation step is load-bearing and currently unsupported.
  2. [§4] §4 (spectrum comparison and stability analysis): The chiral QSL is identified as physical because its MF spin excitations follow the ED dynamical structure factor at S ≤ 3/2 and the solution persists to S ≲ 2. Since ED spectra cannot be obtained at S = 1 or S = 2 on useful clusters, the persistence argument remains internal to the mean-field theory; an independent check (e.g., variational Monte Carlo on the same ansatz or finite-S corrections) is required to substantiate that the same state remains the lowest-energy candidate at integer S.
minor comments (2)
  1. Notation for the singlet and triplet mean-field amplitudes is introduced without a compact table summarizing their self-consistent values across the phase diagram; adding such a table would improve readability.
  2. Figure captions for the dynamical structure factor plots should explicitly state the momentum path and the broadening used, to facilitate direct comparison with the ED data shown in the same panels.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: §3 (mean-field construction) and the paragraph following Eq. (the mixed decoupling): The mixed singlet-triplet Schwinger-boson decoupling is stated to be justified by quantitative agreement with ED for S ≤ 3/2 and exact reproduction of Luttinger-Tisza energies at large S. No error estimate, 1/S expansion, or gauge-fluctuation analysis is supplied for the integer-S regime S ≲ 2 that is the target of the central claim; because the ansatz selection itself rests on spectral matching performed only inside the benchmarked window, the extrapolation step is load-bearing and currently unsupported.

    Authors: We agree that the mixed decoupling is an uncontrolled approximation and that no 1/S expansion or gauge-fluctuation analysis is provided for the S ≲ 2 window. The justification rests on the quantitative match to ED energies for S ≤ 3/2 and the exact reproduction of Luttinger-Tisza results at large S. In the revised manuscript we will add an explicit statement acknowledging the uncontrolled character of the theory in the intermediate-S regime and clarifying that the extrapolation is supported by the dual benchmarks rather than by a controlled expansion. revision: partial

  2. Referee: §4 (spectrum comparison and stability analysis): The chiral QSL is identified as physical because its MF spin excitations follow the ED dynamical structure factor at S ≤ 3/2 and the solution persists to S ≲ 2. Since ED spectra cannot be obtained at S = 1 or S = 2 on useful clusters, the persistence argument remains internal to the mean-field theory; an independent check (e.g., variational Monte Carlo on the same ansatz or finite-S corrections) is required to substantiate that the same state remains the lowest-energy candidate at integer S.

    Authors: The chiral ansatz is selected because its spin excitations provide the closest match to the ED dynamical structure factor within the benchmarked window S ≤ 3/2; the MF solution is then observed to remain stable up to S ≲ 2. We acknowledge that an independent verification (e.g., variational Monte Carlo) would be desirable. Such calculations lie outside the scope of the present work, which develops and benchmarks the mixed Schwinger-boson mean-field theory. We will add a brief remark noting that complementary methods would be valuable for future confirmation. revision: partial

Circularity Check

0 steps flagged

No circularity: ansatz selection and extrapolation rest on independent ED benchmarks

full rationale

The derivation introduces a mixed singlet-triplet Schwinger-boson MF decoupling justified by direct numerical comparison to ED energies (S ≤ 3/2) and exact Luttinger-Tisza semiclassical limits (large S). The physical ansatz is then selected by matching the computed spin excitation spectrum against independent ED dynamical structure factor data. These benchmarks are external to the MF self-consistency equations; the MF parameters are variationally optimized on the Hamiltonian but the selection criterion itself is not defined by those parameters. No quoted step equates a prediction to a fitted input by construction, nor does any load-bearing premise reduce to a self-citation chain. The extrapolation to integer S ≲ 2 is an uncontrolled assumption but does not constitute circularity under the stated criteria.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The approach relies on the Schwinger boson representation and a mean-field decoupling whose validity is justified only by numerical comparison rather than analytic control. Several mean-field amplitudes (hopping and pairing) are determined self-consistently and therefore function as fitted parameters.

free parameters (2)
  • singlet and triplet mean-field amplitudes
    Self-consistently determined hopping and pairing parameters on nearest-neighbor bonds that are optimized to minimize the mean-field energy.
  • Lagrange multiplier for boson constraint
    Enforces the average boson number per site to equal 2S; adjusted during the self-consistent solution.
axioms (2)
  • standard math Schwinger boson representation of spin operators is exact when the local constraint is enforced.
    Invoked at the start of the mean-field construction.
  • domain assumption Mean-field decoupling of quartic boson terms yields a qualitatively correct phase diagram near the Kitaev point.
    Justified by comparison with exact diagonalization for S ≤ 3/2 but not proven for larger S.

pith-pipeline@v0.9.0 · 5757 in / 1620 out tokens · 21792 ms · 2026-05-24T01:13:50.413349+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.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages · 1 internal anchor

  1. [1]

    Savary \ and\ author L

    author author L. Savary \ and\ author L. Balents ,\ 10.1088/0034-4885/80/1/016502 journal journal Reports on Progress in Physics \ volume 80 ,\ pages 016502 ( year 2017 ) NoStop

    work page doi:10.1088/0034-4885/80/1/016502 2017

  2. [2]

    Anyons in an exactly solved model and beyond

    author author A. Kitaev ,\ 10.1016/j.aop.2005.10.005 journal journal Annals of Physics \ volume 321 ,\ pages 2 ( year 2006 ) NoStop

    work page internal anchor Pith review doi:10.1016/j.aop.2005.10.005 2005

  3. [3]

    Kasahara , author T

    author author Y. Kasahara , author T. Ohnishi , author Y. Mizukami , author O. Tanaka , author S. Ma , author K. Sugii , author N. Kurita , author H. Tanaka , author J. Nasu , author Y. Motome , author T. Shibauchi , \ and\ author Y. Matsuda ,\ 10.1038/s41586-018-0274-0 journal journal Nature \ volume 559 ,\ pages 227 ( year 2018 ) NoStop

    work page doi:10.1038/s41586-018-0274-0 2018

  4. [4]

    Yokoi , author S

    author author T. Yokoi , author S. Ma , author Y. Kasahara , author S. Kasahara , author T. Shibauchi , author N. Kurita , author H. Tanaka , author J. Nasu , author Y. Motome , author C. Hickey , author S. Trebst , \ and\ author Y. Matsuda ,\ 10.1126/science.aay5551 journal journal Science \ volume 373 ,\ pages 568 ( year 2021 ) NoStop

    work page doi:10.1126/science.aay5551 2021

  5. [5]

    author author A. M. \ Samarakoon , author A. Banerjee , author S.-S. \ Zhang , author Y. Kamiya , author S. E. \ Nagler , author D. A. \ Tennant , author S.-H. \ Lee , \ and\ author C. D. \ Batista ,\ 10.1103/PhysRevB.96.134408 journal journal Physical Review B \ volume 96 ,\ pages 134408 ( year 2017 ) NoStop

    work page doi:10.1103/physrevb.96.134408 2017

  6. [6]

    author author K. M. \ Taddei , author V. O. \ Garlea , author A. M. \ Samarakoon , author L. D. \ Sanjeewa , author J. Xing , author T. W. \ Heitmann , author C. Dela Cruz , author A. S. \ Sefat , \ and\ author D. Parker ,\ 10.1103/PhysRevResearch.5.013022 journal journal Physical Review Research \ volume 5 ,\ pages 013022 ( year 2023 ) NoStop

    work page doi:10.1103/physrevresearch.5.013022 2023

  7. [7]

    Xu , author J

    author author C. Xu , author J. Feng , author H. Xiang , \ and\ author L. Bellaiche ,\ 10.1038/s41524-018-0115-6 journal journal npj Computational Materials \ volume 4 ,\ pages 57 ( year 2018 ) NoStop

    work page doi:10.1038/s41524-018-0115-6 2018

  8. [8]

    Xu , author J

    author author C. Xu , author J. Feng , author M. Kawamura , author Y. Yamaji , author Y. Nahas , author S. Prokhorenko , author Y. Qi , author H. Xiang , \ and\ author L. Bellaiche ,\ 10.1103/PhysRevLett.124.087205 journal journal Physical Review Letters \ volume 124 ,\ pages 087205 ( year 2020 ) NoStop

    work page doi:10.1103/physrevlett.124.087205 2020

  9. [9]

    Baskaran , author D

    author author G. Baskaran , author D. Sen , \ and\ author R. Shankar ,\ 10.1103/PhysRevB.78.115116 journal journal Physical Review B \ volume 78 ,\ pages 115116 ( year 2008 ) NoStop

    work page doi:10.1103/physrevb.78.115116 2008

  10. [10]

    \ Lee , author N

    author author H.-Y. \ Lee , author N. Kawashima , \ and\ author Y. B. \ Kim ,\ 10.1103/PhysRevResearch.2.033318 journal journal Physical Review Research \ volume 2 ,\ pages 033318 ( year 2020 ) NoStop

    work page doi:10.1103/physrevresearch.2.033318 2020

  11. [11]

    Khait , author P

    author author I. Khait , author P. P. \ Stavropoulos , author H.-Y. \ Kee , \ and\ author Y. B. \ Kim ,\ 10.1103/PhysRevResearch.3.013160 journal journal Physical Review Research \ volume 3 ,\ pages 013160 ( year 2021 ) NoStop

    work page doi:10.1103/physrevresearch.3.013160 2021

  12. [12]

    \ Dong \ and\ author D

    author author X.-Y. \ Dong \ and\ author D. N. \ Sheng ,\ 10.1103/PhysRevB.102.121102 journal journal Physical Review B \ volume 102 ,\ pages 121102 ( year 2020 ) ,\ note arXiv: 1911.12854 NoStop

    work page doi:10.1103/physrevb.102.121102 2020

  13. [13]

    Zhu , author Z.-Y

    author author Z. Zhu , author Z.-Y. \ Weng , \ and\ author D. N. \ Sheng ,\ 10.1103/PhysRevResearch.2.022047 journal journal Physical Review Research \ volume 2 ,\ pages 022047 ( year 2020 ) NoStop

    work page doi:10.1103/physrevresearch.2.022047 2020

  14. [14]

    Ma ,\ 10.1103/PhysRevLett.130.156701 journal journal Physical Review Letters \ volume 130 ,\ pages 156701 ( year 2023 ) NoStop

    author author H. Ma ,\ 10.1103/PhysRevLett.130.156701 journal journal Physical Review Letters \ volume 130 ,\ pages 156701 ( year 2023 ) NoStop

    work page doi:10.1103/physrevlett.130.156701 2023

  15. [15]

    Kos \ and\ author M

    author author P. Kos \ and\ author M. Punk ,\ 10.1103/PhysRevB.95.024421 journal journal Physical Review B \ volume 95 ,\ pages 024421 ( year 2017 ) NoStop

    work page doi:10.1103/physrevb.95.024421 2017

  16. [16]

    Kargarian , author A

    author author M. Kargarian , author A. Langari , \ and\ author G. A. \ Fiete ,\ 10.1103/PhysRevB.86.205124 journal journal Physical Review B \ volume 86 ,\ pages 205124 ( year 2012 ) NoStop

    work page doi:10.1103/physrevb.86.205124 2012

  17. [17]

    Schneider , author J

    author author B. Schneider , author J. C. \ Halimeh , \ and\ author M. Punk ,\ 10.1103/PhysRevB.105.125122 journal journal Physical Review B \ volume 105 ,\ pages 125122 ( year 2022 ) NoStop

    work page doi:10.1103/physrevb.105.125122 2022

  18. [18]

    author author A. Auerbach ,\ 10.1007/978-1-4612-0869-3 title Interacting Electrons and Quantum Magnetism ,\ Graduate Texts in Contemporary Physics \ ( publisher Springer New York ,\ address New York, NY ,\ year 1994 ) NoStop

    work page doi:10.1007/978-1-4612-0869-3 1994

  19. [19]

    Mezio , author C

    author author A. Mezio , author C. N. \ Sposetti , author L. O. \ Manuel , \ and\ author A. E. \ Trumper ,\ 10.1209/0295-5075/94/47001 journal journal EPL (Europhysics Letters) \ volume 94 ,\ pages 47001 ( year 2011 ) NoStop

    work page doi:10.1209/0295-5075/94/47001 2011

  20. [20]

    Mezio , author L

    author author A. Mezio , author L. O. \ Manuel , author R. R. P. \ Singh , \ and\ author A. E. \ Trumper ,\ 10.1088/1367-2630/14/12/123033 journal journal New Journal of Physics \ volume 14 ,\ pages 123033 ( year 2012 ) NoStop

    work page doi:10.1088/1367-2630/14/12/123033 2012

  21. [21]

    Flint \ and\ author P

    author author R. Flint \ and\ author P. Coleman ,\ 10.1103/PhysRevB.79.014424 journal journal Physical Review B \ volume 79 ,\ pages 014424 ( year 2009 ) NoStop

    work page doi:10.1103/physrevb.79.014424 2009

  22. [22]

    Chaloupka , author G

    author author J. Chaloupka , author G. Jackeli , \ and\ author G. Khaliullin ,\ 10.1103/PhysRevLett.110.097204 journal journal Physical Review Letters \ volume 110 ,\ pages 097204 ( year 2013 ) NoStop

    work page doi:10.1103/physrevlett.110.097204 2013

  23. [23]

    Read \ and\ author S

    author author N. Read \ and\ author S. Sachdev ,\ 10.1103/PhysRevLett.66.1773 journal journal Physical Review Letters \ volume 66 ,\ pages 1773 ( year 1991 ) NoStop

    work page doi:10.1103/physrevlett.66.1773 1991

  24. [24]

    Sachdev ,\ 10.1103/PhysRevB.45.12377 journal journal Physical Review B \ volume 45 ,\ pages 12377 ( year 1992 ) NoStop

    author author S. Sachdev ,\ 10.1103/PhysRevB.45.12377 journal journal Physical Review B \ volume 45 ,\ pages 12377 ( year 1992 ) NoStop

    work page doi:10.1103/physrevb.45.12377 1992

  25. [25]

    author author J. M. \ Luttinger ,\ 10.1103/PhysRev.81.1015 journal journal Physical Review \ volume 81 ,\ pages 1015 ( year 1951 ) NoStop

    work page doi:10.1103/physrev.81.1015 1951

  26. [26]

    Litvin ,\ 10.1016/0031-8914(74)90257-2 journal journal Physica \ volume 77 ,\ pages 205 ( year 1974 ) NoStop

    author author D. Litvin ,\ 10.1016/0031-8914(74)90257-2 journal journal Physica \ volume 77 ,\ pages 205 ( year 1974 ) NoStop

    work page doi:10.1016/0031-8914(74)90257-2 1974

  27. [27]

    Merino \ and\ author A

    author author J. Merino \ and\ author A. Ralko ,\ 10.1103/PhysRevB.97.205112 journal journal Physical Review B \ volume 97 ,\ pages 205112 ( year 2018 ) NoStop

    work page doi:10.1103/physrevb.97.205112 2018

  28. [28]

    Lugan , author L

    author author T. Lugan , author L. D. C. \ Jaubert , \ and\ author A. Ralko ,\ 10.1103/PhysRevResearch.1.033147 journal journal Physical Review Research \ volume 1 ,\ pages 033147 ( year 2019 ) NoStop

    work page doi:10.1103/physrevresearch.1.033147 2019

  29. [29]

    Lugan , author L

    author author T. Lugan , author L. D. C. \ Jaubert , author M. Udagawa , \ and\ author A. Ralko ,\ 10.1103/PhysRevB.106.L140404 journal journal Physical Review B \ volume 106 ,\ pages L140404 ( year 2022 ) NoStop

    work page doi:10.1103/physrevb.106.l140404 2022

  30. [30]

    author author J. C. \ Halimeh \ and\ author M. Punk ,\ 10.1103/PhysRevB.94.104413 journal journal Physical Review B \ volume 94 ,\ pages 104413 ( year 2016 ) NoStop

    work page doi:10.1103/physrevb.94.104413 2016

  31. [31]

    Janssen \ and\ author U

    author author L. Janssen \ and\ author U. F. P. \ Seifert ,\ 10.1103/PhysRevB.105.045120 journal journal Physical Review B \ volume 105 ,\ pages 045120 ( year 2022 ) \ NoStop

    work page doi:10.1103/physrevb.105.045120 2022

  32. [32]

    author author P. M. \ Cônsoli , author L. Janssen , author M. Vojta , \ and\ author E. C. \ Andrade ,\ 10.1103/PhysRevB.102.155134 journal journal Physical Review B \ volume 102 ,\ pages 155134 ( year 2020 ) NoStop

    work page doi:10.1103/physrevb.102.155134 2020

  33. [33]

    Journal of Physics: Condensed Matter , abstract =

    author author S. Toth \ and\ author B. Lake ,\ 10.1088/0953-8984/27/16/166002 journal journal Journal of Physics: Condensed Matter \ volume 27 ,\ pages 166002 ( year 2015 ) NoStop

    work page doi:10.1088/0953-8984/27/16/166002 2015

  34. [34]

    Wang ,\ 10.1103/PhysRevB.82.024419 journal journal Physical Review B \ volume 82 ,\ pages 024419 ( year 2010 ) NoStop

    author author F. Wang ,\ 10.1103/PhysRevB.82.024419 journal journal Physical Review B \ volume 82 ,\ pages 024419 ( year 2010 ) NoStop

    work page doi:10.1103/physrevb.82.024419 2010

  35. [35]

    Tchernyshyov , author R

    author author O. Tchernyshyov , author R. Moessner , \ and\ author S. L. \ Sondhi ,\ 10.1209/epl/i2005-10389-2 journal journal Europhysics Letters (EPL) \ volume 73 ,\ pages 278 ( year 2006 ) \ NoStop

    work page doi:10.1209/epl/i2005-10389-2 2006

  36. [36]

    Messio , author C

    author author L. Messio , author C. Lhuillier , \ and\ author G. Misguich ,\ 10.1103/PhysRevB.87.125127 journal journal Physical Review B \ volume 87 ,\ pages 125127 ( year 2013 ) NoStop

    work page doi:10.1103/physrevb.87.125127 2013

  37. [37]

    Hickey , author C

    author author C. Hickey , author C. Berke , author P. P. \ Stavropoulos , author H.-Y. \ Kee , \ and\ author S. Trebst ,\ 10.1103/PhysRevResearch.2.023361 journal journal Physical Review Research \ volume 2 ,\ pages 023361 ( year 2020 ) NoStop

    work page doi:10.1103/physrevresearch.2.023361 2020

  38. [38]

    Ralko \ and\ author J

    author author A. Ralko \ and\ author J. Merino ,\ 10.1103/PhysRevLett.124.217203 journal journal Physical Review Letters \ volume 124 ,\ pages 217203 ( year 2020 ) NoStop

    work page doi:10.1103/physrevlett.124.217203 2020

  39. [39]

    Bieri , author C

    author author S. Bieri , author C. Lhuillier , \ and\ author L. Messio ,\ 10.1103/PhysRevB.93.094437 journal journal Physical Review B \ volume 93 ,\ pages 094437 ( year 2016 ) NoStop

    work page doi:10.1103/physrevb.93.094437 2016

  40. [40]

    Kimchi \ and\ author A

    author author I. Kimchi \ and\ author A. Vishwanath ,\ 10.1103/PhysRevB.89.014414 journal journal Physical Review B \ volume 89 ,\ pages 014414 ( year 2014 ) NoStop

    work page doi:10.1103/physrevb.89.014414 2014

  41. [41]

    Kishimoto , author K

    author author M. Kishimoto , author K. Morita , author Y. Matsubayashi , author S. Sota , author S. Yunoki , \ and\ author T. Tohyama ,\ 10.1103/PhysRevB.98.054411 journal journal Physical Review B \ volume 98 ,\ pages 054411 ( year 2018 ) NoStop

    work page doi:10.1103/physrevb.98.054411 2018

  42. [42]

    Rousochatzakis , author U

    author author I. Rousochatzakis , author U. K. \ Rössler , author J. van den Brink , \ and\ author M. Daghofer ,\ 10.1103/PhysRevB.93.104417 journal journal Physical Review B \ volume 93 ,\ pages 104417 ( year 2016 ) NoStop

    work page doi:10.1103/physrevb.93.104417 2016

  43. [43]

    Koga , author H

    author author A. Koga , author H. Tomishige , \ and\ author J. Nasu ,\ 10.7566/JPSJ.87.063703 journal journal Journal of the Physical Society of Japan \ volume 87 ,\ pages 063703 ( year 2018 ) NoStop

    work page doi:10.7566/jpsj.87.063703 2018

  44. [44]

    Evenbly \ and\ author R

    author author G. Evenbly \ and\ author R. N. C. \ Pfeifer ,\ 10.1103/PhysRevB.89.245118 journal journal Physical Review B \ volume 89 ,\ pages 245118 ( year 2014 ) NoStop

    work page doi:10.1103/physrevb.89.245118 2014

  45. [45]

    Gohlke , author R

    author author M. Gohlke , author R. Verresen , author R. Moessner , \ and\ author F. Pollmann ,\ 10.1103/PhysRevLett.119.157203 journal journal Physical Review Letters \ volume 119 ,\ pages 157203 ( year 2017 ) NoStop

    work page doi:10.1103/physrevlett.119.157203 2017