pith. sign in

arxiv: 2601.20189 · v2 · submitted 2026-01-28 · ❄️ cond-mat.str-el · hep-lat

High-precision ground state parameters of the two-dimensional spin-1/2 Heisenberg model on the square lattice

Anders W. Sandvik This is my paper

Pith reviewed 2026-05-16 10:36 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-lat
keywords Heisenberg antiferromagnetsquare latticequantum Monte Carloground state energysublattice magnetizationchiral perturbation theoryfinite size scalingspin stiffness
0
0 comments X

The pith

The ground state energy density of the square-lattice spin-1/2 Heisenberg antiferromagnet is -0.669441857(7)

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

The paper performs quantum Monte Carlo simulations using the stochastic series expansion method on square lattices up to 96 by 96 sites at very low temperatures. It computes the energy, order parameter, and several other ground state properties, then extrapolates to the infinite-size limit. The resulting energy precision is three orders of magnitude better than prior work, and the finite-size scaling matches chiral perturbation theory including logarithmic corrections. These results provide high-accuracy benchmarks for the model that is central to understanding quantum magnetism in two dimensions.

Core claim

Extensive SSE quantum Monte Carlo simulations on L x L lattices with L up to 96 at temperatures low enough to reach the T=0 limit yield extrapolated ground-state energy density e0 = -0.669441857(7) and sublattice magnetization ms = 0.307447(2). The leading finite-size corrections agree quantitatively with chiral perturbation theory, and a logarithmic correction to the order parameter with exponent gamma = 0.82(4) is confirmed.

What carries the argument

Stochastic series expansion quantum Monte Carlo combined with chiral perturbation theory finite-size scaling forms, including logarithmic terms, for extrapolation from finite L to infinite size.

If this is right

  • The tabulated results for periodic, open, and cylindrical boundaries enable direct comparisons with other numerical techniques.
  • The confirmed value of the logarithmic exponent gamma provides a new benchmark for theoretical calculations.
  • Precise values for spin stiffness, spin-wave velocity, and susceptibilities are now available for testing effective field theories.
  • The method demonstrates that moderately sized lattices suffice when low enough temperatures are reached.

Where Pith is reading between the lines

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

  • Similar high-precision extrapolations could be applied to slightly perturbed models to study quantum phase transitions.
  • The agreement with chiral perturbation theory suggests that higher-order terms in the effective theory could be tested with even larger lattices.
  • These benchmarks may help resolve discrepancies in estimates from other methods like series expansions or density matrix renormalization group.

Load-bearing premise

The chosen simulation temperatures are low enough that the system has reached its true ground state, and the chiral perturbation theory scaling forms capture all important finite-size corrections for lattices up to size 96.

What would settle it

A calculation on a lattice with L much larger than 96, say L=200, that yields an energy density differing from -0.669441857 by more than the reported uncertainty of 7 in the last digit would falsify the extrapolation.

Figures

Figures reproduced from arXiv: 2601.20189 by Anders W. Sandvik.

Figure 1
Figure 1. Figure 1: FIG. 1. Uniform susceptibility versus the system size for three [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Probability vs the lattice size of the system being [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Scatter plot of the coefficients of the cubic and quar [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Residual ∆( [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The squared sublattice magnetization with the fit [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Scatter plot of the coefficients of the linear ( [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Twice the ground state energy per bond for systems [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Distortion field of the order parameter, Eq. (21), vs [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Inverse-size dependence of the sublattice magneti [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
read the original abstract

Several ground state properties of the square-lattice $S=1/2$ Heisenberg antiferromagnet are computed (the energy, order parameter, spin stiffness, spinwave velocity, long-wavelength susceptibility, and staggered susceptibility) using extensive quantum Monte Carlo simulations with the stochastic series expansion method. Moderately sized lattices are studied at temperatures $T$ sufficiently low to realize the $T \to 0$ limit. Results for periodic $L\times L$ lattices with $L \in [6,96]$ are tabulated versus $L$ and extrapolations to infinite system size are carried out. The extrapolated ground state energy density is $e_0=-0.669441857(7)$, which represents an improvement in precision of three orders of magnitude over the previously best result. The leading and subleading finite-size corrections to $e_0$ are in full quantitative agreement with predictions from chiral perturbation theory, thus further supporting the soundness of both the extrapolations and the theory. The extrapolated sublattice magnetization is $m_s=0.307447(2)$, which agrees well with previous estimates but with a much smaller statistical error. The coefficient of the linear in $L^{-1}$ correction to $m^2_s$ agrees with the value from chiral perturbation theory and the presence of a factor $\ln^\gamma(L)$ in the second-order correction is also confirmed, with the previously not known value of the exponent being $\gamma = 0.82(4)$. The finite-size corrections to the staggered susceptibility point to logarithmic corrections also in this quantity. To facilitate benchmarking of methods for which periodic boundary conditions are challenging, results for systems with open and cylindrical boundaries are also listed and their spatially inhomogeneous order parameters are analyzed.

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

0 major / 3 minor

Summary. The manuscript reports extensive stochastic series expansion quantum Monte Carlo simulations of the square-lattice spin-1/2 Heisenberg antiferromagnet on lattices up to L=96 at low temperatures. It provides extrapolated infinite-size values for the ground-state energy density e_0=-0.669441857(7), sublattice magnetization m_s=0.307447(2), and other quantities including spin stiffness and susceptibilities, with finite-size corrections analyzed using chiral perturbation theory forms, including confirmation of logarithmic corrections with exponent γ=0.82(4). Results for open and cylindrical boundaries are also provided to facilitate benchmarking.

Significance. If the results hold, this work delivers a three-order-of-magnitude improvement in the precision of the ground-state energy benchmark for this paradigmatic model, together with quantitative confirmation that extracted correction coefficients match independent chiral PT predictions. This strengthens both the numerical extrapolations and the validity of the scaling assumptions, providing a high-accuracy reference for validating other methods and for studies of quantum antiferromagnets.

minor comments (3)
  1. [Energy extrapolation] The abstract states that the leading and subleading finite-size corrections to e_0 agree with chiral PT, but the main text should explicitly tabulate the fitted coefficients alongside the PT predictions for direct comparison (e.g., in the energy extrapolation section).
  2. [Magnetization analysis] The value γ=0.82(4) for the logarithmic correction to m_s^2 is reported; the manuscript should state the range of L values and the number of points included in that particular fit to allow assessment of its robustness.
  3. [Methods and data tables] Tables listing raw data versus L are referenced; including a brief note on how statistical errors were estimated from the SSE runs (e.g., binning or jackknife) would improve reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment, including the recommendation to accept. We are pleased that the work is recognized for delivering a substantial improvement in precision for the ground-state benchmarks of the square-lattice S=1/2 Heisenberg antiferromagnet and for confirming the chiral perturbation theory predictions.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper computes ground-state quantities via direct SSE QMC simulations on finite L×L lattices (L=6..96) at sufficiently low T to reach the T=0 limit, followed by standard finite-size extrapolations using established chiral PT scaling forms. The central results (e_0, m_s, etc.) are numerical outputs of this procedure; the reported agreement between fitted correction coefficients and independent PT predictions is a consistency check, not a self-definitional reduction. No self-citation chain, ansatz smuggling, or renaming of known results is load-bearing for the quoted precision. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Central claims rest on numerical data plus standard scaling assumptions from chiral perturbation theory; no new entities are postulated.

free parameters (1)
  • finite-size fit coefficients
    Coefficients in the L-dependent extrapolation forms for energy and magnetization squared are determined from the simulation data.
axioms (1)
  • domain assumption Chiral perturbation theory supplies the correct functional form for finite-size corrections including logarithmic terms
    Invoked to guide and validate the extrapolations to infinite volume.

pith-pipeline@v0.9.0 · 5618 in / 1194 out tokens · 22155 ms · 2026-05-16T10:36:08.499875+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 3 Pith papers

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

  1. Parallel Scan Recurrent Neural Quantum States for Scalable Variational Monte Carlo

    cond-mat.str-el 2026-05 conditional novelty 7.0

    PSR-NQS makes recurrent neural quantum states scalable for variational Monte Carlo by using parallel scan recurrence, reaching accurate results on 52x52 two-dimensional lattices.

  2. svPITE: A Python package for the state-vector-based probabilistic imaginary-time evolution algorithm

    quant-ph 2026-05 unverdicted novelty 4.0

    svPITE is a Python package that provides state-vector and shot-based implementations of probabilistic imaginary-time evolution for quantum ground-state preparation along with parameter tuning and interoperability for ...

  3. svPITE: A Python package for the state-vector-based probabilistic imaginary-time evolution algorithm

    quant-ph 2026-05 unverdicted novelty 3.0

    svPITE is a Python package for ground-state preparation via probabilistic imaginary-time evolution, supporting state-vector and shot-based modes with exact-diagonalization benchmarking and interoperability for dynamic...

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages · cited by 2 Pith papers · 1 internal anchor

  1. [1]

    This form without any higher-order corrections fits the data forL≥12 extremely well, per- haps suggesting that there is noL −3 correction

    and extrapolated using a fit including the predicted log correction to theL −2 term (the leading correction be- ing linear inL −1). This form without any higher-order corrections fits the data forL≥12 extremely well, per- haps suggesting that there is noL −3 correction. All data forL≥6 can be fitted with an addedL −5 correction, which leads to the best ex...

    work page 2000

  2. [2]

    The trend with increasingL min is also not indicative of eventual agree- ment for forL min >14 (until the size of the distribution grows too large for meaningful comparisons)

    While the individualx 0 andx 1 windows do have large overlaps with the distribution, the joint (x 0, x1) window 12 is far away from the high-probability region, to the extent that an agreement is statistically very unlikely (noting also that the indicated window corresponds to two stan- dard deviations of the mean coefficients). The trend with increasingL...

    work page

  3. [3]

    The trend of increas- ingly good variational states based on TNs and NNs will only continue and better benchmarks are called for

    is now outdated, considering that some of the emerg- ing many-body methods are now exceeding the statistical precision reported there, at least in the case of the ground state energy forLup to 10 [22]. The trend of increas- ingly good variational states based on TNs and NNs will only continue and better benchmarks are called for. The raw data listed here ...

    work page

  4. [4]

    E. Y. Loh, Jr., J. E. Gubernatis, R. T. Scalettar, S. R. White, D. J. Scalapino, and R. L. Sugar, Sign problem in the numerical simulation of many-electron systems, Phys. Rev. B41, 9301 (1990)

    work page 1990

  5. [5]

    Henelius and A

    P. Henelius and A. W. Sandvik, Sign problem in Monte Carlo simulations of frustrated quantum spin systems, Phys. Rev. B62, 1102 (2000)

    work page 2000

  6. [6]

    Troyer and U.-J

    M. Troyer and U.-J. Wiese, Computational Complex- ity and Fundamental Limitations to Fermionic Quantum Monte Carlo Simulations, Phys. Rev. Lett.94, 170201 (2005)

    work page 2005

  7. [7]

    A. W. Sandvik, Stochastic series expansion method with operator-loop update, Phys. Rev. B59, R14157 (1999)

    work page 1999

  8. [8]

    O. F. Sylju˚ asen and A. W. Sandvik, Quantum Monte Carlo with directed loops, Phys. Rev. E66, 046701 (2002)

    work page 2002

  9. [9]

    A. W. Sandvik, Computational Studies of Quantum Spin Systems, AIP Conf. Proc.1297, 135 (2010)

    work page 2010

  10. [10]

    H. G. Evertz, G. Lana, and M. Marcu, Cluster algorithm for vertex models, Phys. Rev. Lett.70, 875 (1993)

    work page 1993

  11. [11]

    A. W. Sandvik and H. G. Evertz, Loop updates for varia- tional and projector quantum Monte Carlo simulations in the valence-bond basis, Phys. Rev. B98, 024407 (2010)

    work page 2010

  12. [12]

    N. Ma, P. Weinberg, H. Shao, W. Guo, D.-X. Yao, and A. W. Sandvik, Anomalous Quantum-Critical Scal- ing Corrections in Two-Dimensional Antiferromagnets, Phys. Rev. Lett.121, 117202 (2018)

    work page 2018

  13. [13]

    A. W. Sandvik and B. Zhao, Consistent Scaling Expo- nents at the Deconfined Quantum-Critical Point, Chin. Phys. Lett.37, 057502 (2020)

    work page 2020

  14. [14]

    Takahashi, H

    J. Takahashi, H. Shao, B. Zhao, W. Guo, A. W. Sand- vik, SO(5) multicriticality in two-dimensional quantum magnets, arXiv:2405.06607

    work page arXiv

  15. [15]

    S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B48, 10345 (1993)

    work page 1993

  16. [16]

    Schollwoeck, The density-matrix renormalization group in the age of matrix product states, Ann

    U. Schollwoeck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011)

    work page 2011

  17. [17]

    E. M. Stoudenmire and S. R. White, Real-space parallel density matrix renormalization group, Phys. Rev. B87, 155137 (2013)

    work page 2013

  18. [18]

    V. Murg, F. Verstraete, and J. I. Cirac, Exploring frus- trated spin systems using projected entangled pair states, Phys. Rev. B79, 195119 (2009)

    work page 2009

  19. [19]

    W.-Y. Liu, S. Dong, C. Wang, Y. Han, H. An, G.-C. Guo, and L. He, Gapless spin liquid ground state of the spin-1/2J 1-J2 Heisenberg model on square lattices, Phys. Rev. B98, 241109 (2018)

    work page 2018

  20. [20]

    Or´ us, Tensor networks for complex quantum systems, Nat

    R. Or´ us, Tensor networks for complex quantum systems, Nat. Rev. Phys.1, 538 (2019)

    work page 2019

  21. [21]

    Quantum spin liquid phase in the Shastry-Sutherland model revealed by high-precision infinite projected entangled-pair states

    P. Corboz, Y. Zhang, B. Ponsioen, and F. Mila, Quan- tum spin liquid phase in the Shastry-Sutherland model revealed by high-precision infinite projected entangled- pair states, arXiv:2502.14091

    work page internal anchor Pith review Pith/arXiv arXiv

  22. [22]

    Carleo and M

    G. Carleo and M. Troyer, Solving the quantum many- body problem with artificial neural networks, Science 355, 602 (2017)

    work page 2017

  23. [23]

    Lange, A

    H. Lange, A. Van de Walle, A. Abedinnia, and A. Bohrdt, 17 From Architectures to Applications: A Review of Neu- ral Quantum States, Quantum Sci. Technol.9, 040501 (2024)

    work page 2024

  24. [24]

    D. Wu, R. Rossi, F. Vicentini, and G. Carleo, From tensor-network quantum states to tensorial recurrent neural networks, Phys. Rev. Res.5, L032001 (2023)

    work page 2023

  25. [25]

    Chen and M

    A. Chen and M. Heyl, Empowering deep neural quantum states through efficient optimization, Nat. Phys.20, 1476 (2024)

    work page 2024

  26. [26]

    Rende, S

    R. Rende, S. Goldt, F. Becca, and L. L. Viteritti, Fine- tuning neural network quantum states, Phys. Rev. Res. 6043280 (2024)

    work page 2024

  27. [27]

    Bortone, Y

    M. Bortone, Y. Rath, and G. H. Booth, Impact of condi- tional modelling for a universal autoregressive quantum state, Quantum8, 1245 (2024)

    work page 2024

  28. [28]

    J. Wang, J. Surace, I. Fr´ erot, B. Legat, M.-O. Renou, V. Magron, and Ac´ ın, Certifying Ground-State Properties of Many-Body Systems, Phys. Rev. X14031006 (2024)

    work page 2024

  29. [29]

    I. Kull, N. Schuch, B. Dive, and M. Navascu´ es, Lower Bounds on Ground-State Energies of Local Hamiltonians through the Renormalization Group, Phys. Rev. X 14, 021008 (2024)

    work page 2024

  30. [30]

    M. S. Moss, R. Wiersema, M. Hibat-Allah, J. Car- rasquilla, and R. G. Melko, Leveraging recurrence in neural network wavefunctions for large-scale simula- tions of Heisenberg antiferromagnets: the square lattice, arXiv:2502.17144

    work page arXiv

  31. [31]

    Hasenfratz and F

    P. Hasenfratz and F. Niedermayer, Finite size and tem- perature effects in the AF Heisenberg model, Z. Phys. B 92, 91 (1993)

    work page 1993

  32. [32]

    Niedermayer and C

    F. Niedermayer and C. Weiermann, The rotator spec- trum in the delta-regime of the O(n) effective field theory in 3 and 4 dimensions, Nucl. Phys. B842, 248 (2011)

    work page 2011

  33. [33]

    P. W. Anderson, New Approach to the Theory of Su- perexchange Interactions, Phys. Rev.115, 2 (1959)

    work page 1959

  34. [34]

    B. I. Halperin and P. C. Hohenberg, Hydrodynamic The- ory of Spin Waves, Phys. Rev.188, 898 (1969)

    work page 1969

  35. [35]

    Oitmaa and D

    J. Oitmaa and D. D. Betts, Ground state properties of the Heisenberg antiferromagnet and XY magnet in three dimensions, Phys. Lett. A68, 450 (1978)

    work page 1978

  36. [36]

    Oitmaa and D

    J. Oitmaa and D. D. Betts, The ground state of two quantum models of magnetism, Can. J. Phys.56, 897 (1978)

    work page 1978

  37. [37]

    D. A. Huse, Ground-state staggered magnetization of two-dimensional quantum Heisenberg antiferromagnets, Phys. Rev. B37, 2380 (1988)

    work page 1988

  38. [38]

    J. D. Reger and A. P. Young, Monte Carlo simulations of the spin-1/2 Heisenberg antiferromagnet on a square lattice, Phys. Rev. B37, 5978 (1988)

    work page 1988

  39. [39]

    Chakravarty, B

    S. Chakravarty, B. I. Halperin, and D. R. Nelson, Two- dimensional quantum Heisenberg antiferromagnet at low temperatures, Phys. Rev. B39, 2344 (1989)

    work page 1989

  40. [40]

    Liang, Existence of N´ eel order at T=0 in the spin-1/2 antiferromagnetic Heisenberg model on a square lattice, Phys

    S. Liang, Existence of N´ eel order at T=0 in the spin-1/2 antiferromagnetic Heisenberg model on a square lattice, Phys. Rev. B42, 6555 (1990)

    work page 1990

  41. [41]

    M. S. Makivic and H.-Q. Ding, Two-dimensional spin-1/2 Heisenberg antiferromagnet: A quantum Monte Carlo study, Phys. Rev. B43, 3562 (1991)

    work page 1991

  42. [42]

    Manousakis, The spin-1/2 Heisenberg antiferromagnet on a square lattice and its application to the cuprous oxides, Rev

    E. Manousakis, The spin-1/2 Heisenberg antiferromagnet on a square lattice and its application to the cuprous oxides, Rev. Mod. Phys.63, 1 (1991)

    work page 1991

  43. [43]

    K. J. Runge, Quantum Monte Carlo calculation of the long-range order in the Heisenberg antiferromagnet, Phys. Rev. B45, 7229 (1992)

    work page 1992

  44. [44]

    K. J. Runge, Finite-size study of the ground-state energy, susceptibility, and spin-wave velocity for the Heisenberg antiferromagnet, Phys. Rev. B45, 12292 (1992)

    work page 1992

  45. [45]

    Wiese and H.-P

    U.-J. Wiese and H.-P. Ying, A determination of the low energy parameters of the 2-d Heisenberg antiferromag- net, Z. Phys. B93, 147 (1994)

    work page 1994

  46. [46]

    B. B. Beard and U.-J. Wiese, Simulations of Discrete Quantum Systems in Continuous Euclidean Time Phys. Rev. Lett.77, 5130 (1996)

    work page 1996

  47. [47]

    A. W. Sandvik, Finite-size scaling of the ground-state pa- rameters of the two-dimensional Heisenberg model, Phys. Rev. B56, 11678 (1997)

    work page 1997

  48. [48]

    Jiang and U.-J

    F.-J. Jiang and U.-J. Wiese, High-precision determi- nation of low-energy effective parameters for a two- dimensional Heisenberg quantum antiferromagnet Phys. Rev. B83, 155120 (2011)

    work page 2011

  49. [49]

    A. Sen, H. Suwa, and A. W. Sandvik, Velocity of exci- tations in ordered, disordered, and critical antiferromag- nets, Phys. Rev. B92, 195145 (2015)

    work page 2015

  50. [50]

    Neuberger and T

    H. Neuberger and T. Ziman, Finite-size effects in Heisen- berg antiferromagnets, Phys. Rev. B39, 2608 (1989)

    work page 1989

  51. [51]

    D. S. Fisher, Universality, low-temperature properties, and finite-size scaling in quantum antiferromagnets, Phys. Rev. B39, 11783 (1989)

    work page 1989

  52. [52]

    A. W. Sandvik and J. Kurkij¨ arvi, Quantum Monte Carlo simulation method for spin systems, Phys. Rev. B43, 5950 (1991)

    work page 1991

  53. [53]

    E. L. Pollock and D. M. Ceperley, Path-integral com- putation of superfluid densities, Phys. Rev. B36, 8343 (1987)

    work page 1987

  54. [54]

    Lavalle, S

    C. Lavalle, S. Sorella, and A. Parola, Anomalous Finite Size Spectrum in the S=1/2 Two Dimensional Heisenberg Model, Phys. Rev. Lett.80, 1746 (1998)

    work page 1998

  55. [55]

    Niedermayera and P

    F. Niedermayera and P. Weisz, Casimir squared correc- tion to the standard rotator Hamiltonian for the O(n) sigma-model in the delta-regime, J. High Energy Phys. 2018, 70 (2018)

    work page 2018

  56. [56]

    J. Yang, A. W. Sandvik, and L. Wang, Quantum crit- icality and spin liquid phase in the Shastry-Sutherland model, Phys. Rev.105, L060409 (2022)

    work page 2022

  57. [57]

    Sanyal, A

    S. Sanyal, A. Banerjee, K. Damle, and A. W. Sandvik, Antiferromagnetic order in systems with doubletS tot = 1/2 ground states, Phys. Rev.86, 064418 (2012)

    work page 2012