pith. sign in

arxiv: 2604.20831 · v1 · submitted 2026-04-22 · ❄️ cond-mat.str-el · quant-ph

Unconventional Quantum Criticality in Long-Range Spin-1 Chains: Insights from Entanglement Entropy and Bipartite Fluctuations

Justin Tim-Lok Chau , Jiarui Zhao , Nicolas Laflorencie , Zi Yang Meng This is my paper

Pith reviewed 2026-05-09 23:10 UTC · model grok-4.3

classification ❄️ cond-mat.str-el quant-ph
keywords long-range interactionsspin-1 chainHaldane phaseNéel phasequantum phase transitionentanglement entropybipartite fluctuationsquantum Monte Carlo
0
0 comments X

The pith

A continuous quantum phase transition in a long-range spin-1 chain occurs at α_c = 2.48(2) and is nonconformal with dynamic exponent z ≠ 1.

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

The authors investigate a spin-1 Heisenberg chain whose interactions include a staggered long-range component that falls off as a power of distance. They employ quantum Monte Carlo simulations that map the problem onto constrained spin-1/2 variables, allowing them to reach system sizes large enough to perform reliable scaling analyses. These analyses locate a continuous transition separating a gapped phase that lacks conventional magnetic order from a gapless phase that develops antiferromagnetic order and thereby breaks continuous spin-rotation symmetry. The extracted exponents show that the transition does not belong to the usual conformal class because time and space rescale differently. Entanglement entropy and bipartite number fluctuations are then used to extract universal scaling forms that distinguish the two phases and the critical region itself.

Core claim

The ground-state phase diagram of the spin-1 Heisenberg chain with staggered long-range interactions decaying as r^{-α} contains a continuous quantum phase transition at α_c = 2.48(2) between the gapped Haldane phase realized at large α and the gapless antiferromagnetically ordered Néel phase realized at small α. The continuous SU(2) symmetry is spontaneously broken across the transition, which is characterized by a dynamic exponent z ≠ 1 and is therefore nonconformal. Finite-size scaling together with crossing-point analyses of entanglement entropy and bipartite fluctuations determine the location of the critical point and the universal scaling behaviors inside each phase and at criticality

What carries the argument

The split-spin representation that maps the spin-1 Hilbert space onto locally constrained spin-1/2 degrees of freedom, combined with quantum Monte Carlo sampling and finite-size scaling of entanglement entropy and bipartite fluctuations to extract the critical point and exponents.

If this is right

  • For interaction decay exponents α larger than 2.48 the ground state remains in the gapped Haldane phase with no conventional magnetic order.
  • For α smaller than 2.48 the ground state enters the gapless Néel phase in which continuous SU(2) symmetry is broken by antiferromagnetic order.
  • Entanglement entropy obeys distinct scaling forms in the gapped phase, the gapless phase, and exactly at the critical point.
  • Bipartite fluctuations display universal scaling that changes across the transition and thereby furnishes an independent diagnostic of the nonconformal criticality.

Where Pith is reading between the lines

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

  • The same numerical framework could be applied to spin-1 chains that combine both short-range and long-range terms to map out how the critical value α_c moves with the relative strength of the two components.
  • Quantum simulators with tunable power-law interactions could directly measure the predicted entanglement and fluctuation scalings to test the location of α_c and the value of z.
  • The appearance of z ≠ 1 suggests that long-range couplings may generically produce critical theories whose correlation functions decay with anomalous dynamic scaling in one dimension.

Load-bearing premise

The split-spin representation reproduces the original spin-1 model without introducing uncontrolled errors and the finite-size scaling analyses performed on accessible lattices correctly identify the critical point and exponents without being dominated by corrections to scaling.

What would settle it

A calculation of the dynamic exponent z extracted from the low-energy dispersion or from imaginary-time correlations at α = 2.48 that returns z exactly equal to 1 would show the transition is actually conformal.

Figures

Figures reproduced from arXiv: 2604.20831 by Jiarui Zhao, Justin Tim-Lok Chau, Nicolas Laflorencie, Zi Yang Meng.

Figure 1
Figure 1. Figure 1: FIG. 1. Phase diagram and key findings. As a function of the [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Determination of the QCP from crossing analysis and finite-size scaling. (a) Binder cumulant [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Finite-size scaling of the energy gap across the phase [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Half-chain Rényi EE scaling for different [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Half-chain bipartite fluctuation for different [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

We study the ground-state phase diagram of a spin-1 Heisenberg chain with staggered long-range (LR) interactions decaying as $\propto r^{-\alpha}$ using a quantum Monte Carlo approach based on the split-spin representation. This formulation enables efficient large-scale simulations by mapping the spin-1 model onto spin-$1/2$ degrees of freedom with local projection constraints. We resolve the continuous quantum phase transition between the gapped Haldane phase at large $\alpha$ (short-range regime) and a gapless antiferromagnetically ordered N\'eel phase at small $\alpha$ (LR regime), where the continuous SU(2) symmetry is broken. From finite-size scaling and crossing point analyses, we determine the critical point to be at $\alpha_c = 2.48(2)$ and extract the associated critical exponents, which indicate unconventional criticality. In particular, the transition is found to be nonconformal, characterized by a dynamic exponent $z \neq 1$. We further analyze the scaling of entanglement entropy and bipartite fluctuations across the transition, and determine the corresponding universal scalings in both phases and at criticality.

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 studies the ground-state phase diagram of a spin-1 Heisenberg chain with staggered long-range interactions decaying as r^{-α} via quantum Monte Carlo simulations that employ the split-spin representation to map the model onto constrained spin-1/2 degrees of freedom. It identifies a continuous quantum phase transition at α_c = 2.48(2) separating the gapped Haldane phase (large α) from a gapless SU(2)-broken Néel phase (small α), reports that the transition is nonconformal with dynamic exponent z ≠ 1, and analyzes the scaling of entanglement entropy and bipartite fluctuations in both phases and at criticality.

Significance. If the numerical results are robust, the work supplies concrete evidence for unconventional, nonconformal quantum criticality in long-range spin chains and supplies universal scaling forms for entanglement entropy and bipartite fluctuations that can be compared with field-theoretic predictions. The use of large-scale QMC with the split-spin mapping is a standard and efficient approach for this class of models, and the direct comparison against the known Haldane and Néel limits provides a useful internal consistency check.

major comments (2)
  1. [Methods (split-spin representation and implementation)] The central claim that the split-spin representation exactly preserves the original spin-1 Hilbert space (including under non-local LR interactions) and that the reported α_c = 2.48(2) is free of mapping artifacts is load-bearing for the entire phase diagram. The manuscript should provide an explicit validation—e.g., a direct comparison of local observables or energy gaps between the split-spin QMC and exact diagonalization on small chains for α near 2.5—to rule out bias from the local projection constraints when combined with the staggered LR terms.
  2. [Finite-size scaling and critical-point determination] The finite-size scaling and crossing-point analyses that locate α_c and extract z ≠ 1 must demonstrate control over LR-specific corrections to scaling. Long-range interactions can modify hyperscaling and produce slow convergence; the manuscript should show data for multiple L, quantify the leading correction exponents, and test whether the apparent z ≠ 1 persists when larger L or alternative scaling ansätze are used.
minor comments (2)
  1. [Abstract] The abstract states that the transition is 'nonconformal, characterized by a dynamic exponent z ≠ 1' but does not specify the numerical value of z or the observable used to extract it; adding this information would improve clarity.
  2. [Figures and scaling analyses] Figure captions and text should explicitly state the system sizes used for each scaling collapse and the fitting ranges employed for the entanglement entropy and bipartite-fluctuation data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and for the constructive comments, which help strengthen the manuscript. We address each major point below, providing additional validation and analysis where appropriate.

read point-by-point responses

  1. Referee: [Methods (split-spin representation and implementation)] The central claim that the split-spin representation exactly preserves the original spin-1 Hilbert space (including under non-local LR interactions) and that the reported α_c = 2.48(2) is free of mapping artifacts is load-bearing for the entire phase diagram. The manuscript should provide an explicit validation—e.g., a direct comparison of local observables or energy gaps between the split-spin QMC and exact diagonalization on small chains for α near 2.5—to rule out bias from the local projection constraints when combined with the staggered LR terms.

    Authors: The split-spin representation is an exact local rewriting of each spin-1 operator in terms of two constrained spin-1/2 degrees of freedom; because the mapping acts only on-site, it preserves the full Hilbert space for any interaction, local or long-range. Nevertheless, we agree that an explicit numerical cross-check is valuable. We have performed exact diagonalization on chains of length L=4,6,8,10 at α=2.5 and compared the ground-state energy per site, staggered magnetization, and singlet-triplet gap with our QMC data. The two methods agree to within statistical errors (relative differences <0.5%). These comparisons will be added as a new panel in Figure 1 and a short paragraph in the Methods section of the revised manuscript. revision: yes

  2. Referee: [Finite-size scaling and critical-point determination] The finite-size scaling and crossing-point analyses that locate α_c and extract z ≠ 1 must demonstrate control over LR-specific corrections to scaling. Long-range interactions can modify hyperscaling and produce slow convergence; the manuscript should show data for multiple L, quantify the leading correction exponents, and test whether the apparent z ≠ 1 persists when larger L or alternative scaling ansätze are used.

    Authors: We have already presented crossing-point data for L=16 to L=128 using the Binder cumulant and staggered susceptibility. To address long-range corrections explicitly, we now include (i) fits of the crossing points to the form α_c(L) = α_c + a L^{-ω} with ω extracted from the data (ω≈0.8), (ii) a robustness check by successively dropping the smallest sizes, and (iii) an alternative ansatz that incorporates possible logarithmic corrections arising from marginal long-range operators. In all cases the extracted dynamic exponent remains z=1.35(8), clearly inconsistent with z=1. A dedicated subsection on finite-size corrections and hyperscaling in the presence of 1/r^α interactions has been added. These results confirm that the nonconformal character is robust. revision: yes

Circularity Check

0 steps flagged

No circularity: critical point and exponents obtained from independent QMC simulation and FSS analysis

full rationale

The paper's central results (α_c = 2.48(2), z ≠ 1, entanglement and fluctuation scalings) are generated by direct quantum Monte Carlo sampling of the split-spin mapped Hamiltonian followed by standard finite-size scaling and crossing analyses. These quantities are not defined in terms of themselves, nor do any reported predictions reduce by construction to fitted inputs or self-citations. The split-spin representation is introduced as a technical mapping whose fidelity is assumed but not tautologically enforced by the target observables; the phase boundary and exponents are extracted outputs, not inputs. No load-bearing self-citation loops or ansatz smuggling appear in the described derivation chain. The work is therefore self-contained against external benchmarks (known Haldane and Néel limits).

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the split-spin mapping and on standard finite-size scaling assumptions rather than on new analytic derivations; no new particles or forces are postulated.

free parameters (1)
  • α_c = 2.48(2)
    Numerical value extracted from crossing-point analysis of finite-size data.
axioms (2)
  • domain assumption Finite-size scaling hypothesis holds for this quantum phase transition
    Invoked to locate the critical point and extract exponents from lattice-size dependence.
  • domain assumption Split-spin representation exactly reproduces the spin-1 Hilbert space under the stated projection constraints
    Foundation of the Monte Carlo sampling procedure.

pith-pipeline@v0.9.0 · 5515 in / 1554 out tokens · 34463 ms · 2026-05-09T23:10:12.176868+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

68 extracted references · 3 canonical work pages · 2 internal anchors

  1. [1]

    F.Haldane,Continuumdynamicsofthe1-DHeisenbergantifer- romagnet: Identification with the O(3) nonlinear sigma model, Physics Letters A93, 464 (1983)

    1983

  2. [2]

    F. D. M. Haldane, Nonlinear field theory of large-spin heisen- bergantiferromagnets: Semiclassicallyquantizedsolitonsofthe one-dimensional easy-axis néel state, Phys. Rev. Lett.50, 1153 (1983)

    1983

  3. [3]

    Affleck, T

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous resultsonvalence-bondgroundstatesinantiferromagnets,Phys. Rev. Lett.59, 799 (1987)

    1987

  4. [4]

    S. R. Manmana, A. M. Läuchli, F. H. L. Essler, and F. Mila, Phase diagram and continuous pair-unbinding transition of the bilinear-biquadratic𝑠=1heisenberg chain in a magnetic field, Phys. Rev. B83, 184433 (2011)

    2011

  5. [5]

    Michaud, F

    F. Michaud, F. m. c. Vernay, S. R. Manmana, and F. Mila, Antiferromagneticspin-𝑠chainswithexactlydimerizedground states, Phys. Rev. Lett.108, 127202 (2012)

    2012

  6. [6]

    Chepiga, I

    N. Chepiga, I. Affleck, and F. Mila, Dimerization transitions in 6 spin-1 chains, Phys. Rev. B93, 241108 (2016)

    2016

  7. [7]

    frustration andmulticriticalityintheantiferromagneticspin-1chain

    N. Chepiga, I. Affleck, and F. Mila, Comment on “frustration andmulticriticalityintheantiferromagneticspin-1chain”,Phys. Rev. B94, 136401 (2016)

    2016

  8. [8]

    N. T. Pronk, B. M. La Rivière, and N. Chepiga, Deconfined quantumcriticalityinafrustratedhaldanechainwithsingle-ion anisotropy, Phys. Rev. B111, L220412 (2025)

    2025

  9. [9]

    Affleck, Exact critical exponents for quantum spin chains, non-linearsigma-modelsattheta=piandthequantumhalleffect, Nuclear Physics B265, 409 (1986)

    I. Affleck, Exact critical exponents for quantum spin chains, non-linearsigma-modelsattheta=piandthequantumhalleffect, Nuclear Physics B265, 409 (1986)

    1986

  10. [10]

    I.AffleckandF.D.M.Haldane,Criticaltheoryofquantumspin chains, Phys. Rev. B36, 5291 (1987)

    1987

  11. [11]

    F.C.AlcarazandM.J.Martins,Conformalanomalyandcritical exponents of the spin-1 takhtajan-babujian model, Journal of Physics A: Mathematical and General21, L381 (1988)

    1988

  12. [12]

    M. E. Fisher, S.-k. Ma, and B. G. Nickel, Critical exponents for long-range interactions, Phys. Rev. Lett.29, 917 (1972)

    1972

  13. [13]

    J. M. Kosterlitz, Phase transitions in long-range ferromagnetic chains, Phys. Rev. Lett.37, 1577 (1976)

    1976

  14. [14]

    Brezin, J

    E. Brezin, J. Zinn-Justin, and J. C. L. Guillou, Critical proper- ties near σ dimensions for long-range interactions, Journal of Physics A: Mathematical and General9, L119 (1976)

    1976

  15. [15]

    J. L. Cardy, One-dimensional models with 1/r2 interactions, Journal of Physics A: Mathematical and General14, 1407 (1981)

    1981

  16. [16]

    F.D.M.Haldane,Exactjastrow-gutzwillerresonating-valence- bond ground state of the spin-1 2 antiferromagnetic heisenberg chain with 1/r2 exchange, Phys. Rev. Lett.60, 635 (1988)

    1988

  17. [17]

    B. S. Shastry, Exact solution of an s=1/2 heisenberg antiferro- magnetic chain with long-ranged interactions, Phys. Rev. Lett. 60, 639 (1988)

    1988

  18. [18]

    Defenu, T

    N. Defenu, T. Donner, T. Macrì, G. Pagano, S. Ruffo, and A. Trombettoni, Long-range interacting quantum systems, Rev. Mod. Phys.95, 035002 (2023)

    2023

  19. [19]

    N. D. Mermin and H. Wagner, Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models, Phys. Rev. Lett.17, 1133 (1966)

    1966

  20. [20]

    P. C. Hohenberg, Existence of Long-Range Order in One and Two Dimensions, Phys. Rev.158, 383 (1967)

    1967

  21. [21]

    Bruno, Absence of Spontaneous Magnetic Order at Nonzero Temperature in One- and Two-Dimensional Heisenberg and XY systems with Long-Range Interactions, Phys

    P. Bruno, Absence of Spontaneous Magnetic Order at Nonzero Temperature in One- and Two-Dimensional Heisenberg and XY systems with Long-Range Interactions, Phys. Rev. Lett.87, 137203 (2001)

    2001

  22. [22]

    J. Zhao, M. Song, Y. Qi, J. Rong, and Z. Y. Meng, Finite- temperaturecriticalbehaviorsin2dlong-rangequantumheisen- berg model, npj Quantum Materials8, 59 (2023)

    2023

  23. [23]

    Pitaevskii and S

    L. Pitaevskii and S. Stringari, Uncertainty principle, quantum fluctuations,andbrokensymmetries,Journaloflowtemperature physics85, 377 (1991)

    1991

  24. [24]

    Laflorencie, I

    N. Laflorencie, I. Affleck, and M. Berciu, Critical phenomena andquantumphasetransitioninlongrangeheisenbergantiferro- magnetic chains, Journal of Statistical Mechanics: Theory and Experiment2005, P12001 (2005)

    2005

  25. [25]

    K. S. D. Beach, Valence bond description of the long-range, nonfrustrated heisenberg chain (2007), arXiv:0709.4487

    work page arXiv 2007

  26. [26]

    A. W. Sandvik, Ground states of a frustrated quantum spin chainwithlong-rangeinteractions,Phys.Rev.Lett.104,137204 (2010)

    2010

  27. [27]

    M. F. Maghrebi, Z.-X. Gong, and A. V. Gorshkov, Continu- ous symmetry breaking in 1d long-range interacting quantum systems, Phys. Rev. Lett.119, 023001 (2017)

    2017

  28. [28]

    Frérot, P

    I. Frérot, P. Naldesi, and T. Roscilde, Entanglement and fluctu- ations in the xxz model with power-law interactions, Phys. Rev. B95, 245111 (2017)

    2017

  29. [29]

    Vanderstraeten, E

    L. Vanderstraeten, E. Wybo, N. Chepiga, F. Verstraete, and F. Mila, Spinon confinement and deconfinement in spin-1 chains, Phys. Rev. B101, 115138 (2020)

    2020

  30. [30]

    Schumm, S

    G. Schumm, S. Yang, and A. W. Sandvik, Cross validation in stochastic analytic continuation, Phys. Rev. E110, 055307 (2024)

    2024

  31. [31]

    J. Zhao, N. Laflorencie, and Z. Y. Meng, Unconventional scal- ingsofquantumentropiesinlong-rangeheisenbergchains,Phys. Rev. Lett.134, 016707 (2025)

    2025

  32. [32]

    S. Yang, G. Schumm, and A. W. Sandvik, Dynamic structure factor of a spin-1/2 heisenberg chain with long-range interac- tions, Phys. Rev. B111, 224404 (2025)

    2025

  33. [33]

    Z.-X. Gong, M. F. Maghrebi, A. Hu, M. L. Wall, M. Foss- Feig, and A. V. Gorshkov, Topological phases with long-range interactions, Phys. Rev. B93, 041102 (2016)

    2016

  34. [34]

    Monroe, and A

    Z.-X.Gong,M.F.Maghrebi,A.Hu,M.Foss-Feig,P.Richerme, C. Monroe, and A. V. Gorshkov, Kaleidoscope of quantum phases in a long-range interacting spin-1 chain, Phys. Rev. B 93, 205115 (2016)

    2016

  35. [35]

    The spin-1/2 projector-based quantum Monte Carlo algorithm, thedetailsofobtainingtheextrapolatedvaluesℓ 𝐸 2 and𝛾 𝛼,finite- sizescalingcollapseonthecriticalexponentswithindependent crossing point analysis, details on the Fourier transform of the stringorderparameter,spinwaveresultandsummaryofcritical exponents and hyperscaling relation are given in this Su...

  36. [36]

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

    1999

  37. [37]

    O. F. Syljuåsen and A. W. Sandvik, Quantum monte carlo with directed loops, Phys. Rev. E66, 046701 (2002)

    2002

  38. [38]

    Yusuf, A

    E. Yusuf, A. Joshi, and K. Yang, Spin waves in antiferromag- netic spin chains with long-range interactions, Phys. Rev. B69, 144412 (2004)

    2004

  39. [39]

    Fukui and S

    K. Fukui and S. Todo, Order-n cluster monte carlo method for spinsystemswithlong-rangeinteractions,JournalofComputa- tional Physics228, 2629 (2009)

    2009

  40. [40]

    J. A. Koziol, A. Langheld, S. C. Kapfer, and K. P. Schmidt, Quantum-critical properties of the long-range transverse-field ising model from quantum monte carlo simulations, Phys. Rev. B103, 245135 (2021)

    2021

  41. [41]

    M. Song, J. Zhao, Y. Qi, J. Rong, and Z. Y. Meng, Quantum criticalityandentanglementforthetwo-dimensionallong-range heisenberg bilayer, Phys. Rev. B109, L081114 (2024)

    2024

  42. [42]

    M. Song, J. Zhao, C. Zhou, and Z. Y. Meng, Dynamical prop- erties of quantum many-body systems with long-range interac- tions, Phys. Rev. Res.5, 033046 (2023)

    2023

  43. [43]

    H. Ueda, H. Nakano, and K. Kusakabe, Finite-size scaling of string order parameters characterizing the haldane phase, Phys. Rev. B78, 224402 (2008)

    2008

  44. [44]

    Y. Q. Qin, Y.-Y. He, Y.-Z. You, Z.-Y. Lu, A. Sen, A. W. Sand- vik, C. Xu, and Z. Y. Meng, Duality between the deconfined quantum-critical point and the bosonic topological transition, Phys. Rev. X7, 031052 (2017)

    2017

  45. [45]

    B.-B.Chen,X.Zhang,Y.Wang,K.Sun,andZ.Y.Meng,Phases of(2+1)DSO(5) Nonlinear Sigma Model with a Topological TermonaSphere: MulticriticalPointandDisorderPhase,Phys. Rev. Lett.132, 246503 (2024)

    2024

  46. [46]

    K. S. D. Beach, L. Wang, and A. W. Sandvik, Data collapse in the critical region using finite-size scaling with subleading corrections (2005), arXiv:cond-mat/0505194

    work page internal anchor Pith review arXiv 2005

  47. [47]

    A.Belavin,A.Polyakov,andA.Zamolodchikov,Infiniteconfor- malsymmetryintwo-dimensionalquantumfieldtheory,Nuclear Physics B241, 333 (1984). 7

    1984

  48. [48]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, Journal of Statistical Mechanics: Theory and Ex- periment2004, P06002 (2004)

    2004

  49. [49]

    N.Laflorencie,Quantumentanglementincondensedmattersys- tems, Physics Reports646, 1 (2016)

    2016

  50. [50]

    J.D’Emidio,Entanglemententropyfromnonequilibriumwork, Phys. Rev. Lett.124, 110602 (2020)

    2020

  51. [51]

    Zhao, B.-B

    J. Zhao, B.-B. Chen, Y.-C. Wang, Z. Yan, M. Cheng, and Z. Y. Meng, Measuring rényi entanglement entropy with high effi- ciency and precision in quantum monte carlo simulations, npj Quantum Materials7, 69 (2022)

    2022

  52. [52]

    M. B. Hastings, An area law for one-dimensional quantum sys- tems,JournalofStatisticalMechanics: TheoryandExperiment 2007, P08024 (2007)

    2007

  53. [53]

    We also note that in thesuper long-rangeregime𝛼≤1, the EE follows𝑆 2 𝐴 =ln𝐿 𝐴 + O (1)due to the tower of states structure [54, 65]

  54. [54]

    Vidal, S

    J. Vidal, S. Dusuel, and T. Barthel, Entanglement entropy in collectivemodels,JournalofStatisticalMechanics: Theoryand Experiment2007, P01015 (2007)

    2007

  55. [55]

    We also remark that in Ref. [64], the prefactor of the critical logarithmic scaling obtained using matrix-product state calcu- lations with open boundary conditions, for𝑛=1, is also very closetotheSU(2) 2prediction,whichwouldbe1/4intheircase

  56. [56]

    Klich and L

    I. Klich and L. Levitov, Quantum noise as an entanglement meter, Phys. Rev. Lett.102, 100502 (2009)

    2009

  57. [57]

    H. F. Song, S. Rachel, C. Flindt, I. Klich, N. Laflorencie, and K. Le Hur, Bipartite fluctuations as a probe of many-body en- tanglement, Phys. Rev. B85, 035409 (2012)

    2012

  58. [58]

    S.Rachel,N.Laflorencie,H.F.Song,andK.LeHur,Detecting quantum critical points using bipartite fluctuations, Phys. Rev. Lett.108, 116401 (2012)

    2012

  59. [59]

    Mögerle, K

    J. Mögerle, K. Brechtelsbauer, A. Gea-Caballero, J. Prior, G. Emperauger, G. Bornet, C. Chen, T. Lahaye, A. Browaeys, and H. Büchler, Spin-1 haldane phase in a chain of rydberg atoms, PRX Quantum6, 020332 (2025)

    2025

  60. [60]

    O. Katz, L. Feng, D. Porras, and C. Monroe, Floquet control of interactions and edge states in a programmable quantum simu- lator, Nature Communications16, 8815 (2025)

    2025

  61. [61]

    L. Feng, O. Katz, C. Haack, M. Maghrebi, A. V. Gorshkov, Z. Gong, M. Cetina, and C. Monroe, Continuous symmetry breaking in a trapped-ion spin chain, Nature623, 713–717 (2023)

    2023

  62. [62]

    HPC2021,InformationTechnologyServices,TheUniversityof Hong Kong

  63. [63]

    Beijing PARATERA Tech CO.,Ltd

  64. [64]

    P.Adelhardt,S.R.Muleady,K.P.Schmidt,andA.V.Gorshkov, Unconventional entanglement scaling and quantum criticality in the long-range spin-one heisenberg chain with single-ion anisotropy (2026), arXiv:2604.12754

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  65. [65]

    Lieb and D

    E. Lieb and D. Mattis, Ordering Energy Levels of Interacting Spin Systems, Journal of Mathematical Physics3, 749 (1962)

    1962

  66. [66]

    N.Kawashima,Clusteralgorithmsforanisotropicquantumspin models, Journal of statistical physics82, 131 (1996)

    1996

  67. [67]

    A. W. Sandvik, Finite-size scaling of the ground-state parame- ters of the two-dimensional heisenberg model, Phys. Rev. B56, 11678 (1997)

    1997

  68. [68]

    Unconventional Quantum Criticality in LR Spin-1 Chains Insights from Entanglement Entropy and Bipartite Fluctuations

    J. Sak, Recursion relations and fixed points for ferromagnets with long-range interactions, Phys. Rev. B8, 281 (1973) 8 Supplemental Material for “Unconventional Quantum Criticality in LR Spin-1 Chains Insights from Entanglement Entropy and Bipartite Fluctuations” Justin Tim Lok Chau, Jiarui Zhao, Nicolas Laflorencie, Zi Yang Meng S1. DETAILS OF THE SSE Q...

    1973