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REVIEW 3 major objections 4 minor 77 references

Continuously varying critical exponents in an exactly solvable long-range cluster XY mode

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

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Pith's one-line read In an exactly solvable antiferromagnetic cluster XY chain with interactions decaying as $1/r^\alpha$, the critical exponents $\nu$ and $z$ vary continuously with $\alpha$, and the identity $\nu z=1$ still holds.

desk verdict The analytic z(α)=α−1 result is solid and new, but the numerical verification is circular and Table I contradicts the analytic prediction at α=2; still worth refereeing. read the letter →

arxiv 2502.04165 v1 pith:BAOCDOJI submitted 2025-02-06 cond-mat.str-el cond-mat.stat-mech

Tian-Cheng Yi , Chengxiang Ding , Maoxin Liu , Liangsheng Li , Wen-Long You This is my paper · ORCID
classification cond-mat.str-elcond-mat.stat-mech
keywords long-rangeinteractionsclusterXYmodelcriticalexponentsdynamicexponentcorrelationlengthJordan-Wignertransformationfidelitysusceptibilityentanglemententropy
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

The paper studies a spin-1/2 cluster XY chain in a transverse field whose interactions decay as $J_m = m^{-\alpha}$. Because the model maps exactly to free fermions, the finite-size gap can be computed and the dynamic critical exponent $z$ read off; the result is $z = \alpha-1$ for $1<\alpha<2$ and $z=1$ for $\alpha\ge 2$. Together with $\nu z = 1$, this gives a correlation-length exponent $\nu$ that also varies continuously with $\alpha$, unlike the long-range antiferromagnetic Ising chain where the exponents are fixed. The exponents are verified by scaling collapse of the derivative of the correlation function and of fidelity susceptibility, and the central charge $c$ extracted from entanglement entropy also decreases with $\alpha$ toward $0.5$. If correct, this provides a rare exactly solvable example in which long-range cluster interactions tune the universality class.

What carries the argument

The central object is the free-fermion representation of the cluster XY Hamiltonian obtained by Jordan-Wigner transformation, whose Bogoliubov quasiparticle energies are $\varepsilon_k = 2\sqrt{\delta_k^2 + \epsilon_k^2}$ with $\epsilon_k = \sum_{m=1}^{M} m^{-\alpha}\cos(m k) + h$ and $\delta_k = \gamma \sum_{m=1}^{M} m^{-\alpha}\sin(m k)$. The gap closes at $k=0$ or $k=\pi$ when $h$ reaches $h_{c1} = -\sum_{m} m^{-\alpha}$ or $h_{c2} = -\sum_{m} (-1)^m m^{-\alpha}$. The dynamic exponent is extracted from the finite-size gap $\Delta_N \sim |\sum_{m=1}^{N/2} m^{1-\alpha}|/N$, where the truncation $M=N/2$ is the longest possible interaction range under periodic boundary conditions; the Euler-Maclaurin asymptotic expansion of this sum produces the piecewise formula for $z$.

What would settle it

For $1<\alpha<2$, compute the exact gap at $h_{c1}$ from the free-fermion spectrum for a sequence of increasing $N$ with the interaction range held at $M=N/2$, and fit $z$ from $\Delta_N \propto N^{-z}$; if the fitted $z$ does not approach $\alpha-1$ as $N\to\infty$, the claim fails. A complementary check is to construct the low-energy continuum theory by taking $N\to\infty$ before expanding around $k=0$: if the resulting velocity diverges and the gap instead closes as $N^{-1}$, the order of limits changes the exponent.

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Extended reading notes

Core claim

The central claim is that in the long-range antiferromagnetic cluster XY model $H = \sum_{j,m} J_m[(1+\gamma)/2 \sigma_j^x \sigma_{j+m}^x + (1-\gamma)/2 \sigma_j^y \sigma_{j+m}^y] \prod_{p=j+1}^{j+m-1}\sigma_p^z - h\sum_j \sigma_j^z$ with $J_m = m^{-\alpha}$, the critical exponents are continuously varying functions of $\alpha$. The derivation starts from the size dependence of the gap at the critical field $h_{c1}$: $\Delta_N \sim N^{-z}$ with $N^{-z} \sim (1/N)\left|\sum_{m=1}^{N/2} m^{1-\alpha}\right|$. Euler-Maclaurin expansion of the generalized harmonic number yields $z = \alpha-1$ for $1<\alpha<2$ and $z=1$ for $\alpha\ge2$, with logarithmic corrections at $\alpha=2$, and the relation $\nu z = 1$ holds throughout. The same exponents are obtained from scaling collapse of the field derivative of the farthest correlation function and of fidelity susceptibility, and the central charge $c$ from half-chain entanglement entropy varies with $\alpha$, converging to $0.5$ as $\alpha\to\infty$.

Load-bearing premise

The derivation relies on identifying the finite-size gap computed with the interaction range truncated at half the system size with the true thermodynamic dynamical exponent, without proving that the thermodynamic limit and the low-momentum limit commute for $1<\alpha<2$.

Editorial extensions

If this is right

  • For all $\alpha>1$, the identity $\nu z = 1$ holds exactly, so measuring either exponent determines the other.
  • In the regime $\alpha\ge 2$, the model falls into the standard short-range XY/Ising universality class with $\nu=1$ and $z=1$.
  • In the regime $1<\alpha<2$, tuning the interaction decay exponent continuously changes both $\nu$ and $z$, giving a one-parameter family of critical behaviors.
  • The central charge $c$ obtained from entanglement entropy at criticality is also $\alpha$-dependent, approaching $0.5$ in the large-$\alpha$ limit.
  • The exponents are confirmed numerically by scaling collapse of the correlation-function derivative and fidelity susceptibility, so the prediction is directly checkable in finite-size simulations.

Reading between the lines

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

  • A natural next test is whether the $\alpha$-dependent exponents persist in the true thermodynamic limit if the momentum cutoff is taken before the system size; if they do, the critical theory for $1<\alpha<2$ is nonlocal and has a scale-dependent effective velocity.
  • The variation of the central charge $c$ within a free-fermion model hints that the critical theory may be a rescaled free-fermion CFT with a cutoff-dependent Fermi velocity, rather than a genuinely different conformal field theory.
  • The same gap-scaling derivation could be applied to other cluster-type models with Jordan-Wigner strings, predicting similarly tunable exponents and providing a family of solvable long-range models for quantum simulation.
  • In a Rydberg-atom array with programmable power-law interactions and cluster terms, measuring the gap scaling or correlation length as a function of $\alpha$ would directly test whether $z=\alpha-1$ appears in a real system.
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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 studies a spin-1/2 long-range cluster XY chain with algebraically decaying couplings, maps it to free fermions via the Jordan-Wigner transformation, and derives the critical fields hc1 and hc2 and the relation νz=1. The main claim is that the dynamic exponent z (and therefore ν via νz=1) varies continuously with the decay exponent α, with z=α−1 for 1<α<2 and z=1 for α≥2. The authors then present finite-size scaling analyses of correlation-function derivatives and fidelity susceptibility, using data collapse with adjustable exponents, and compute the entanglement entropy to extract a central charge c that also varies with α. The free-fermion mapping and the formulas for the critical fields are clean, but the derivation of the α-dependent z contains a technical error that undermines the central claim.

Significance. If correct, the result would be notable: an exactly solvable one-dimensional spin model with continuously varying critical exponents and a varying effective central charge is of interest for long-range interacting quantum simulators. The manuscript's strengths are its transparent free-fermion solution, explicit formulas for the critical fields, and the detailed numerical scaling study. However, the load-bearing step that produces z=α−1 is based on an invalid linearization of sin(mk), and the numerical verification treats the exponents being verified as fitting parameters. As it stands, the central claim is not supported and the paper would require a substantially revised derivation and a re-evaluation of the numerical evidence.

major comments (3)
  1. [Sec. II, Eqs. (18)–(20) and Supplemental Eqs. (A.1)–(B.7)] The derivation of the central result uses an invalid uniform expansion sin(mk)≈mk: for k∼1/N and m up to M=N/2, mk is of order unity. A correct evaluation gives δ_k∼N^{2−α} for α<2 (or δ_∞(k)∼k^{α−2} for 2<α<3), so the gap does not scale as N^{−(α−1)}. This invalidates z=α−1.
  2. [Sec. III, Eq. (24), Fig. 4, and Table I] The verification section treats the predicted exponents as fitting parameters, and the fitted values deviate from the analytic predictions by up to 14% (α=2.0, ν=1.1438 vs ν=1). The collapse therefore does not independently confirm the claimed exponents.
  3. [Sec. III, Eq. (30) and Fig. 7] The CFT scaling form used to extract c is not justified for α<2, where the single-particle spectrum has no conventional low-energy continuum and the finite-size gap does not close.
minor comments (4)
  1. [Eq. (22)] The determinant in Eq. (22) uses the index j in the last row without defining it; the row index should presumably be i+r−1 rather than j−1.
  2. [Fig. 1 caption] The caption mentions dashed lines indicating the location of the critical point as α→∞, but no dashed lines are visible in the figure.
  3. [Sec. II, text before Eq. (2)] The text notes that the ground-state energy diverges for α≤1, but the subsequent analysis and figures use α>1; this restriction should be stated explicitly in the model definition.
  4. [Abstract] The phrase 'To optimize scaling behavior, we verify these critical exponents ... by adjusting fitting parameters' is self-contradictory, since verification by adjusting the exponents being verified is circular; this phrasing should be revised.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the z(alpha) derivation is a self-contained exact free-fermion calculation; the data-collapse fits are transparently labeled fitting parameters.

full rationale

The central claim, z = alpha - 1 for 1 < alpha < 2 and z = 1 for alpha >= 2, is derived from the finite-size gap of the exactly diagonalized free-fermion model. Equation (20) and Supplemental Eqs. (A.1)-(B.7) evaluate the gap from the explicitly computed dispersion epsilon_k = 2*sqrt(delta_k^2 + epsilon_k^2), and the Euler-Maclaurin analysis of f(N) = (1/N) sum_{m=1}^{N/2} m^{1-alpha} is a direct asymptotic evaluation of that gap, not a restatement of the input. The scaling verification in Sec. III fixes nu from the analytic relation nu = 1/z and uses beta_P and beta_F, which the paper explicitly labels as 'fitting parameter(s)'; these auxiliary exponents are not dressed up as predictions, and the horizontal scaling variable N^{1/nu}(h - h_c) genuinely tests nu. The collapse is therefore a consistency check rather than an independent derivation, but it is not circular. The truncation M = N/2 and the question of whether this finite-size sequence reproduces the thermodynamic dynamical exponent for alpha < 2 is a limit-interchange correctness concern, not a circularity, and no load-bearing self-citation chain was found; citations to prior work by the authors are used for standard or independently established results, such as the alpha = infinity nearest-neighbor XY limit.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central exact derivation has no free parameters beyond the model constants J, γ, h, α. The free parameters listed (βP, βF, ν_fit, c) enter only the numerical verification and are fitted to achieve data collapse. The axioms are the standard tools (Jordan-Wigner, finite-size scaling) plus the specific assumption that the truncated finite-size gap defines the true thermodynamic exponents.

free parameters (4)
  • βP = 0.5093 (α=2.0, γ=1.0), 0.3643 (α=1.8), 0.0959 (α=1.6)
    Fitted parameter in Eq. (24) chosen to improve data collapse of the correlation derivative; not derived.
  • βF = 0.7056 (α=2.0), 0.5402 (α=1.8), 0.2449 (α=1.6)
    Fitted parameter in Eq. (28) for fidelity susceptibility collapse; not derived.
  • ν_fit = 1.1438 (α=2.0), 1.3351 (α=1.8), 1.7157 (α=1.6)
    Effective correlation-length exponent used in the data collapse (Table I, Fig. 4). These differ from the analytic ν from the gap derivation, so they are effectively fitted to achieve collapse.
  • central charge c = values shown in Fig. 7(b), converging to 0.5 as α increases
    Extracted from the log-law fit of entanglement entropy in Eq. (30).
assumptions (5)
  • domain assumption The Jordan-Wigner transformation with PBC/APBC maps the long-range cluster XY spin chain to an exactly quadratic fermionic Hamiltonian.
    Used in Sec. II, Eq. (6). The transformation is standard, but its validity for cluster terms with strings over long distances is assumed.
  • domain assumption The gap expansion near k=0 and k=π is linear in |k|, giving νz=1.
    Equations (16)-(19). For α<2 the sum ∑ m^{1-α} diverges with the cutoff, so the expansion is only formal for the finite-size system.
  • ad hoc to paper The finite-size gap scaling Δ_N ~ N^{-z} with M=N/2 correctly defines the thermodynamic dynamical exponent.
    Eq. (20) and Supplemental Eq. (A.1). This is the load-bearing premise that the order of limits (N→∞ vs k→0) does not alter z.
  • domain assumption The scaling forms for P and χF in Eqs. (24) and (28) hold with a single correlation-length exponent ν.
    Standard finite-size scaling, but applied here to a system with z<1 where the validity is not discussed.
  • domain assumption The entanglement entropy at criticality follows SL ~ (c/3) log(N/π sin(πL/N)) for all α.
    Eq. (30). This CFT formula is used even for α<2 where z≠1 and the system is not Lorentz invariant.

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Pith. "Pith review of Continuously varying critical exponents in an exactly solvable long-range cluster XY mode." pith.science (2026). https://pith.science/paper/BAOCDOJI

@misc{pith2026250204165,
  author       = {Pith},
  title        = {Pith review of: Continuously varying critical exponents in an exactly solvable long-range cluster XY mode},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BAOCDOJI}},
  note         = {Machine review of arXiv:2502.04165}
}
abstract

We investigate a generalized antiferromagnetic cluster XY model in a transverse magnetic field, where long-range interactions decay algebraically with distance. This model can be exactly solvable within a free fermion framework. By analyzing the gap, we explicitly derive the critical exponents $\nu$ and $z$, finding that the relationship $\nu z = 1$ still holds. However, the values of $\nu$ and $z$ depend on the decaying exponent $\alpha$, in contrast to those for the quantum long-range antiferromagnetic Ising chain. To optimize scaling behavior, we verify these critical exponents using correlation functions and fidelity susceptibility, achieving excellent data collapse across various system sizes by adjusting fitting parameters. Finally, we compute the entanglement entropy at the critical point to determine the central charge $c$, and find it also varies with $\alpha$. This study provides insights into the unique effect of long-range cluster interactions on the critical properties of quantum spin systems.

Figures

Figures reproduced from arXiv: 2502.04165 by the authors.

Figure 1
Figure 1. FIG. 1. Critical lines for the power-law-decaying exponent [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. To obtain the critical exponent z, we selected a range of N values from Nmin to Nmax for fitting. In this model, the critical exponent is no longer constant and will change with α. When α > 3, ν = 1, z = 1. In this case, the model come back to the traditional XY model with only nearest-neighbor inter￾actions. Notably, the case of α = 2 represents a special point where the correlation function is characterized by an … view at source ↗
Figure 4
Figure 4. FIG. 4. Critical behavior of the correlation function for [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (5 more)
Figure 3
Figure 3. Figure 3: FIG. 3. Correlation function [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Critical behavior of the fidelity susceptibility [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Critical behavior of the correlation function [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The half-chain entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 1
Figure 1. Figure 1: FIG. 1. The impact of finite-size effects on the extraction of the critical exponent [PITH_FULL_IMAGE:figures/full_fig_p009_1.png]

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Reference graph

Works this paper leans on

77 extracted references · 61 canonical work pages

  1. [1]

    Sachdev, Quantum Phase Transitions(Cambridge University Press, Cambridge, 2000)

    S. Sachdev, Quantum Phase Transitions(Cambridge University Press, Cambridge, 2000)

    work page 2000

  2. [2]

    Pfeuty, Ann Phys 57, 79 (1970)

    P. Pfeuty, Ann Phys 57, 79 (1970)

    work page 1970

  3. [3]

    Pfeuty and R

    P. Pfeuty and R. Elliott, J. Phys. C: Solid State Phys. 4, 2370 (1971)

    work page 1971

  4. [4]

    Ising, Zeitschrift fur Physik 31, 253 (1925)

    E. Ising, Zeitschrift fur Physik 31, 253 (1925). 7

    work page 1925

  5. [5]

    X. Wang, X. Feng, T. Hartung, K. Jansen, and P. Stornati, Phys. Rev. A 108, 022612 (2023)

    work page 2023

  6. [6]

    Imoto, Y

    T. Imoto, Y . Susa, R. Miyazaki, T. Kadowaki, and Y . Matsuzaki, arXiv:2402.19114 (2024)

    arXiv 2024

  7. [7]

    B. H. Elfeky, M. C. Dartiailh, S. M. Farzaneh, and J. Shabani, Phys. Rev. Res. 3, 033108 (2021)

    work page 2021

  8. [8]

    Dutta, G

    A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, and D. Sen, Quantum phase transitions in trans- verse field spin models: from statistical physics to quantum in- formation (Cambridge University Press, 2015)

    work page 2015

Show all 77 references
  1. [9]

    Carrasquilla and R

    J. Carrasquilla and R. G. Melko, Nat. Phys 13, 431 (2017)

    2017

  2. [10]

    Wang, Phys

    L. Wang, Phys. Rev. B 94, 195105 (2016)

    2016

  3. [11]

    M.-Y . Mao, Z. Cheng, L. Li, N. Wu, and W.-L. You, Phys. Rev. A 109, 042428 (2024)

    2024

  4. [12]

    Castellano, S

    C. Castellano, S. Fortunato, and V . Loreto, Rev. Mod. Phys.81, 591 (2009)

    2009

  5. [13]

    H. A. Kramers and G. H. Wannier, Phys. Rev. 60, 252 (1941)

    1941

  6. [14]

    H. A. Kramers and G. H. Wannier, Phys. Rev. 60, 263 (1941)

    1941

  7. [15]

    Onsager, Phys

    L. Onsager, Phys. Rev. 65, 117 (1944)

    1944

  8. [16]

    W. Zhu, C. Han, E. Huffman, J. S. Hofmann, and Y .-C. He, Phys. Rev. X 13, 021009 (2023)

    2023

  9. [17]

    Defenu, T

    N. Defenu, T. Donner, T. Macr `ı, G. Pagano, S. Ruffo, and A. Trombettoni, Rev. Mod. Phys.95, 035002 (2023)

    2023

  10. [18]

    Koffel, M

    T. Koffel, M. Lewenstein, and L. Tagliacozzo, Phys. Rev. Lett. 109, 267203 (2012)

    2012

  11. [19]

    Hauke and L

    P. Hauke and L. Tagliacozzo, Phys. Rev. Lett. 111, 207202 (2013)

    2013

  12. [20]

    Eisert, M

    J. Eisert, M. van den Worm, S. R. Manmana, and M. Kastner, Phys. Rev. Lett. 111, 260401 (2013)

    2013

  13. [21]

    M. G. Nezhadhaghighi and M. A. Rajabpour, Phys. Rev. B 88, 045426 (2013)

    2013

  14. [22]

    Juh ´asz, I

    R. Juh ´asz, I. A. Kov´acs, and F. Igl´oi, Europhysics Letters 107, 47008 (2014)

    2014

  15. [23]

    V odola, L

    D. V odola, L. Lepori, E. Ercolessi, and G. Pupillo, New J. Phys 18, 015001 (2015)

    2015

  16. [24]

    A. S. Buyskikh, M. Fagotti, J. Schachenmayer, F. Essler, and A. J. Daley, Phys. Rev. A93, 053620 (2016)

    2016

  17. [25]

    Fey and K

    S. Fey and K. P. Schmidt, Phys. Rev. B 94, 075156 (2016)

    2016

  18. [26]

    Sun, Phys

    G. Sun, Phys. Rev. A 96, 043621 (2017)

    2017

  19. [27]

    Z. Zhu, G. Sun, W.-L. You, and D.-N. Shi, Phys. Rev. A 98, 023607 (2018)

    2018

  20. [28]

    Adelhardt, J

    P. Adelhardt, J. A. Koziol, A. Schellenberger, and K. P. Schmidt, Phys. Rev. B 102, 174424 (2020)

    2020

  21. [29]

    J. A. Koziol, A. Langheld, S. C. Kapfer, and K. P. Schmidt, Phys. Rev. B 103, 245135 (2021)

    2021

  22. [30]

    Adelhardt, J

    P. Adelhardt, J. A. Koziol, A. Langheld, and K. P. Schmidt, Entropy 26, 401 (2024)

    2024

  23. [31]

    H.-K. Tang, I. Yudhistira, U. Chattopadhyay, M. Ulybyshev, P. Sengupta, F. F. Assaad, and S. Adam, Phys. Rev. B 110, 155120 (2024)

    2024

  24. [32]

    Chakraborty and N

    D. Chakraborty and N. Angelinos, Phys. Rev. A 110, 042408 (2024)

    2024

  25. [33]

    Solfanelli and N

    A. Solfanelli and N. Defenu, Phys. Rev. E 110, 044121 (2024)

    2024

  26. [34]

    B. J. P. Pernack, M. V . Fistul, and I. M. Eremin, Phys. Rev. B 110, 184502 (2024)

    2024

  27. [35]

    Smacchia, L

    P. Smacchia, L. Amico, P. Facchi, R. Fazio, G. Florio, S. Pas- cazio, and V . Vedral, Phys. Rev. A84, 022304 (2011)

    2011

  28. [36]

    Ding, Phys

    C. Ding, Phys. Rev. E 100, 042131 (2019)

    2019

  29. [37]

    Guo, X.-J

    Z.-X. Guo, X.-J. Yu, X.-D. Hu, and Z. Li, Phys. Rev. A 105, 053311 (2022)

    2022

  30. [38]

    Schwartz, D

    I. Schwartz, D. Cogan, E. R. Schmidgall, Y . Don, L. Gantz, O. Kenneth, N. H. Lindner, and D. Gershoni, Science 354, 434 (2016)

    2016

  31. [39]

    M. V . Larsen, X. Guo, C. R. Breum, J. S. Neergaard-Nielsen, and U. L. Andersen, Science 366, 369 (2019)

    2019

  32. [40]

    Raussendorf and J

    R. Raussendorf and J. Harrington, Phys. Rev. Lett. 98, 190504 (2007)

    2007

  33. [41]

    Raussendorf, J

    R. Raussendorf, J. Harrington, and K. Goyal, New J. Phys 9, 199 (2007)

    2007

  34. [42]

    D. L. Zhou, B. Zeng, Z. Xu, and C. P. Sun, Phys. Rev. A 68, 062303 (2003)

    2003

  35. [43]

    C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, Phys. Rev. A 73, 052306 (2006)

    2006

  36. [44]

    M. A. Nielsen, Phys. Rev. Lett. 93, 040503 (2004)

    2004

  37. [45]

    M. A. Nielsen, Rep. Math. Phys 57, 147 (2006)

    2006

  38. [46]

    A. Y . Kitaev, Annals Phys.303, 2 (2003)

    2003

  39. [47]

    S. Cao, B. Wu, F. Chen, M. Gong, Y . Wu, Y . Ye, C. Zha, H. Qian, C. Ying, S. Guo, et al., Nature 619, 738 (2023)

    2023

  40. [48]

    Ebadi, T

    S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, et al., Nature 595, 227 (2021)

    2021

  41. [49]

    Semeghini, H

    G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Sama- jdar, et al., Science 374, 1242 (2021)

    2021

  42. [50]

    Zeybek, R

    Z. Zeybek, R. Mukherjee, and P. Schmelcher, Phys. Rev. Lett. 131, 203003 (2023)

    2023

  43. [51]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner,et al., Nature 551, 579 (2017)

    2017

  44. [52]

    Scholl, M

    P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V . Lienhard, L.-P. Henry, T. C. Lang, T. Lahaye, et al., Nature 595, 233 (2021)

    2021

  45. [53]

    Bluvstein, A

    D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Semeghini, S. Ebadi, T. T. Wang, A. A. Michailidis, N. Maskara, W. W. Ho, et al., Science 371, 1355 (2021)

    2021

  46. [54]

    Browaeys and T

    A. Browaeys and T. Lahaye, Nat. Phys 16, 132 (2020)

    2020

  47. [55]

    Labuhn, D

    H. Labuhn, D. Barredo, S. Ravets, S. De L ´es´eleuc, T. Macr`ı, T. Lahaye, and A. Browaeys, Nature 534, 667 (2016)

    2016

  48. [56]

    C. Chen, G. Bornet, M. Bintz, G. Emperauger, L. Leclerc, V . S. Liu, P. Scholl, D. Barredo, J. Hauschild, S. Chatterjee, et al., Nature 616, 691 (2023)

    2023

  49. [57]

    Sbierski, M

    B. Sbierski, M. Bintz, S. Chatterjee, M. Schuler, N. Y . Yao, and L. Pollet, Phys. Rev. B 109, 144411 (2024)

    2024

  50. [58]

    Deger and T.-C

    A. Deger and T.-C. Wei, Quantum Inf Process 18, 1 (2019)

    2019

  51. [59]

    Sinha, M

    A. Sinha, M. M. Rams, and J. Dziarmaga, Phys. Rev. B 99, 094203 (2019)

    2019

  52. [60]

    Chepiga and F

    N. Chepiga and F. Mila, Nature Communications 12, 414 (2021)

    2021

  53. [61]

    E. Lieb, T. Schultz, and D. Mattis, Annals of Physics 16, 407 (1961)

    1961

  54. [62]

    Defenu, G

    N. Defenu, G. Morigi, L. Dell’Anna, and T. Enss, Phys. Rev. B 100, 184306 (2019)

    2019

  55. [63]

    See the Supplemental Material for a detailed discussion of the impact of finite-size effects on the fitted critical exponents

  56. [64]

    Yi, Y .-R

    T.-C. Yi, Y .-R. Ding, J. Ren, Y .-M. Wang, and W.-L. You, Acta Phys. Sin. 67, 140303 (2018)

    2018

  57. [65]

    Yi, W.-L

    T.-C. Yi, W.-L. You, N. Wu, and A. M. Ole´s, Phys. Rev. B 100, 024423 (2019)

    2019

  58. [66]

    Liu, T.-C

    Z.-A. Liu, T.-C. Yi, J.-H. Sun, Y .-L. Dong, and W.-L. You, Phys. Rev. E 102, 032127 (2020)

    2020

  59. [67]

    Liu, Y .-L

    Z.-A. Liu, Y .-L. Dong, N. Wu, Y . Wang, and W.-L. You, Physica A 579, 126122 (2021)

    2021

  60. [68]

    Zhao, T.-C

    Z. Zhao, T.-C. Yi, M. Xue, and W.-L. You, Phys. Rev. A 105, 063306 (2022)

    2022

  61. [69]

    M. E. Fisher and M. N. Barber, Phys. Rev. Lett. 28, 1516 (1972)

    1972

  62. [70]

    W. W. Cheng and J.-M. Liu, Phys. Rev. A 82, 012308 (2010). 8

    2010

  63. [71]

    Lv, T.-C

    T. Lv, T.-C. Yi, L. Li, G. Sun, and W.-L. You, Phys. Rev. A105, 013315 (2022)

    2022

  64. [72]

    Zanardi, H

    P. Zanardi, H. T. Quan, X. Wang, and C. P. Sun, Phys. Rev. A 75, 032109 (2007)

    2007

  65. [73]

    You, Y .-W

    W.-L. You, Y .-W. Li, and S.-J. Gu, Phys. Rev. E 76, 022101 (2007)

    2007

  66. [74]

    Campos Venuti and P

    L. Campos Venuti and P. Zanardi, Phys. Rev. Lett. 99, 095701 (2007)

    2007

  67. [75]

    Calabrese and J

    P. Calabrese and J. Cardy, J. Phys. A: Math. Theor. 42, 504005 (2009)

    2009

  68. [76]

    J. Ren, Z. Wang, W. Chen, and W.-L. You, Phys. Rev. E 105, 034128 (2022)

    2022

  69. [77]

    Continuously varying critical exponents in an exactly solvable long-range cluster XY model

    S. O. Skrøvseth and K. Olaussen, Phys. Rev. A 72, 022318 (2005). 9 Supplemental Material for “Continuously varying critical exponents in an exactly solvable long-range cluster XY model” Appendix A: Finite-size effect on the fitted critical exponents In this supplementary mater...

    2005

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Reviewed August 8, 2026 · model on record in the stance chip above.