Pith. sign in

REVIEW 3 major objections 3 minor 1 cited by

Spontaneous Symmetry Breaking in Two-dimensional Long-range Heisenberg Model

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

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

Pith's one-line read This paper claims that the two-dimensional Heisenberg model with interactions decaying as 1/r^{d+σ} has a finite-temperature long-range-ordered phase for every σ ≤ 2, including the marginal σ = 2 case, through a single continuous transition

desk verdict Central claim collides with a rigorous theorem; the LR-SRW model is nice, but the σ=2 LRO conclusion is not supported. read the letter →

arxiv 2512.01956 v2 pith:RAOO2XZT submitted 2025-12-01 cond-mat.stat-mech

Tianning Xiao , Dingyun Yao , Lode Polle , Zhijie Fan , Youjin Deng This is my paper · ORCID
classification cond-mat.stat-mech
keywords long-rangeHeisenbergmodelspontaneoussymmetrybreakingmarginalcaseσ=2transversefluctuationsfinite-sizescalingsimplerandomwalkcontinuousphasetransitionlow-dimensionalstatisticalmechanics
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

Long-range interactions change the rules for when magnets can order. Simulating the classical Heisenberg model on two-dimensional lattices up to L = 8192, this paper argues that genuine spontaneous symmetry breaking survives at finite temperature for every decay exponent σ ≤ 2, and that the long-range-to-short-range crossover sits exactly at σ* = 2. The marginal case σ = 2 is the delicate one: instead of falling into the short-range class, the system orders, with the magnetization approaching its infinite-size value only logarithmically in system size. To understand why, the paper introduces a long-range random walk with a fixed total length and shows it reproduces the transverse-fluctuation scaling of the spin model in two and three dimensions. If the paper is right, the usual no-order arguments for two-dimensional continuous symmetry need a clear boundary: they apply for σ > 2 but not at σ = 2.

What carries the argument

The load-bearing object is the fixed-length long-range simple random walk—a walk with power-law jump probabilities whose total walk length is held at O(L^d) to respect extensivity. Its height-field correlation gives D_k = L^2/χ_k ∼ L^{2−σ} for σ < 2, ∼ ln L for σ = 2, and ∼ L^0 for σ > 2, which is the same three-regime scaling the paper measures in the transverse-fluctuation channel of the Heisenberg model. The walk replaces the quadratic free-field description as the fluctuation bookkeeper at σ = 2, avoiding the logarithmic amplitude divergence that would forbid order.

What would settle it

At σ = 2 and β = 4, measure ⟨M²⟩ and the transverse susceptibility for L from 1024 to 16384. If the intercept c1 in ⟨M²⟩ = c1 + c2/ln(L/L0) keeps decreasing when L0 is fixed independently (e.g., from the susceptibility data), or if the transverse susceptibility grows faster than L²/ln L, the claimed long-range order at the marginal point fails. Equivalently, a direct real-space check: in the ordered phase g(r) should approach a positive g0 with corrections ~1/ln r; a decay of g(r) − g0 inconsistent with 1/ln r would rule out the claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that the 2D long-range Heisenberg model orders at finite temperature for all σ ≤ 2 through a single continuous transition. In the ordered phase, the spin correlation function saturates to a positive constant: g(r) ≃ g0 + a r^{σ−d} for σ < 2, and g(r) ≃ g0 + a/ln r at σ = 2. For σ > 2 the model is asymptotically free, with no finite-temperature transition and an exponentially diverging correlation length, so the threshold separating long-range and short-range behavior is σ* = 2. The argument is carried by a long-range simple random walk constrained to a total length of order L^d, whose transverse susceptibility mirrors the fluctuation susceptibility of the spin mo

Load-bearing premise

The σ = 2 ordering claim rests on an assumed finite-size form, ⟨M²⟩ = c1 + c2/ln(L/L0), with L0 and a subtraction coefficient b chosen per temperature; if subleading corrections change the extrapolated intercept, the positive magnetization could vanish in the thermodynamic limit.

Editorial extensions

If this is right

  • The long-range/short-range threshold in 2D is σ* = 2: systems with σ ≤ 2 have finite-temperature long-range order, while those with σ > 2 do not.
  • At the marginal point, order is real but slow to develop: magnetization approaches a positive constant as 1/ln(L/L0) and correlations saturate as g0 + a/ln r.
  • Both the 2D long-range XY and Heisenberg models show a single continuous transition, with no separate quasi-long-range phase.
  • The fixed-length random-walk scaling form gives a concrete diagnostic for transverse-fluctuation physics in other long-range continuous-symmetry models in 2D and 3D.
  • A geometric continuous-symmetry model (the long-range uniform forest) shows the same ordering at σ = 2, suggesting the criterion extends beyond spin Hamiltonians.

Reading between the lines

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

  • If the σ = 2 ordering claim is correct, the standard no-order theorem for low-dimensional continuous symmetry fails at the marginal point not because of power counting but because it treats free-field fluctuation amplitudes as extensive; the paper implicitly locates the breakdown at that step.
  • The random-walk analogy suggests a model-independent test: at a candidate marginal point, transverse susceptibilities should follow χ_k ≈ A L²/ln(L/L0) over a wide range of L, and the fitted A should connect to the spin stiffness; comparing A across models would separate universal from model-dependent content.
  • A natural next test would be quantum versions of the 2D long-range Heisenberg model at σ = 2, where quantum fluctuations might shift or sharpen the threshold; the walk-based scaling form gives a concrete finite-size ansatz to check.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper presents Monte Carlo simulations of the two-dimensional classical long-range Heisenberg model with interactions decaying as 1/r^{d+σ}. The authors claim that for all σ ≤ 2, including the marginal case σ = 2, the system undergoes a single continuous finite-temperature transition into a long-range-ordered phase. At σ = 2 they report ⟨M²⟩ extrapolating to a positive value with a logarithmic finite-size correction and a correlation function g(r) ≈ g₀ + a/ln r. They introduce a fixed-length long-range simple random walk and conjecture a scaling form for its correlation function, Eq. (1)/(S6), from which they propose a general criterion for the existence of low-temperature long-range order in O(n) systems in any dimension d. The paper also reports supporting data for the 2D LR uniform forest model at σ = 2.

Significance. If the σ = 2 claim were correct, it would overturn the extended Mermin–Wagner theorem as cited and would establish a new threshold σ* = 2 separating long-range and short-range behavior, with substantial implications for low-dimensional long-range systems. The numerical effort is considerable: simulations up to L = 8192 using an accelerated Luijten–Blöte cluster algorithm, with several complementary observables (β_L, ξ/L, ⟨M²⟩, χ_k), and the fixed-length LR-SRW comparison is a nice heuristic device. However, the central σ = 2 conclusion is in direct conflict with a rigorous theorem, and the manuscript does not address that theorem's actual proof. The proposed general criterion is at present an analogy, not a derivation. These issues are load-bearing, so the paper as written does not constitute a reliable advance.

major comments (3)
  1. [§3, Fig. 2(c); Discussion] The claimed LRO at σ = 2 contradicts the rigorous extended Mermin–Wagner theorem (Ref. [25]). That theorem is proved from the Bogoliubov inequality, not from a Gaussian free field (GFF) description. For the Hamiltonian H = −∑ J_{ij} S_i·S_j with J(r) ~ 1/r^{d+σ} and d = σ = 2, the Fourier transform satisfies J(0)−J(k) ≍ k² ln(1/k) for small k, so the standard spin-wave bound makes ⟨M²⟩ vanish in the thermodynamic limit unless a hypothesis of the theorem is violated. The manuscript's rebuttal—that the theorem 'essentially relies on the GFF description' and that GFF magnitudes violate extensivity—misidentifies the proof mechanism; the theorem does not assume extensivity of a height field. No violated hypothesis is demonstrated. Thus the σ = 2 conclusion is internally inconsistent with a cited rigorous result.
  2. [SM Eq. (S2), Fig. S2(c1,c2), Fig. 2(c)] The extrapolation to positive ⟨M²⟩ at σ = 2 rests on the assumed form ⟨M²⟩ = c₁ + c₂/ln(L/L₀), with L₀ = e^{−5.94}, e^{−7.1}, e^{−7.4} fitted per temperature, and on the subtracted observable M_r² = ⟨M²⟩ − b⟨M_k²⟩ with b = 153, 182, 182 chosen so that the curve remains positive. No error bars are shown, and the sensitivity of the intercept c₁ to L₀ and b is not reported. A different but equally plausible correction form (e.g., a power law in L) could change the intercept. Since the theorem already forbids LRO at σ = 2, the burden is on these fits to demonstrate a clean, stable extrapolation; the current presentation does not meet that burden.
  3. [§4, Eq. (1), SM Eq. (S6)] The general criterion is inferred from the fixed-L LR-SRW model, but Eq. (1)/(S6) is a conjectured scaling form, introduced and tested only on that model. The logical step from a fixed-length random walk to the low-temperature phase of an O(n) spin model is not justified. Moreover, the assertion that a Goldstone-mode contribution g(r) ~ r^{2−d} = const for d = 2 and σ > 2 'disrupts' LRO is not self-evident: a constant plateau in the connected correlation can coexist with long-range order, and the paper itself states that both GFF and fixed-L LR-SRW fail for σ > 2 in 2D. The proposed criterion is therefore not established by the presented evidence.
minor comments (3)
  1. [Figs. 1–3, SM Figs. S1–S6] Most figures lack error bars and fit ranges. Statistical uncertainties for β_c, ν, ω, and the intercepts in Fig. 2(c) should be reported, along with the number of independent samples and autocorrelation estimates.
  2. [§3, Fig. 2(b)] The correlation-length exponent ν ≈ 8 at σ = 2 is inferred from 'bending-up' in a semi-log plot, without a fit range or uncertainty. An exponent this large needs a careful scaling analysis before it can support a continuous transition.
  3. [Eq. (1), Discussion] The notation 'fixed-L' is used for both the total walk length ℒ and the linear size L. Also, the statement that g(r) ~ r^{2−d} 'does not decay' for d = 2 is confusing because r^{2−d} = r^0 = 1; the intended meaning should be stated explicitly.

Circularity Check

2 steps flagged · score 4.0 of 10

Moderate partial circularity: the σ=2 'prediction' of LRO is the fitted intercept of a 1/ln(L/L0) form imported from the authors' self-built fixed-L LR-SRW, and the general criterion is read off from that same model; the underlying Heisenberg MC evidence is nevertheless independent.

  1. fitted input called prediction [Fig. 2(c) caption and main text 'Long-range Heisenberg model'; SM Eq. (S2), Fig. S2(c1,c2)]
    "The main plot demonstrates that the squared magnetization ⟨M2⟩ converges to positive values: 0.25, 0.61, 0.81, respectively, following a logarithmic decay: ∼ 1/ln(L/L0) ... With b ≈ 153, 182, and 182 respectively, M2r converge to non-vanishing values ... [SM Eq. (S2):] ⟨M2⟩ = c1 + c2/ln(L/L0) for σ = 2."

    The reported order parameter M ≈ 0.51, 0.78, 0.90 is exactly √c1 of the fitted intercepts in Eq. (S2), with L0 = e^{−5.94}, e^{−7.1}, e^{−7.4} and b = 153/182/182 tuned per temperature. The 1/ln(L/L0) form is not derived from the Heisenberg Hamiltonian at σ=2 — the paper concedes this logarithmic scaling of χk 'can no longer be derived from GFFs' — but is imported from the authors' LR-SRW Eq. (1). The LRO claim at σ=2 is therefore the fitted constant of an assumed ansatz being presented as a predicted order parameter.

  2. self definitional [Main text 'Goldstone-mode physics and LR-SRW' and 'Proposal of a general criterion'; Eq. (1); Discussion]
    "The length constraint reflects the extensivity of statistical systems and avoids the amplitude divergence of the GFFs for σ ≥ 2. ... Since the magnitudes of GFFs exhibit logarithmic divergence, violating the extensivity of statistical systems, we suggest that the prediction of Eq. (1) could be more reliable. ... Based on whether or not the correlation function g(r) would vanish in Eq. (1), we propose a criterion for determining the existence of finite-T LRO."

    The criterion is read off from Eq. (1), which is the measured correlation of the authors' own fixed-L LR-SRW — a model whose defining constraint (total length fixed at O(L^d)) is imposed precisely to remove the GFF amplitude divergence, and is justified by the same 'extensivity' premise used to dismiss the rigorous extended M-W theorem. The conclusion is therefore built into the model by construction: no divergence in the input model ⇒ 'LRO could survive'. Eq. (1) is only 'numerically confirm[ed] ... for the fixed-L LR-SRW', and the spin-system confirmation at σ=2 uses the identical 1/ln(L/L0) form, making the general prediction a self-consistency loop rather than an independent derivation.

full rationale

The paper's genuinely self-contained result is the Monte Carlo evidence for the 2D LR-Heisenberg model itself: pseudo-critical βL converging to βc = 1.27(2) with L^{−ω} corrections (Fig. 2a), ξ/L crossings at three σ values (Fig. S1), and χk scaling (Fig. 3a). These data do not reduce to the fixed-L LR-SRW, so the central claim (SSB for σ ≤ 2) is not circular. The circularity is partial and located in two places. First, at the marginal point σ=2 the LRO conclusion is obtained by fitting ⟨M²⟩ = c1 + c2/ln(L/L0) with per-temperature L0 and a per-temperature subtraction constant b in M²r; the quoted order-parameter values are the fitted intercepts, and the fitting form is the same 1/ln ansatz that the criterion is supposed to predict. Second, the proposed general criterion is inferred from Eq. (1), the correlation function of the authors' own fixed-L LR-SRW, whose fixed-length constraint is chosen to 'avoid the amplitude divergence of the GFFs' — the same extensivity argument used to reject the extended M-W theorem; so the criterion's prediction at σ=2 is contained in the model's construction. Cross-references to the same group's XY results [39,40] are consistency checks, not load-bearing derivations. Whether the σ=2 LRO actually survives rigorous Bogoliubov-inequality bounds (Bruno) is a correctness question, not a circularity question, but it underscores that the extrapolation-based 'prediction' is fragile.

Assumptions & free parameters 4 free parameters · 4 assumptions · 1 invented entities

The central claim depends on several fitted parameters (L0, b, rescaling constants), a conjectured scaling form for a new random-walk model, and a field-theoretic GFF description that the paper itself argues fails at σ=2. The marginal-case extrapolation is therefore the least secure part of the ledger.

free parameters (4)
  • b (subtraction coefficient in M_r²) = b=153,182,182 for β=2,4,8 at σ=2; b=19,40 for σ=1.75,1.875
    Chosen per temperature so that M² > M_r² and finite-size corrections are suppressed; the positive extrapolated intercept of M_r² is used to conclude LRO.
  • L0 (logarithmic scale in fits) = L0=e^{−5.94}, e^{−7.1}, e^{−7.4} for β=2,4,8 at σ=2
    Fitted in the ⟨M²⟩ vs 1/ln(L/L0) extrapolation; the intercept c1 depends on L0, so this is a free parameter entering the LRO conclusion.
  • Rescaling constants (a,b) for σ=3 correlation length = a=2.07, b=−1.09
    Used to collapse σ=3 data onto the nearest-neighbor exponential divergence to demonstrate SR universality; fitted to the data.
  • LR-SRW total-length prefactor C = C=1/4 (2D), C=1 (3D)
    The total walk length is set to C L^d by hand; no derivation is given. It affects amplitudes of the correlation function but not the scaling forms.
assumptions (4)
  • domain assumption Gaussian free field describes transverse Goldstone fluctuations in the ordered phase.
    Used in the SM to derive g(r)~r^{σ−d} for σ<2 and to identify the failure at σ=2; this is an effective field-theory approximation, not proven for the lattice model.
  • ad hoc to paper The fixed-length constraint on the LR-SRW (total walk length O(L^d)) correctly enforces extensivity.
    Introduced without derivation; the LRO criterion for σ=2 follows from the resulting g(r)~1/ln r behavior.
  • ad hoc to paper The conjectured scaling form g(r,L) ~ ... in Eq. (1) / Eq. (S6) holds.
    Conjectured and tested numerically only on the LR-SRW; the general criterion is a direct consequence, so the criterion inherits this assumption.
  • domain assumption A single second-order transition can be identified by ξ/L crossings and power-law convergence of β_L.
    Standard finite-size scaling methodology; assumes no additional transitions and that equilibrium is reached.
invented entities (1)
  • Fixed-length long-range simple random walk (fixed-L LR-SRW / Lévy flight)
    purpose: Model of Goldstone-mode fluctuations in low-T LR spin systems; its connected correlation function is used to infer the LRO criterion.
    The model is validated only by comparing to the same Heisenberg simulations that motivate it; it makes no independent, outside-the-paper falsifiable prediction.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Spontaneous Symmetry Breaking in Two-dimensional Long-range Heisenberg Model." pith.science (2026). https://pith.science/paper/RAOO2XZT

@misc{pith2026251201956,
  author       = {Pith},
  title        = {Pith review of: Spontaneous Symmetry Breaking in Two-dimensional Long-range Heisenberg Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RAOO2XZT}},
  note         = {Machine review of arXiv:2512.01956}
}
abstract

Algebraically decaying interactions $\sim 1/r^{d+\sigma}$ can lead to nontrivial universality beyond short-range (SR) theories and spontaneous symmetry breaking in low-dimensional systems. We perform large-scale Monte Carlo simulations for the classical long-range (LR) Heisenberg model in two dimensions (2D) up to linear size $L=8192$. We show that the system enters a long-range-ordered phase through a single continuous phase transition for all $\sigma \leq 2$, including the marginal case $\sigma=2$. In contrast, for $\sigma > 2$ it recovers the SR asymptotically free behavior with no finite-temperature transition. This places the LR--SR crossover threshold at $\sigma_* = 2$. To characterize the ordered phase, we introduce an LR simple random walk with a fixed total length $\mathcal{L} \sim\mathcal{O}(L^d)$. This fixed-$\mathcal L$ walk reproduces the finite-size scaling of the Goldstone-mode fluctuations in the LR Heisenberg model in both two and three dimensions, including the logarithmic scaling at $\sigma = 2$. These results further motivate a general criterion for the existence of finite-temperature long-range order in LR systems with continuous symmetry in any spatial dimension.

Figures

Figures reproduced from arXiv: 2512.01956 by the authors.

Figure 1
Figure 1. FIG. 1. The Phase diagram of the 2D LR-Heisenberg model, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Existence of the finite-T phase transition and emergenc [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The faithful characterization of the Goldstone-mode phy [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Perturbative Renormalization and Universality Diagram for Long-Range Quantum Criticality

    cond-mat.stat-mech 2026-06 unverdicted novelty 7.0 of 10

    A controlled ε-δ expansion around the LR-SR boundary yields two-loop expressions for ν, η_ω and η_k in long-range quantum O(n) models together with a proposed universality diagram.

Reference graph

Works this paper leans on

88 extracted references · 1 linked inside Pith · cited by 1 Pith paper

  1. [25]

    Picco, Critical behavior of the Ising model with long range interactions (2012), arXiv:1207.1018 [cond-mat, physics:hep-th]

    M. Picco, Critical behavior of the Ising model with long range interactions (2012), arXiv:1207.1018 [cond-mat, physics:hep-th]

    arXiv 2012

  2. [1]

    6 T Disorder ξ ∼ exp(bβ ) ξ ∼ | T − Tc|− ν LRO g(r) = g0 + arσ − d g(r) = g0 + a/ ln r Tc for CG FIG. 1. The Phase diagram of the 2D LR-Heisenberg model, which includes: mean-field regime for σ ≤ 1 with critical behaviors governed by the Gaussian fixed point, nonclassical regime for 1 < σ ≤ 2 with σ-dependent critical exponents, and SR regime for σ > 2 with...

  3. [2]

    D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982)

    1982

  4. [3]

    27(2) (a) βL : ξ/L = 1 σ = 2 NN

    5 βL βc = 1. 27(2) (a) βL : ξ/L = 1 σ = 2 NN

  5. [4]

    magneti- zation density

    5 1. 0 1. 5 2. 0 2. 5 3. 0 aβ + b 100 101 102 103 ξ ∼ e2πβ ∼ t− ν (b) β < β c Size 32 64 128 256 512 1024 2048 4096 8192 NN σ = 3 σ = 2 NN σ = 3 σ = 2 0 0.05 0.1 0.15 1/ ln(L/L 0) 0 0.2 0.4 0.6 0.8 1 ⟨M 2⟩ (c) β > β c β =2 β =4 β =8 0.6 1.0 1.4 1.8 2.2 0 1 2 ξ/L vs. β 0 0.1 0.2 0.3 0.4L− ω -0.5 0 0.5 1 M 2 r FIG. 2. Existence of the finite-T phase transiti...

  6. [5]

    von Klitzing, Rev

    K. von Klitzing, Rev. Mod. Phys. 58, 519 (1986)

    1986

  7. [6]

    Cui-Zu, Z

    C. Cui-Zu, Z. Jinsong, F. Xiao, S. Jie, Z. Zuocheng, G. Minghua, L. Kang, O. Yunbo, W. Pang, W. Li-Li, J. Zhong-Qing, F. Yang, J. Shuaihua, C. Xi, J. Jinfeng, D. Xi, F. Zhong, Z. Shou-Cheng, H. Ke, W. Yayu, L. Li, M. Xu-Cun, and X. Qi-Kun, Science 340, 167 (2013)

    2013

  8. [7]

    C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005)

    2005

Show all 88 references
  1. [8]

    B. B. Andrei, H. Taylor L., and Z. Shou-Cheng, Science 314, 1757 (2006)

    2006

  2. [9]

    Markus, W

    K. Markus, W. Steffen, B. Christoph, R. Andreas, B. Hartmut, M. Laurens W., Q. Xiao-Liang, and Z. Shou- Cheng, Science 318, 766 (2007)

    2007

  3. [10]

    N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966)

    1966

  4. [11]

    J. M. Kosterlitz, Rev. Mod. Phys. 89, 040501 (2017)

    2017

  5. [12]

    F. J. Dyson, Communications in Mathematical Physics 12, 91 (1969)

    1969

  6. [13]

    Maleev, Soviet Journal of Experimental and Theoret- ical Physics 43, 1240 (1976)

    S. Maleev, Soviet Journal of Experimental and Theoret- ical Physics 43, 1240 (1976)

    1976

  7. [14]

    Fr¨ ohlich, R

    J. Fr¨ ohlich, R. Israel, E. H. Lieb, and B. Simon, Commu- nications in Mathematical Physics 62, 1 (1978)

    1978

  8. [15]

    Defenu, T

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

    2023

  9. [16]

    M. E. Fisher, S.-k. Ma, and B. G. Nickel, Phys. Rev. Lett. 29, 917 (1972)

    1972

  10. [17]

    Sak, Physical Review B 8, 281 (1973)

    J. Sak, Physical Review B 8, 281 (1973)

    1973

  11. [18]

    Honkonen and M

    J. Honkonen and M. Y. Nalimov, J. Phys. A: Math. Theor. 22, 751 (1989)

    1989

  12. [19]

    Honkonen, J

    J. Honkonen, J. Phys. A: Math. Theor. 23, 825 (1990)

    1990

  13. [20]

    Defenu, A

    N. Defenu, A. Trombettoni, and A. Codello, Phys. Rev. E 92, 052113 (2015)

    2015

  14. [21]

    Luijten and H

    E. Luijten and H. W. J. Bl¨ ote, Phys. Rev. Lett. 89, 025703 (2002)

    2002

  15. [22]

    M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, Phys. Rev. E 89, 062120 (2014)

    2014

  16. [23]

    Horita, H

    T. Horita, H. Suwa, and S. Todo, Physical Review E 95, 012143 (2017)

    2017

  17. [24]

    Shiratani and S

    S. Shiratani and S. Todo, Phys. Rev. E 110, 064106 (2024)

    2024

  18. [26]

    Blanchard, M

    T. Blanchard, M. Picco, and M. A. Rajabpour, Europhys. Lett. 101, 56003 (2013)

    2013

  19. [27]

    Grassberger, J

    P. Grassberger, J. Stat. Phys. 153, 289 (2013)

    2013

  20. [28]

    Bruno, Physical Review Letters 87, 137203 (2001)

    P. Bruno, Physical Review Letters 87, 137203 (2001)

    2001

  21. [29]

    Laflorencie, I

    N. Laflorencie, I. Affleck, and M. Berciu, Journal of Statistical Mechanics: Theory and Experiment 2005, P12001 (2005)

    2005

  22. [30]

    M. F. Maghrebi, Z.-X. Gong, and A. V. Gorshkov, Phys. Rev. Lett. 119, 023001 (2017)

    2017

  23. [31]

    Zhu and A.-M

    R.-G. Zhu and A.-M. Wang, Phys. Rev. B 74, 012406 (2006)

    2006

  24. [32]

    Cavallo, F

    A. Cavallo, F. Cosenza, and L. De Cesare, Phys. Rev. B 66, 174439 (2002). 6

    2002

  25. [33]

    Yusuf, A

    E. Yusuf, A. Joshi, and K. Yang, Phys. Rev. B 69, 144412 (2004)

    2004

  26. [34]

    Cavallo, F

    A. Cavallo, F. Cosenza, and L. De Cesare, Physica A: Sta- tistical Mechanics and its Applications 332, 301 (2004)

    2004

  27. [35]

    Cescatti, M

    F. Cescatti, M. Ib´ a˜ nez Berganza, A. Vezzani, and R. Bu- rioni, Phys. Rev. B 100, 054203 (2019)

    2019

  28. [36]

    J. R. de Sousa, The European Physical Journal B - Con- densed Matter and Complex Systems 43, 93 (2005)

    2005

  29. [37]

    Sbierski, M

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

    2024

  30. [38]

    J. Zhao, M. Song, Y. Qi, J. Rong, and Z. Y. Meng, npj Quantum Materials 8, 59 (2023)

    2023

  31. [39]

    Tiwari, H

    A. Tiwari, H. Ahn, B. Kumar, J. Saini, P. K. Srivas- tava, B. Singh, C. Lee, and S. Ghosh, Phys. Rev. B 109, L020407 (2024)

    2024

  32. [40]

    Giachetti, N

    G. Giachetti, N. Defenu, S. Ruffo, and A. Trombettoni, Phys. Rev. Lett. 127, 156801 (2021)

    2021

  33. [41]

    Giachetti, A

    G. Giachetti, A. Trombettoni, S. Ruffo, and N. Defenu, Phys. Rev. B 106, 014106 (2022)

    2022

  34. [42]

    D. Yao, T. Xiao, C. Zhang, Y. Deng, and Z. Fan, Phys. Rev. B 112, 144429 (2025)

    2025

  35. [43]

    T. Xiao, D. Yao, C. Zhang, Z. Fan, and Y. Deng, Chinese Physics Letters 42, 070002 (2025)

    2025

  36. [44]

    Z. Zhou, J. Grimm, S. Fang, Y. Deng, and T. M. Garoni, Phys. Rev. Lett. 121, 185701 (2018)

    2018

  37. [45]

    Y. Deng, T. M. Garoni, J. Grimm, and Z. Zhou, Journal of Statistical Mechanics: Theory and Experiment 2022, 053208 (2022)

    2022

  38. [46]

    Y. Deng, T. M. Garoni, J. Grimm, and Z. Zhou, Journal of Statistical Mechanics: Theory and Experiment 2024, 023203 (2024)

    2024

  39. [47]

    Bouchaud and A

    J.-P. Bouchaud and A. Georges, Physics reports 195, 127 (1990)

    1990

  40. [48]

    Zaburdaev, S

    V. Zaburdaev, S. Denisov, and J. Klafter, Rev. Mod. Phys. 87, 483 (2015)

    2015

  41. [49]

    Luijten and H

    E. Luijten and H. W. Bl¨ ote, International Journal of Modern Physics C (IJMPC) 06, 359 (1995)

    1995

  42. [50]

    Luijten and H

    E. Luijten and H. W. J. Bl¨ ote, Physical Review B 56, 8945 (1997)

    1997

  43. [51]

    D. Yao, C. Zhang, Z. Y. Xie, Z. Fan, and Y. Deng, Phys. Rev. B 111, 214403 (2025)

    2025

  44. [52]

    Caracciolo and A

    S. Caracciolo and A. Pelissetto, Nuclear Physics B 420, 141 (1994)

    1994

  45. [53]

    Falcioni and A

    M. Falcioni and A. Treves, Nuclear Physics B 265, 671 (1986)

    1986

  46. [54]

    Billoire, Phys

    A. Billoire, Phys. Rev. B 54, 990 (1996)

    1996

  47. [55]

    Y. Sun, M. Hu, Y. Deng, and J.-P. Lv, Phys. Rev. Lett. 131, 207101 (2023)

    2023

  48. [56]

    H. Chen, J. Salas, and Y. Deng, SciPost Phys. 16, 121 (2024)

    2024

  49. [57]

    Bauerschmidt, N

    R. Bauerschmidt, N. Crawford, T. Helmuth, and A. Swan, Communications in Mathematical Physics 381, 1223 (2021)

    2021

  50. [58]

    Jos´ e, Phys

    J. Jos´ e, Phys. Rev. D 14, 2826 (1976)

    1976

  51. [59]

    Fr¨ ohlich, Communications in Mathematical Physics 47, 233 (1976)

    J. Fr¨ ohlich, Communications in Mathematical Physics 47, 233 (1976)

    1976

  52. [60]

    Br´ ezin and J

    E. Br´ ezin and J. Zinn-Justin, Phys. Rev. Lett. 36, 691 (1976)

    1976

  53. [61]

    Spontaneous Symmetry Breaking in Two-dimensional Long-range Heisenber g Model

    A. Polyakov, Physics Letters B 59, 79 (1975). 1 Supplemental Material for “Spontaneous Symmetry Breaking in Two-dimensional Long-range Heisenber g Model” 2D LR-HEISENBERG MODEL A T σ ≤ 2 In this section, we present supplemental results on the critical properties and low-temper...

    1975

  54. [62]

    6 1. 0 1. 4 1. 8 2. 2 0 1 2 3 ξ/L (a) σ = 1. 75 64 128 256 512 1024 2048

  55. [63]

    8 1. 0 1. 4 1. 8 2. 2 β 0 1 2 3 (b) σ = 1. 875

  56. [64]

    6 1. 0 1. 4 1. 8 2. 2 0 1 2 (c) σ = 2 0 0.1 0.2 1.05 1.1 1.15 1.2 βc = 1. 063(3) ξ/L = 0. 65 βL vs. L− 0.9 0 0.03 0.06 0.09 1.1 1.15 1.2 1.25 βc = 1. 134(8) ξ/L = 0. 8 βL vs. L− 0.8 0 0.1 0.2 0.3

  57. [65]

    27(2) ξ/L = 1 βL vs

    8 βc = 1. 27(2) ξ/L = 1 βL vs. L− 0.73 FIG. S1. Demonstration of the second-order phase transition for σ = 1 .75 (a), 1 .875 (b) and 2 (c). Main plots display the second-order correlation length ratio (to the system size) ξ/L as a function of the inverse temperature β. The cle...

  58. [66]

    0 0. 2 0. 4 0. 6 L− 0.25

  59. [67]

    00 ⟨M 2⟩ (a1) σ = 1. 75

  60. [68]

    0 0. 2 0. 4 0. 6 0. 8 L− 0.125

  61. [69]

    00 0. 05 0. 10 0. 15 1/ ln(L/L 0)

  62. [70]

    0 (c1) σ = 2 β =2 β =4 β =8

  63. [71]

    0 M 2 r (a2) σ = 1. 75

  64. [72]

    00 0. 15 0. 30 L− 0.6

  65. [73]

    0 0. 1 0. 2 0. 3 0. 4 L− 0.41 − 1 0 1 (c2) σ = 2 FIG. S2. Demonstration of LRO for σ = 1 .75, 1.875 and 2. The upper figures display the L-dependence of the magnetization ⟨M 2⟩. For σ = 1.75 (a1) and 1 .875 (b1), ⟨M 2⟩ converges to positive values as ∼ Lσ−2; for σ = 2 (c1), ⟨M ...

  66. [74]

    0 0. 1 0. 2 0. 3 0. 4 0. 5 r/L 0 2 4 6g(r) (a) σ = 3 512 1024 2048 4096 8192

  67. [75]

    0 0. 1 0. 2 0. 3 0. 4 0. 5 r/L 0 2 4g(r) ln(r/ 0. 1) (b) σ = 2

  68. [77]

    4 g(r)r0.25 (c) σ = 1. 75 FIG. S3. FSS analysis of the correlation function for different σ in 2D fixed- L LR-SR W. The rescaled correlations – specifically g(r), g(r) ln(r/0.1), and g(r)r0.25 for σ = 3 (a), 2 (b), and 1 .75 (c), respectively – are plotted as a function of the sc...

  69. [79]

    4 g(r)r (a) σ = 3 32 64 128 256

  70. [80]

    0 0. 1 0. 2 0. 3 0. 4 0. 5 r/L

  71. [81]

    1) (b) σ = 2

    6 g(r)r ln(r/ 0. 1) (b) σ = 2

  72. [82]

    0 0. 1 0. 2 0. 3 0. 4 0. 5 r/L 0 0.05 0.1 0.15 0.2 g(r)r1.25 (c) σ = 1. 75 FIG. S4. FSS analysis of the correlation function for the 3D fixed- L LR-SR W with different σ. Rescaled correlation functions – g(r)r, g(r)r ln(r/0.1), g(r)r1.25 for σ = 3 (a), 2 (b) and 1 .75 (c) respec...

  73. [83]

    0 0. 1 0. 2 0. 3 0. 4 0. 5 L− 0.45 − 0. 5

  74. [84]

    7 M 2 r (a) w = 15 w = 20 w = 25 101 102 103 L 100 300 500 L2 χ k (b) FIG. S6. Evidence of LRO and Goldstone modes in 2D LR-UF model at σ = 2 in the low-temperature phase ( w = 15, 20, and 25; note that wc ≈ 7 from Fig. S5). (a) FSS of the rescaled magnetization M 2 r = ⟨M 2⟩ ...

  75. [85]

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

    1972

  76. [86]

    Kardar, Statistical Physics of Fields (Cambridge University Press, Cambridge, 2007)

    M. Kardar, Statistical Physics of Fields (Cambridge University Press, Cambridge, 2007)

    2007

  77. [87]

    H. Chen, J. Salas, and Y. Deng, Anomalous criticality coexi sts with giant cluster in the uniform forest model, SciPost Phys. 16, 121 (2024)

    2024

  78. [88]

    Bauerschmidt, N

    R. Bauerschmidt, N. Crawford, T. Helmuth, and A. Swan, Random spanning forests and hyperbolic symmetry, Communi- cations in Mathematical Physics 381, 1223 (2021)

    2021

  79. [89]

    Bauerschmidt, N

    R. Bauerschmidt, N. Crawford, and T. Helmuth, Percolation tran sition for random forests in d ≥ 3, Inventiones mathemat- icae 237, 445 (2024)

    2024

  80. [90]

    P. Hou, S. Fang, J. Wang, H. Hu, and Y. Deng, Geometric properti es of the fortuin-kasteleyn representation of the ising model, Phys. Rev. E 99, 042150 (2019)

    2019

Pith tools

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