pith. sign in

arxiv: 2606.22398 · v1 · pith:VXBZ5GRAnew · submitted 2026-06-21 · ❄️ cond-mat.dis-nn

Percolation of Zero-Weight Paths and the Shape of the Phase Boundary in the Two-Dimensional Random-Bond Ising Model

Amirhossein Manouchehri , Kirill Shtengel This is my paper

Pith reviewed 2026-06-26 09:49 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn
keywords random-bond Ising modelzero-weight percolationferromagnetic phase boundaryNishimori pointtwo-dimensional disordercritical exponentsgeometric transitionphase diagram
0
0 comments X

The pith

Zero-weight percolation is incompatible with ferromagnetic order and fixes a vertical phase boundary below the Nishimori point in the 2D random-bond Ising model.

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

The paper establishes that a purely geometric transition, the percolation of paths containing equal numbers of +J and -J bonds, cannot coexist with ferromagnetic order. Because this zero-weight percolation depends only on the fixed disorder realization and not on temperature, the ferromagnetic phase boundary must run vertically in the (p,T) plane below the Nishimori point. Numerical work locates the percolation threshold at p_c = 0.1000(2) with associated exponents, a value below earlier estimates of the ferromagnetic critical disorder, yet the authors conclude that this geometric criterion governs the loss of order.

Core claim

The onset of zero-weight percolation, the emergence of a system-spanning path with equal numbers of +J and -J bonds, is incompatible with ferromagnetic ordering. Its purely geometrical character makes the transition independent of temperature, so the ferromagnetic phase boundary is vertical below the Nishimori point. Finite-size scaling yields p_c = 0.1000(2) together with u = 1.26(1), eta/ u = 0.85(1), u/ u = 0.264(5), and fractal dimension d_f ≈ 1.11.

What carries the argument

Zero-weight percolation, the geometric emergence of a spanning path that balances an equal number of +J and -J bonds.

If this is right

  • The ferromagnetic phase boundary runs vertically at p = 0.1000(2) for all temperatures below the Nishimori point.
  • Loss of ferromagnetism occurs exactly when zero-weight paths first percolate, independent of thermal fluctuations.
  • The reported critical exponents characterize the geometric transition that limits ferromagnetic order.
  • This geometric criterion supplies a temperature-independent explanation for the stability limit of the ferromagnetic phase.

Where Pith is reading between the lines

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

  • The same geometric balance argument could be tested in three-dimensional or quantum versions of the random-bond Ising model.
  • If the incompatibility holds, any future calculation claiming ferromagnetic order above p_c at low T would need to reconcile with the presence of balanced paths.
  • The approach offers a way to locate phase boundaries in other disordered systems by identifying analogous zero-cost or balanced configurations.

Load-bearing premise

A percolating zero-weight path necessarily prevents ferromagnetic ordering at any temperature below the Nishimori point.

What would settle it

A numerical or experimental observation of spontaneous magnetization persisting on a disorder realization that already contains a percolating zero-weight path at a temperature below the Nishimori point.

Figures

Figures reproduced from arXiv: 2606.22398 by Amirhossein Manouchehri, Kirill Shtengel.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic phase diagram of the two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic illustration of the geometrical argument [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison of the percolation probability for the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Log-log plot of the mean spanning-path length [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Left and center: A pair of bonds configurations (the [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
read the original abstract

We explore the connection between the low-temperature boundary of the ferromagnetic phase in the two-dimensional $\pm J$ random-bond Ising model, where antiferromagnetic bonds occur with probability $p$ and a geometric transition dubbed ``zero-weight percolation''. We argue that the onset of this percolation characterized by the emergence of a percolating path containing an equal number of $+J$ and $-J$ bonds is incompatible with ferromagnetic ordering. Due to its purely geometrical nature, this percolation criterion is a property of a disorder realization and is independent of the temperature, which in turn suggests that the ferromagnetic phase boundary is vertical below the Nishimori point in the $(p,T)$ plane. Using a dynamic-programming algorithm combined with finite-size scaling, we identify the critical disorder at which zero-weight paths first percolate as $p_c = 0.1000(2)$, and we extract the associated critical exponents $\nu = 1.26(1)$, $\beta/\nu = 0.85(1)$, $\gamma/\nu = 0.264(5)$, and fractal dimension $d_f \approx 1.11$. The value of $p_c$ is below the previously reported values of the critical disorder strength corresponding to the loss of the ferromagnetic order, both at zero temperature and the Nishimori point. Nevertheless, we argue that the percolation transition studied in this paper is behind the loss of ferromagnetism and thus provides a new, purely geometrical perspective on the stability of ferromagnetic order in disordered spin systems.

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 / 1 minor

Summary. The manuscript explores the connection between the low-temperature boundary of the ferromagnetic phase in the 2D ±J random-bond Ising model and a geometric 'zero-weight percolation' transition. It argues that the emergence of a percolating path with equal numbers of +J and -J bonds is incompatible with ferromagnetic ordering; because this transition is purely geometric and temperature-independent, the ferromagnetic phase boundary must be vertical below the Nishimori point in the (p,T) plane. Using dynamic programming and finite-size scaling, the authors report pc = 0.1000(2) together with exponents ν = 1.26(1), β/ν = 0.85(1), γ/ν = 0.264(5) and fractal dimension df ≈ 1.11, while noting that this pc lies below previously published estimates for the loss of ferromagnetism.

Significance. If the asserted incompatibility holds, the work supplies a new, temperature-independent geometric criterion for the stability of ferromagnetic order in disordered spin systems. The numerical component is a clear strength: the dynamic-programming algorithm combined with finite-size scaling yields precise pc and exponent estimates with explicit error bars. The geometric perspective, if rigorously justified, would be a useful addition to the literature on the random-bond Ising model.

major comments (2)
  1. [Abstract] Abstract: the central claim that 'the onset of this percolation ... is incompatible with ferromagnetic ordering' is stated as an argument but supplies no derivation, proof sketch, or explicit mechanism (e.g., domain-wall energetics, degeneracy counting, or correlation-function argument) showing why a percolating zero-weight path must destroy spontaneous magnetization at any finite T below the Nishimori point. This step is load-bearing for the vertical-boundary conclusion.
  2. [Abstract] Abstract and numerical-results section: the reported pc = 0.1000(2) is explicitly stated to lie below all previously published values of the critical disorder strength for the ferromagnetic boundary (both at T = 0 and at the Nishimori point), yet the manuscript concludes that the percolation transition 'is behind the loss of ferromagnetism.' This numerical tension must be reconciled for the incompatibility argument to determine the magnetic phase boundary.
minor comments (1)
  1. The abstract refers to a 'dynamic-programming algorithm' without a concise description of its state representation or complexity; adding one sentence would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of the numerical work and the geometric perspective, as well as for identifying two load-bearing points that require clarification. We address both major comments below and will revise the manuscript to strengthen the incompatibility argument and reconcile the numerical values.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that 'the onset of this percolation ... is incompatible with ferromagnetic ordering' is stated as an argument but supplies no derivation, proof sketch, or explicit mechanism (e.g., domain-wall energetics, degeneracy counting, or correlation-function argument) showing why a percolating zero-weight path must destroy spontaneous magnetization at any finite T below the Nishimori point. This step is load-bearing for the vertical-boundary conclusion.

    Authors: We agree that the manuscript presents the incompatibility as an argument without a self-contained derivation. In the revised manuscript we will add a dedicated subsection (likely in the introduction or a new 'Mechanism' section) that supplies an explicit mechanism: a percolating zero-weight path defines a domain wall whose energy cost is identically zero for any temperature, because the number of +J and -J bonds is equal; flipping the wall therefore maps one ground-state configuration to another with identical energy and opposite magnetization on one side of the wall. This degeneracy precludes spontaneous magnetization at any finite T. The argument relies only on the geometry of the disorder realization and holds below the Nishimori point where the Nishimori symmetry does not enforce order. We will also note that the same geometric object controls the T=0 boundary, thereby justifying the vertical phase boundary. revision: yes

  2. Referee: [Abstract] Abstract and numerical-results section: the reported pc = 0.1000(2) is explicitly stated to lie below all previously published values of the critical disorder strength for the ferromagnetic boundary (both at T = 0 and at the Nishimori point), yet the manuscript concludes that the percolation transition 'is behind the loss of ferromagnetism.' This numerical tension must be reconciled for the incompatibility argument to determine the magnetic phase boundary.

    Authors: The manuscript already flags this numerical discrepancy. We maintain that zero-weight percolation supplies the microscopic geometric mechanism for the loss of ferromagnetism and therefore sets the true location of the boundary; earlier estimates may reflect different observables (e.g., Binder cumulants or magnetization histograms) or stronger finite-size corrections that systematically overestimate pc. In the revision we will (i) tabulate the literature values with their reported uncertainties and observables, (ii) add a paragraph discussing why a purely geometric, T-independent criterion can lie below finite-T or T=0 magnetic estimates, and (iii) emphasize that our pc provides a lower bound consistent with the vertical-boundary scenario. If additional references are supplied by the referee we will incorporate them. revision: yes

Circularity Check

0 steps flagged

No significant circularity; geometric percolation threshold computed independently of magnetic boundary

full rationale

The paper computes the zero-weight percolation threshold p_c = 0.1000(2) directly from disorder realizations using a dynamic-programming algorithm and finite-size scaling, yielding exponents without reference to magnetic observables. The incompatibility argument is stated as a geometric property of the disorder configuration that is temperature-independent, but this assertion does not reduce any equation or numerical output to a prior fit, self-citation, or definitional equivalence. No load-bearing step equates the claimed vertical phase boundary to its own input by construction, and the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that zero-weight percolation precludes ferromagnetic order; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Zero-weight percolation is incompatible with ferromagnetic ordering
    This premise directly links the geometric transition to the magnetic phase boundary and is invoked to conclude the boundary is vertical.

pith-pipeline@v0.9.1-grok · 5819 in / 1255 out tokens · 26932 ms · 2026-06-26T09:49:33.533647+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

56 extracted references · 20 linked inside Pith

  1. [1]

    Stauffer and A

    D. Stauffer and A. Aharony,Introduction To Percolation Theory(Taylor & Francis, 1994)

    1994

  2. [2]

    Melchert and A

    O. Melchert and A. K. Hartmann, New J. Phys.10, 043039 (2008), arXiv:0711.4069

    Pith/arXiv arXiv 2008

  3. [3]

    Nishimori,Statistical Physics of Spin Glasses and Information Processing(Oxford University Press, 2001)

    H. Nishimori,Statistical Physics of Spin Glasses and Information Processing(Oxford University Press, 2001)

    2001

  4. [4]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J. Math. Phys.43, 4452 (2002), arXiv:quant-ph/0110143

    Pith/arXiv arXiv 2002

  5. [5]

    C. Wang, J. Harrington, and J. Preskill, Ann. Phys.303, 31 (2003), arXiv:quant-ph/0207088

    Pith/arXiv arXiv 2003

  6. [6]

    Ozeki and H

    Y. Ozeki and H. Nishimori, J. Phys. Soc. Jpn.56, 3265 (1987)

    1987

  7. [7]

    Le Doussal and A

    P. Le Doussal and A. B. Harris, Phys. Rev. Lett.61, 625 (1988)

    1988

  8. [8]

    Ozeki and H

    Y. Ozeki and H. Nishimori, J. Phys. Soc. Jpn.56, 1568 (1987)

    1987

  9. [9]

    Kitatani, J

    H. Kitatani, J. Phys. Soc. Japan61, 4049 (1992)

    1992

  10. [10]

    Amoruso and A

    C. Amoruso and A. K. Hartmann, Phys. Rev. B70, 134425 (2004), arXiv:cond-mat/0401464

    Pith/arXiv arXiv 2004

  11. [11]

    Fisch and A

    R. Fisch and A. K. Hartmann, Phys. Rev. B75, 174415 (2007), arXiv:cond-mat/0603729

    Pith/arXiv arXiv 2007

  12. [12]

    Kawashima and H

    N. Kawashima and H. Rieger, Europhys. Lett.39, 85 (1997), arXiv:cond-mat/9612116

    Pith/arXiv arXiv 1997

  13. [13]

    Melchert and A

    O. Melchert and A. K. Hartmann, Phys. Rev. B79, 184402 (2009), arXiv:0902.2938

    Pith/arXiv arXiv 2009

  14. [14]

    Honecker, M

    A. Honecker, M. Picco, and P. Pujol, Phys. Rev. Lett.87, 047201 (2001), arXiv:cond-mat/0010143

    Pith/arXiv arXiv 2001

  15. [15]

    Hasenbusch, F

    M. Hasenbusch, F. P. Toldin, A. Pelissetto, and E. Vicari, Phys. Rev. E77, 051115 (2008), arXiv:0803.0444

    Pith/arXiv arXiv 2008

  16. [16]

    Merz and J

    F. Merz and J. T. Chalker, Phys. Rev. B65, 054425 (2002), arXiv:cond-mat/0106023

    Pith/arXiv arXiv 2002

  17. [17]

    Freund and P

    H. Freund and P. Grassberger, J. Phys. A: Math. Gen. 22, 4045 (1989)

    1989

  18. [18]

    Fajen, A

    H. Fajen, A. K. Hartmann, and A. P. Young, Phys. Rev. E102, 012131 (2020), arXiv:1905.04220

    arXiv 2020

  19. [19]

    Nishimori, Prog

    H. Nishimori, Prog. Theor. Phys.66, 1169 (1981)

    1981

  20. [20]

    Y. Liu, D. Wang, X. Wang, D.-X. Yao, and L.-H. Tang, arXiv:2505.04298 (2025)

    arXiv 2025

  21. [21]

    S. L. A. de Queiroz, Phys. Rev. B79, 174408 (2009), arXiv:0902.4153

    Pith/arXiv arXiv 2009

  22. [22]

    Nishimori and M

    H. Nishimori and M. Ohzeki, Physica A389, 2907 (2010), arXiv:0911.5561

    Pith/arXiv arXiv 2010

  23. [23]

    Picco, A

    M. Picco, A. Honecker, and P. Pujol, J. Stat. Mech: The- ory Exp.2006, P09006 (2006), arXiv:cond-mat/0606312

    Pith/arXiv arXiv 2006

  24. [24]

    Wan, X.-D

    Z.-Q. Wan, X.-D. Dai, and G.-Y. Zhu, Phys. Rev. Re- search8, 023059 (2026), arXiv:2511.02907

    Pith/arXiv arXiv 2026

  25. [25]

    Delfino, J

    G. Delfino, J. Stat. Mech: Theory Exp.2025, 043203 (2025), arXiv:2412.07383

    arXiv 2025

  26. [26]

    Coniglio, F

    A. Coniglio, F. di Liberto, G. Monroy, and F. Peruggi, Phys. Rev. B44, 12605 (1991)

    1991

  27. [27]

    Machta, C

    J. Machta, C. M. Newman, and D. L. Stein, J. Stat. Phys. 130, 113 (2007), arXiv:0707.0073

    Pith/arXiv arXiv 2007

  28. [28]

    Harada, Phys

    K. Harada, Phys. Rev. E84, 056704 (2011), arXiv:1102.4149

    Pith/arXiv arXiv 2011

  29. [29]

    Houdayer and A

    J. Houdayer and A. Hartmann, Phys. Rev. B70, 014418 (2004)

    2004

  30. [30]

    Stauffer, Phys

    D. Stauffer, Phys. Rep.54, 1 (1979). 10

    1979

  31. [31]

    Grinstein, C

    G. Grinstein, C. Jayaprakash, and M. Wortis, Phys. Rev. B19, 260 (1979)

    1979

  32. [32]

    Bendisch, U

    J. Bendisch, U. Derigs, and A. Metz, Discrete Appl. Math. 52, 139 (1994)

    1994

  33. [33]

    J. A. Blackman, J. R. Gonçalves, and J. Poulter, Phys. Rev. E58, 1502 (1998)

    1998

  34. [34]

    Nishimori, J

    H. Nishimori, J. Phys. Soc. Jpn.55, 3305 (1986)

    1986

  35. [35]

    Melchert, L

    O. Melchert, L. Apolo, and A. K. Hartmann, Phys. Rev. E81, 051108 (2010), arXiv:1003.1591

    Pith/arXiv arXiv 2010

  36. [36]

    Claussen, L

    G. Claussen, L. Apolo, O. Melchert, and A. K. Hartmann, Phys. Rev. E86, 056708 (2012), arXiv:1207.2893

    Pith/arXiv arXiv 2012

  37. [37]

    B. V. Kagan and B. Spivak, Fiz. Tverd. Tela31, 293 (1989), [Sov. Phys. Solid State 31, 518 (1989)]

    1989

  38. [38]

    Medina, M

    E. Medina, M. Kardar, Y. Shapir, and X. R. Wang, Phys. Rev. Lett.62, 941 (1989)

    1989

  39. [39]

    Kim and D

    H. Kim and D. A. Huse, Phys. Rev. B83, 052405 (2011), arXiv:1010.0053

    Pith/arXiv arXiv 2011

  40. [40]

    C. L. Baldwin, C. R. Laumann, and B. Spivak, Phys. Rev. B97, 014203 (2018), arXiv:1709.03516

    Pith/arXiv arXiv 2018

  41. [41]

    C. K. Thomas and H. G. Katzgraber, Phys. Rev. E84, 040101 (2011). Appendix A: Algorithm for Identifying Zero-Weight Paths To identify zero-weight paths, we employ a constrained state-space search that combines frontier expansion with memory-assisted dynamic-relaxation updates. For a cho- sen starting site on the dual lattice, the algorithm deter- mines al...

    2011

  42. [42]

    Afrontier stateconsists of a current lattice site together with the partial path used to reach it

    Overview of the Search Procedure For each disorder realization, candidate spanning paths are sought starting from sites on one open boundary of the dual lattice. Afrontier stateconsists of a current lattice site together with the partial path used to reach it. The search proceeds iteratively:

  43. [43]

    For each frontier state, all edges already used by the partial path are marked unavailable

  44. [44]

    A dynamic-relaxation solver is then applied on the remaining lattice to identify all sites reachable from the current position with zero net bond weight

  45. [45]

    The new segment is concatenated with the existing partial path

    Each newly reachable site defines a candidate path extension. The new segment is concatenated with the existing partial path

  46. [46]

    The resulting path is accepted only if it satisfies the global simple-path constraint (no repeated directed edge and no immediate reverse reuse of a previously traversed edge)

  47. [47]

    The procedure repeats until no new states are generated or a spanning path is found

    Accepted extensions generate the next frontier. The procedure repeats until no new states are generated or a spanning path is found. If a spanning path additionally intersects the central window W defined in Sec. II, the sample is classified as cross-percolating

  48. [48]

    The table stores whether the state(i, j, k)is reachable under the currently imposed forbidden edge set

    Dynamic-Relaxation Step For a fixed frontier state, reachability is computed using a three-index table T(i, j, k),(A1) where( i, j)denotes a dual-lattice site andk∈ [−L, L]is the cumulative bond sum along the candidate extension. The table stores whether the state(i, j, k)is reachable under the currently imposed forbidden edge set. The initial condition i...

  49. [49]

    After each candidate extension is reconstructed, the full path is checked explicitly

    Simple-Path Constraint Because the zero-weight condition is global, local reach- ability alone is insufficient: concatenated segments must also remain simple paths. After each candidate extension is reconstructed, the full path is checked explicitly. A path is rejected if:

  50. [50]

    any directed edge appears more than once, or

  51. [51]

    These checks enforce self-avoidance at the level of tra- versed edges and prevent backtracking loops

    both orientations of the same edge appear in the path. These checks enforce self-avoidance at the level of tra- versed edges and prevent backtracking loops. 11

  52. [52]

    Search Bounds and Validation Two practical bounds are imposed:

  53. [53]

    The number of frontier-expansion iterations is lim- ited toL

  54. [54]

    The second bound follows from parity

    The total path length is limited toL2/2. The second bound follows from parity. For evenL, a zero-weight path must contain equal numbers of+1and −1edges and therefore has even length. Consequently, sites separated from the starting point by an odd graph distance cannot be reached by zero-weight paths, so at most half of theL2 dual lattice sites are reachab...

  55. [55]

    Although the formal cost grows polynomially with system size, the practical runtime was substantially smaller than the worst-case bound for all lattices studied

    Computational Cost The worst-case runtime depends on the number of fron- tier states generated and on the density of admissible zero-weight extensions. Although the formal cost grows polynomially with system size, the practical runtime was substantially smaller than the worst-case bound for all lattices studied. This made simulations up toL = 48 computati...

  56. [56]

    Pseudocode Input: weighted dual lattice, start boundary, target boundary, window W for each boundary seed: initialize frontier with trivial path while frontier nonempty: for each frontier state: forbid previously used edges run zero-sum reachability solver reconstruct candidate extensions reject nonsimple paths add valid new states to next frontier if spa...