Square ice Coulomb phase as a percolated vertex lattice

Benjamin Canals; Johann Coraux; Nicod\`eme Rougier; Nicolas Rougemaille

REVIEW 4 major objections 5 minor 38 references

Square ice Coulomb phase as a percolated vertex lattice

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The Coulomb phase of square ice is statistically indistinguishable from an unconstrained random tiling of type I and type II vertices at 37% and 63% fractions, so vertex maps alone can identify it.

desk verdict Useful diagnostic for Coulomb phases from vertex statistics, but the main comparison is partly fitted and sampler mixing is not shown. read the letter →

arxiv 2411.16533 v1 pith:U6BU6FG2 submitted 2024-11-25 cond-mat.str-el cond-mat.dis-nn

Johann Coraux , Nicodème Rougier , Benjamin Canals , Nicolas Rougemaille This is my paper · ORCID

classification cond-mat.str-elcond-mat.dis-nn

keywords squareiceCoulombphasevertexstatisticspercolationtheoryartificialspinloopflipalgorithmclustersizedistributionmonopoledensity

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 square ice is a canonical example of a two-dimensional Coulomb phase, a massively degenerate ground state held together by the two-in/two-out ice rule. This paper asks whether that ground-state manifold has its own fingerprints in the purely local arrangement of ice-rule-satisfying vertices, and answers yes. Using a loop-flip shuffling algorithm, it finds that for large lattices the average vertex populations approach 38% type I and 62% type II, vertex-vertex correlations are weak (a few percent, tending to about 0.06 at long distances), and the size distribution of type I clusters matches random site percolation once the fractions are set to 37/63. The match survives the addition of a few percent of magnetic monopoles, and it correctly identifies Coulomb-phase micro-states in two published artificial square ice experiments while flagging detuned lattices. If this picture holds, Coulomb-phase physics can be read off directly from a real-space vertex map, without computing magnetic correlations.

What carries the argument

The machinery is the vertex map: each spin configuration is converted into a lattice of labels recording whether each vertex is type I or type II, with the six ice-rule configurations split into type I (two realizations) and type II (four realizations). Configurations are sampled with a shuffling algorithm that flips spins along closed loops or boundary-to-boundary strings, preserving the two-in/two-out constraint, and the resulting vertex statistics are compared with those of an unconstrained random tiling of type I and type II vertices, which is a site percolation problem on the square lattice. The comparison quantities are the average vertex fractions $\rho_I$ and $\rho_{II}$, the pairwise vertex correlator $\langle\sigma_i\sigma_j\rangle$ (with $\sigma_i=\pm1$ labeling vertex type), and the edge-connected cluster-size distribution of type I vertices, whose random-tiling limit is governed by the square-lattice percolation threshold $p_c\simeq0.593$.

What would settle it

Enumerate all ice-rule-satisfying configurations on a small lattice (say $8\times8$ or $10\times10$) and compute the exact average fraction of type I vertices and the exact type I cluster-size distribution; if these exact values deviate from the 37/63 random-percolation prediction beyond finite-size error, the equivalence is an artifact of the loop-flip sampling rather than a property of the Coulomb phase.

Watch

Extended reading notes

Core claim

The paper's central claim is that the square ice Coulomb phase is well approximated by an unconstrained random gas of its two vertex types. In the large-lattice limit, the ice-rule constraint renormalizes the degeneracy-based expectations of 1/3 and 2/3 to about 38% type I and 62% type II vertices; pairwise vertex correlations are small and approach the value of a random tiling at large distances; and the cluster-size distribution of type I vertices is well described by percolation theory on a square lattice, with the effective type I fraction slightly lowered to 37%. Because 62% exceeds the square-lattice percolation threshold of about 0.593, type II vertices form a percolating network while type I clusters remain finite. This vertex-level description remains reasonably accurate up to a few percent of magnetic monopoles, provided the vertex fractions are adjusted downward to account for the monopoles. The paper also shows that two sets of artificial square ice data previously identified as Coulomb phases match this ideal vertex gas, whereas a deliberately detuned set does not, making the type I cluster-size distribution a practical diagnostic for the presence of a Coulomb phase.

Load-bearing premise

The results rest on the assumption that after a number of loop and string flips comparable to the lattice size, the shuffling algorithm produces an unbiased sample of the ice ground-state manifold; the paper checks that the initial configuration does not matter, but gives no proof that the sample is representative.

Editorial extensions

If this is right

Coulomb-phase micro-states can be recognized from vertex data alone: populations near 38%/62%, a first-neighbor vertex correlator near 0.06, and type I cluster sizes following the percolation power-law form.
The effective 37/63 fractions, rather than the degeneracy-based 1/3 and 2/3, are the correct null model for vertex statistics of ice-rule states; the local spin constraint acts like a small shift in the effective vertex concentration.
Type II vertices percolate and type I clusters do not, so the size distribution of type I clusters is a sensitive indicator of whether a measured artificial spin ice is in the Coulomb phase.
Dilute magnetic monopoles do not invalidate the vertex-gas picture; the fraction of type I and type II vertices should simply be reduced linearly with monopole density when comparing with experiments.
For artificial square ice, tuning the geometric parameters (height offset or vertex hole diameter) closer to degeneracy should push the measured vertex statistics onto the percolation curve.

Reading between the lines

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

A stronger equivalence, not claimed in the paper, would be that the vertex projection of the ice manifold converges exactly to Bernoulli percolation with a fixed effective occupation probability in the thermodynamic limit; the numerics support approximation, not exactness.
A sharper test of the claim would be exact enumeration of all ice-rule configurations on small lattices; agreement with the 37/63 percolation cluster distribution would show the equivalence is intrinsic to the manifold rather than a property of the loop-flip sampler.
The same vertex-percolation criterion could be applied to thermally active artificial square ice once monopole densities are reduced below a few percent, extending the diagnostic to systems the paper notes are currently too monopole-rich.
The ideal-vertex-gas description may carry over to other two-dimensional Coulomb phases, such as kagome ice, whose vertex classes could define a similar random-tiling limit.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this paper gives a clean, inexpensive diagnostic for spotting Coulomb phases from vertex maps, and the 38/62 vertex population split is a genuinely useful number. But the central comparison with random tilings is partly fitted, and the sampler's representativeness is asserted rather than demonstrated.

What's new and good: they compute vertex populations, vertex-vertex correlations, and type-I cluster size distributions in the square ice manifold using loop/string flips, then compare with unconstrained random tilings. The 38/62 limit (not the naive 1/3–2/3 from degeneracy) is correct and worth knowing. The near-ideal gas picture, with small but nonzero vertex correlations, and the match of cluster statistics to square-lattice percolation around rho = 0.37, are plausible and clearly presented. The re-analysis of two artificial square ice experiments is suggestive and will likely push others to adopt the diagnostic.

Soft spots, in order of severity:

1. The central comparison is partly circular. They measure rho_I = 38%, then use 37% in the random tiling to get better large-cluster agreement. They are transparent about the one-percent nudge, but it means the percolation match is not parameter-free. A proper goodness-of-fit test with the fitted rho would tell us how much this matters.

2. The sampler's mixing is not established. They check initial-condition independence and that rho_I plateaus after n^2 flips, but that doesn't guarantee convergence for large clusters, which are the slow modes at the ice point. This is not a fatal flaw—the plateau is likely robust—but autocorrelation analysis or a comparison with an independent sampler would make the case much stronger.

3. The experimental comparisons in Fig. 5 are visual. The agreement in (a) and (b) looks good, and the failure in (c) is telling, but the lattices are small (20x20, 30x30) and the vertex populations in Table I deviate substantially from ideal, so quantitative measures would help.

Overall: the core observation is probably right, the paper is honest about its limitations, and it gives experimentalists a new tool. It doesn't reorganize the theory, but it is a solid, useful contribution. I would accept it for peer review, with requests for code/data release and a quantitative fit before acceptance.

Referee Report

4 major / 5 minor

Summary. The paper studies the square ice Coulomb phase through the statistics of ice-rule-satisfying vertices (type I and type II) rather than spin textures. Using a loop/string flip Monte Carlo algorithm, the authors measure the vertex populations (about 38% type I and 62% type II for large lattices), the pairwise vertex correlations, and the size distribution of type I vertex clusters. These are compared with a completely random, constraint-free vertex tiling at the same fractions. The central claim, stated in Section II, is that the square ice manifold is well approximated by a random 37%/63% vertex tiling for large lattices, with cluster sizes described by percolation theory, and that this description remains reasonable at dilute monopole densities. The paper then re-analyzes two published artificial square ice experiments and argues that the vertex cluster-size distribution can serve as a criterion to identify Coulomb-phase micro-states.

Significance. If the central claim is correct, the paper offers a simple, real-space characterization of the square ice Coulomb phase that is complementary to magnetic correlation functions and structure factors, and it proposes a readily measurable observable for artificial spin ice experiments. The study's strength is its clear formulation of an independent null model: an unconstrained random vertex tiling with matched type I/II fractions. The paper also provides a substantial numerical dataset (up to 4x10^4 micro-states for the 100x100 lattice) and directly confronts the model with experimental data from two different artificial square ice platforms. However, the numerical evidence for the central claim is not yet conclusive: the Monte Carlo sampler's representativeness is not established, the comparison fraction is adjusted post hoc, and the quality of the cluster-size agreement is assessed visually rather than quantitatively. These issues are load-bearing because the paper's main conclusion rests on the equivalence between the ice manifold and the random tiling.

major comments (4)

[Section III, first paragraph and Fig. 2(a)] The shuffling algorithm's representativeness of the full ice manifold is not established. The paper states that the initial condition does not change the results and that rho_I reaches a plateau after n^2 flips, but this does not demonstrate convergence. The loop/string flip dynamics at the ice point is critical, with large-scale loop modes that can have very long autocorrelation times. If the Markov chain under-samples or over-samples configurations with particular large-cluster structures, the measured 38/62 populations and the cluster-size distributions in Fig. 3(b) would be sampling artifacts rather than intrinsic manifold properties. I request convergence diagnostics beyond the mean rho_I: e.g., autocorrelation times for the largest cluster size and the vertex-vertex correlation function, a comparison of independent chains started from ordered and random initial configurations, and for small lattice sizes a check against exact enumeration or a provably uniform sampler (such as a worm or loop algorithm with known stationary distribution). Without such evidence, the quantitative basis for the paper's central claim is incomplete.
[Section II and Fig. 3(b)] The central claim is stated as a 37%/63% random tiling, but the measured vertex fractions are 38%/62%, and the 37%/63% value is introduced post hoc because it gives 'a much better agreement' for the 100x100 cluster-size distribution. This means the comparison fraction is a fitted parameter, not a prediction. The paper does not provide a quantitative measure of agreement (e.g., a chi-square or Kolmogorov-Smirnov statistic over the cluster-size range) that would justify preferring 37% over 38% or show that the residual difference is statistically significant rather than within the Monte Carlo noise. Without such an analysis, the claim that the square ice manifold is 'well approximated' by a specific random tiling is not properly supported. Please report the best-fit rho_I and its uncertainty for each lattice size and boundary condition, and clarify whether 37% is a universal value or a lattice-size-dependent effective parameter.
[Table I, footnote [37], and Fig. 4] The monopole-density corrections to the vertex gas populations, rho_gas_I = rho_limit_I - 0.4 rho_m and rho_gas_II = rho_limit_II - 0.6 rho_m, are calibrated from the linear dependence measured in the same simulations (inset of Fig. 4). These prefactors are then used to compare experimental vertex populations with the 'random vertex tiling containing a monopole density rho_m'. This is partially circular: the benchmark is adjusted using information from the very system it is meant to explain. Moreover, the prefactors 0.4 and 0.6 are not derived from any constraint or first-principles argument (one might naively expect 0.38 and 0.62 from the zero-monopole fractions). Please clarify the origin of these values, provide their uncertainty, and test whether the conclusions in Table I are robust within that uncertainty.
[Fig. 5 and Section IV] The experimental comparisons in Figs. 5(a)-(c) are visual and lack error bars on the experimental cluster-size distributions and on the shuffling-algorithm predictions. The conclusion that lattices 1-10 are compatible with the Coulomb-phase vertex gas while lattices 11-12 are not would be strengthened by a quantitative discrepancy measure, especially for the large-cluster tail where finite-size effects and counting statistics are most severe. I also note that in Fig. 5 the comparison is made with 'predictions from the shuffling algorithm' rather than with the random-tiling model; the text should clarify whether the shuffling algorithm with injected monopoles is being used, and if so, how the injected monopole configuration is sampled.

minor comments (5)

[Introduction] There are several typographical errors, including 'contraint' (should be 'constraint') and 'exemple' (should be 'example'). These should be corrected.
[Fig. 2(a) caption] The caption says 'after n^2 loops and strings have been flipped, n^2 being the number of vertices in the lattice.' Clarify whether the horizontal axis is the number of flips in units of n^2 or the raw number, and define n consistently (the text later uses n x n vertices, which is fine).
[Section III, paragraph on cluster size distributions] The description of the power-law and the 'ad hoc function of s, for instance an exponential' is vague. If the paper claims a specific analytical form from percolation theory, it should give the expected exponents and the fitted range of s; otherwise, the statement is not testable.
[Table I, footnote [38]] The phrase 'monopole are counted as 0' should be 'monopoles are counted as 0'. More importantly, the text should explain why monopoles are excluded from the vertex correlator and whether this choice affects the comparison.
[Section IV, last paragraph] The sentence 'the monopole fractions is large' contains a subject-verb agreement error ('fractions is'). Additionally, the discussion of thermally active systems is interesting but would benefit from a quantitative statement about how far below the percolation threshold the type II fraction falls.

Circularity Check

2 steps flagged · score 4.0 of 10

The 37/63 random-tiling benchmark is calibrated to the simulation output, and the monopole 'gas' expectations in Table I are fitted from the same simulation, so parts of the agreement are tuned rather than predicted.

fitted input called prediction [Section III, discussion of Fig. 3(b)]
"In fact, differences between the two distributions become discernable for large cluster sizes: For example, as show in Fig. 3(b), the distribution of type I vertices in a 100×100 lattice slightly deviates from the 38%/62% random distribution, and replacing the initial 38%/62% fractions by 37%/63% gives a much better agreement. In other words, the local constraint present in the spin model translates into a change of about 1% in the vertex fractions used in the random tiling approach."

The 37/63 fraction used in the random-tiling benchmark is not derived from first principles or from percolation theory; it is chosen post hoc because it makes the random tiling's cluster-size distribution match the very shuffling-algorithm data the paper is explaining. The subsequent claim that 'a percolation problem on a square lattice accounts well for the vertex distribution in the square ice manifold' therefore rests on a composition parameter tuned to the target data. The mildness of the adjustment (1% away from the measured 38/62) gives the fit some content, but the agreement is still partly manufactured rather than a parameter-free prediction.
fitted input called prediction [Table I, footnote [37]]
"The two vertex populations expected in a random vertex tiling containing a monopole density ρm are calculated such that ρgas I = ρlimit I − 0.4ρm and ρgas II = ρlimit II − 0.6ρm, where ρlimit is the limit value of the associated vertex fraction for a 20 × 20 or 30 × 30 lattice size. The 0.4 and 0.6 prefactors are deduced from the linear dependence of ρI against ρm (see inset of Fig. 4)."

The 'expected' gas populations in Table I are labeled as coming from a random vertex tiling, but the 0.4 and 0.6 prefactors are fitted to the linear dependence observed in the shuffling-algorithm simulations with injected monopoles. These fitted values are then used as the benchmark against which experimental vertex populations are judged to be 'systematically lower/larger' than expected. This is a transparent calibration of the model to itself, but it is not an independent random-tiling or percolation prediction; the experimental comparison therefore tests agreement with a simulation-fitted line rather than with an external theoretical prediction.

full rationale

The paper's broad structure is not circular: the constraint-free random tiling is an independent reference model, and comparing the square ice manifold's vertex correlations and cluster-size distributions to that reference is a legitimate way to test the 'ideal vertex gas' picture. The central claim that the manifold is well approximated by a random tiling retains real content because the measured 38/62 population is an independent input and the cluster-size distribution shape is not fully determined by the density alone. However, two load-bearing numerical inputs are fitted to the simulation output rather than predicted. First, the 37/63 fraction is obtained by adjusting the measured 38/62 specifically to improve agreement with the cluster-size distribution that the paper then says percolation theory accounts for; this is a one-parameter fit, not a parameter-free confirmation. Second, the 0.4/0.6 monopole prefactors used to define 'expected' gas populations in Table I are deduced from the same shuffling-algorithm simulations, so the experimental comparison is calibrated to the model rather than to an independent random-tiling calculation. The unproven ergodicity of the loop-flip sampler is a correctness/robustness concern, not a circularity, and the self-citations to prior experimental work are not load-bearing for the main numerical result. Overall, the derivation chain contains partial fitted-input circularity, but the central claim is not fully reducible to its inputs, warranting a score of 4.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim rests on two fitted parameters (the vertex fractions and the monopole correction factors) and on the unproven representativeness of the loop-flip sampler. No new physical entities are introduced.

free parameters (2)

type I vertex fraction rho_I = 0.38 (large lattices), adjusted to 0.37 for cluster-size comparison
Measured from loop-flip simulations; no analytic derivation is given. It is the only input parameter of the random tiling baseline and the percolation comparison.
monopole correction prefactors = 0.4 and 0.6
Used in the footnote of Table I to convert a nominal monopole density into expected gas populations rho_gas_I and rho_gas_II; fitted to the linear dependence shown in the inset of Fig. 4.

assumptions (3)

domain assumption Loop and string flips sample the square ice ground-state manifold uniformly (or at least representatively)
Invoked in Section III, first paragraph; convergence is checked numerically but no ergodicity or stationary distribution proof is offered.
domain assumption All six ice-rule vertices are equally weighted in the ground-state manifold
The shuffling algorithm generates configurations from the equal-weight set of ice-rule satisfying tilings; this is the standard six-vertex model assumption but is not derived.
standard math Binary type I/type II vertex map with edge-connected clusters is equivalent to site percolation on a square lattice
Used throughout Section III; the random tiling with independent vertex types and fixed fraction is exactly square-lattice site percolation, whose threshold and cluster distributions are external known results.

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Cite this review

Pith. "Pith review of Square ice Coulomb phase as a percolated vertex lattice." pith.science (2026). https://pith.science/paper/U6BU6FG2

@misc{pith2026241116533,
  author       = {Pith},
  title        = {Pith review of: Square ice Coulomb phase as a percolated vertex lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U6BU6FG2}},
  note         = {Machine review of arXiv:2411.16533}
}

read the original abstract

The square ice is a canonical example of a Coulomb phase in two dimensions: Its ground state is extensively degenerate and satisfies a local constraint on the spin arrangement (the so-called ice rule). In this paper, we use a loop flip algorithm to explore the properties of this ground state that we analyze not in terms of a spin texture, but rather in terms of a spatial distribution of ice-rule satisfying vertices. More specifically, we determine for various lattice sizes the average vertex populations characterizing the ice manifold, the pairwise vertex correlations, and the size distribution of vertex clusters. Comparing these results to those obtained from random, constraint-free vertex tilings, the square ice manifold is found to resemble an almost ideal vertex gas, and the cluster size distribution of ice-rule satisfying vertices is well approximated by percolation theory. Remarkably, this description remains reasonably accurate when monopoles are present in a dilute amount, allowing a direct comparison with experiments. Revising former experimental results on two artificial square ice systems, we illustrate the interest of our approach to spot the presence of a Coulomb phase from a vertex analysis.

Figures

Figures reproduced from arXiv: 2411.16533 by the authors.

**Figure 2.** FIG. 2. (a) Fraction [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Typical vertex map obtained from the shuf [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Statistics of the type I cluster size distributions [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5. Statistics of the type I cluster size distributions [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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Reference graph

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B. Ding, C. Li, M. Zhang, G. Lu, and F. Ji, Numer- ical analysis of percolation cluster size distribution in two-dimensional and three-dimensional lattices. Eur. J. B 87, 179 (2014)

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The 0.4 and 0.6 prefactors are deduced from the linear dependence of ρI against ρm (see inset of Fig

The two vertex populations expected in a random vertex tiling containing a monopole density ρm are calculated such that ρgas I = ρlimit I − 0.4ρm and ρgas II = ρlimit II − 0.6ρm, where ρlimit is the limit value of the associated vertex fraction for a 20 × 20 or 30 × 30 lattice...
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In the image analysis of the two experimental situ- ations, monopole are counted as 0 when calculating the vertex correlator. 8

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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