arxiv: 2605.10017 · v1 · submitted 2026-05-11 · ❄️ cond-mat.stat-mech · cond-mat.mes-hall· cond-mat.mtrl-sci· math-ph· math.MP

Recognition: no theorem link

Families of planar lattices with arbitrarily high T_{rm c} for the ferromagnetic Ising model

Davidson Noby Joseph , Connor M. Walsh , Igor Boettcher

Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:24 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.mes-hallcond-mat.mtrl-scimath-phmath.MP

keywords planar latticesIsing modelcritical temperatureferromagnetictessellationsApollonian latticescoordination numberphase transitions

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The pith

Certain families of planar lattices achieve arbitrarily high critical temperatures for the ferromagnetic Ising model.

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

The paper constructs families of periodic lattices on the plane by repeatedly triangulating existing faces, starting from the triangular lattice. This process increases the maximum number of neighbors any site can have, which in turn raises the critical temperature Tc of the Ising ferromagnet. The authors derive that Tc grows logarithmically with this maximum coordination number qmax, specifically with prefactor 2 over ln 2. They introduce a conjectured optimal bound Tc star of qmax, which these Apollonian lattices appear to meet asymptotically. If correct, this shows that Tc in two-dimensional Ising systems can be made arbitrarily large by suitable choice of lattice geometry.

Core claim

We construct families of periodic tessellations of the plane with arbitrarily high critical temperature, Tc, for the classical ferromagnetic Ising model. Motivated by exact bounds that tie large Tc to large maximal coordination number qmax, we create the lattices through iterative triangulation and obtain explicit expressions for their Tc. Tc scales asymptotically as Tc/J ~ (2/ln 2) ln qmax. We conjecture that the function Tc*(qmax) is optimal for all periodic tessellations of the plane, and show that the family of Apollonian lattices saturates this bound.

What carries the argument

Iterative triangulation of the triangular lattice to generate Apollonian lattices with arbitrarily large maximal coordination number qmax, from which explicit Tc values are derived.

If this is right

The critical temperature can be increased without limit by constructing lattices with sufficiently high qmax.
A universal scaling law Tc/J ∼ (2 / ln 2) ln qmax holds for the constructed families.
Apollonian lattices achieve the conjectured maximal Tc for any given qmax among all periodic planar lattices.
Explicit formulas for Tc in these lattices provide concrete examples for studying high-temperature ferromagnetism in two dimensions.

Where Pith is reading between the lines

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

Realizing these high-qmax lattices in physical systems such as topoelectric circuits could allow Ising models to remain ordered at higher temperatures than on standard lattices.
The logarithmic scaling may extend to other two-dimensional spin models or to questions of optimality in non-periodic planar graphs.
Lattice designs that maximize Tc for given qmax could improve thermal robustness in Ising-based optimization hardware.

Load-bearing premise

The exact bounds that require large maximal coordination number qmax to achieve large Tc apply to these iteratively triangulated periodic lattices.

What would settle it

An independent numerical computation of the critical temperature for one of the constructed lattices with large but finite qmax that deviates from the derived explicit expression or exceeds the conjectured Tc*(qmax).

Figures

Figures reproduced from arXiv: 2605.10017 by Connor M. Walsh, Davidson Noby Joseph, Igor Boettcher.

**Figure 1.** Figure 1: FIG. 1. The procedure of iterative triangulation is applied to a single triangle and to two example lattices. The top panel shows [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Plots of lattice critical temperatures [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Examples of the iterative triangulation procedure [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. The decimation of the spin [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The twelve base lattices with varying maximal coordination number [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 1.** Figure 1: Lattices constructed with n ≥ 4 iterations are denoted by Apollonian–qmax and Laves-CaVO–qmax, respectively. For the Apollonian lattices, qmax = 2n 6, whereas for the LavesCaVO family we have qmax = 2n 8. For large n, the computed critical temperatures are observed to grow linearly in n, as predicted by Eq. (10). VII. ASYMPTOTICS OF ITERATIVE TRIANGULATION In this section, we establish the asymptotic equa… view at source ↗

**Figure 6.** Figure 6: FIG. 6. The star-triangle transformation allows for the deci [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: If more precision is needed, one should implement Eq. (F8), truncating at a moderate value of n. For instance, choosing n = 30 gives results with a relative error less than 10−4 for all q ≤ 100. [1] B. M. McCoy and J.-M. Maillard, The importance of the Ising model, Prog. Theor. Phys. 127, 791 (2012). [2] M. E. Fisher, Simple Ising models still thrive!: A review of some recent progress, Phys. A: Stat. Mech… view at source ↗

read the original abstract

We construct families of periodic tessellations of the plane with arbitrarily high critical temperature, $T_{\rm c}$, for the classical ferromagnetic Ising model. Our approach is motivated by recently found exact bounds, which imply that large values of $T_{\rm c}$ require large values of the maximal coordination number of the lattice, $q_{\rm max}$. We create such lattices through iterative triangulation and derive explicit expressions for their $T_{\rm c}$. Furthermore, we show that $T_{\rm c}$ for these families scales asymptotically as $T_{\rm c}/J\sim A \ln q_{\rm max}$ with a universal prefactor $A=2/\ln 2$. We introduce a function $T_{\rm c}^*(q_{\rm max})$ that we conjecture to be optimal for all periodic tessellations of the plane. We show that the family of so-called Apollonian lattices, which are derived from the Triangular lattice through iterative triangulation, saturates this bound. The lattices discussed in this work are relevant for theoretical questions of optimality in network systems and may be realized experimentally in Coherent Ising Machines or topoelectric circuits in the future.

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.

Desk Editor's Note private letter to a colleague

The paper builds explicit planar lattice families with Ising Tc growing unbounded as (2/ln2) ln q_max and conjectures optimality for Apollonian ones, but the upper bound claim stays unproven.

read the letter

The main takeaway is that these authors construct families of periodic planar lattices where the ferromagnetic Ising Tc can be driven arbitrarily high by raising the maximum coordination number q_max, and they derive that Tc scales as (2/ln2) ln q_max for all their families with a universal prefactor. They start from the triangular lattice and use iterative triangulation to generate new periodic tessellations, then obtain explicit expressions for Tc on each family. The Apollonian lattices reach the top of their set and saturate the conjectured Tc*(q_max) function they propose as optimal across all periodic planar graphs. This adds concrete examples to earlier bounds that tied high Tc to high q_max, and the scaling result with its specific constant is a clear addition. The constructions are useful for testing connectivity effects on 2D phase transitions and could guide experimental realizations. The main limitation is that the optimality conjecture lacks a matching general upper bound or proof; it rests on saturation by one family, so it remains possible that other lattices exceed Tc* at fixed q_max. The scaling and explicit formulas stand independently of that conjecture. This work is for people studying Ising models on graphs, bounds on critical temperatures, or network design for optimization. A reader interested in exact or asymptotic results on lattices would get direct value from the families. It deserves peer review because the constructions and scaling are concrete enough to warrant referee time even if the conjecture needs strengthening.

Referee Report

1 major / 2 minor

Summary. The paper constructs families of periodic planar lattices via iterative triangulation, starting from lattices such as the triangular lattice, yielding arbitrarily large maximal coordination number q_max. Explicit expressions for the ferromagnetic Ising critical temperature Tc are derived for these families, from which the asymptotic scaling Tc/J ∼ (2/ln 2) ln q_max is obtained. A function Tc^*(q_max) is defined and conjectured to be the optimal (supremum) Tc achievable by any periodic planar tessellation at fixed q_max; the Apollonian subfamily is shown to saturate this conjectured bound.

Significance. If the explicit Tc expressions and scaling derivation are correct, the work supplies concrete, constructible examples of planar periodic lattices with unbounded Tc, directly illustrating the link between high q_max and high Tc implied by recent bounds. The universal prefactor and conjectured optimality function provide a quantitative benchmark for lattice design, with relevance to network optimization questions and possible experimental platforms such as coherent Ising machines or topoelectric circuits.

major comments (1)

The section introducing Tc^*(q_max): the optimality conjecture for all periodic tessellations rests on saturation by the Apollonian family and the absence of counterexamples within the constructed families, but no general upper-bound proof or exhaustive comparison to other known lattices is supplied; this makes the saturation claim conditional rather than definitive, though the existence of arbitrarily high-Tc families is independent of the conjecture.

minor comments (2)

The abstract and main text refer to 'explicit expressions' for Tc; the manuscript should clarify whether these are closed-form solutions or recursive relations that must be solved numerically for each member of the family.
The asymptotic analysis yielding the prefactor 2/ln 2 should include a brief statement of the large-q_max approximation steps and any error terms to allow independent verification of the scaling.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment and constructive comment. We address the point below and will revise the manuscript accordingly to clarify the conjectural status of the optimality claim.

read point-by-point responses

Referee: The section introducing Tc^*(q_max): the optimality conjecture for all periodic tessellations rests on saturation by the Apollonian family and the absence of counterexamples within the constructed families, but no general upper-bound proof or exhaustive comparison to other known lattices is supplied; this makes the saturation claim conditional rather than definitive, though the existence of arbitrarily high-Tc families is independent of the conjecture.

Authors: We agree with the referee that the optimality of Tc^*(q_max) is presented as a conjecture rather than a proven result. It is supported by the observation that the Apollonian subfamily saturates the proposed bound while other constructed families do not exceed it, but we supply neither a general upper-bound proof for arbitrary periodic planar tessellations nor an exhaustive comparison against all known lattices. Such a proof remains an open question. As the referee notes, the core results—the explicit constructions of families with arbitrarily high Tc and the asymptotic scaling Tc/J ∼ (2/ln 2) ln q_max—are independent of this conjecture. In the revised version we will rephrase the relevant section to state more explicitly that Tc^*(q_max) is conjectured to be optimal, and we will add a clarifying sentence underscoring the independence of the main findings from the conjecture. revision: yes

Circularity Check

0 steps flagged

No significant circularity: explicit lattice constructions and derived scaling are independent of the labeled conjecture.

full rationale

The paper constructs families of periodic planar lattices via iterative triangulation starting from known lattices such as the triangular lattice, then derives explicit (closed-form or recursive) expressions for the Ising critical temperature Tc on these lattices. The asymptotic scaling Tc/J ∼ (2/ln 2) ln q_max is obtained by direct analysis of those expressions as q_max grows, without parameter fitting or redefinition of inputs as outputs. The function Tc*(q_max) is introduced explicitly as a conjecture for the supremum over all periodic tessellations and is not used to derive the scaling or the existence of high-Tc families; the Apollonian saturation is shown within the constructed family. No self-definitional loop, fitted-input prediction, or load-bearing self-citation chain appears in the central derivation chain. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on prior exact bounds for motivation and the new iterative triangulation construction; no free parameters are fitted as the prefactor is stated to be universal and derived.

axioms (1)

domain assumption Recently found exact bounds imply that large Tc requires large q_max
Explicitly stated as the motivation for the construction approach in the abstract.

pith-pipeline@v0.9.0 · 5529 in / 1428 out tokens · 116495 ms · 2026-05-12T03:24:18.568463+00:00 · methodology

discussion (0)

Reference graph

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The Laves-Star lattice in Fig. 1 is constructed from the Triangular lattice through iterative triangulation, and we indeed confirm for the Laves-Star lattice that T ′ c J = 1 artanh g(tc) = 5.007.(7) The next member in the family, obtained from the Laves-Star lattice through iterative triangulation, is the Compass-Rose lattice from Fig. 1, for which we co...

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1 gn(t∆c ) + 2 ln 2 ln q q∆max −2 ln 1 + 1 nln 2 ln q q∆max #−1 .(F7) Applyingh n on both sides gives t∗ c(q) = lim n→∞ hn

Rapidly convergent expression We first show that t∗ c(q) = lim n→∞ hn " 1 gn(tΛc ) + 2 ln 2 ln q 6 −2 ln 1 + 1 nln 2 ln q 6 #−1! ,(F2) which converges much faster innthan the expression in Eq. (93) and thus allows for efficient numerical compu- tation ofT ∗ c . First, we outline a proof of the equivalence of Eqs. (F2) and (93), and then show why the repre...

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T aylor Polynomial approximation We now present numerical values for the first six co- efficients of the Taylor series expansion ofT ∗ c (q) about q= 15. By expanding aboutq= 15, we obtain an ex- pression that gives good accuracy for 6≤q≤24. We write T ∗ c (q) J = ∞X k=0 ak(q−15) k.(F24) 4 5 6 7 T ∗ c /J Exact Taylor (to order 4) Taylor (to order 6) 10 15...

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