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arxiv: 1907.04748 · v1 · pith:YWJ52GFMnew · submitted 2019-07-10 · ❄️ cond-mat.stat-mech · math-ph· math.MP

Spin-spin correlations in central rows of Ising models with holes

Helen Au-Yang , Jacques H.H. Perk This is my paper

Pith reviewed 2026-05-24 23:32 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MP
keywords Ising modelspin-spin correlationsToeplitz determinantsgenerating functionscritical behaviorstrips with holessquare-root singularities
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The pith

Spin-spin correlations in central rows of Ising strips with holes are Toeplitz determinants whose generating functions become square roots of degree-(m+1) polynomials in the infinite vertical limit.

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

The paper studies spin-spin correlations along central rows in a geometry of alternating horizontal Ising strips of width m and layers of finite strings of length n separated by N=1. It establishes that these correlations are given exactly by Toeplitz determinants. The associated generating functions are ratios of two polynomials; when the vertical extent of the system is taken to infinity the ratios become square roots of polynomials whose degree is set by the strip width m. Near the critical temperature the decay of the correlations follows the power laws of the ordinary two-dimensional Ising model, while farther from criticality the dependence on m and n can differ.

Core claim

The spin-spin correlations in the central rows of each strip and of each strings layer can be written as Toeplitz determinants. Their generating functions are ratios of two polynomials; in the limit of infinite vertical size these become square roots of polynomials of degree m+1. The asymptotic behaviors near the critical temperature are two-dimensional Ising-like, although in regions not very close to criticality the behavior may be different for different m and n.

What carries the argument

Toeplitz determinant (a determinant whose matrix entries are constant along each diagonal) whose generating function is a ratio of polynomials that reduces to the square root of a degree-(m+1) polynomial when the vertical size diverges.

If this is right

  • The similarity of specific heats for different N reported in earlier work follows from the structure of these central-row correlations.
  • The leading critical exponents remain those of the two-dimensional Ising model regardless of the strip width m.
  • Away from criticality the correlation length and amplitude depend on the particular integers m and n.
  • The same Toeplitz representation applies to the central row inside each layer of strings.

Where Pith is reading between the lines

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

  • The polynomial degree being exactly m+1 suggests that the effective number of degrees of freedom that survive the vertical limit is fixed by the strip width alone.
  • The rational-to-square-root transition may supply a practical route to closed-form expressions for other defect geometries that admit a transfer-matrix description.
  • Direct comparison of the predicted Toeplitz determinants against transfer-matrix numerics on cylinders of increasing height would test the infinite-size reduction step.

Load-bearing premise

The lattice consists of alternating infinite horizontal Ising strips of width m separated by N=1 from layers of strings of length n, and the analysis requires taking the infinite vertical size limit to obtain the square-root generating functions.

What would settle it

Compute the spin-spin correlation directly on a large but finite realization of the strip-and-string geometry and check whether it equals the numerical value of the corresponding Toeplitz determinant.

Figures

Figures reproduced from arXiv: 1907.04748 by Helen Au-Yang, Jacques H.H. Perk.

Figure 1
Figure 1. Figure 1: Part of special layered Ising model for case with strip width m = 4 and string length n = 4. The full model has horizontal size ¯p and vertical size p(m + n). Pair correlation is calculated both for mid-strip rows and mid-string rows. To be specific, we consider the Ising model consisting of a periodic array of p horizontal strips of width m and length ¯p, which are connected by ¯p vertical strings of leng… view at source ↗
Figure 2
Figure 2. Figure 2: (Color online) The root γj+1 in (164) is plotted for m = 2j = 2, 4, 6, 8, 12, 16,, as a function of z. For z = 0 (T = ∞), we have γj+1 = 0, and for z = 1 (T = 0), we find γj+1 = 1. As shown in the figure, as z increases from zero, γj+1 increases and approaches 1. In fact, γj+1 ≈ 1 for a large region of z near z = 1. This means that at sufficient low temperature, the exponential decay term is almost irrelev… view at source ↗
read the original abstract

In our previous works on infinite horizontal Ising strips of width $m$ alternating with layers of strings of Ising chains of length $n$, we found the surprising result that the specific heats are not much different for different values of $N$, the separation of the strings. For this reason, we study here for $N=1$ the spin-spin correlation in the central row of each strip, and also the central row of a strings layer. We show that these can be written as a Toeplitz determinants. Their generating functions are ratios of two polynomials, which in the limit of infinite vertical size become square roots of polynomials whose degrees are $m+1$ where $m$ is the size of the strips. We find the asymptotic behaviors near the critical temperature to be two-dimensional Ising-like. But in regions not very close to criticality the behavior may be different for different $m$ and $n$. Finally, in the appendix we shall present results for generating functions in more general models.

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

0 major / 3 minor

Summary. The manuscript studies spin-spin correlations in the central rows of infinite horizontal Ising strips of width m alternating with layers of strings of length n at separation N=1. It claims these correlations can be expressed as Toeplitz determinants whose generating functions are ratios of two polynomials; in the infinite-vertical-size limit these become square roots of polynomials of degree m+1. From this structure the near-critical asymptotics are shown to match the two-dimensional Ising form, while behavior farther from criticality may depend on m and n. An appendix gives generating-function results for more general models.

Significance. If the explicit constructions hold, the work supplies concrete Toeplitz representations and the required square-root singularity structure for a geometry with holes, extending the authors' earlier specific-heat calculations. The reduction to degree-(m+1) polynomials and the appendix on general models are strengths that facilitate further analytic or numerical checks.

minor comments (3)
  1. [Abstract] Abstract, line 3: 'written as a Toeplitz determinants' contains a grammatical mismatch; correct to 'written as Toeplitz determinants' and verify consistency in §2 and §3.
  2. [§3] The transition from finite to infinite vertical size (leading to the square-root form) is central to the asymptotics claim; a brief statement of the limiting procedure in the main text before the appendix would improve readability.
  3. [§2] Notation for the separation parameter N is introduced in the abstract but its explicit value N=1 is used without a dedicated sentence in the model definition; add one clarifying sentence in §2.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and for recommending minor revision. The report provides a concise summary of the manuscript but lists no specific major comments. We therefore have no individual points to rebut or revise at this stage, though we remain ready to address any minor editorial suggestions.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper explicitly constructs the spin-spin correlations for the N=1 strip-with-holes geometry as Toeplitz determinants from the underlying Ising transfer-matrix or Pfaffian definitions. The generating functions as ratios of polynomials, their infinite-vertical-size limit to square-root forms of degree m+1, and the resulting near-Tc asymptotics are obtained directly from those determinant expressions without reduction to fitted parameters, self-definitions, or load-bearing self-citations. Prior works are cited only for context on specific heats; the central claims here are shown by direct calculation for the stated geometry and are independent of those citations. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of the Ising model and Toeplitz determinants already established in the literature; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • standard math Standard algebraic properties of Toeplitz determinants and their generating functions for Ising correlations
    Invoked to express the central-row correlations as determinants (abstract).

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 2 internal anchors

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