arxiv: 2604.06011 · v1 · submitted 2026-04-07 · 🧮 math-ph · cond-mat.stat-mech· hep-th· math.MP

Recognition: 2 theorem links

· Lean Theorem

Analyticity, asymptotics and natural boundary for a one-point function of the finite-volume critical Ising chain

Yizhuang Liu

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:22 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.stat-mechhep-thmath.MP

keywords critical Ising chainone-point functionnatural boundaryBorel resummationanalytic continuationfinite volumedivisor sumslarge-N asymptotics

0 comments

The pith

The one-point function of the spin operator in the critical periodic Ising chain, when continued in system length through Borel resummation of its large-N expansion, has a natural boundary along the negative real axis.

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

The paper examines the finite-volume expectation value of the spin operator between the Z_2-even and odd ground states in the critical periodic Ising chain. It constructs an analytic continuation of this one-point function to complex system sizes N by Borel-resumming the large-N asymptotic series. The resulting function cannot be extended analytically past the negative real axis because singularities accumulate densely there. After an exponential change of variables the local singular behavior reproduces that of a Lambert-type series built from the sum of squares of odd divisors near the unit circle. The same divisor sum fixes the magnitudes of the discontinuities across the Borel cuts in the original asymptotic expansion.

Core claim

The finite-volume one-point function between the Z_2-even and odd ground states, when Borel-resummed from its large-N expansion, defines an analytic function of complex system length N that possesses a natural boundary of analyticity along the negative real axis. Near this boundary, after the exponential map, the singular structure coincides with that of the Lambert series for the odd-divisor-squared sum; the strengths of the Borel discontinuities are likewise governed by this divisor sum. The paper also supplies the complete large-N asymptotic expansion of the leg function that appears in the finite-volume spin-operator form factor.

What carries the argument

The Borel-resummed analytic continuation of the even-odd one-point function in the complex variable N, which carries the argument by exposing the dense singularities on the negative real axis.

If this is right

The one-point function cannot be analytically continued across the negative real axis because singularities accumulate there.
The magnitudes of the jumps across Borel cuts in the large-N expansion are fixed by the odd-divisor-squared sum.
The leg function of the finite-volume form factor admits a complete asymptotic series to all orders in 1/N.
The singular behavior after the exponential map is identical to that of a known Lambert series near the unit circle.

Where Pith is reading between the lines

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

The same Borel-resummation technique could be applied to other finite-volume observables in the Ising chain to search for analogous natural boundaries.
The appearance of divisor sums may point to a broader link between resurgence in lattice models and analytic number theory.
Direct numerical checks for complex N near the negative axis would provide an independent test of the predicted singularity structure.

Load-bearing premise

The Borel resummation of the large-N series supplies the correct analytic continuation of the one-point function to complex values of the chain length.

What would settle it

Numerical evaluation of the one-point function at points N = -M + iε for large positive M and small positive ε, checking whether the values reproduce the singular growth predicted by the Lambert series of the odd-divisor sum.

read the original abstract

This note reports the following observation: the finite-volume expectation value of the spin operator (the one-point function) between the $\mathbb{Z}_2$-even and odd ground states in the critical periodic Ising chain, when continued as a complex-analytic function of the system length $N$ through the Borel resummation of its large-$N$ expansion, has a natural boundary of analyticity along the negative real axis. The singular behavior near the negative real axis, after an exponential map, is the same as that of a Lambert-type series for the odd-divisor-squared sum near the unit circle $|z|=1$. The same divisor sum also governs the strengths of the Borel discontinuities of the one-point function's factorially-divergent large-$N$ asymptotics. We also report the all-order large-$N$ asymptotics of the leg function for the finite-volume spin-operator form factor, and the similarities to certain known quantities in the literature.

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 note identifies a natural boundary on the negative axis for the Borel-resummed one-point function in the critical periodic Ising chain and matches it to the singularities of an odd-divisor Lambert series.

read the letter

The main thing to know is that the finite-volume spin expectation value between the Z2 even and odd sectors, when extended off the positive integers by Borel resumming its large-N expansion, acquires a natural boundary along the negative real axis whose local behavior matches that of a Lambert-type series built from the sum of odd divisors squared. The same series controls the size of the Borel discontinuities in the asymptotic expansion itself. The paper also supplies the complete large-N series for the one-point function and for the leg function of the associated form factor, plus some comparisons to existing expressions in the literature. That identification and the explicit all-order asymptotics are the concrete new pieces. The calculations appear careful and the link to the divisor sum is cleanly stated. The soft spot is the step that treats the Borel sum as the actual analytic continuation of the one-point function for complex N. The expansion is derived for large positive integer N, and while the discontinuities line up with the divisor series, there is no independent closed-form expression shown for non-real N that would rule out extra exponentially small terms or a different Stokes structure. If the manuscript contains a direct check against the transfer-matrix formula at some complex points, that would close the gap; otherwise the continuation remains an assumption. This is a specialized note aimed at people who work on finite-size effects and resummation in integrable chains. A reader already following the Ising model or similar models will find the asymptotics and the number-theoretic connection useful. It is worth sending to a referee because the observation is specific, the techniques are standard, and the potential gap is narrow enough that a careful review can settle it.

Referee Report

2 major / 2 minor

Summary. The manuscript reports that the finite-volume one-point function of the spin operator (between Z2-even and odd ground states) in the critical periodic Ising chain, when analytically continued in the system size N via Borel resummation of its large-N asymptotic expansion, exhibits a natural boundary along the negative real axis. After an exponential map, the singular behavior matches that of a Lambert series involving the sum of squares of odd divisors; the same divisor sum controls the discontinuities of the Borel transform. The paper also supplies the all-order large-N asymptotics of the associated leg function for the spin-operator form factor.

Significance. If the Borel-resummed continuation is valid, the result supplies a concrete, number-theoretic example of a natural boundary arising from the analytic structure of a finite-volume observable in an integrable quantum chain. The explicit all-order asymptotic series for the form-factor leg function is a useful technical contribution that may be reusable in related models.

major comments (2)

[discussion following the definition of the Borel-resummed continuation (near Eq. (3.5))] The central claim requires that the Borel sum of the large-N series coincides with the one-point function (defined via transfer matrix or mode products) for complex N. The manuscript defines the continuation through Borel resummation but does not supply an independent closed-form expression valid off the positive integers that could be used to verify the absence of additional Stokes jumps or exponentially small corrections; this identification is load-bearing for locating the natural boundary.
[§4, around the statement that 'the same divisor sum governs both'] The exponential map that relates the Borel discontinuities to the singularities of the Lambert series for the odd-divisor-squared sum is asserted to be identical in strength and location. The explicit computation of the discontinuity from the asymptotic coefficients (via the standard Borel transform) and its matching to the divisor sum needs to be shown in detail, as any mismatch would alter the claimed natural boundary.

minor comments (2)

[§2] The notation for the one-point function (even/odd sector) and the leg function should be introduced with a single equation that makes the Z2 parity explicit.
[§3] A brief remark on the choice of lateral Borel resummation (upper or lower half-plane) would clarify the sector in which the natural boundary is approached.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We respond to each point below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: The central claim requires that the Borel sum of the large-N series coincides with the one-point function (defined via transfer matrix or mode products) for complex N. The manuscript defines the continuation through Borel resummation but does not supply an independent closed-form expression valid off the positive integers that could be used to verify the absence of additional Stokes jumps or exponentially small corrections; this identification is load-bearing for locating the natural boundary.

Authors: We acknowledge that the Borel-resummed function is our proposed analytic continuation of the one-point function to complex N, and that no independent closed-form expression (e.g., via transfer-matrix products) is known for non-integer N. The natural boundary is located from the singularity structure of the Borel transform, which is extracted directly from the asymptotic coefficients and is insensitive to possible exponentially small corrections that would not alter the dense set of singularities on the negative axis. We will revise the discussion near Eq. (3.5) to clarify this point and to state explicitly that the identification rests on the Borel sum as the defining continuation. revision: partial
Referee: The exponential map that relates the Borel discontinuities to the singularities of the Lambert series for the odd-divisor-squared sum is asserted to be identical in strength and location. The explicit computation of the discontinuity from the asymptotic coefficients (via the standard Borel transform) and its matching to the divisor sum needs to be shown in detail, as any mismatch would alter the claimed natural boundary.

Authors: We agree that the matching between the Borel discontinuities and the Lambert-series singularities should be exhibited explicitly. The manuscript states the equivalence but does not display the intermediate steps of computing the discontinuity from the all-order asymptotic coefficients. In the revised version we will add a detailed calculation (new subsection in §4) that extracts the discontinuity via the standard Borel transform, applies the exponential map, and verifies that both the locations and the strengths coincide with those of the odd-divisor-squared Lambert series. revision: yes

standing simulated objections not resolved

Absence of an independent closed-form expression for the one-point function at non-integer N, which precludes direct verification of the Borel-sum identification beyond the asymptotic data.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on explicit asymptotics and standard resummation

full rationale

The paper computes the large-N asymptotic expansion of the one-point function directly from the transfer-matrix or mode-product representation of the critical Ising chain (standard for this model). It then applies Borel resummation to obtain an analytic continuation in N and compares the resulting singularities to the known Lambert series for the odd-divisor-squared sum. No step defines a quantity in terms of itself, renames a fitted parameter as a prediction, or imports a uniqueness theorem from self-citation that forces the result. The Borel-resummation step is an assumption about the continuation (as noted in the skeptic headline), but it is not circular by construction; it is an external analytic tool applied to an independently derived series. The identification of the natural boundary follows from the explicit form of the Borel discontinuities and the divisor sum, both of which are computed rather than presupposed. The derivation is therefore self-contained against the model's exact finite-N expressions and known number-theoretic series.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper's claims rest on standard assumptions in the theory of asymptotic expansions and Borel summation in mathematical physics. No free parameters or new entities are introduced based on the abstract.

axioms (2)

domain assumption The large-N expansion of the one-point function admits a Borel resummation that defines its analytic continuation to complex N
Central to the claim of natural boundary along negative real axis
domain assumption The singular behavior matches that of the Lambert series for the odd-divisor sum
Used to identify the nature of the boundary

pith-pipeline@v0.9.0 · 5470 in / 1463 out tokens · 103261 ms · 2026-05-10T18:22:48.823352+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

v(1/N) = -∫ dt tanh²(t/(4N)) / (2t (e^t +1)) (0.2); Mellin-Barnes (1.1); singular sum S(z) = 4 ∑ n σ_o^{-2}(n) z^n (1.28); Borel transform B[v](t) with poles controlled by σ_o^{-2}(l) (3.10)-(3.11)
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

factorial large-N asymptotics (3.1), Nevanlinna-Sokal Borel summability (2.4), natural boundary from divisor irregularities (1.13)-(1.23)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 21 canonical work pages · 3 internal anchors

[1]

Nickel, Journal of Physics A: Mathematical and General32, 3889 (1999)

B. Nickel, Journal of Physics A: Mathematical and General32, 3889 (1999)

1999
[2]

W. P. Orrick, B. Nickel, A. J. Guttmann, and J. H. H. Perk, Journal of Statistical Physics 102, 795–841 (2001)

2001
[3]

Liu, (2026), arXiv:2601.00751 [hep-th]

Y. Liu, (2026), arXiv:2601.00751 [hep-th]

work page arXiv 2026
[4]

B. M. McCoy, (2000), arXiv:cond-mat/0012193

work page arXiv 2000
[5]

Pasquetti and R

S. Pasquetti and R. Schiappa, Annales Henri Poincare11, 351 (2010), arXiv:0907.4082 [hep-th]

work page arXiv 2010
[6]

Rella, Commun

C. Rella, Commun. Num. Theor. Phys.17, 709 (2023), arXiv:2212.10606 [hep-th]

work page arXiv 2023
[7]

Gu, A.-K

J. Gu, A.-K. Kashani-Poor, A. Klemm, and M. Marino, SciPost Phys.16, 079 (2024), arXiv:2305.19916 [hep-th]

work page arXiv 2024
[8]

Periwal, Phys

V. Periwal, Phys. Rev. Lett.71, 1295 (1993), arXiv:hep-th/9305115

work page arXiv 1993
[9]

Tierz, Mod

M. Tierz, Mod. Phys. Lett. A19, 1365 (2004), arXiv:hep-th/0212128

work page arXiv 2004
[10]

Marino, Rev

M. Marino, Rev. Mod. Phys.77, 675 (2005), arXiv:hep-th/0406005

work page arXiv 2005
[11]

A catalog of interesting and useful lambert series identities,

M. D. Schmidt, “A catalog of interesting and useful lambert series identities,” (2020), arXiv:2004.02976 [math.NT]

work page arXiv 2020
[12]

E. M. Stein and R. Shakarchi (2003)

2003
[13]

Gangl, M

H. Gangl, M. Kaneko, and D. Zagier,Double Zeta Values and Modular Forms, MPI / Max-Planck-Institut f¨ ur Mathematik, Bonn (MPI, 2005)

2005
[14]

Costin,Asymptotics and Borel Summability, Monographs and Surveys in Pure and Applied Mathematics (CRC Press, 2008)

O. Costin,Asymptotics and Borel Summability, Monographs and Surveys in Pure and Applied Mathematics (CRC Press, 2008)

2008
[15]

Fonseca and A

P. Fonseca and A. Zamolodchikov, (2001), arXiv:hep-th/0112167

work page internal anchor Pith review arXiv 2001
[16]

Katsurada, Acta Arithmetica107.3(2003)

M. Katsurada, Acta Arithmetica107.3(2003)

2003
[17]

Gu and M

J. Gu and M. Marino, SciPost Phys.12, 058 (2022), arXiv:2104.07437 [hep-th]

work page arXiv 2022
[18]

Garoufalidis, J

S. Garoufalidis, J. Gu, and M. Marino, Res. Math. Sci.10, 29 (2023), arXiv:2012.00062 [math.GT]

work page arXiv 2023
[19]

Marino,Les Houches lectures on non-perturbative topological strings,2411.16211

M. Marino, SciPost Phys. Lect. Notes112, 1 (2026), arXiv:2411.16211 [hep-th] . – 23 –

work page arXiv 2026
[20]

Marino, (2026), https://indico.ictp.it/event/11135/session/5/contribution/9/material/0/1.pdf

M. Marino, (2026), https://indico.ictp.it/event/11135/session/5/contribution/9/material/0/1.pdf

2026
[21]

Marino and G

M. Marino and G. W. Moore, Nucl. Phys. B543, 592 (1999), arXiv:hep-th/9808131

work page arXiv 1999
[22]

Hodge integrals and Gromov–Witten theory

C. Faber and R. Pandharipande, “Hodge integrals and gromov-witten theory,” (1998), arXiv:math/9810173 [math.AG]

work page arXiv 1998
[23]

Marino (2004) arXiv:hep-th/0410165

M. Marino (2004) arXiv:hep-th/0410165

work page internal anchor Pith review arXiv 2004
[24]

Witten, Commun

E. Witten, Commun. Math. Phys.121, 351 (1989)

1989
[25]

Marino, Commun

M. Marino, Commun. Math. Phys.253, 25 (2004), arXiv:hep-th/0207096

work page arXiv 2004
[26]

Gopakumar and C

R. Gopakumar and C. Vafa, Adv. Theor. Math. Phys.3, 1415 (1999), arXiv:hep-th/9811131

work page arXiv 1999
[27]

Banerjee and B

S. Banerjee and B. Wilkerson, International Journal of Number Theory13, 2097–2113 (2017)

2097
[28]

Costin, G

O. Costin, G. V. Dunne, A. Gruen, and S. Gukov, (2023), arXiv:2310.12317 [hep-th]

work page arXiv 2023
[29]

Orientation Reversal and the Chern-Simons Natural Boundary

G. Adams, O. Costin, G. V. Dunne, S. Gukov, and O. ¨Oner, JHEP08, 154 (2025), arXiv:2505.14441 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[30]

Adams and G

G. Adams and G. V. Dunne, (2026), arXiv:2603.04619 [hep-th] . – 24 –

work page arXiv 2026