Nonlocal behaviors of spin correlations in the Haldane-Shastry model

Maowen Tang; Yan He

REVIEW 3 major objections 4 minor 20 references

Nonlocal behaviors of spin correlations in the Haldane-Shastry model

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The Haldane-Shastry model's spin correlations split into a local continuum part and nonlocal factors, and the ring-frustration factor is exactly $\cos(\pi\alpha)$, matching the Heisenberg model.

desk verdict Exact HS correlations confirm the nonlocal factor R(alpha)=cos(pi alpha), but the odd-N sign must be handled via an envelope definition before the claim is rigorous. read the letter →

arxiv 1908.08638 v1 pith:2DLTEQKU submitted 2019-08-23 cond-mat.stat-mech

Maowen Tang , Yan He This is my paper · ORCID

classification cond-mat.stat-mech

keywords Haldane-Shastrymodelspincorrelationsnonlocalfactorsringfrustrationoddchainsinverse-squareexchangecontinuumlimitfinite-sizescaling

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 paper sets out to split the spin-correlation function of a closed quantum spin chain into a local piece, which survives the infinite-size continuum limit, and nonlocal factors that encode the ring geometry. It carries out this split for the Haldane-Shastry model, a one-dimensional antiferromagnet with inverse-square exchange couplings, by computing the ground-state correlations exactly for both even and odd numbers of spins. The central result is that the factor measuring ring frustration, the extra cost of accommodating a staggered spin pattern on an odd ring, is $R(\alpha)=\cos(\pi\alpha)$, exactly the same function previously found for the Heisenberg model. The paper also derives an analytic even-chain factor, $R^{(E)}(\alpha)=\pi\alpha/\sin(\pi\alpha)$, which it shows is almost identical to the numerical fit. If this is right, the odd-versus-even difference in spin correlations is governed by a geometric factor common to different spin models, which can be studied exactly in the simpler Haldane-Shastry ground state.

What carries the argument

The load-bearing object is the nonlocal factor, the ratio of a finite-ring correlation at fixed $\alpha=r/N$ to the continuum correlation $C_\infty(r)$; the paper's claim is that this ratio depends only on $\alpha$. The calculation is carried by the Gutzwiller-Jastrow ground state of the Haldane-Shastry model, $\Psi(x_1,\dots,x_M)=\prod_i z_{x_i}\prod_{i<j}(z_{x_i}-z_{x_j})^2$ with $z_x=e^{2\pi i x/N}$. Norms and correlation numerators are evaluated by rewriting the squared wave function as a confluent alternant, a determinant whose rows pair each site value with its derivative, then expanding that determinant via Laplace's theorem and using the unit-root identity $\sum_{x=1}^N z_x^n=\delta_{n,0}$ to remove all but finitely many terms. The factorization emerges from an integral representation of the even-$N$ correlation after a $1/M$ expansion, in which the leading term becomes the local piece and the leftover prefactor in $\alpha$ becomes the nonlocal factor.

What would settle it

Calculate, by exact diagonalization of small Haldane-Shastry rings, the ratio $C_{r,N}/C_\infty(r)$ for two different separations with the same fraction $\alpha=r/N$, for example $r=3,N=12$ and $r=4,N=16$. The factorization predicts equal ratios up to finite-size corrections, so any systematic $r$-dependence at fixed $\alpha$ would falsify it. In the same data, compare $C_{r,2M+1}/C_{r,2M}$ with $\cos(\pi\alpha)$ at fixed $\alpha$ for increasing $M$; a persistent drift away from the cosine, especially near $\alpha=1/2$, would falsify the central claim.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that Haldane-Shastry spin correlations factorize: $C_{r,N}=C_\infty(r)\times R^{(E)}(\alpha)$ for even $N$, with an additional factor $R(\alpha)$ for odd $N$, where $\alpha=r/N$ is the fixed fraction of the ring between the two spins. The continuum correlation is $C_\infty(r)=\frac{1}{2\pi r}\int_0^{\pi r}dt\,\frac{\sin t}{t}$. Expanding the exact even-$N$ correlation for large $M$ gives $R^{(E)}(\alpha)=\pi\alpha/\sin(\pi\alpha)$, which the authors show is almost indistinguishable from the numerical fit $1+0.428\sinh^2(1.969\alpha)$. The odd-$N$ correlation carries an extra factor $\cos(M\theta')\approx\cos(\pi\alpha)$, so the ring-frustration ratio between odd and even chains is $R(\alpha)=R^{(O)}(\alpha)/R^{(E)}(\alpha)=\cos(\pi\alpha)$. The paper identifies this cosine factor as exactly the same one previously extracted numerically for the Heisenberg model, and notes that the XY model shows the same form.

Load-bearing premise

The load-bearing premise is that, in the limit of an infinitely long chain with the spin separation held at a fixed fraction $\alpha$ of the chain, the ratio of the finite-ring correlation to the infinite-chain correlation depends only on $\alpha$ and not on the absolute separation $r$; the paper asserts this cancellation before demonstrating it.

Editorial extensions

If this is right

For an odd Haldane-Shastry ring, the frustration factor $\cos(\pi\alpha)$ vanishes at $\alpha=1/2$, the largest possible separation on the ring, so correlations near half the ring are suppressed relative to an even chain.
The analytic form $\pi\alpha/\sin(\pi\alpha)$ reproduces the numerically fitted even-chain nonlocal factor, giving a closed expression where only a numerical fit was previously available.
All four degenerate ground states of the odd-$N$ chain give the same spin correlations, so the nonlocal factors are properties of the model rather than of a particular ground-state choice.
Because the Haldane-Shastry ground state is far simpler than the Bethe-ansatz state of the Heisenberg model, the same cosine frustration factor can now be derived analytically rather than inferred from small-system numerics.

Reading between the lines

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

If the cosine frustration factor is universal across one-dimensional antiferromagnetic rings, the odd-even difference in spin correlations is fixed by ring geometry alone; a direct test would be to extract $R(\alpha)$ for spin chains beyond those studied here, such as spin-1 or dimerized rings.
The factorization suggests a practical scheme for small-ring numerics: divide out the nonlocal factor to expose universal continuum correlation data, which could improve finite-size extrapolations of critical exponents.
The paper treats only ground states, so a natural extension is to ask whether the same factorization holds in thermal or excited states of the Haldane-Shastry model; the ratio definition could be checked at finite temperature for small rings.
The place to look first for a breakdown is near $\alpha=1/2$, because the subleading terms dropped in the $1/M$ expansion need not vanish uniformly there; a careful finite-size study at fixed $\alpha$ close to $1/2$ would test the factorization most sharply.

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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

Short version: this paper gives exact finite-size spin correlations for the Haldane-Shastry chain and shows that the ring-frustration nonlocal factor is R(alpha)=cos(pi alpha), the same as the Heisenberg chain. That is a real result, and it is not circular: the HS ground state is known, the correlations are evaluated analytically, and the odd/even comparison is direct.

What is good: the wavefunction and norm are handled cleanly; the correlation formulas in Eqs. (30) and (38) are exact and simple enough to check. The paper also does not oversell. It compares with earlier numerical XY and Heisenberg values honestly, and it makes clear that the HS ground state is much simpler than the Bethe ansatz state. The analytic even-N factor pi alpha/sin(pi alpha) matches the numerical fit, so the confidence in that piece is decent.

Soft spots, in order of importance. First, the odd-N limit is not well-defined as written. Equation (38) carries an oscillatory sign (-1)^r via Eq. (39), and "set aside the unimportant minus sign" is not a mathematical operation. For fixed alpha with r growing, the sequence C_{r,2M+1} has no limit; R(alpha) in Eq. (6) only makes sense if you define it on the envelope, that is, on |C_{r,2M+1}| or after stripping the parity factor. The stress-test note is right about this. It is repairable, but it is a genuine hole in the definitions. Second, the Laplace expansion that produces Eq. (25) is summarized in a sentence. The norm has a full appendix; the correlation numerator does not. A referee should ask for that derivation or at least a clear statement of which terms vanish. Third, Eq. (37) is called analytic but comes from dropping the O(1/M) term in Eq. (35); the discarded series is not controlled. It matches the numerical fit, so it is probably fine, but it should be labeled as approximate. The two fitted constants in Eq. (34) have no uncertainties; that is minor.

Bottom line: the central identification R(alpha)=cos(pi alpha) is likely correct and is a useful data point for the possible universality of nonlocal factors in one-dimensional spin chains. The paper needs a revision that fixes the envelope definition and tightens the expansion steps. I would send it to a serious referee, with a clear note about the sign issue.

Referee Report

3 major / 4 minor

Summary. The paper introduces the concept of nonlocal factors for spin correlation functions, defined as ratios of correlations in the nonlocal scaling limit (r and N large with alpha = r/N fixed) to the local continuum correlation. The authors apply this to the Haldane-Shastry model, for which the ground state and spin correlations are computed exactly for both even and odd numbers of spins from the Gutzwiller-Jastrow wavefunction. They derive analytically that the ring-frustration nonlocal factor is R(alpha) = cos(pi alpha), identical to that of the Heisenberg model, and they give an approximate analytic form for the even-N factor R(E)(alpha) = pi alpha / sin(pi alpha) together with a numerical fit.

Significance. If the stated result is made rigorous, the paper provides a rare analytical example of factorization of spin correlations into local and nonlocal contributions in an exactly solvable model. The central claim, equality of the ring-frustration factor R(alpha) with the Heisenberg model result, is derived directly from the exact ground-state wavefunction without fitting parameters, and the explicit correlation formulas are a useful starting point for further analysis. The paper would be of interest to the statistical mechanics and strongly correlated electron communities. The main caveat is that the odd-N correlation limit as currently defined is not well-posed, so the central claim needs a precise redefinition before it can be accepted.

major comments (3)

[Section IV, Eqs. (4), (38)-(41)] The limit defining C(O)(r, alpha) in Eq. (4) does not exist for generic alpha because Eq. (38) contains the factor (-1)^r, and for fixed alpha = r/N with N = 2M+1, r grows with M so (-1)^r oscillates without a limit. The phrase 'Set aside the unimportant minus sign' after Eq. (39) silently replaces the correlation by its absolute value, which is a different observable. Consequently, the ratio R(alpha) = C(O)/C(E) in Eq. (6) is not defined as written. The result can be repaired by defining R(alpha) on the envelope of the oscillating correlation, i.e., on |C(O)|, and proving that the envelope has a well-defined limit; this redefinition must be stated explicitly and used consistently in all subsequent formulas and in Figure 1.
[Section III and Appendix A, Eqs. (24)-(26), (A5)] The Laplace-expansion step that selects the nonzero contributions in the confluent alternant is asserted rather than derived. Equations (25) and (A5) contain nontrivial combinatorial factors such as (2M-1)!! and 2M!!, but the manuscript does not show how these arise from the expansion or how the summation over x_j inside the determinant is handled. Since formulas (30) and (38) are the exact input for the nonlocal factors, this is a load-bearing technical step; please expand the derivation or give a reference that contains the full computation.
[Section IV, Eqs. (35)-(37)] The derivation of R(E)(alpha) = pi alpha / sin(pi alpha) discards O(1/M) terms in Eq. (35) without proving that they vanish uniformly in alpha, particularly near alpha = 1/2. The numerical fit in Eq. (34) and the approximate analytic form in Eq. (37) visibly differ, so the manuscript should clarify that Eq. (37) is an approximate result and state the size of the correction in the M -> infinity limit. This issue does not by itself invalidate the ring-frustration factor R(alpha), which is obtained from the ratio C_{r,2M+1}/C_{r,2M}, but it weakens the claimed analytic derivation of R(E).

minor comments (4)

[Section IV, after Eq. (39)] Replace 'Set aside the unimportant minus sign' with a precise statement that the nonlocal factor is defined through the absolute value or envelope of the correlation, as the current wording changes the observable being studied.
[Eq. (32)] The notation 'sin t/2M sin t/2M' is ambiguous; it should be written as sin(t/2)/sin(t/(2M)) or with explicit parentheses so that the change of integration variable is clear.
[Figure 1 caption] The caption should state whether the plotted ratio C_{r,2M+1}/C_{r,2M} uses absolute values, given that the odd-N correlation carries an oscillatory sign.
[Eq. (7)] The expression for R(alpha) of the transverse Ising model is typeset ambiguously as 'cos πα/2 − sin πα/2'; please insert parentheses to distinguish cos(pi alpha/2) - sin(pi alpha/2) from other readings.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Haldane-Shastry nonlocal factors are derived analytically from the exact ground-state wavefunction, not from the inputs they are compared with.

full rationale

The paper's central derivation is self-contained. The ground-state wavefunction in Eq. (12) is the known Haldane-Shastry Gutzwiller-Jastrow form, and the spin correlations in Eqs. (30) and (38) are obtained analytically from it via confluent alternants and root-of-unity sums, with no parameter fitted to the target quantities. The even-N nonlocal factor is extracted from the analytic large-M expansion of Eq. (32), giving R(E)(α)=πα/sin(πα) in Eq. (37); the numerical fit in Eq. (34) is only an illustrative comparison, not an input to the derivation. The ring-frustration factor R(α)=cos(πα) in Eq. (41) follows from the ratio of the odd-N correlation Eq. (38) to the even-N correlation Eq. (30), again without fitting. The comparison to the Heisenberg model uses the earlier numerical result cited as [18] only as an external benchmark, not as the source of the HS result. The self-citation to the authors' prior proposal [13] is contextual and does not carry the derivation. The main mathematical caveat is that the factor (-1)^r in Eq. (39) makes the strict large-M limit of C(O)(r,α) in Eq. (4) ill-defined, so the claim should be read as an envelope/absolute-value statement; this is a rigor concern, not a circular reduction of the result to its own inputs.

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

The central claim rests on known exact ground-state wavefunctions of the Haldane-Shastry model, on algebraic identities used to evaluate determinants, and on a factorization assumption about nonlocal correlation ratios. Two constants (0.428 and 1.969) are fitted to numerical data for the even-N nonlocal factor, but they are not needed for the ring-frustration claim.

free parameters (2)

amplitude constant in R(E) fit = 0.428
Fitted to the ratio C_{r,2M}/C_inf(r) computed at M = 1000 in Figure 1(a), Eq (34). Not used for the central ring-frustration claim.
exponent coefficient in R(E) fit = 1.969
Second free parameter in Eq (34), fitted to the same numerical data. Not used for the central ring-frustration claim.

assumptions (3)

domain assumption Haldane-Shastry ground state for even and odd N has the Gutzwiller-Jastrow form Psi = prod z_{x_i} prod (z_{x_i}-z_{x_j})^2, with four degenerate states for odd N.
Invoked in Sec II, Eq (12) and (17); the paper cites [19] and attributes a note from H.-H. Tu, but does not prove the odd-N degeneracy or wavefunction in detail.
standard math Confluent-alternant identities of Appendix B and the root-of-unity sum rule sum_x z_x^n = N delta_{n0}.
Used in Appendix A and Section III to evaluate the norm and correlation numerators; these are standard algebraic facts, but the identities are derived via a derivative trick that is only sketched.
ad hoc to paper Nonlocal-factor factorization: after the large-N limit, C(O)/C_inf and C(E)/C_inf depend only on alpha = r/N, so the r dependence cancels in Eq (5).
This is the defining assumption of the nonlocal-factor program introduced in prior work; the paper verifies it for the Haldane-Shastry model but does not prove it in general.

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Cite this review

Pith. "Pith review of Nonlocal behaviors of spin correlations in the Haldane-Shastry model." pith.science (2026). https://pith.science/paper/2DLTEQKU

@misc{pith2026190808638,
  author       = {Pith},
  title        = {Pith review of: Nonlocal behaviors of spin correlations in the Haldane-Shastry model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2DLTEQKU}},
  note         = {Machine review of arXiv:1908.08638}
}

read the original abstract

The nonlocal factors of spin correlations are introduced for lattice spin models. Based on this concept, we investigate the nonlocal behavior of the Haldane-Shastry model with or without ring frustration. The ground state and spin correlations of the Haldane-Shastry model are calculated for both even and odd number of spins, then the nonlocal factors can be deduced analytically. It is found that the nonlocal factor due the ring frustration is the same as the Heisenberg model.

Figures

Figures reproduced from arXiv: 1908.08638 by the authors.

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

R. Z. Bariev, Zh. Eksp. Teor. Fiz. 77, 1217 (1979) [Sov. Phys. JETP 50, 613 (1979)]

work page 1979
[2]

G. G. Cabrera and R. Jullien, Phys. Rev. Lett. 57, 393 (1986)

work page 1986
[3]

G. G. Cabrera and R. Jullien, Phys. Rev. B 35, 7062 (1987)

work page 1987
[4]

Karbach and K.-H

M. Karbach and K.-H. Mutter, J. Phys. A: Math. Gen. 28, 4469 (1995)

work page 1995
[5]

Cador, etc., Angew

O. Cador, etc., Angew. Chem. Int. Edit. 43, 5196, (2004)

work page 2004
[6]

Klaus Barwinkel,etc., Phys. Rev. B 68, 054422, (2003)

work page 2003
[7]

Baker, etc, Proc

Michael L. Baker, etc, Proc. Nat. Acad. Sci. 109, 19113, (2012)

work page 2012
[8]

Campostrini, A

M. Campostrini, A. Pelissetto, and E. Vicari, Phys. Rev. E 91, 042123 (2015)

work page 2015

Show all 20 references

[9]

Campostrini, A

M. Campostrini, A. Pelissetto, and E. Vicari, J. Stat. Me ch. P11015 (2015)

2015
[10]

J.-J. Dong, P. Li and Q.-H. Chen, J. Stat. Mech. 113102 (2 016)
[11]

He and H

Y. He and H. Guo, J. Stat. Mech. 093101 (2017)

2017
[12]

Dong, Z.-Y.Zheng and P

J.-J. Dong, Z.-Y.Zheng and P. Li, Phys. Rev. E 97, 012133 (2018)

2018
[13]

Li and Y

P. Li and Y. He, Phys. Rev. E 99, 032135 (2019)

2019
[14]

F. D. M. Haldane, Phys. Rev. Lett. 60, 635 (1988)

1988
[15]

B. S. Shastry, Phys. Rev. Lett. 60, 639 (1988)

1988
[16]

Di Francesco, P

P. Di Francesco, P. Mathieu, D. Senechal Conformal Field Theory , Springer-Verlag, New York, 1997, chapter 3

1997
[17]

T. A. Kaplan, P. Horsch, and J. Borysowicz, Phys. Rev. B 35, 1877 (1987)

1987
[18]

K. A. Hallberg, P. Horsch, and G. Martinez, Phys. Rev. B 52, R719 (1995)

1995
[19]

Kuramoto and Y

Y. Kuramoto and Y. Kato Dynamics of One-Dimensional Quantum Systems , Cambridge University Press, Cambridge, 2008, chapter 4

2008
[20]

B. M. McCoy, Advanded Statistical Mechanics , Oxford University Press, Oxford, 2010

2010

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

R. Z. Bariev, Zh. Eksp. Teor. Fiz. 77, 1217 (1979) [Sov. Phys. JETP 50, 613 (1979)]

work page 1979

[2] [2]

G. G. Cabrera and R. Jullien, Phys. Rev. Lett. 57, 393 (1986)

work page 1986

[3] [3]

G. G. Cabrera and R. Jullien, Phys. Rev. B 35, 7062 (1987)

work page 1987

[4] [4]

Karbach and K.-H

M. Karbach and K.-H. Mutter, J. Phys. A: Math. Gen. 28, 4469 (1995)

work page 1995

[5] [5]

Cador, etc., Angew

O. Cador, etc., Angew. Chem. Int. Edit. 43, 5196, (2004)

work page 2004

[6] [6]

Klaus Barwinkel,etc., Phys. Rev. B 68, 054422, (2003)

work page 2003

[7] [7]

Baker, etc, Proc

Michael L. Baker, etc, Proc. Nat. Acad. Sci. 109, 19113, (2012)

work page 2012

[8] [8]

Campostrini, A

M. Campostrini, A. Pelissetto, and E. Vicari, Phys. Rev. E 91, 042123 (2015)

work page 2015