Extended mermin-wagner theorem for classical and quantum long-range system

[peter0627ustc]

Released by

Youjin Deng, USTC

Contact email

yjdeng@ustc.edu.cn

Method

Proof / Monte Carlo

Challenge issue

Focused Track-A challenge. This issue isolates the marginal-point consistency question raised in Track A of #86. At $\sigma=2$, a published extension of the Mermin–Wagner theorem appears inconsistent with recent large-scale Monte Carlo results for the two-dimensional long-range XY model. The challenge has two complementary directions: proof and Monte Carlo. Determine whether the discrepancy comes from a hypothesis or limiting procedure, a finite-size/logarithmic effect, a gap in the proof, or a genuine counterexample.

The apparent inconsistency at $\sigma=2$

Consider a $d$-dimensional XY or Heisenberg system with monotonically decreasing interactions

$$ J(R)\propto R^{-\alpha}, \qquad \alpha=d+\sigma. $$

Bruno's extension of the Mermin–Wagner theorem (Phys. Rev. Lett. 87, 137203 (2001); arXiv:cond-mat/0105129) states, in Corollary 2, that one- and two-dimensional Heisenberg or XY systems with monotonically decreasing interactions cannot have finite-temperature ferro- or antiferromagnetic long-range order when

$$ \alpha\ge 2d. $$

For $d=2$, the equality case $\alpha=4$ is exactly $\sigma=2$. The theorem therefore appears to exclude finite-temperature ferromagnetic long-range order at this marginal point.

In contrast, the large-scale Monte Carlo study of the two-dimensional long-range XY model by Yao et al. (arXiv:2411.01811, Phys. Rev. B 112, 144429 (2025)) reports a second-order transition into a ferromagnetically ordered phase for $\sigma\le2$. At $\sigma=2$, the low-temperature correlation function is fitted as

$$ g(r)\sim \frac{a}{\ln r}+g_0, $$

with a nonzero asymptote $g_0$, interpreted as long-range order. On their face, the theorem and the numerical conclusion disagree at the same boundary point.

Conflict-of-interest disclosure: the $\sigma=2$ numerical result under examination is from the releaser's group. This challenge is an adversarial test of our own result. A falsification of the numerical interpretation, identification of a missing theorem hypothesis, a valid counterexample, or a statistically justified conclusion that accessible sizes cannot decide the question are all successful scientific outcomes.

Required model and convention audit

Before comparing conclusions, state the Hamiltonian and finite-volume prescription explicitly. The core classical model is the two-dimensional XY model

$$ H=-\frac12\sum_{i\ne j}J_{ij},\mathbf S_i\cdot\mathbf S_j, \qquad \mathbf S_i=(\cos\theta_i,\sin\theta_i), $$

with $J_{ij}>0$ and asymptotic decay $J(R)\sim R^{-4}$ at $\sigma=2$.

The analysis must distinguish at least the following conventions rather than treating them as automatically equivalent:

• the infinite-lattice interaction used in the theorem;
• periodic image sums versus minimum-image distance on an $L\times L$ torus;
• bare coupling versus a size-dependent normalization $c(\sigma,L)$;
• strict positivity and monotonicity of $J(R)$ on the discrete lattice;
• the order of the thermodynamic, zero-field, low-momentum, and low-temperature limits.

If two conventions are claimed to have the same thermodynamic limit, quantify the finite-size difference at $\sigma=2$ instead of assuming it away.

This is primarily a Track-A problem. The directly conflicting numerical system is the classical two-dimensional long-range XY model. Quantum systems enter through the scope of the extended theorem and any independent proof, not as a required separate numerical track.

Direction 1 — proof and theorem-to-lattice audit

Re-derive the marginal step in Bruno's argument for the discrete two-dimensional lattice. In particular, determine the small-$k$ behavior of the relevant dispersion-like quantity

$$ \widetilde E(k)=\sum_R J(R),[1-\cos(k\cdot R)] $$

for $J(R)\sim R^{-4}$. The claimed marginal scaling is

$$ \widetilde E(k)\sim C k^2\ln(1/k), $$

which would make

$$ \int_{|k|<\Lambda}\frac{d^2k}{\widetilde E(k)} $$

diverge as a log-log and thereby exclude spontaneous order.

The deliverable must identify every assumption needed to turn this infrared divergence into the no-order conclusion. It should check the lattice remainder, the coefficient and sign, boundary terms, and uniformity of the limits. A valid independent derivation is preferred; a line-by-line proof audit that isolates an unsupported step is also acceptable.

The proof direction should state explicitly which steps apply to classical spins, quantum spins, or both. A valid independent proof or corrected theorem at the boundary is a complete outcome; no separate quantum numerical implementation is required.

Proof validation

• Reproduce the known non-marginal scaling for $d&lt;\alpha&lt;d+2$ and for $\alpha&gt;d+2$ before analyzing $\alpha=d+2$.
• Check the discrete lattice sum against direct high-precision numerical evaluation of $\widetilde E(k)$ over several decades in $k$.
• Repeat for both the infinite-lattice interaction and the finite periodic convention used in the numerical work.
• State explicitly which conclusion applies to classical systems, quantum systems, or both under the assumptions used in the derivation.

Direction 2 — independent Monte Carlo test

Independently reproduce the low-temperature correlation function of the two-dimensional long-range XY model at $\sigma=2$, with control points on both sides of the boundary. The primary task is not merely to obtain a visually nonzero plateau, but to distinguish among at least these hypotheses:

1. $g(r)\to g_0&gt;0$ with inverse-logarithmic corrections;
2. $g(r)\to0$ asymptotically through a logarithmically slow decay;
3. current sizes and temperatures cannot discriminate the two.

Use a preregistered finite-size analysis with raw per-size observables, seeds, autocorrelation information, equilibration tests, and code revision. Fit both ordered and no-order ansätze, report a model-selection score such as AIC/BIC or held-out predictive error, and repeat after changing the fit window and dropping the smallest sizes.

Monte Carlo validation

• Reproduce a short-range two-dimensional XY benchmark, including the expected absence of conventional ferromagnetic long-range order.
• Reproduce at least one published $\sigma&lt;2$ ordered control point and one $\sigma&gt;2$ short-range control point using the same implementation and normalization as the marginal run.
• Compare minimum-image and periodic-image interactions at matched sizes.
• Measure the magnetization, correlation function, susceptibility/Binder-type ratios, and a stiffness or winding observable where the algorithm permits it.
• Report a result as inconclusive if the inferred $g_0$ changes qualitatively under a registered fit-window, correction-to-scaling, or interaction-convention check.

Fairness and analysis contract

1. Day-0 scaffold: provide either a tested long-range XY Monte Carlo sampler or a reproducible script for evaluating the lattice sum $\widetilde E(k)$, with small-size fixtures and an explicit interaction convention.
2. Raw-data escrow: release per-size raw observables, seeds, autocorrelation/equilibration diagnostics, fit inputs, and the exact code revision before the final extrapolation.
3. Locked analysis: register fit windows, correction ansätze, exclusion rules, and model-selection criteria before inspecting the largest-size conclusion.
4. Independent adjudication: the final compatibility assessment should be reviewed by someone who is not an author of the numerical paper under test.
5. No one-fit verdict: if the conclusion changes under a reasonable registered alternative fit or order of limits, report the boundary as unresolved.

What counts as a resolution

A successful result must provide one of the following:

• a proof that Bruno's hypotheses apply to the simulated lattice model and that the apparent nonzero $g_0$ is a finite-size/logarithmic artifact;
• a precise hypothesis or limiting procedure that makes the theorem inapplicable to the numerical model;
• a demonstrable gap in the marginal proof, together with a corrected statement;
• a reproducible counterexample satisfying the stated theorem hypotheses;
• or an uncertainty-controlled demonstration that presently accessible sizes cannot distinguish order from logarithmically slow decay.

A broad statement that “the theorem and numerics disagree” without a convention audit, or a single preferred fit without stability tests, is not sufficient.

Deliverables

1. A theorem-to-model hypothesis table covering symmetry, classical/quantum scope, dimension, interaction sign/monotonicity, summability, boundary convention, normalization, and limit order.
2. For the proof direction, a reproducible derivation and numerical check of $\widetilde E(k)$ at $d=2$, $\alpha=4$.
3. For the Monte Carlo direction, finite-size data and competing fits for $g(r)$ and $g_0$, including stability and model-selection results.
4. A concise verdict: compatible, theorem inapplicable, proof gap/counterexample, or unresolved, with the exact assumptions attached.
5. Public code, raw data, and the locked analysis record.

References

• P. Bruno, Phys. Rev. Lett. 87, 137203 (2001); arXiv:cond-mat/0105129.
• D. Yao, T. Xiao, C. Zhang, Y. Deng, and Z. Fan, arXiv:2411.01811, Phys. Rev. B 112, 144429 (2025).
• Related discussion: quantum.harness issue #86.

[JunkaiWang-TheoPhy]

@yjdeng Dear Prof.Deng— I have started an independent theorem/data audit for this challenge and have reviewed the public Zenodo record, its version/file history, the arXiv source, and the available supplementary materials. Thank you for making the run-level results public.

For the marginal $\sigma=2$ analysis, I could not locate synchronized bin/block means, jackknife replicas, cross-observable covariance matrices, or autocorrelation time series. The public data.dat contains run-level means and marginal error bars together with seed and N_sample. For the low-temperature slices $\beta=1,2,4,8$, only $L=512$ appears to have enough equal-length independent runs to estimate a run-to-run covariance (nine $2\times10^5$-sample runs, in addition to one longer run); most other sizes have only one aggregated run.

This matters because the discriminating test links $M^2$, $M^2_{k_{\min}}$, and $\xi/L$. A simultaneous fit that ignores their cross-covariance is only a diagonal-error pseudo-likelihood, so its absolute $\chi^2$ and information-criterion differences may not be reliable.

Would it be possible to release the following minimal data for the $\sigma=2$ low-temperature points?

1. Synchronized bin/block means or jackknife replicas for $M^2$ and $M^2_{k_{\min}}$, ideally also the replicas used to construct $\xi/L$;
2. The bin length, number of bins, thermalization length, and measurement interval;
3. A note on whether the same bin boundaries were used for all observables and how the quoted $\xi/L$ uncertainty was jackknifed;
4. If already available, the covariance matrices and integrated autocorrelation-time estimates.

Per-sweep configurations are not necessary. A compact table with one row per $(\beta,L,\text{seed},\text{bin})$ would be sufficient. If the original bin files are no longer available, several independent same-length replicas at selected $L$ values—especially the largest sizes—would still be useful.

With these data, I propose a preregistered covariance-aware comparison of

$$ M^2(L)=g_0+\frac{a_1}{\ell}+\frac{a_2}{\ell^2}+\cdots $$

against

$$ M^2(L)=A\ell^{-p}\left(1+\frac{b_1}{\ell}+\frac{b_2}{\ell^2}+\cdots\right), \qquad \ell=\ln(L/L_0), $$

jointly with $M^2_{k_{\min}}$, using a shared $p(T)$, varying $L_{\min}$, held-out largest-size prediction, and bootstrap calibration. The analysis scripts and fit protocol would be made public. This request is not a claim that the published data are incorrect; it is intended to determine whether the present finite-size range can statistically distinguish a positive $g_0$ from logarithmically slow decay.

Thank you very much if you'd like to help! ;-D

Best,
J.K. Wang
July 28th

[JunkaiWang-TheoPhy]

Dear Prof. Deng and Issue #158 participants,

I have now completed an interim theorem-to-model audit, a high-precision
lattice-kernel check, and a reanalysis of the public summary data. I would
like to report the result before the requested synchronized bins/covariances
become available.

Interim conclusion

For the stated two-dimensional classical XY model at

$$ d=2,\qquad \sigma=2,\qquad J(R)\propto |R|^{-4}, $$

I find that the apparent theorem/model loopholes can be closed. In
particular, the conclusion is stronger than the vanishing of a field-defined
one-point magnetization:

$$ \boxed{ \lim_{L\to\infty} \left\langle |M_L|^2\right\rangle_{\Lambda_L}=0 \qquad (T>0). } $$

Thus a positive thermodynamic intercept is excluded for this model. This
does not by itself prove the sharper asymptotic law

$$ g(r)\sim[\log r]^{-p(T)}. $$

That remains a spin-wave/RG candidate whose nonlinear and vortex corrections
still need to be controlled.

1. Marginal lattice kernel

With the paper's normalization

$$ \sum_{R\ne0}J(R)=4, $$

the infinite square-lattice coupling is

$$ J(R)=\frac{c_\infty}{|R|^4}, \qquad c_\infty=\frac{6}{\pi^2G} =0.6637008046138535\ldots, $$

where $G$ is Catalan's constant. Splitting the lattice sum at
$|R|\asymp |k|^{-1}$ gives

$$
E(k)

\sum_{R\ne0}J(R)[1-\cos(k\cdot R)]

\frac{\pi c_\infty}{2}|k|^2\log\frac1{|k|}
+O(|k|^2).
$$

The predicted logarithmic coefficient is

$$
\rho_{\log}

\frac{\pi c_\infty}{2}
=1.0425387859782584\ldots.
$$

An independent high-precision axial lattice-sum calculation gives

$$ \rho_{\log}^{\rm fit} =1.0425387750974222, $$

a relative difference of approximately $1.04\times10^{-8}$. I therefore do
not find a sign, coefficient, or lattice-remainder failure in the marginal
$k^2\log(1/k)$ dispersion.

The infrared integral consequently diverges:

$$ \int\frac{d^2k}{E(k)} \sim \int_0^\epsilon \frac{dk}{k\log(1/k)} =\infty, $$

with finite-size growth proportional to $\log\log L$.

2. Direct classical argument

The quantum/classical scope issue can be bypassed. For the finite classical
XY Gibbs measure, define

$$
D_k

\sum_xe^{ik\cdot x}
\frac{\partial}{\partial\theta_x},
\qquad
A_k

\sum_xe^{-ik\cdot x}\sin\theta_x.
$$

Periodic Gibbs integration by parts and Cauchy--Schwarz give

$$ \left\langle|A_k|^2\right\rangle \left\langle D_{-k}D_kH\right\rangle \ge T\left|\left\langle D_kA_k\right\rangle\right|^2. $$

For positive coupling,

$$ \left\langle D_kA_k\right\rangle=Nm_{L,h}, \qquad \left\langle D_{-k}D_kH\right\rangle \le N[h+E_L(k)]. $$

After summing over momenta and using Parseval,

$$ 1 \ge Tm_{L,h}^2 \frac1N\sum_k\frac1{h+E_L(k)}. $$

Taking $L\to\infty$ at fixed $h&gt;0$, followed by $h\downarrow0$, yields

$$ m_h^2 \lesssim \frac1{T\log\log(1/h)} \longrightarrow0. $$

No quantum commutator or $S\to\infty$ limit is needed.

There is also an independent all-Gibbs-state result. With

$$ p(R)=J(R)/4, $$

the associated random walk satisfies

$$ 1-\widehat p(k)=E(k)/4 $$

and is recurrent. Theorem 2 of
Ioffe--Shlosman--Velenik then implies
that every infinite-volume Gibbs state is $SO(2)$-invariant.

3. Bridge to the measured zero-field $M^2$

For a fixed block $Q_R$, define

$$
B_{R,x}

\frac1{|Q_R|}
\sum_{u\in Q_R}S_{x+u}.
$$

On the torus,

$$ M_L=\frac1{L^2}\sum_xB_{R,x}. $$

Jensen's inequality and translation invariance give

$$ \left\langle|M_L|^2\right\rangle_L \le \left\langle|B_{R,0}|^2\right\rangle_L. $$

Take a local weak torus limit at fixed $R$. Its translation-ergodic
components are Gibbs states and are therefore $SO(2)$-invariant. The mean
ergodic theorem then gives

$$ \lim_{R\to\infty} \left\langle|B_{R,0}|^2\right\rangle=0. $$

It follows that

$$ \lim_{L\to\infty} \left\langle|M_L|^2\right\rangle_L=0. $$

This closes the possible distinction between a theorem formulated through
spontaneous magnetization and a simulation extrapolation of the zero-field
squared magnetization.

4. Minimum-image thermodynamic limit

Extending the normalized minimum-image interaction by zero outside its
representative square gives

$$ \sum_R \left|J_L^{\rm MI}(R)-J_\infty(R)\right| =O(L^{-2}), $$

because the omitted two-dimensional $R^{-4}$ tail is $O(L^{-2})$ and
$c_L-c_\infty=O(L^{-2})$. Hence local weak limits of the torus measures are
Gibbs states of the fixed infinite-volume $R^{-4}$ interaction.

At the lowest momentum,

$$ E_\infty(2\pi/L) \asymp L^{-2}\log L, $$

so the absolute $O(L^{-2})$ convention difference produces a relative

$$ O(1/\log L) $$

finite-size correction. This can strongly affect inverse-log intercept
fits, but it does not produce a distinct thermodynamic phase.

5. Public-summary-data reanalysis

I used the public
Zenodo record, choosing at each
$(\beta,L)$ the row with the largest reported sample count. The analysis
protocol was locked before this extended run, although it should be described
as retrospectively locked rather than a true preregistration because earlier
exploratory fits had already inspected the data.

On the registered window $L\ge64$, I compared

$$ \mathrm{O2}:\quad M^2=g_0+\frac{a_1}{\ell}+\frac{a_2}{\ell^2} $$

with

$$ \mathrm{D2}:\quad M^2=A\ell^{-p}+B\ell^{-p-1}, \qquad \ell=\log L. $$

The D2 results are:

$\beta$	$p$	D2 $\chi^2/\mathrm{dof}$	D2 GOF	O2 $\chi^2/\mathrm{dof}$
1	0.31641	2.987	0.00146	80.62
2	0.10219	1.203	0.288	77.51
4	0.04418	0.808	0.595	40.91
8	0.02054	0.435	0.901	33.82

Thus D2 is acceptable for $\beta=2,4,8$ but still needs additional
corrections at $\beta=1$; O2 fails absolute GOF very strongly at all four
temperatures.

I also checked the exact shifted-log form used in the paper,

$$ \mathrm{OP}:\quad M^2=g_0+\frac{a}{\log(L/L_0)}, $$

against the equal-parameter-count form

$$ \mathrm{DP}:\quad M^2=A[\log(L/L_0)]^{-p}. $$

At $L_{\min}=64$,

$$
\Delta\mathrm{AICc}

\mathrm{AICc}{\rm OP}-\mathrm{AICc}{\rm DP}
$$

is respectively

$$ 37.47,\quad37.66,\quad4.91,\quad12.94 $$

for $\beta=1,2,4,8$, favoring decay on this window. However, when
$L_{\min}$ is increased to $192$--$384$, the preference becomes weak and can
change sign because only a few points and a very small logarithmic lever arm
remain. My interpretation is therefore not that the public data alone prove
the exact DP asymptotic form, but that they do not establish a positive
thermodynamic intercept.

The final doubling-size effective logarithmic exponents remain positive:

$\beta$	largest pair	$p_{\rm eff}$
1	$4096\to8192$	$0.27291\pm0.00469$
2	$4096\to8192$	$0.09263\pm0.00153$
4	$2048\to4096$	$0.04170\pm0.00024$
8	$2048\to4096$	$0.01970\pm0.00014$

They show no clear drift toward zero over the available range.

In 2,000-replica synthetic tests for each temperature and truth model, AICc
selected the generating O2/D2 family in $100%$ of replicas. Importantly,
when D2 was true, the wrong O2 fit still produced a nominally significant
positive $g_0$ in $100%$ of replicas, but passed absolute GOF in $0%$ and
was selected by AICc in $0%$. This shows why the significance of a fitted
intercept is not sufficient without absolute GOF, window stability, and
non-nested model comparison.

6. Remaining data limitation

The public table contains marginal errors but not synchronized bin means,
joint jackknife replicas, or the covariance of $M^2$ and
$M^2_{k_{\min}}$. A sensitivity scan over

$$
\rho

\operatorname{Corr}(M^2,M^2_{k_{\min}})
\in[-0.8,0.8]
$$

continues to favor the tested joint decay form, but this is not a substitute
for the actual covariance. The residual construction

$$ M_r^2=M^2-bM^2_{k_{\min}} $$

also requires this covariance and the uncertainty in the data-selected
coefficient $b$.

Therefore my present conclusion is:

The theorem-to-model audit excludes a positive thermodynamic
magnetization, including the zero-field torus $M^2$ observable. The public
summary data favor logarithmic decay over the published ordered ansatz on
reasonably populated windows, while largest-only fits remain
correction-sensitive. Synchronized binned data are still needed for the
final covariance-aware numerical comparison and for identifying the precise
asymptotic correlation law.

This is not a claim that the published Monte Carlo measurements themselves
are incorrect. The issue is the thermodynamic extrapolation and the
identifiability of an extremely slow decay. I would be very grateful for
corrections to any step above, especially if the simulated interaction or
normalization differs from the convention I audited. I have retained the
analysis scripts, protocol, numerical tables, and proof details in a
reproducibility bundle and can make them available for review.

Best regards,

J.K. Wang

[JunkaiWang-TheoPhy]

Update: the full reproducible implementation is now public in PR #260:

#260

Direct entry points:

• Reproduction guide and verdict
• Classical no-order proof and the bridge from zero-field $$\langle |M_L|^2\rangle$$ to SSB
• Kernel audit and public-data analysis
• Pre-registered analysis protocol for the requested synchronized-bin data

The PR includes a pinned Zenodo download with SHA-256 verification, executable analysis scripts, committed tables/figures, and tests. On the presently auditable model and public data, the defensible verdict is:

1. finite-temperature ferromagnetic LRO is excluded by a direct classical Bogoliubov argument (independently supported by the recurrence theorem);
2. the minimum-image, size-normalized interaction converges to the same infinite-volume $$1/r^4$$ model;
3. the measured marginal kernel has the required positive $$k^2\log(1/k)$$ coefficient;
4. the public aggregate data do not identify a positive thermodynamic $$M^2$$ intercept robustly;
5. the stronger claim $$g(r)\sim[\log r]^{-p(T)}$$ remains a low-temperature candidate rather than a theorem and should be tested with synchronized bin means/covariances.

The remaining data request is therefore deliberately small: synchronized bin means (or jackknife replicas) for $$M^2$$, $$M^2_{k_{\min}}$$, and preferably $$\xi/L$$ at each $$(T,L)$$, plus binning/autocorrelation metadata. Per-sweep configurations are not needed.

Corrections to the model convention, proof assumptions, or data interpretation are very welcome.