[challenge]: Does a critical long-range lattice model exhibit global conformal symmetry?

Released by

Youjin Deng, USTC

Contact email

yjdeng@ustc.edu.cn

Method

Long-range cluster Monte Carlo / multipoint correlation functions / optional conformal-data extraction

Challenge issue

Test global conformal symmetry directly in a critical long-range spin model. The challenge should not assume either the Sak boundary (\sigma_=2-\eta_{\rm SR}) or the geometric boundary (\sigma_=2). Instead, it asks which conformal predictions survive in the long-range regime and how their finite-size behavior changes across the disputed crossover window.

The open question

Consider an Ising or (O(N)) model with

$$
J(r)\sim r^{-(d+\sigma)}.
$$

Long-range fixed points are nonlocal: the continuum action contains a fractional kinetic term and generally lacks a conserved, local (d)-dimensional stress tensor. This does not by itself rule out global conformal symmetry. Perturbative defect constructions, bootstrap calculations, large-(N) analyses, and shadow/OPE relations provide evidence for a nonlocal CFT description, but a direct nonperturbative lattice test of special conformal covariance remains limited.

The location of the long-range/short-range crossover must be treated as an open input:

• Sak scenario: (\sigma_*=2-\eta_{\rm SR});
• geometric scenario: (\sigma_*=2).

Issue #86 presents numerical and field-theory evidence for both possibilities. Its marginal-point consistency discussion also illustrates why equality cases, interaction conventions, logarithmic corrections, and orders of limits must be audited rather than hidden inside a presumed boundary.

Accordingly, this challenge separates two questions:

1. At a point safely inside the interacting long-range regime under both scenarios, do lattice multipoint functions approach global-conformal predictions?
2. When (\sigma) approaches the disputed crossover window, do apparent violations reflect a genuine loss of conformality, crossover drift, logarithmic corrections, or insufficient scale separation?

A conformal-compatible result, a controlled violation, and a statistically justified “unresolved at accessible sizes” are all valid outcomes.

Core model

Use Ising spins on an (L\times L) square torus,

$$
s_x=\pm1,\qquad
H=-\sum_{x<y}\frac{s_xs_y}{r_L(x,y)^{2+\sigma}},
$$

with minimum-image distance, one coupling per unordered pair, bare (J=1), no distance cutoff, and periodic boundary conditions.

The core point is

$$
d=2,\qquad \sigma=1.2.
$$

It lies above the long-range Gaussian boundary (\sigma=d/2=1) and below both proposed long-range/short-range crossover values. It is chosen to test an interacting long-range fixed point without deciding the boundary dispute in advance.

For this convention, reproduce the published critical anchor

$$
T_c=6.83427(1)
$$

from Angelini, Parisi, and Ricci-Tersenghi before interpreting conformal observables. Critical temperatures from Ewald, image-summed, normalized, or (c(\sigma,L))-rescaled Hamiltonians must not be mixed with this convention.

Direct Möbius-covariance test

For four points define

$$
u=\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2},
\qquad
v=\frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2},
$$

and measure

$$
\mathcal Q(1,2,3,4)=
\frac{\langle s_1s_2s_3s_4\rangle}
{\langle s_1s_2\rangle\langle s_3s_4\rangle}.
$$

For identical scalar primaries in a globally conformal theory, the continuum limit of (\mathcal Q) depends only on ((u,v)). The central test therefore compares lattice geometries with the same cross ratios that are related by a non-affine Möbius transformation, not merely by translation, rotation, reflection, or dilation.

One pinned pair is

$$
\begin{aligned}
A_R/R&={(0,0),(5,0),(5,5),(0,5)},\
B_R/R&={(0,0),(5,0),(6,2),(5,5)},
\end{aligned}
$$

for which ((u,v)=(1/4,1/4)). Define

$$
\delta_{\rm M}(L,R)=
\frac{2[\mathcal Q(A_R)-\mathcal Q(B_R)]}
{\mathcal Q(A_R)+\mathcal Q(B_R)}.
$$

Use at least one independent geometry pair at another cross-ratio point. The relevant limit is simultaneous:

$$
R\rightarrow\infty,\qquad R/L\rightarrow0.
$$

A fixed-(R) limit retains lattice artifacts, while a fixed (R/L) limit probes a torus rather than the infinite plane.

Monte Carlo implementation

Use a Swendsen–Wang/Fukui–Todo long-range cluster method. Translation and square-lattice symmetry averages are encouraged. FK connectivity gives improved estimators for both two- and four-spin correlators, but raw spin-product estimators must be retained as an independent check.

Ratios and paired residuals should be formed after blocked averaging. Propagate covariance with jackknife or bootstrap; symmetry-related measurements reduce variance but do not count as independent Monte Carlo samples.

Validation

Before a conformality claim:

• compare (L=2,4) energies and selected correlators with exact enumeration;
• verify raw and FK-improved correlators agree within joint confidence intervals;
• reproduce the nearest-neighbor 2D Ising critical point and show that the same geometry residuals extrapolate toward zero;
• run a generalized-free synthetic positive control in which equality at fixed ((u,v)) is exact;
• reproduce the (\sigma=1.2) critical temperature and the expected two-point scaling within uncertainty.

Failure of a critical anchor is evidence that the fixed point has not been isolated, not evidence against conformal symmetry.

Measurement and analysis

Use the size ladder

$$
L=64,128,256,512,
$$

with (L=1024) as a stretch target. Choose several integer (R) values for each (L), retaining only geometries whose diameter is at most (0.15L).

Fit (\delta_{\rm M}(L,R)) and the independent residual with at least two preregistered correction forms, for example

$$
\delta(L,R)=\delta_\infty+aR^{-\omega}+b(R/L)^\sigma
$$

and a less theory-dependent alternative with a free or analytic correction exponent. Report covariance-aware goodness of fit and stability after dropping the smallest (L) and (R).

Do not tune the critical temperature to minimize the Möbius residual. Criticality must be located with an independent Binder or correlation-ratio analysis.

Exploratory crossover scan

After freezing the core analysis at (\sigma=1.2), select a small number of additional (\sigma) values spanning the window between (2-\eta_{\rm SR}) and (2). Determine each critical point independently and repeat a reduced version of the geometry test.

This scan is exploratory. It should compare, rather than assume:

• drift toward a short-range conformal theory near (2-\eta_{\rm SR});
• persistence of long-range behavior until (\sigma=2);
• logarithmic or marginal corrections that can imitate either crossover;
• finite-size failure to distinguish the two scenarios.

The challenge does not define a measured nonzero residual at one size as evidence for a new scale-but-not-conformal phase. Such a claim requires stable continuum behavior and successful short-range, generalized-free, criticality, and estimator controls.

Fairness and five-day contract

1. Day-0 scaffold: provide a tested long-range cluster sampler, frozen coupling generator, geometry fixtures, and exact small-size tests.
2. Raw-data release: retain block-level correlators, seeds, autocorrelation estimates, coupling convention, and code revision.
3. Locked analysis: register fit windows, correction forms, exclusion rules, and the equivalence margin before the largest-size result is inspected.
4. No one-fit verdict: correction-form or fit-window dependence forces an inconclusive result.
5. Boundary neutrality: final interpretation must state separately what follows from the data and what depends on choosing the Sak or geometric crossover scenario.

If the Day-0 scaffold is unavailable, the five-day floor is validation plus a documented finite-size trend; no continuum conformality verdict is required.

Adjudication

Allowed conclusions are:

1. Conformal-compatible at the stated resolution: both independent residuals extrapolate to an equivalence interval around zero under all registered analyses.
2. Controlled violation of the tested prediction: a residual remains nonzero across seeds, critical-temperature brackets, geometry pairs, and correction models while all controls pass.
3. Inconclusive: accessible scales, crossover drift, or correction-model dependence prevent discrimination.

The observable tests global special-conformal covariance of scalar four-point functions. It does not directly establish the existence or absence of a local stress tensor or full Virasoro symmetry.

What is reproduction and what is new

• Critical temperatures, two-point exponents, exact-enumeration checks, and nearest-neighbor controls are reproduction milestones.
• The primary new result is a continuum test of Möbius-related four-point geometries at an interacting long-range fixed point.
• The exploratory extension is the evolution of that test across the disputed crossover window without presupposing (\sigma_*).
• Public code, raw data, geometry fixtures, and the locked analysis record are required for a citable positive, negative, or inconclusive result.

References

1. M. F. Paulos et al., “Conformal Invariance in the Long-Range Ising Model,” arXiv:1509.00008.
2. C. Behan et al., “A scaling theory for the long-range to short-range crossover and an infrared duality,” arXiv:1703.05325.
3. C. Behan et al., “Analytic and numerical bootstrap for the long-range Ising model,” arXiv:2311.02742.
4. M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, “Relations between Short Range and Long Range Ising models,” arXiv:1401.6805.
5. K. Fukui and S. Todo, “Order-(N) Cluster Monte Carlo Method for Spin Systems with Long-range Interactions,” arXiv:0802.0272.
6. Quantum Harness issue #86 and its marginal-point consistency discussion.

Begin with the Day-0 scaffold: the long-range cluster sampler, frozen coupling generator, geometry fixtures, and exact small-size tests. Validate L=2,4 against exact enumeration, reproduce the sigma=1.2 critical anchor, then run the registered Möbius-residual analyses; completion requires controls, raw data, covariance-aware fits, and a conformal-compatible, controlled-violation, or inconclusive result.