[challenge]: Where does long-range universality end? Three adversarial tests of the sigma*=7/4 vs 2 dispute (merged #84+#86+#100)

Released by

Kun Chen, Institute of Theoretical Physics, Chinese Academy of Sciences

Contact email

chenkun0228@gmail.com

Method

MPS / QMC / VMC-NQS

Challenge issue

Consolidated challenge. The former issues #84 (classical 2D equilibrium test) and #100 (Kibble-Zurek dynamics) are merged here as Tracks A and C. A team picks one track. The three tracks test the same long-range/short-range crossover with independent methods.

The dispute

For interactions (J(r)\sim r^{-(d+\sigma)}), two crossover criteria compete:

• Sak: (\sigma_*=2-\eta_{\rm SR}). Evidence includes Luijten-Blöte MC (cond-mat/0112472), Horita-Suwa-Todo (arXiv:1605.09496), crossover field theory (arXiv:1703.05325), conformal bootstrap (arXiv:2311.02742), and the 1D quantum-chain QMC of Shiratani-Todo (arXiv:2305.14121).
• Geometric: (\sigma_*=2). Recent evidence comes from large-scale classical simulations (arXiv:2512.04805, (L\le8192)) and two-loop RG calculations for classical (arXiv:2602.07818) and quantum (arXiv:2606.22407) models.

Conflict-of-interest disclosure: the (\sigma_*=2) predictions are from the releaser's group. This challenge is an adversarial test of our own claim. Falsification and a statistically justified "inconclusive" result are both successful scientific outcomes.

Fairness and five-day contract

1. Day-0 scaffold is mandatory. Organizers must provide at least one tested starting point for the chosen track: a long-range cluster sampler, an MPO/QMC template, or a TDVP driver, plus input conventions and small-size fixtures. Without it, only the validation floor below is promised.
2. Raw-data escrow: per-size/per-ramp raw observables, seeds, autocorrelation information, convergence metadata, and code revision are deposited before extrapolation.
3. Locked analysis: fit windows, both correction ansätze, exclusion rules, and model-selection criteria are registered by the end of day 3.
4. Blind adjudication: a helpdesk-assigned adjudicator with no stake in either prediction receives anonymized curves and applies the registered rules.
5. No one-fit thermodynamic claim: a conclusion that changes after dropping the smallest size, changing the fit window, or changing the registered correction ansatz is reported as inconclusive.

Track A - classical 2D long-range Ising (Monte Carlo)

Use a square (L\times L) torus with minimum-image distance:

[
H=-\sum_{i<j}\frac{c(\sigma,L)}{r_{ij}^{2+\sigma}}s_i s_j,\qquad
\sum_{j\ne i}\frac{c(\sigma,L)}{r_{ij}^{2+\sigma}}=4 .
]

This normalization is the one used in arXiv:2512.04805. Do not mix its (\beta_c) values with bare-(J=1), Ewald, or unnormalized conventions.

Implement either the Fukui-Todo (O(N)) cluster method (arXiv:0802.0272) or the Clock Monte Carlo method (Michel-Tan-Deng, PRE 99, 010105(R)). An independent implementation of the other algorithm is a stretch cross-check.

Pinned observables:

[
Q_m=\frac{\langle M^2\rangle^2}{\langle M^4\rangle},\qquad
R_p=\langle R_2\rangle-2\langle R_0\rangle ,
]

where (R_2) is the event that an FK cluster wraps both torus directions and (R_0) is the no-wrapping event.

Validation

• Exact NN anchor: with nearest-neighbor coupling (J=1), reproduce (\beta_c=\ln(1+\sqrt2)/2), (R_p=0) at criticality, and (\eta=1/4). The high-precision SR value is (Q_m=0.856216(1)).
• Published LR control: at (\sigma=2.5), reproduce the published (\beta_c=0.369446(2)), (R_{p,c}=0.001(2)), (Q_{m,c}=0.857(1)), and (\eta=0.250(1)). For an LR graph, (R_p\to0) in the SR regime is a numerical universality statement, not the finite-(L) exact duality identity of the NN planar model.

Measurement

Use (\sigma\in{1.75,1.875,2.0,2.5}). Published critical points in the pinned normalization are

[
\beta_c={0.329136(1),,0.336985(2),,0.344439(2),,0.369446(2)}.
]

The required size ladder is (L=64,128,256,512); (L=1024) is a stretch target after autocorrelation and wall-clock audits. Reproduce finite-size (R_p(L)) and (Q_m(L)) curves and their crossings. The values (R_{p,c}=-0.207(9)) and (\eta=0.293(3)) at (\sigma=1.875) are (L\to\infty) estimates from fits reaching (L=8192), not finite-size targets for this school.

Adjudication

(R_p) and (Q_m) are primary; (\eta) is secondary. Fit both a power-law correction and a log/marginal correction, report AIC/BIC (or another preregistered score), and repeat after dropping the smallest size. School-scale data are allowed to conclude only that the scenarios are unresolved.

Track B - 1D quantum long-range transverse-field Ising chain

[
H=-\sum_{i<j}\frac{Z_iZ_j}{|i-j|^{1+\sigma}}-\Gamma\sum_iX_i .
]

The pinned convention is bare (J=1) with periodic image sums evaluated using the Hurwitz (\zeta) function, matching Shiratani-Todo.

Validation

• NN limit: (\Gamma_c=1), (z=1).
• Mean-field check: (z=\sigma/2) for (\sigma\le2/3); (z(2/3)=1/3).
• Published table: reproduce at least two (\Gamma_c) entries from Table II of arXiv:2305.14121, including (\Gamma_c(7/4)=1.5609(3)).

Measurement

A team runs one core method:

• DMRG/MPS with a documented sum-of-exponentials MPO and tail-error sweep, (L=64)-256. Locate (\Gamma_c) from a pinned correlation/Binder-ratio crossing, then estimate (z) from (\Delta(L,\Gamma_c)\sim L^{-z}); report ground- and excited-state truncation errors separately.
• SSE/continuous-time QMC from an organizer-provided tested sampler, (L\le512). Estimate (z) from the tuned imaginary-time/spatial aspect ratio (or another preregistered estimator) and reproduce the corresponding correlation-ratio crossing.

Measure (z) and (\gamma/\nu) at (\sigma\in{1.6,7/4,1.8,2.0}); obtain (\gamma/\nu) from the registered zero-momentum structure-factor/susceptibility scaling. At (\sigma=7/4), the published extrapolations give (z=0.91(2)) (power correction) and (0.98(3)) (log correction); reproducing this sensitivity is a validation target, not permission to select the preferred fit. Compare with the two-loop curve of arXiv:2606.22407 and the FRG curve of arXiv:1704.00528.

Sum-of-exponentials bias must be quantified by increasing the number of poles and/or comparison with the transfer-matrix-function analysis of arXiv:2606.20522. An NQS route is allowed only as a stretch cross-check with a preverified scaffold, small-(L) ED tests, multiple optimization seeds, and explicit ansatz/sample convergence; it is not by itself a reliable five-day route to an excited-state gap.

Five-day floor

If the advanced sampler/MPO scaffold is unavailable or fails its tests, the accepted deliverable stops at NN validation, two published (\Gamma_c) reproductions, and a documented finite-size drift study. No boundary verdict is then expected.

Track C - Kibble-Zurek dynamics of the same chain

For a linear NN validation quench, pin

[
\Gamma(t)/J=-t/\tau_Q,\quad t\le0,\quad \hbar=1,
]

ending at (\Gamma=0). In this convention the exact result of Dziarmaga (cond-mat/0509490) is

[
n_{\rm kink}=\frac{1}{2\pi\sqrt{2J\tau_Q}},\qquad \mu=\frac12 .
]

Validation and cross-method gate

• Reproduce both the exponent and prefactor in the NN chain.
• For (L\le20) and fast/intermediate ramps, compare TDVP with ED.
• Measure defects with both the domain-wall operator and a correlation-length/correlation-function estimator; explain their finite-size difference.
• Reproduce a selected student-scale curve from Jaschke et al. (arXiv:1612.07437) before claiming a new trend.

Measurement

• Guaranteed floor: (\mu(\sigma)) at (\sigma\in{1.0,1.25,1.5}), with finite-time-scaling collapse and bond-dimension/time-step convergence.
• Disputed-window stretch: pool 3-4 points in (7/4<\sigma<2). The expected separation may be only (0.005)-(0.024); single-point discrimination is not promised.

For every (\sigma), use at least 6 logarithmically spaced (\tau_Q) values. Register before fitting how the fast-quench plateau and the finite-size adiabatic tail are excluded; a claimed exponent needs at least 4 retained consecutive rates and stability under removal of either endpoint. Count kinks at the registered ramp endpoint. Later-time coarsening can alter defect statistics; arXiv:2512.02112 is a protocol warning from a different Rydberg realization, not a quantitative anchor for this Hamiltonian. Likewise, arXiv:2506.06841 establishes a fast-quench plateau in other realizations and is used only to motivate excluding that regime. The 61-ion experiment (arXiv:2208.03060) reported an average exponent near (0.42) at interaction exponent (\alpha\approx1), with thermodynamic extrapolation (0.39\pm0.11); its protocol and exponent convention differ, so it is context rather than a direct error-bar target.

What is reproduction and what is new

• NN/SR anchors, published (\Gamma_c), old KZ curves, and the sub-(7/4) floor are reproduction milestones.
• A publishable edge requires an independently implemented, uncertainty-controlled result in the disputed window, or a rigorous demonstration that five-day-accessible scales cannot distinguish the two scenarios.
• Prior-art claims must be rechecked at submission time. A negative/inconclusive result is citable only with released raw data, code, and the locked analysis record.

References

Key sources are linked inline. The main numerical anchors are arXiv:2512.04805, arXiv:2305.14121, and cond-mat/0509490.

Start by selecting one of Tracks A, B, or C and checking whether the mandatory tested scaffold, input conventions, and small-size fixtures are available. Run the specified NN and published validation anchors before attempting new measurements; done means releasing raw observables, convergence metadata, code revision, and the locked analysis record.