Pin down the upper critical dimension of spin glasses in a field
Determine the exact upper critical dimension D_U of spin glasses in an external field by tuning 1D long-range models across an effective dimension and locating where mean-field critical exponents break down.
End goal
Determine the exact upper critical dimension D_U of spin-glass models in an external magnetic field — resolving the value within finite-size-scaling uncertainty and testing the analytic prediction D_U ≤ 8.
Overview
Whether the de Almeida–Thouless spin-glass transition survives in finite dimensions — and the exact upper critical dimension D_U above which mean-field exponents hold — is a long-standing open problem. Direct attack is blocked twice over: renormalization-group expansions around mean-field theory break down for spin glasses in a field, and equilibrating a finite-dimensional spin glass takes exponentially long near T_c (D = 5, 6 are already at the edge of feasibility).
This is a proposal-stage task. The proposal frames the problem and points to a tractable route — a one-dimensional long-range spin glass with couplings J_ij ~ |i-j|^(-σ), where the decay exponent σ acts as a continuous dial on the effective dimension, so the transition can be probed across a range of D on a single line at far lower cost than literal high-dimensional lattices. It does not yet specify a step-by-step workflow with verifiable per-step targets: D_U is unknown, so there is no ground truth to score an agent against. The suggested direction below sketches how such a study would proceed, and any eventual estimate would be tested against the analytic prediction D_U ≤ 8 from a recent loop expansion around the Bethe solution [Angelini et al., 2022]. A full, measurable workflow will be added once the author specifies it.
Proposal document
Tools allowed
3Constraints
Software
Hardware
Datasets
- Synthetic disorder realizations
Gaussian long-range couplings J_ij ~ |i-j|^(-σ) — generated, not measured; thousands of independent realizations per (N, σ, h, T).
- Reference: Angelini et al., PRL 128, 075702 (2022)
Analytic loop expansion around the Bethe solution at zero temperature predicting D_U ≤ 8 (surprisingly above the classical D_U = 6) — the prediction this task tests numerically.
Suggested approach
A direction, not yet a workflow
Problem proposal — workflow to be specified by the author. This entry captures an open problem and the route the author suggested for attacking it. Because the upper critical dimension DU is unknown, there is no ground-truth answer and no scientist-set target scores — so the task is not yet scored on the leaderboard. A full, measurable workflow will be added once the author provides one.
- 1
Map the problem onto a one-dimensional long-range spin glass: draw random couplings J_ij ~ |i-j|^(-σ) on a line, where the decay exponent σ tunes the effective dimension D. This probes a continuous range of D on a single geometry, avoiding the exponential cost of literal high-dimensional lattices.
- 2
Equilibrate toward the spin-glass phase with heavy Monte Carlo (e.g. replica exchange / population annealing). Equilibration time grows exponentially near T_c — the central bottleneck — which is why direct simulations at D = 5, 6 remain ambiguous.
- 3
Apply finite-size scaling across σ to find where mean-field critical exponents break down, then map that threshold back to the upper critical dimension D_U.
- 4
Compare any estimate against the analytic prediction D_U ≤ 8 [Angelini et al., PRL 128, 075702 (2022)], which already sits above the classically assumed D_U = 6.
Workflow — awaiting author specification
A verifiable step-by-step protocol — with per-step targets and simulations — will appear here once the author submits it through the review loop.
Evaluation criteria
Frontier · verifiable. There is no ground-truth value of DU, so a result is judged by method quality and internal consistency — not by target scores. These are the criteria a verification would apply; the exact thresholds are to be finalized by the author.
- Equilibration: standard spin-glass equilibration diagnostics pass — independent χ_SG estimators agree, and results converge from hot vs. annealed starts — before any observable is trusted.
- Finite-size scaling: the data collapse of χ_SG and ξ_L/L is of good statistical quality, and the extracted exponents (ν, η) carry controlled error bars.
- Known-limit recovery: the pipeline reproduces the fully-connected Sherrington–Kirkpatrick mean-field result in the limit where it must hold.
- Consistency with the analytic bound: the final D_U estimate and its confidence interval are consistent with the loop-expansion prediction D_U ≤ 8 [Angelini et al., 2022].
- Reproducibility: the conclusion is stable across independent disorder realizations and random seeds.