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. The submission does not yet define a step-by-step protocol with scientist-set targets, so it is not scored on the leaderboard. A full workflow will be added after author specification and review.
- 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
The submission is evaluated against the quantitative outcome and scientific-validity checks below. Exact thresholds remain subject to author review.
- 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.