USW
Materials ScienceStatistical Physics · Spin GlassesExpertUnder reviewFrontier · verifiable

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.

Gangmin Son
Korea Institute for Advanced Study (KIAS)
Upper Critical Dimension of Spin Glass Models in a Field
Proposed task · KIAS · 2026
Weeks of HPC · σ-sweep with thousands of disorder samples per point
registered 2026-06-12
spin-glassstatistical-physicscritical-phenomenafinite-size-scalingAT-lineopen-problemfrontier

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 task takes the tractable route. A one-dimensional long-range spin glass with couplings J_ij ~ |i-j|^(-σ) lets σ act as a continuous dial on the effective dimension, so the transition can be probed across a range of D on a single line. The agent must build and validate the model, equilibrate it with replica-exchange Monte Carlo under strict equilibration tests, perform a finite-size-scaling collapse to extract critical exponents, and locate the threshold σ_U that maps to D_U. The outcome is open-ended — the exact value is unknown — but every step is quantitatively checkable, and the final estimate is tested against the analytic prediction D_U ≤ 8 from a recent loop expansion around the Bethe solution [Angelini et al., 2022].

Tools allowed

3
1D Long-Range Spin-Glass Model·TerminalParallel Tempering MC·HPCFinite-Size Scaling Analysis·Terminal

Constraints

Software

Replica-exchange / population-annealing Monte Carlo (custom C / CUDA)Finite-size-scaling & data-collapse analysis (Python / SciPy)Disorder-averaging job orchestration

Hardware

GPU / HPC cluster for population annealing over thousands of disorder samplesMemory for dense long-range coupling matrices at large N

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.

Workflow

5-step expected workflow

The steps a scientist would expect to run — open-ended, with no ground-truth targets. Open any simulation to test it live.

Frontier problem — an expected procedure, not a measured protocol. No one has solved this yet, so there is no ground-truth answer and no scientist-set target scores. The steps below are the procedure a scientist would expect to run; the per-step figures are conventional method-hygiene diagnostics (equilibration, scaling-collapse quality), not measured targets. The final outcome is judged only by internal scaling quality and consistency with the analytic prediction DU ≤ 8.

  1. 1

    Build & validate the long-range model

    Step 1 / 5

    Construct the 1D long-range spin glass and verify it reproduces known limits before any production runs.

    Protocol

    1. aDraw Gaussian couplings J_ij with variance ~ |i-j|^(-2σ) on a ring of N spins, with field h.
    2. bMap σ to the effective dimension D and choose a σ-grid bracketing the expected σ_U.
    3. cValidate against the fully-connected (Sherrington–Kirkpatrick) mean-field limit.

    Targets

    Coupling-variance fidelity≤2% deviation
    σ-grid coverage≥6σ points
    MFT-limit recovery≤5% error
    Expected output

    A validated coupling generator producing disorder realizations across a grid of (N, σ, h, T).

    Simulations · click to test

    output carries into step 2
  2. 2

    Equilibrate with replica-exchange Monte Carlo

    Step 2 / 5

    Reach true equilibrium at each (N, σ, h, T) and prove it with the standard spin-glass equilibration tests.

    Protocol

    1. aRun parallel tempering / population annealing across a temperature ladder.
    2. bVerify equilibration via agreement of two independent χ_SG estimators and convergence from hot vs. annealed starts (the symmetry P(q) = P(-q) holds only at h = 0).
    3. cAverage over thousands of independent disorder realizations.

    Targets

    Equilibration pass rate≥95%
    Disorder samples / point≥1000realizations
    Swap acceptance≥0.3fraction
    Expected output

    Equilibrated ensembles with passed diagnostics for every (N, σ, h, T).

    Simulations · click to test

    output carries into step 3
  3. 3

    Locate the AT transition

    Step 3 / 5

    Measure spin-glass observables and find the transition temperature from finite-size crossings.

    Protocol

    1. aCompute the spin-glass susceptibility χ_SG and the correlation length ξ_L / L.
    2. bLocate T_c(σ, h) from finite-size crossings, combining ξ_L/L with field-appropriate dimensionless ratios (e.g. R_12) — in a field a single clean crossing may be absent.

    Targets

    T_c relative error≤1%
    Crossing spread≤2.5% of T_c
    Expected output

    T_c(σ, h) for each σ with statistical error bars.

    Simulations · click to test

    output carries into step 4
  4. 4

    Finite-size scaling & exponents

    Step 4 / 5

    Collapse the data and extract the critical exponents that distinguish mean-field from non-mean-field behavior.

    Protocol

    1. aCollapse χ_SG and ξ_L/L onto scaling functions for each σ.
    2. bExtract the correlation-length exponent ν and anomalous dimension η with error bars.
    3. cQuantify the goodness of each data collapse.

    Targets

    Scaling-collapse quality≤1.5reduced χ²
    Exponent precision≤5% error on ν
    Expected output

    ν(σ), η(σ) with uncertainties and a collapse-quality score per σ.

    Simulations · click to test

    output carries into step 5
  5. 5

    Map σ_U → D_U and report the bound

    Step 5 / 5

    Convert the threshold to a dimension and check it against the literature.

    Protocol

    1. aLocate σ_U where exponents cross from mean-field to non-mean-field values.
    2. bConvert σ_U to D_U via the dictionary D = (2 − η)/(2σ − 1); note the in-field boundary (D ≈ 8) differs from the η-driven zero-field value (σ = 2/3 → D = 6).
    3. cReport D_U with a confidence interval and test consistency with the analytic prediction D_U ≤ 8 [Angelini et al., 2022].

    Targets

    Determined D_U≤8dimensions
    Determination tightness≤1Δ dimensions
    Consistency with prediction≥100%
    Expected output

    An estimate (or bound) of D_U with a quantified confidence interval and a consistency verdict against prior evidence.

    Simulations · click to test