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A Machine Learning Approach to Duality in Statistical Physics
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Formulates discovery of statistical-physics dualities as an optimization over a neural-network observable map and dual couplings, matching correlation-function features via MCMC, and rediscovers Kramers-Wannier duality for the 2d Ising model.
Problem
Finding dual Hamiltonians and dual observable maps for lattice statistical mechanics models is currently done by hand; there is no general systematic method, especially for deformed models such as the Ising model with a plaquette coupling.
Approach
Parameterize the original-to-dual observable map with a small neural network (hard attention over neighbouring link products) and learn dual couplings jointly by minimizing a loss that matches spatially averaged local correlation features between original and dual models, with gradients for dual couplings estimated via MCMC and control variates.
Key results
- On 8x8 lattices the method recovers the Kramers-Wannier dual branch, numerically reproducing the known temperature mapping sinh(2β)sinh(2β~)=1 alongside original and antiferromagnetic branches.
- For the plaquette model (300 experiments), no new dual theories were found when κ≠0; runs mostly recovered the original frame.
- Control-variate variance reduction improved convergence, and approximate duals along β+κ=const matched theoretical dual moments on 24x24 lattices.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| learn a neural observable map and dual Hamiltonian | 2D Ising model · 2D Ising model | duality recovery | Helped |
| use a dual-lattice mapping rather than the original-link mapping | 2D Ising model · 2D Ising model | feature-matching loss | Helped |
| add a plaquette coupling and search within the same Hamiltonian form | 2D Ising plaquette model · 2D Ising plaquette model with κ ≠ 0 | new dual theory recovery | Hurt |
| allow dual couplings ˜β and ˜κ to vary (ablation) | 2D Ising plaquette model · 2D Ising model with target κ = 0 | recovery of expected Kramers-Wannier duals | Helped |
| use variance reduction with a constant-baseline control variate (ablation) | Neural duality framework · 2D Ising plaquette model, β = 0.25 | convergence | Helped |
| remove variance reduction from gradient estimation (ablation) | Neural duality framework · 2D Ising plaquette model, β = 0.25 | convergence | Hurt |
| increase lattice size beyond 8 × 8 (ablation) | Neural duality framework · 2D Ising model, β = 0.25 | recovery rate | Helped |
| train in the post-transition regime (ablation) | Neural duality framework · 2D Ising model, β > βc | duality recovery | Hurt |
| use a rotation and reflection to relate horizontal and vertical link assignments (ablation) | Neural duality framework · 2D Ising model | duality recovery | Helped |
| include approximate duals along the line ˜β + ˜κ = const in the ordered regime (ablation) | 2D Ising plaquette model · Moments across approximate duals | moment agreement | No clear effect |
| use approximate duals from the learned framework on larger lattices without retraining (ablation) | Neural duality framework · 2D Ising model, 24 × 24 lattice | moment agreement | Helped |
Surprising findings
- For the plaquette model with κ≠0 the optimizer found no new duals, and the algorithm fails to find the dual near and beyond the phase transition β_c≈0.44.
- Clusters of approximate duals lie along β~+κ~=const when the dual is deep in the ordered phase, since observables then depend essentially on a single spin-flip probability.
Limitations
- Gradient estimation for dual couplings requires expensive MCMC equilibration each step, and convergence is noisy.
- Method performs poorly near and above the phase transition and in the symmetry-broken phase.
- Features for matching are hand-constructed, and no new dualities were discovered.
Experimental setup
Datasets: MCMC samples from 2d Ising model MCMC samples from 2d plaquette Ising model
Benchmarks: Kramers-Wannier duality line sinh(2β)sinh(2β~)=1 plaquette model approximate duals
Metrics: recovered dual coupling vs theory loss fraction of runs recovering target within tolerance moment matching on 24x24 lattices
Baselines: Betzler & Krippendorf (2020) contrastive divergence variant without variance reduction
Builds on: Kramers-Wannier duality Boltzmann machine learning Gumbel-Softmax Metropolis MCMC
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