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Preserving Hamiltonian Locality in Real-Space Coarse-Graining via Kernel Projection

Sun Haoyuan

TL;DR

The paper tackles critical slowing down in simulations of lattice criticality by introducing the Energy-Constrained Mapping Kernel (ECMK), a spatial projection method that generates large-scale Ising configurations from compact seeds while enforcing a NN energy constraint at criticality. By projecting generated fields onto the nearest-neighbor energy manifold and training with a physics-guided loss, ECMK preserves universal critical properties such as $G(r) \sim r^{-1/4}$, a stable Binder cumulant, and isotropic structure factors, enabling ultra-large lattice generation without iterative Monte Carlo equilibration. The approach achieves significant GPU-accelerated speedups (e.g., ~31× over Wolff and ~68× over Metropolis at $L=13{,}824$) and demonstrates robust scalability from seed to $L=13{,}824^2$, with potential applications to other spin models and lattice theories. This work provides a practical inverse coarse-graining framework that retains critical universality while bypassing temporal relaxation, offering a scalable path to exploring critical phenomena on ultra-large scales.

Abstract

Numerical simulations of critical lattice systems are fundamentally limited by critical slowing down, as long-range correlations are typically established through slow temporal equilibration. A physically constrained generative framework that replaces temporal relaxation with a spatial projection mechanism for critical systems is proposed. Using the two-dimensional Ising model at criticality as a benchmark, we introduce an energy-constrained kernel that synthesizes large-scale configurations from compact equilibrated seeds by enforcing Hamiltonian-level observables. The generated configurations are projected onto the nearest-neighbor energy manifold, ensuring thermodynamic consistency while retaining universal critical properties. We show that the resulting configurations reproduce scale-invariant spin correlations, Binder cumulants, and isotropic structure factors for lattice sizes exceeding 10,000, without iterative Monte Carlo equilibration. While not a strict renormalization group transformation, and motivated by renormalization ideas, the method provides a practical inverse mapping that retains universal features of criticality and enables efficient GPU-parallel generation of ultra-large critical ensembles.

Preserving Hamiltonian Locality in Real-Space Coarse-Graining via Kernel Projection

TL;DR

The paper tackles critical slowing down in simulations of lattice criticality by introducing the Energy-Constrained Mapping Kernel (ECMK), a spatial projection method that generates large-scale Ising configurations from compact seeds while enforcing a NN energy constraint at criticality. By projecting generated fields onto the nearest-neighbor energy manifold and training with a physics-guided loss, ECMK preserves universal critical properties such as , a stable Binder cumulant, and isotropic structure factors, enabling ultra-large lattice generation without iterative Monte Carlo equilibration. The approach achieves significant GPU-accelerated speedups (e.g., ~31× over Wolff and ~68× over Metropolis at ) and demonstrates robust scalability from seed to , with potential applications to other spin models and lattice theories. This work provides a practical inverse coarse-graining framework that retains critical universality while bypassing temporal relaxation, offering a scalable path to exploring critical phenomena on ultra-large scales.

Abstract

Numerical simulations of critical lattice systems are fundamentally limited by critical slowing down, as long-range correlations are typically established through slow temporal equilibration. A physically constrained generative framework that replaces temporal relaxation with a spatial projection mechanism for critical systems is proposed. Using the two-dimensional Ising model at criticality as a benchmark, we introduce an energy-constrained kernel that synthesizes large-scale configurations from compact equilibrated seeds by enforcing Hamiltonian-level observables. The generated configurations are projected onto the nearest-neighbor energy manifold, ensuring thermodynamic consistency while retaining universal critical properties. We show that the resulting configurations reproduce scale-invariant spin correlations, Binder cumulants, and isotropic structure factors for lattice sizes exceeding 10,000, without iterative Monte Carlo equilibration. While not a strict renormalization group transformation, and motivated by renormalization ideas, the method provides a practical inverse mapping that retains universal features of criticality and enables efficient GPU-parallel generation of ultra-large critical ensembles.
Paper Structure (9 sections, 5 equations, 4 figures, 2 tables)

This paper contains 9 sections, 5 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. The Energy-Constrained Mapping Kernel
  4. Loss Function and Physical Constraints
  5. Results and Physical Verifications
  6. Morphological Analysis and Fractal Visualization
  7. Verification of Physical Observables
  8. Computational Efficiency and Scalability
  9. Conclusion

Figures (4)

  • Figure 1: Schematic architecture of the ECMK.
  • Figure 2: Representative spin configuration of the 2D Ising model at the critical temperature $\beta_c$, generated via the ECMK framework ($L=13,824$ lattice). This exhibits characteristic multi-scale domains and fractal structures of the model. One can observe the nested hierarchy of spin clusters, qualitatively demonstrating the scale-invariance property of the system near the phase transition.
  • Figure 3: Spin correlation and spectral density at $L=124,416$, generated by tiling ECMK, after 4 expanding stages from the seed $L=512$
  • Figure 4: Comparison of static structure factors $S(\mathbf{k})$ across different synthesis methods.