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

TL;DR

This paper introduces a kernel-based spatial projection method for critical lattice systems that efficiently generates large-scale configurations preserving critical properties without slow equilibration.

Contribution

It proposes a novel energy-constrained kernel projection framework that maintains universal critical features in large-scale configurations, bypassing traditional slow Monte Carlo methods.

Findings

Reproduces scale-invariant correlations and structure factors for large lattices

Ensures thermodynamic consistency through energy manifold projection

Enables GPU-parallel generation of critical ensembles without iterative equilibration

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.